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from_articles Package

region_deterministic_bnm_g2d_a

What:
Reproduces Figure 13, first column, of Sanz-Leon P., Knock, S. A., Spiegler, A. and Jirsa V. Mathematical framework for large-scale brain network modelling in The Virtual Brain. Neuroimage, 2014, (in review)
Needs:
A working installation of tvb
Run:

python region_deterministic_bnm_g2d_a.py -s True -f True

#Subsequent calls can be made with: python region_deterministic_bnm_g2d_a.py -f True

tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.beta(a, b, size=None)

Draw samples from a Beta distribution.

The Beta distribution is a special case of the Dirichlet distribution, and is related to the Gamma distribution. It has the probability distribution function

\[f(x; a,b) = \frac{1}{B(\alpha, \beta)} x^{\alpha - 1} (1 - x)^{\beta - 1},\]

where the normalization, B, is the beta function,

\[B(\alpha, \beta) = \int_0^1 t^{\alpha - 1} (1 - t)^{\beta - 1} dt.\]

It is often seen in Bayesian inference and order statistics.

Note

New code should use the beta method of a default_rng() instance instead; see random-quick-start.

a : float or array_like of floats
Alpha, positive (>0).
b : float or array_like of floats
Beta, positive (>0).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if a and b are both scalars. Otherwise, np.broadcast(a, b).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized beta distribution.

Generator.beta: which should be used for new code.

tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.binomial(n, p, size=None)

Draw samples from a binomial distribution.

Samples are drawn from a binomial distribution with specified parameters, n trials and p probability of success where n an integer >= 0 and p is in the interval [0,1]. (n may be input as a float, but it is truncated to an integer in use)

Note

New code should use the binomial method of a default_rng() instance instead; see random-quick-start.

n : int or array_like of ints
Parameter of the distribution, >= 0. Floats are also accepted, but they will be truncated to integers.
p : float or array_like of floats
Parameter of the distribution, >= 0 and <=1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if n and p are both scalars. Otherwise, np.broadcast(n, p).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized binomial distribution, where each sample is equal to the number of successes over the n trials.
scipy.stats.binom : probability density function, distribution or
cumulative density function, etc.

Generator.binomial: which should be used for new code.

The probability density for the binomial distribution is

\[P(N) = \binom{n}{N}p^N(1-p)^{n-N},\]

where \(n\) is the number of trials, \(p\) is the probability of success, and \(N\) is the number of successes.

When estimating the standard error of a proportion in a population by using a random sample, the normal distribution works well unless the product p*n <=5, where p = population proportion estimate, and n = number of samples, in which case the binomial distribution is used instead. For example, a sample of 15 people shows 4 who are left handed, and 11 who are right handed. Then p = 4/15 = 27%. 0.27*15 = 4, so the binomial distribution should be used in this case.

[1]Dalgaard, Peter, “Introductory Statistics with R”, Springer-Verlag, 2002.
[2]Glantz, Stanton A. “Primer of Biostatistics.”, McGraw-Hill, Fifth Edition, 2002.
[3]Lentner, Marvin, “Elementary Applied Statistics”, Bogden and Quigley, 1972.
[4]Weisstein, Eric W. “Binomial Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/BinomialDistribution.html
[5]Wikipedia, “Binomial distribution”, https://en.wikipedia.org/wiki/Binomial_distribution

Draw samples from the distribution:

>>> n, p = 10, .5  # number of trials, probability of each trial
>>> s = np.random.binomial(n, p, 1000)
# result of flipping a coin 10 times, tested 1000 times.

A real world example. A company drills 9 wild-cat oil exploration wells, each with an estimated probability of success of 0.1. All nine wells fail. What is the probability of that happening?

Let’s do 20,000 trials of the model, and count the number that generate zero positive results.

>>> sum(np.random.binomial(9, 0.1, 20000) == 0)/20000.
# answer = 0.38885, or 38%.
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.chisquare(df, size=None)

Draw samples from a chi-square distribution.

When df independent random variables, each with standard normal distributions (mean 0, variance 1), are squared and summed, the resulting distribution is chi-square (see Notes). This distribution is often used in hypothesis testing.

Note

New code should use the chisquare method of a default_rng() instance instead; see random-quick-start.

df : float or array_like of floats
Number of degrees of freedom, must be > 0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if df is a scalar. Otherwise, np.array(df).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized chi-square distribution.
ValueError
When df <= 0 or when an inappropriate size (e.g. size=-1) is given.

Generator.chisquare: which should be used for new code.

The variable obtained by summing the squares of df independent, standard normally distributed random variables:

\[Q = \sum_{i=0}^{\mathtt{df}} X^2_i\]

is chi-square distributed, denoted

\[Q \sim \chi^2_k.\]

The probability density function of the chi-squared distribution is

\[p(x) = \frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2},\]

where \(\Gamma\) is the gamma function,

\[\Gamma(x) = \int_0^{-\infty} t^{x - 1} e^{-t} dt.\]
[1]NIST “Engineering Statistics Handbook” https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm
>>> np.random.chisquare(2,4)
array([ 1.89920014,  9.00867716,  3.13710533,  5.62318272]) # random
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.choice(a, size=None, replace=True, p=None)

Generates a random sample from a given 1-D array

New in version 1.7.0.

Note

New code should use the choice method of a default_rng() instance instead; see random-quick-start.

a : 1-D array-like or int
If an ndarray, a random sample is generated from its elements. If an int, the random sample is generated as if a were np.arange(a)
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
replace : boolean, optional
Whether the sample is with or without replacement
p : 1-D array-like, optional
The probabilities associated with each entry in a. If not given the sample assumes a uniform distribution over all entries in a.
samples : single item or ndarray
The generated random samples
ValueError
If a is an int and less than zero, if a or p are not 1-dimensional, if a is an array-like of size 0, if p is not a vector of probabilities, if a and p have different lengths, or if replace=False and the sample size is greater than the population size

randint, shuffle, permutation Generator.choice: which should be used in new code

Sampling random rows from a 2-D array is not possible with this function, but is possible with Generator.choice through its axis keyword.

Generate a uniform random sample from np.arange(5) of size 3:

>>> np.random.choice(5, 3)
array([0, 3, 4]) # random
>>> #This is equivalent to np.random.randint(0,5,3)

Generate a non-uniform random sample from np.arange(5) of size 3:

>>> np.random.choice(5, 3, p=[0.1, 0, 0.3, 0.6, 0])
array([3, 3, 0]) # random

Generate a uniform random sample from np.arange(5) of size 3 without replacement:

>>> np.random.choice(5, 3, replace=False)
array([3,1,0]) # random
>>> #This is equivalent to np.random.permutation(np.arange(5))[:3]

Generate a non-uniform random sample from np.arange(5) of size 3 without replacement:

>>> np.random.choice(5, 3, replace=False, p=[0.1, 0, 0.3, 0.6, 0])
array([2, 3, 0]) # random

Any of the above can be repeated with an arbitrary array-like instead of just integers. For instance:

>>> aa_milne_arr = ['pooh', 'rabbit', 'piglet', 'Christopher']
>>> np.random.choice(aa_milne_arr, 5, p=[0.5, 0.1, 0.1, 0.3])
array(['pooh', 'pooh', 'pooh', 'Christopher', 'piglet'], # random
      dtype='<U11')
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.dirichlet(alpha, size=None)

Draw samples from the Dirichlet distribution.

Draw size samples of dimension k from a Dirichlet distribution. A Dirichlet-distributed random variable can be seen as a multivariate generalization of a Beta distribution. The Dirichlet distribution is a conjugate prior of a multinomial distribution in Bayesian inference.

Note

New code should use the dirichlet method of a default_rng() instance instead; see random-quick-start.

alpha : sequence of floats, length k
Parameter of the distribution (length k for sample of length k).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n), then m * n * k samples are drawn. Default is None, in which case a vector of length k is returned.
samples : ndarray,
The drawn samples, of shape (size, k).
ValueError
If any value in alpha is less than or equal to zero

Generator.dirichlet: which should be used for new code.

The Dirichlet distribution is a distribution over vectors \(x\) that fulfil the conditions \(x_i>0\) and \(\sum_{i=1}^k x_i = 1\).

The probability density function \(p\) of a Dirichlet-distributed random vector \(X\) is proportional to

\[p(x) \propto \prod_{i=1}^{k}{x^{\alpha_i-1}_i},\]

where \(\alpha\) is a vector containing the positive concentration parameters.

The method uses the following property for computation: let \(Y\) be a random vector which has components that follow a standard gamma distribution, then \(X = \frac{1}{\sum_{i=1}^k{Y_i}} Y\) is Dirichlet-distributed

[1]David McKay, “Information Theory, Inference and Learning Algorithms,” chapter 23, http://www.inference.org.uk/mackay/itila/
[2]Wikipedia, “Dirichlet distribution”, https://en.wikipedia.org/wiki/Dirichlet_distribution

Taking an example cited in Wikipedia, this distribution can be used if one wanted to cut strings (each of initial length 1.0) into K pieces with different lengths, where each piece had, on average, a designated average length, but allowing some variation in the relative sizes of the pieces.

>>> s = np.random.dirichlet((10, 5, 3), 20).transpose()
>>> import matplotlib.pyplot as plt
>>> plt.barh(range(20), s[0])
>>> plt.barh(range(20), s[1], left=s[0], color='g')
>>> plt.barh(range(20), s[2], left=s[0]+s[1], color='r')
>>> plt.title("Lengths of Strings")
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.exponential(scale=1.0, size=None)

Draw samples from an exponential distribution.

Its probability density function is

\[f(x; \frac{1}{\beta}) = \frac{1}{\beta} \exp(-\frac{x}{\beta}),\]

for x > 0 and 0 elsewhere. \(\beta\) is the scale parameter, which is the inverse of the rate parameter \(\lambda = 1/\beta\). The rate parameter is an alternative, widely used parameterization of the exponential distribution [3]_.

The exponential distribution is a continuous analogue of the geometric distribution. It describes many common situations, such as the size of raindrops measured over many rainstorms [1]_, or the time between page requests to Wikipedia [2]_.

Note

New code should use the exponential method of a default_rng() instance instead; see random-quick-start.

scale : float or array_like of floats
The scale parameter, \(\beta = 1/\lambda\). Must be non-negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if scale is a scalar. Otherwise, np.array(scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized exponential distribution.

Generator.exponential: which should be used for new code.

[1]Peyton Z. Peebles Jr., “Probability, Random Variables and Random Signal Principles”, 4th ed, 2001, p. 57.
[2]Wikipedia, “Poisson process”, https://en.wikipedia.org/wiki/Poisson_process
[3]Wikipedia, “Exponential distribution”, https://en.wikipedia.org/wiki/Exponential_distribution
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.f(dfnum, dfden, size=None)

Draw samples from an F distribution.

Samples are drawn from an F distribution with specified parameters, dfnum (degrees of freedom in numerator) and dfden (degrees of freedom in denominator), where both parameters must be greater than zero.

The random variate of the F distribution (also known as the Fisher distribution) is a continuous probability distribution that arises in ANOVA tests, and is the ratio of two chi-square variates.

Note

New code should use the f method of a default_rng() instance instead; see random-quick-start.

dfnum : float or array_like of floats
Degrees of freedom in numerator, must be > 0.
dfden : float or array_like of float
Degrees of freedom in denominator, must be > 0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if dfnum and dfden are both scalars. Otherwise, np.broadcast(dfnum, dfden).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Fisher distribution.
scipy.stats.f : probability density function, distribution or
cumulative density function, etc.

Generator.f: which should be used for new code.

The F statistic is used to compare in-group variances to between-group variances. Calculating the distribution depends on the sampling, and so it is a function of the respective degrees of freedom in the problem. The variable dfnum is the number of samples minus one, the between-groups degrees of freedom, while dfden is the within-groups degrees of freedom, the sum of the number of samples in each group minus the number of groups.

[1]Glantz, Stanton A. “Primer of Biostatistics.”, McGraw-Hill, Fifth Edition, 2002.
[2]Wikipedia, “F-distribution”, https://en.wikipedia.org/wiki/F-distribution

An example from Glantz[1], pp 47-40:

Two groups, children of diabetics (25 people) and children from people without diabetes (25 controls). Fasting blood glucose was measured, case group had a mean value of 86.1, controls had a mean value of 82.2. Standard deviations were 2.09 and 2.49 respectively. Are these data consistent with the null hypothesis that the parents diabetic status does not affect their children’s blood glucose levels? Calculating the F statistic from the data gives a value of 36.01.

Draw samples from the distribution:

>>> dfnum = 1. # between group degrees of freedom
>>> dfden = 48. # within groups degrees of freedom
>>> s = np.random.f(dfnum, dfden, 1000)

The lower bound for the top 1% of the samples is :

>>> np.sort(s)[-10]
7.61988120985 # random

So there is about a 1% chance that the F statistic will exceed 7.62, the measured value is 36, so the null hypothesis is rejected at the 1% level.

tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.gamma(shape, scale=1.0, size=None)

Draw samples from a Gamma distribution.

Samples are drawn from a Gamma distribution with specified parameters, shape (sometimes designated “k”) and scale (sometimes designated “theta”), where both parameters are > 0.

Note

New code should use the gamma method of a default_rng() instance instead; see random-quick-start.

shape : float or array_like of floats
The shape of the gamma distribution. Must be non-negative.
scale : float or array_like of floats, optional
The scale of the gamma distribution. Must be non-negative. Default is equal to 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if shape and scale are both scalars. Otherwise, np.broadcast(shape, scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized gamma distribution.
scipy.stats.gamma : probability density function, distribution or
cumulative density function, etc.

Generator.gamma: which should be used for new code.

The probability density for the Gamma distribution is

\[p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},\]

where \(k\) is the shape and \(\theta\) the scale, and \(\Gamma\) is the Gamma function.

The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant.

[1]Weisstein, Eric W. “Gamma Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html
[2]Wikipedia, “Gamma distribution”, https://en.wikipedia.org/wiki/Gamma_distribution

Draw samples from the distribution:

>>> shape, scale = 2., 2.  # mean=4, std=2*sqrt(2)
>>> s = np.random.gamma(shape, scale, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> import scipy.special as sps  
>>> count, bins, ignored = plt.hist(s, 50, density=True)
>>> y = bins**(shape-1)*(np.exp(-bins/scale) /  
...                      (sps.gamma(shape)*scale**shape))
>>> plt.plot(bins, y, linewidth=2, color='r')  
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.geometric(p, size=None)

Draw samples from the geometric distribution.

Bernoulli trials are experiments with one of two outcomes: success or failure (an example of such an experiment is flipping a coin). The geometric distribution models the number of trials that must be run in order to achieve success. It is therefore supported on the positive integers, k = 1, 2, ....

The probability mass function of the geometric distribution is

\[f(k) = (1 - p)^{k - 1} p\]

where p is the probability of success of an individual trial.

Note

New code should use the geometric method of a default_rng() instance instead; see random-quick-start.

p : float or array_like of floats
The probability of success of an individual trial.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if p is a scalar. Otherwise, np.array(p).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized geometric distribution.

Generator.geometric: which should be used for new code.

Draw ten thousand values from the geometric distribution, with the probability of an individual success equal to 0.35:

>>> z = np.random.geometric(p=0.35, size=10000)

How many trials succeeded after a single run?

>>> (z == 1).sum() / 10000.
0.34889999999999999 #random
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.get_state()

Return a tuple representing the internal state of the generator.

For more details, see set_state.

legacy : bool, optional
Flag indicating to return a legacy tuple state when the BitGenerator is MT19937, instead of a dict.
out : {tuple(str, ndarray of 624 uints, int, int, float), dict}

The returned tuple has the following items:

  1. the string ‘MT19937’.
  2. a 1-D array of 624 unsigned integer keys.
  3. an integer pos.
  4. an integer has_gauss.
  5. a float cached_gaussian.

If legacy is False, or the BitGenerator is not MT19937, then state is returned as a dictionary.

set_state

set_state and get_state are not needed to work with any of the random distributions in NumPy. If the internal state is manually altered, the user should know exactly what he/she is doing.

tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.gumbel(loc=0.0, scale=1.0, size=None)

Draw samples from a Gumbel distribution.

Draw samples from a Gumbel distribution with specified location and scale. For more information on the Gumbel distribution, see Notes and References below.

Note

New code should use the gumbel method of a default_rng() instance instead; see random-quick-start.

loc : float or array_like of floats, optional
The location of the mode of the distribution. Default is 0.
scale : float or array_like of floats, optional
The scale parameter of the distribution. Default is 1. Must be non- negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if loc and scale are both scalars. Otherwise, np.broadcast(loc, scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Gumbel distribution.

scipy.stats.gumbel_l scipy.stats.gumbel_r scipy.stats.genextreme weibull Generator.gumbel: which should be used for new code.

The Gumbel (or Smallest Extreme Value (SEV) or the Smallest Extreme Value Type I) distribution is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. The Gumbel is a special case of the Extreme Value Type I distribution for maximums from distributions with “exponential-like” tails.

The probability density for the Gumbel distribution is

\[p(x) = \frac{e^{-(x - \mu)/ \beta}}{\beta} e^{ -e^{-(x - \mu)/ \beta}},\]

where \(\mu\) is the mode, a location parameter, and \(\beta\) is the scale parameter.

The Gumbel (named for German mathematician Emil Julius Gumbel) was used very early in the hydrology literature, for modeling the occurrence of flood events. It is also used for modeling maximum wind speed and rainfall rates. It is a “fat-tailed” distribution - the probability of an event in the tail of the distribution is larger than if one used a Gaussian, hence the surprisingly frequent occurrence of 100-year floods. Floods were initially modeled as a Gaussian process, which underestimated the frequency of extreme events.

It is one of a class of extreme value distributions, the Generalized Extreme Value (GEV) distributions, which also includes the Weibull and Frechet.

The function has a mean of \(\mu + 0.57721\beta\) and a variance of \(\frac{\pi^2}{6}\beta^2\).

[1]Gumbel, E. J., “Statistics of Extremes,” New York: Columbia University Press, 1958.
[2]Reiss, R.-D. and Thomas, M., “Statistical Analysis of Extreme Values from Insurance, Finance, Hydrology and Other Fields,” Basel: Birkhauser Verlag, 2001.

Draw samples from the distribution:

>>> mu, beta = 0, 0.1 # location and scale
>>> s = np.random.gumbel(mu, beta, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, density=True)
>>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)
...          * np.exp( -np.exp( -(bins - mu) /beta) ),
...          linewidth=2, color='r')
>>> plt.show()

Show how an extreme value distribution can arise from a Gaussian process and compare to a Gaussian:

>>> means = []
>>> maxima = []
>>> for i in range(0,1000) :
...    a = np.random.normal(mu, beta, 1000)
...    means.append(a.mean())
...    maxima.append(a.max())
>>> count, bins, ignored = plt.hist(maxima, 30, density=True)
>>> beta = np.std(maxima) * np.sqrt(6) / np.pi
>>> mu = np.mean(maxima) - 0.57721*beta
>>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)
...          * np.exp(-np.exp(-(bins - mu)/beta)),
...          linewidth=2, color='r')
>>> plt.plot(bins, 1/(beta * np.sqrt(2 * np.pi))
...          * np.exp(-(bins - mu)**2 / (2 * beta**2)),
...          linewidth=2, color='g')
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.hypergeometric(ngood, nbad, nsample, size=None)

Draw samples from a Hypergeometric distribution.

Samples are drawn from a hypergeometric distribution with specified parameters, ngood (ways to make a good selection), nbad (ways to make a bad selection), and nsample (number of items sampled, which is less than or equal to the sum ngood + nbad).

Note

New code should use the hypergeometric method of a default_rng() instance instead; see random-quick-start.

ngood : int or array_like of ints
Number of ways to make a good selection. Must be nonnegative.
nbad : int or array_like of ints
Number of ways to make a bad selection. Must be nonnegative.
nsample : int or array_like of ints
Number of items sampled. Must be at least 1 and at most ngood + nbad.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if ngood, nbad, and nsample are all scalars. Otherwise, np.broadcast(ngood, nbad, nsample).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized hypergeometric distribution. Each sample is the number of good items within a randomly selected subset of size nsample taken from a set of ngood good items and nbad bad items.
scipy.stats.hypergeom : probability density function, distribution or
cumulative density function, etc.

Generator.hypergeometric: which should be used for new code.

The probability density for the Hypergeometric distribution is

\[P(x) = \frac{\binom{g}{x}\binom{b}{n-x}}{\binom{g+b}{n}},\]

where \(0 \le x \le n\) and \(n-b \le x \le g\)

for P(x) the probability of x good results in the drawn sample, g = ngood, b = nbad, and n = nsample.

Consider an urn with black and white marbles in it, ngood of them are black and nbad are white. If you draw nsample balls without replacement, then the hypergeometric distribution describes the distribution of black balls in the drawn sample.

Note that this distribution is very similar to the binomial distribution, except that in this case, samples are drawn without replacement, whereas in the Binomial case samples are drawn with replacement (or the sample space is infinite). As the sample space becomes large, this distribution approaches the binomial.

[1]Lentner, Marvin, “Elementary Applied Statistics”, Bogden and Quigley, 1972.
[2]Weisstein, Eric W. “Hypergeometric Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/HypergeometricDistribution.html
[3]Wikipedia, “Hypergeometric distribution”, https://en.wikipedia.org/wiki/Hypergeometric_distribution

Draw samples from the distribution:

>>> ngood, nbad, nsamp = 100, 2, 10
# number of good, number of bad, and number of samples
>>> s = np.random.hypergeometric(ngood, nbad, nsamp, 1000)
>>> from matplotlib.pyplot import hist
>>> hist(s)
#   note that it is very unlikely to grab both bad items

Suppose you have an urn with 15 white and 15 black marbles. If you pull 15 marbles at random, how likely is it that 12 or more of them are one color?

>>> s = np.random.hypergeometric(15, 15, 15, 100000)
>>> sum(s>=12)/100000. + sum(s<=3)/100000.
#   answer = 0.003 ... pretty unlikely!
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.laplace(loc=0.0, scale=1.0, size=None)

Draw samples from the Laplace or double exponential distribution with specified location (or mean) and scale (decay).

The Laplace distribution is similar to the Gaussian/normal distribution, but is sharper at the peak and has fatter tails. It represents the difference between two independent, identically distributed exponential random variables.

Note

New code should use the laplace method of a default_rng() instance instead; see random-quick-start.

loc : float or array_like of floats, optional
The position, \(\mu\), of the distribution peak. Default is 0.
scale : float or array_like of floats, optional
\(\lambda\), the exponential decay. Default is 1. Must be non- negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if loc and scale are both scalars. Otherwise, np.broadcast(loc, scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Laplace distribution.

Generator.laplace: which should be used for new code.

It has the probability density function

\[f(x; \mu, \lambda) = \frac{1}{2\lambda} \exp\left(-\frac{|x - \mu|}{\lambda}\right).\]

The first law of Laplace, from 1774, states that the frequency of an error can be expressed as an exponential function of the absolute magnitude of the error, which leads to the Laplace distribution. For many problems in economics and health sciences, this distribution seems to model the data better than the standard Gaussian distribution.

[1]Abramowitz, M. and Stegun, I. A. (Eds.). “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing,” New York: Dover, 1972.
[2]Kotz, Samuel, et. al. “The Laplace Distribution and Generalizations, ” Birkhauser, 2001.
[3]Weisstein, Eric W. “Laplace Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/LaplaceDistribution.html
[4]Wikipedia, “Laplace distribution”, https://en.wikipedia.org/wiki/Laplace_distribution

Draw samples from the distribution

>>> loc, scale = 0., 1.
>>> s = np.random.laplace(loc, scale, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, density=True)
>>> x = np.arange(-8., 8., .01)
>>> pdf = np.exp(-abs(x-loc)/scale)/(2.*scale)
>>> plt.plot(x, pdf)

Plot Gaussian for comparison:

>>> g = (1/(scale * np.sqrt(2 * np.pi)) *
...      np.exp(-(x - loc)**2 / (2 * scale**2)))
>>> plt.plot(x,g)
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.logistic(loc=0.0, scale=1.0, size=None)

Draw samples from a logistic distribution.

Samples are drawn from a logistic distribution with specified parameters, loc (location or mean, also median), and scale (>0).

Note

New code should use the logistic method of a default_rng() instance instead; see random-quick-start.

loc : float or array_like of floats, optional
Parameter of the distribution. Default is 0.
scale : float or array_like of floats, optional
Parameter of the distribution. Must be non-negative. Default is 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if loc and scale are both scalars. Otherwise, np.broadcast(loc, scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized logistic distribution.
scipy.stats.logistic : probability density function, distribution or
cumulative density function, etc.

Generator.logistic: which should be used for new code.

The probability density for the Logistic distribution is

\[P(x) = P(x) = \frac{e^{-(x-\mu)/s}}{s(1+e^{-(x-\mu)/s})^2},\]

where \(\mu\) = location and \(s\) = scale.

The Logistic distribution is used in Extreme Value problems where it can act as a mixture of Gumbel distributions, in Epidemiology, and by the World Chess Federation (FIDE) where it is used in the Elo ranking system, assuming the performance of each player is a logistically distributed random variable.

[1]Reiss, R.-D. and Thomas M. (2001), “Statistical Analysis of Extreme Values, from Insurance, Finance, Hydrology and Other Fields,” Birkhauser Verlag, Basel, pp 132-133.
[2]Weisstein, Eric W. “Logistic Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/LogisticDistribution.html
[3]Wikipedia, “Logistic-distribution”, https://en.wikipedia.org/wiki/Logistic_distribution

Draw samples from the distribution:

>>> loc, scale = 10, 1
>>> s = np.random.logistic(loc, scale, 10000)
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, bins=50)

# plot against distribution

>>> def logist(x, loc, scale):
...     return np.exp((loc-x)/scale)/(scale*(1+np.exp((loc-x)/scale))**2)
>>> lgst_val = logist(bins, loc, scale)
>>> plt.plot(bins, lgst_val * count.max() / lgst_val.max())
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.lognormal(mean=0.0, sigma=1.0, size=None)

Draw samples from a log-normal distribution.

Draw samples from a log-normal distribution with specified mean, standard deviation, and array shape. Note that the mean and standard deviation are not the values for the distribution itself, but of the underlying normal distribution it is derived from.

Note

New code should use the lognormal method of a default_rng() instance instead; see random-quick-start.

mean : float or array_like of floats, optional
Mean value of the underlying normal distribution. Default is 0.
sigma : float or array_like of floats, optional
Standard deviation of the underlying normal distribution. Must be non-negative. Default is 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if mean and sigma are both scalars. Otherwise, np.broadcast(mean, sigma).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized log-normal distribution.
scipy.stats.lognorm : probability density function, distribution,
cumulative density function, etc.

Generator.lognormal: which should be used for new code.

A variable x has a log-normal distribution if log(x) is normally distributed. The probability density function for the log-normal distribution is:

\[p(x) = \frac{1}{\sigma x \sqrt{2\pi}} e^{(-\frac{(ln(x)-\mu)^2}{2\sigma^2})}\]

where \(\mu\) is the mean and \(\sigma\) is the standard deviation of the normally distributed logarithm of the variable. A log-normal distribution results if a random variable is the product of a large number of independent, identically-distributed variables in the same way that a normal distribution results if the variable is the sum of a large number of independent, identically-distributed variables.

[1]Limpert, E., Stahel, W. A., and Abbt, M., “Log-normal Distributions across the Sciences: Keys and Clues,” BioScience, Vol. 51, No. 5, May, 2001. https://stat.ethz.ch/~stahel/lognormal/bioscience.pdf
[2]Reiss, R.D. and Thomas, M., “Statistical Analysis of Extreme Values,” Basel: Birkhauser Verlag, 2001, pp. 31-32.

Draw samples from the distribution:

>>> mu, sigma = 3., 1. # mean and standard deviation
>>> s = np.random.lognormal(mu, sigma, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 100, density=True, align='mid')
>>> x = np.linspace(min(bins), max(bins), 10000)
>>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))
...        / (x * sigma * np.sqrt(2 * np.pi)))
>>> plt.plot(x, pdf, linewidth=2, color='r')
>>> plt.axis('tight')
>>> plt.show()

Demonstrate that taking the products of random samples from a uniform distribution can be fit well by a log-normal probability density function.

>>> # Generate a thousand samples: each is the product of 100 random
>>> # values, drawn from a normal distribution.
>>> b = []
>>> for i in range(1000):
...    a = 10. + np.random.standard_normal(100)
...    b.append(np.product(a))
>>> b = np.array(b) / np.min(b) # scale values to be positive
>>> count, bins, ignored = plt.hist(b, 100, density=True, align='mid')
>>> sigma = np.std(np.log(b))
>>> mu = np.mean(np.log(b))
>>> x = np.linspace(min(bins), max(bins), 10000)
>>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))
...        / (x * sigma * np.sqrt(2 * np.pi)))
>>> plt.plot(x, pdf, color='r', linewidth=2)
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.logseries(p, size=None)

Draw samples from a logarithmic series distribution.

Samples are drawn from a log series distribution with specified shape parameter, 0 < p < 1.

Note

New code should use the logseries method of a default_rng() instance instead; see random-quick-start.

p : float or array_like of floats
Shape parameter for the distribution. Must be in the range (0, 1).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if p is a scalar. Otherwise, np.array(p).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized logarithmic series distribution.
scipy.stats.logser : probability density function, distribution or
cumulative density function, etc.

Generator.logseries: which should be used for new code.

The probability density for the Log Series distribution is

\[P(k) = \frac{-p^k}{k \ln(1-p)},\]

where p = probability.

The log series distribution is frequently used to represent species richness and occurrence, first proposed by Fisher, Corbet, and Williams in 1943 [2]. It may also be used to model the numbers of occupants seen in cars [3].

[1]Buzas, Martin A.; Culver, Stephen J., Understanding regional species diversity through the log series distribution of occurrences: BIODIVERSITY RESEARCH Diversity & Distributions, Volume 5, Number 5, September 1999 , pp. 187-195(9).
[2]Fisher, R.A,, A.S. Corbet, and C.B. Williams. 1943. The relation between the number of species and the number of individuals in a random sample of an animal population. Journal of Animal Ecology, 12:42-58.
[3]D. J. Hand, F. Daly, D. Lunn, E. Ostrowski, A Handbook of Small Data Sets, CRC Press, 1994.
[4]Wikipedia, “Logarithmic distribution”, https://en.wikipedia.org/wiki/Logarithmic_distribution

Draw samples from the distribution:

>>> a = .6
>>> s = np.random.logseries(a, 10000)
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s)

# plot against distribution

>>> def logseries(k, p):
...     return -p**k/(k*np.log(1-p))
>>> plt.plot(bins, logseries(bins, a)*count.max()/
...          logseries(bins, a).max(), 'r')
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.multinomial(n, pvals, size=None)

Draw samples from a multinomial distribution.

The multinomial distribution is a multivariate generalization of the binomial distribution. Take an experiment with one of p possible outcomes. An example of such an experiment is throwing a dice, where the outcome can be 1 through 6. Each sample drawn from the distribution represents n such experiments. Its values, X_i = [X_0, X_1, ..., X_p], represent the number of times the outcome was i.

Note

New code should use the multinomial method of a default_rng() instance instead; see random-quick-start.

n : int
Number of experiments.
pvals : sequence of floats, length p
Probabilities of each of the p different outcomes. These must sum to 1 (however, the last element is always assumed to account for the remaining probability, as long as sum(pvals[:-1]) <= 1).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
out : ndarray

The drawn samples, of shape size, if that was provided. If not, the shape is (N,).

In other words, each entry out[i,j,...,:] is an N-dimensional value drawn from the distribution.

Generator.multinomial: which should be used for new code.

Throw a dice 20 times:

>>> np.random.multinomial(20, [1/6.]*6, size=1)
array([[4, 1, 7, 5, 2, 1]]) # random

It landed 4 times on 1, once on 2, etc.

Now, throw the dice 20 times, and 20 times again:

>>> np.random.multinomial(20, [1/6.]*6, size=2)
array([[3, 4, 3, 3, 4, 3], # random
       [2, 4, 3, 4, 0, 7]])

For the first run, we threw 3 times 1, 4 times 2, etc. For the second, we threw 2 times 1, 4 times 2, etc.

A loaded die is more likely to land on number 6:

>>> np.random.multinomial(100, [1/7.]*5 + [2/7.])
array([11, 16, 14, 17, 16, 26]) # random

The probability inputs should be normalized. As an implementation detail, the value of the last entry is ignored and assumed to take up any leftover probability mass, but this should not be relied on. A biased coin which has twice as much weight on one side as on the other should be sampled like so:

>>> np.random.multinomial(100, [1.0 / 3, 2.0 / 3])  # RIGHT
array([38, 62]) # random

not like:

>>> np.random.multinomial(100, [1.0, 2.0])  # WRONG
Traceback (most recent call last):
ValueError: pvals < 0, pvals > 1 or pvals contains NaNs
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.multivariate_normal(mean, cov, size=None, check_valid='warn', tol=1e-8)

Draw random samples from a multivariate normal distribution.

The multivariate normal, multinormal or Gaussian distribution is a generalization of the one-dimensional normal distribution to higher dimensions. Such a distribution is specified by its mean and covariance matrix. These parameters are analogous to the mean (average or “center”) and variance (standard deviation, or “width,” squared) of the one-dimensional normal distribution.

Note

New code should use the multivariate_normal method of a default_rng() instance instead; see random-quick-start.

mean : 1-D array_like, of length N
Mean of the N-dimensional distribution.
cov : 2-D array_like, of shape (N, N)
Covariance matrix of the distribution. It must be symmetric and positive-semidefinite for proper sampling.
size : int or tuple of ints, optional
Given a shape of, for example, (m,n,k), m*n*k samples are generated, and packed in an m-by-n-by-k arrangement. Because each sample is N-dimensional, the output shape is (m,n,k,N). If no shape is specified, a single (N-D) sample is returned.
check_valid : { ‘warn’, ‘raise’, ‘ignore’ }, optional
Behavior when the covariance matrix is not positive semidefinite.
tol : float, optional
Tolerance when checking the singular values in covariance matrix. cov is cast to double before the check.
out : ndarray

The drawn samples, of shape size, if that was provided. If not, the shape is (N,).

In other words, each entry out[i,j,...,:] is an N-dimensional value drawn from the distribution.

Generator.multivariate_normal: which should be used for new code.

The mean is a coordinate in N-dimensional space, which represents the location where samples are most likely to be generated. This is analogous to the peak of the bell curve for the one-dimensional or univariate normal distribution.

Covariance indicates the level to which two variables vary together. From the multivariate normal distribution, we draw N-dimensional samples, \(X = [x_1, x_2, ... x_N]\). The covariance matrix element \(C_{ij}\) is the covariance of \(x_i\) and \(x_j\). The element \(C_{ii}\) is the variance of \(x_i\) (i.e. its “spread”).

Instead of specifying the full covariance matrix, popular approximations include:

  • Spherical covariance (cov is a multiple of the identity matrix)
  • Diagonal covariance (cov has non-negative elements, and only on the diagonal)

This geometrical property can be seen in two dimensions by plotting generated data-points:

>>> mean = [0, 0]
>>> cov = [[1, 0], [0, 100]]  # diagonal covariance

Diagonal covariance means that points are oriented along x or y-axis:

>>> import matplotlib.pyplot as plt
>>> x, y = np.random.multivariate_normal(mean, cov, 5000).T
>>> plt.plot(x, y, 'x')
>>> plt.axis('equal')
>>> plt.show()

Note that the covariance matrix must be positive semidefinite (a.k.a. nonnegative-definite). Otherwise, the behavior of this method is undefined and backwards compatibility is not guaranteed.

[1]Papoulis, A., “Probability, Random Variables, and Stochastic Processes,” 3rd ed., New York: McGraw-Hill, 1991.
[2]Duda, R. O., Hart, P. E., and Stork, D. G., “Pattern Classification,” 2nd ed., New York: Wiley, 2001.
>>> mean = (1, 2)
>>> cov = [[1, 0], [0, 1]]
>>> x = np.random.multivariate_normal(mean, cov, (3, 3))
>>> x.shape
(3, 3, 2)

The following is probably true, given that 0.6 is roughly twice the standard deviation:

>>> list((x[0,0,:] - mean) < 0.6)
[True, True] # random
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.negative_binomial(n, p, size=None)

Draw samples from a negative binomial distribution.

Samples are drawn from a negative binomial distribution with specified parameters, n successes and p probability of success where n is > 0 and p is in the interval [0, 1].

Note

New code should use the negative_binomial method of a default_rng() instance instead; see random-quick-start.

n : float or array_like of floats
Parameter of the distribution, > 0.
p : float or array_like of floats
Parameter of the distribution, >= 0 and <=1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if n and p are both scalars. Otherwise, np.broadcast(n, p).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized negative binomial distribution, where each sample is equal to N, the number of failures that occurred before a total of n successes was reached.

Generator.negative_binomial: which should be used for new code.

The probability mass function of the negative binomial distribution is

\[P(N;n,p) = \frac{\Gamma(N+n)}{N!\Gamma(n)}p^{n}(1-p)^{N},\]

where \(n\) is the number of successes, \(p\) is the probability of success, \(N+n\) is the number of trials, and \(\Gamma\) is the gamma function. When \(n\) is an integer, \(\frac{\Gamma(N+n)}{N!\Gamma(n)} = \binom{N+n-1}{N}\), which is the more common form of this term in the the pmf. The negative binomial distribution gives the probability of N failures given n successes, with a success on the last trial.

If one throws a die repeatedly until the third time a “1” appears, then the probability distribution of the number of non-“1”s that appear before the third “1” is a negative binomial distribution.

[1]Weisstein, Eric W. “Negative Binomial Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/NegativeBinomialDistribution.html
[2]Wikipedia, “Negative binomial distribution”, https://en.wikipedia.org/wiki/Negative_binomial_distribution

Draw samples from the distribution:

A real world example. A company drills wild-cat oil exploration wells, each with an estimated probability of success of 0.1. What is the probability of having one success for each successive well, that is what is the probability of a single success after drilling 5 wells, after 6 wells, etc.?

>>> s = np.random.negative_binomial(1, 0.1, 100000)
>>> for i in range(1, 11): 
...    probability = sum(s<i) / 100000.
...    print(i, "wells drilled, probability of one success =", probability)
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.noncentral_chisquare(df, nonc, size=None)

Draw samples from a noncentral chi-square distribution.

The noncentral \(\chi^2\) distribution is a generalization of the \(\chi^2\) distribution.

Note

New code should use the noncentral_chisquare method of a default_rng() instance instead; see random-quick-start.

df : float or array_like of floats

Degrees of freedom, must be > 0.

Changed in version 1.10.0: Earlier NumPy versions required dfnum > 1.

nonc : float or array_like of floats
Non-centrality, must be non-negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if df and nonc are both scalars. Otherwise, np.broadcast(df, nonc).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized noncentral chi-square distribution.

Generator.noncentral_chisquare: which should be used for new code.

The probability density function for the noncentral Chi-square distribution is

\[P(x;df,nonc) = \sum^{\infty}_{i=0} \frac{e^{-nonc/2}(nonc/2)^{i}}{i!} P_{Y_{df+2i}}(x),\]

where \(Y_{q}\) is the Chi-square with q degrees of freedom.

[1]Wikipedia, “Noncentral chi-squared distribution” https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution

Draw values from the distribution and plot the histogram

>>> import matplotlib.pyplot as plt
>>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000),
...                   bins=200, density=True)
>>> plt.show()

Draw values from a noncentral chisquare with very small noncentrality, and compare to a chisquare.

>>> plt.figure()
>>> values = plt.hist(np.random.noncentral_chisquare(3, .0000001, 100000),
...                   bins=np.arange(0., 25, .1), density=True)
>>> values2 = plt.hist(np.random.chisquare(3, 100000),
...                    bins=np.arange(0., 25, .1), density=True)
>>> plt.plot(values[1][0:-1], values[0]-values2[0], 'ob')
>>> plt.show()

Demonstrate how large values of non-centrality lead to a more symmetric distribution.

>>> plt.figure()
>>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000),
...                   bins=200, density=True)
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.noncentral_f(dfnum, dfden, nonc, size=None)

Draw samples from the noncentral F distribution.

Samples are drawn from an F distribution with specified parameters, dfnum (degrees of freedom in numerator) and dfden (degrees of freedom in denominator), where both parameters > 1. nonc is the non-centrality parameter.

Note

New code should use the noncentral_f method of a default_rng() instance instead; see random-quick-start.

dfnum : float or array_like of floats

Numerator degrees of freedom, must be > 0.

Changed in version 1.14.0: Earlier NumPy versions required dfnum > 1.

dfden : float or array_like of floats
Denominator degrees of freedom, must be > 0.
nonc : float or array_like of floats
Non-centrality parameter, the sum of the squares of the numerator means, must be >= 0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if dfnum, dfden, and nonc are all scalars. Otherwise, np.broadcast(dfnum, dfden, nonc).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized noncentral Fisher distribution.

Generator.noncentral_f: which should be used for new code.

When calculating the power of an experiment (power = probability of rejecting the null hypothesis when a specific alternative is true) the non-central F statistic becomes important. When the null hypothesis is true, the F statistic follows a central F distribution. When the null hypothesis is not true, then it follows a non-central F statistic.

[1]Weisstein, Eric W. “Noncentral F-Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/NoncentralF-Distribution.html
[2]Wikipedia, “Noncentral F-distribution”, https://en.wikipedia.org/wiki/Noncentral_F-distribution

In a study, testing for a specific alternative to the null hypothesis requires use of the Noncentral F distribution. We need to calculate the area in the tail of the distribution that exceeds the value of the F distribution for the null hypothesis. We’ll plot the two probability distributions for comparison.

>>> dfnum = 3 # between group deg of freedom
>>> dfden = 20 # within groups degrees of freedom
>>> nonc = 3.0
>>> nc_vals = np.random.noncentral_f(dfnum, dfden, nonc, 1000000)
>>> NF = np.histogram(nc_vals, bins=50, density=True)
>>> c_vals = np.random.f(dfnum, dfden, 1000000)
>>> F = np.histogram(c_vals, bins=50, density=True)
>>> import matplotlib.pyplot as plt
>>> plt.plot(F[1][1:], F[0])
>>> plt.plot(NF[1][1:], NF[0])
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.normal(loc=0.0, scale=1.0, size=None)

Draw random samples from a normal (Gaussian) distribution.

The probability density function of the normal distribution, first derived by De Moivre and 200 years later by both Gauss and Laplace independently [2]_, is often called the bell curve because of its characteristic shape (see the example below).

The normal distributions occurs often in nature. For example, it describes the commonly occurring distribution of samples influenced by a large number of tiny, random disturbances, each with its own unique distribution [2]_.

Note

New code should use the normal method of a default_rng() instance instead; see random-quick-start.

loc : float or array_like of floats
Mean (“centre”) of the distribution.
scale : float or array_like of floats
Standard deviation (spread or “width”) of the distribution. Must be non-negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if loc and scale are both scalars. Otherwise, np.broadcast(loc, scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized normal distribution.
scipy.stats.norm : probability density function, distribution or
cumulative density function, etc.

Generator.normal: which should be used for new code.

The probability density for the Gaussian distribution is

\[p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }} e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} },\]

where \(\mu\) is the mean and \(\sigma\) the standard deviation. The square of the standard deviation, \(\sigma^2\), is called the variance.

The function has its peak at the mean, and its “spread” increases with the standard deviation (the function reaches 0.607 times its maximum at \(x + \sigma\) and \(x - \sigma\) [2]_). This implies that normal is more likely to return samples lying close to the mean, rather than those far away.

[1]Wikipedia, “Normal distribution”, https://en.wikipedia.org/wiki/Normal_distribution
[2]P. R. Peebles Jr., “Central Limit Theorem” in “Probability, Random Variables and Random Signal Principles”, 4th ed., 2001, pp. 51, 51, 125.

Draw samples from the distribution:

>>> mu, sigma = 0, 0.1 # mean and standard deviation
>>> s = np.random.normal(mu, sigma, 1000)

Verify the mean and the variance:

>>> abs(mu - np.mean(s))
0.0  # may vary
>>> abs(sigma - np.std(s, ddof=1))
0.1  # may vary

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, density=True)
>>> plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) *
...                np.exp( - (bins - mu)**2 / (2 * sigma**2) ),
...          linewidth=2, color='r')
>>> plt.show()

Two-by-four array of samples from N(3, 6.25):

>>> np.random.normal(3, 2.5, size=(2, 4))
array([[-4.49401501,  4.00950034, -1.81814867,  7.29718677],   # random
       [ 0.39924804,  4.68456316,  4.99394529,  4.84057254]])  # random
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.pareto(a, size=None)

Draw samples from a Pareto II or Lomax distribution with specified shape.

The Lomax or Pareto II distribution is a shifted Pareto distribution. The classical Pareto distribution can be obtained from the Lomax distribution by adding 1 and multiplying by the scale parameter m (see Notes). The smallest value of the Lomax distribution is zero while for the classical Pareto distribution it is mu, where the standard Pareto distribution has location mu = 1. Lomax can also be considered as a simplified version of the Generalized Pareto distribution (available in SciPy), with the scale set to one and the location set to zero.

The Pareto distribution must be greater than zero, and is unbounded above. It is also known as the “80-20 rule”. In this distribution, 80 percent of the weights are in the lowest 20 percent of the range, while the other 20 percent fill the remaining 80 percent of the range.

Note

New code should use the pareto method of a default_rng() instance instead; see random-quick-start.

a : float or array_like of floats
Shape of the distribution. Must be positive.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if a is a scalar. Otherwise, np.array(a).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Pareto distribution.
scipy.stats.lomax : probability density function, distribution or
cumulative density function, etc.
scipy.stats.genpareto : probability density function, distribution or
cumulative density function, etc.

Generator.pareto: which should be used for new code.

The probability density for the Pareto distribution is

\[p(x) = \frac{am^a}{x^{a+1}}\]

where \(a\) is the shape and \(m\) the scale.

The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution useful in many real world problems. Outside the field of economics it is generally referred to as the Bradford distribution. Pareto developed the distribution to describe the distribution of wealth in an economy. It has also found use in insurance, web page access statistics, oil field sizes, and many other problems, including the download frequency for projects in Sourceforge [1]_. It is one of the so-called “fat-tailed” distributions.

[1]Francis Hunt and Paul Johnson, On the Pareto Distribution of Sourceforge projects.
[2]Pareto, V. (1896). Course of Political Economy. Lausanne.
[3]Reiss, R.D., Thomas, M.(2001), Statistical Analysis of Extreme Values, Birkhauser Verlag, Basel, pp 23-30.
[4]Wikipedia, “Pareto distribution”, https://en.wikipedia.org/wiki/Pareto_distribution

Draw samples from the distribution:

>>> a, m = 3., 2.  # shape and mode
>>> s = (np.random.pareto(a, 1000) + 1) * m

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, _ = plt.hist(s, 100, density=True)
>>> fit = a*m**a / bins**(a+1)
>>> plt.plot(bins, max(count)*fit/max(fit), linewidth=2, color='r')
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.permutation(x)

Randomly permute a sequence, or return a permuted range.

If x is a multi-dimensional array, it is only shuffled along its first index.

Note

New code should use the permutation method of a default_rng() instance instead; see random-quick-start.

x : int or array_like
If x is an integer, randomly permute np.arange(x). If x is an array, make a copy and shuffle the elements randomly.
out : ndarray
Permuted sequence or array range.

Generator.permutation: which should be used for new code.

>>> np.random.permutation(10)
array([1, 7, 4, 3, 0, 9, 2, 5, 8, 6]) # random
>>> np.random.permutation([1, 4, 9, 12, 15])
array([15,  1,  9,  4, 12]) # random
>>> arr = np.arange(9).reshape((3, 3))
>>> np.random.permutation(arr)
array([[6, 7, 8], # random
       [0, 1, 2],
       [3, 4, 5]])
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.poisson(lam=1.0, size=None)

Draw samples from a Poisson distribution.

The Poisson distribution is the limit of the binomial distribution for large N.

Note

New code should use the poisson method of a default_rng() instance instead; see random-quick-start.

lam : float or array_like of floats
Expectation of interval, must be >= 0. A sequence of expectation intervals must be broadcastable over the requested size.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if lam is a scalar. Otherwise, np.array(lam).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Poisson distribution.

Generator.poisson: which should be used for new code.

The Poisson distribution

\[f(k; \lambda)=\frac{\lambda^k e^{-\lambda}}{k!}\]

For events with an expected separation \(\lambda\) the Poisson distribution \(f(k; \lambda)\) describes the probability of \(k\) events occurring within the observed interval \(\lambda\).

Because the output is limited to the range of the C int64 type, a ValueError is raised when lam is within 10 sigma of the maximum representable value.

[1]Weisstein, Eric W. “Poisson Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/PoissonDistribution.html
[2]Wikipedia, “Poisson distribution”, https://en.wikipedia.org/wiki/Poisson_distribution

Draw samples from the distribution:

>>> import numpy as np
>>> s = np.random.poisson(5, 10000)

Display histogram of the sample:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 14, density=True)
>>> plt.show()

Draw each 100 values for lambda 100 and 500:

>>> s = np.random.poisson(lam=(100., 500.), size=(100, 2))
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.power(a, size=None)

Draws samples in [0, 1] from a power distribution with positive exponent a - 1.

Also known as the power function distribution.

Note

New code should use the power method of a default_rng() instance instead; see random-quick-start.

a : float or array_like of floats
Parameter of the distribution. Must be non-negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if a is a scalar. Otherwise, np.array(a).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized power distribution.
ValueError
If a < 1.

Generator.power: which should be used for new code.

The probability density function is

\[\begin{split}P(x; a) = ax^{a-1}, 0 \le x \le 1, a>0.\end{split}\]

The power function distribution is just the inverse of the Pareto distribution. It may also be seen as a special case of the Beta distribution.

It is used, for example, in modeling the over-reporting of insurance claims.

[1]Christian Kleiber, Samuel Kotz, “Statistical size distributions in economics and actuarial sciences”, Wiley, 2003.
[2]Heckert, N. A. and Filliben, James J. “NIST Handbook 148: Dataplot Reference Manual, Volume 2: Let Subcommands and Library Functions”, National Institute of Standards and Technology Handbook Series, June 2003. https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/powpdf.pdf

Draw samples from the distribution:

>>> a = 5. # shape
>>> samples = 1000
>>> s = np.random.power(a, samples)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, bins=30)
>>> x = np.linspace(0, 1, 100)
>>> y = a*x**(a-1.)
>>> normed_y = samples*np.diff(bins)[0]*y
>>> plt.plot(x, normed_y)
>>> plt.show()

Compare the power function distribution to the inverse of the Pareto.

>>> from scipy import stats 
>>> rvs = np.random.power(5, 1000000)
>>> rvsp = np.random.pareto(5, 1000000)
>>> xx = np.linspace(0,1,100)
>>> powpdf = stats.powerlaw.pdf(xx,5)  
>>> plt.figure()
>>> plt.hist(rvs, bins=50, density=True)
>>> plt.plot(xx,powpdf,'r-')  
>>> plt.title('np.random.power(5)')
>>> plt.figure()
>>> plt.hist(1./(1.+rvsp), bins=50, density=True)
>>> plt.plot(xx,powpdf,'r-')  
>>> plt.title('inverse of 1 + np.random.pareto(5)')
>>> plt.figure()
>>> plt.hist(1./(1.+rvsp), bins=50, density=True)
>>> plt.plot(xx,powpdf,'r-')  
>>> plt.title('inverse of stats.pareto(5)')
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.rand(d0, d1, ..., dn)

Random values in a given shape.

Note

This is a convenience function for users porting code from Matlab, and wraps random_sample. That function takes a tuple to specify the size of the output, which is consistent with other NumPy functions like numpy.zeros and numpy.ones.

Create an array of the given shape and populate it with random samples from a uniform distribution over [0, 1).

d0, d1, ..., dn : int, optional
The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned.
out : ndarray, shape (d0, d1, ..., dn)
Random values.

random

>>> np.random.rand(3,2)
array([[ 0.14022471,  0.96360618],  #random
       [ 0.37601032,  0.25528411],  #random
       [ 0.49313049,  0.94909878]]) #random
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.randint(low, high=None, size=None, dtype=int)

Return random integers from low (inclusive) to high (exclusive).

Return random integers from the “discrete uniform” distribution of the specified dtype in the “half-open” interval [low, high). If high is None (the default), then results are from [0, low).

Note

New code should use the integers method of a default_rng() instance instead; see random-quick-start.

low : int or array-like of ints
Lowest (signed) integers to be drawn from the distribution (unless high=None, in which case this parameter is one above the highest such integer).
high : int or array-like of ints, optional
If provided, one above the largest (signed) integer to be drawn from the distribution (see above for behavior if high=None). If array-like, must contain integer values
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
dtype : dtype, optional

Desired dtype of the result. Byteorder must be native. The default value is int.

New in version 1.11.0.

out : int or ndarray of ints
size-shaped array of random integers from the appropriate distribution, or a single such random int if size not provided.
random_integers : similar to randint, only for the closed
interval [low, high], and 1 is the lowest value if high is omitted.

Generator.integers: which should be used for new code.

>>> np.random.randint(2, size=10)
array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0]) # random
>>> np.random.randint(1, size=10)
array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0])

Generate a 2 x 4 array of ints between 0 and 4, inclusive:

>>> np.random.randint(5, size=(2, 4))
array([[4, 0, 2, 1], # random
       [3, 2, 2, 0]])

Generate a 1 x 3 array with 3 different upper bounds

>>> np.random.randint(1, [3, 5, 10])
array([2, 2, 9]) # random

Generate a 1 by 3 array with 3 different lower bounds

>>> np.random.randint([1, 5, 7], 10)
array([9, 8, 7]) # random

Generate a 2 by 4 array using broadcasting with dtype of uint8

>>> np.random.randint([1, 3, 5, 7], [[10], [20]], dtype=np.uint8)
array([[ 8,  6,  9,  7], # random
       [ 1, 16,  9, 12]], dtype=uint8)
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.randn(d0, d1, ..., dn)

Return a sample (or samples) from the “standard normal” distribution.

Note

This is a convenience function for users porting code from Matlab, and wraps standard_normal. That function takes a tuple to specify the size of the output, which is consistent with other NumPy functions like numpy.zeros and numpy.ones.

Note

New code should use the standard_normal method of a default_rng() instance instead; see random-quick-start.

If positive int_like arguments are provided, randn generates an array of shape (d0, d1, ..., dn), filled with random floats sampled from a univariate “normal” (Gaussian) distribution of mean 0 and variance 1. A single float randomly sampled from the distribution is returned if no argument is provided.

d0, d1, ..., dn : int, optional
The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned.
Z : ndarray or float
A (d0, d1, ..., dn)-shaped array of floating-point samples from the standard normal distribution, or a single such float if no parameters were supplied.

standard_normal : Similar, but takes a tuple as its argument. normal : Also accepts mu and sigma arguments. Generator.standard_normal: which should be used for new code.

For random samples from \(N(\mu, \sigma^2)\), use:

sigma * np.random.randn(...) + mu

>>> np.random.randn()
2.1923875335537315  # random

Two-by-four array of samples from N(3, 6.25):

>>> 3 + 2.5 * np.random.randn(2, 4)
array([[-4.49401501,  4.00950034, -1.81814867,  7.29718677],   # random
       [ 0.39924804,  4.68456316,  4.99394529,  4.84057254]])  # random
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.random(size=None)

Return random floats in the half-open interval [0.0, 1.0). Alias for random_sample to ease forward-porting to the new random API.

tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.random_integers(low, high=None, size=None)

Random integers of type np.int_ between low and high, inclusive.

Return random integers of type np.int_ from the “discrete uniform” distribution in the closed interval [low, high]. If high is None (the default), then results are from [1, low]. The np.int_ type translates to the C long integer type and its precision is platform dependent.

This function has been deprecated. Use randint instead.

Deprecated since version 1.11.0.

low : int
Lowest (signed) integer to be drawn from the distribution (unless high=None, in which case this parameter is the highest such integer).
high : int, optional
If provided, the largest (signed) integer to be drawn from the distribution (see above for behavior if high=None).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
out : int or ndarray of ints
size-shaped array of random integers from the appropriate distribution, or a single such random int if size not provided.
randint : Similar to random_integers, only for the half-open
interval [low, high), and 0 is the lowest value if high is omitted.

To sample from N evenly spaced floating-point numbers between a and b, use:

a + (b - a) * (np.random.random_integers(N) - 1) / (N - 1.)
>>> np.random.random_integers(5)
4 # random
>>> type(np.random.random_integers(5))
<class 'numpy.int64'>
>>> np.random.random_integers(5, size=(3,2))
array([[5, 4], # random
       [3, 3],
       [4, 5]])

Choose five random numbers from the set of five evenly-spaced numbers between 0 and 2.5, inclusive (i.e., from the set \({0, 5/8, 10/8, 15/8, 20/8}\)):

>>> 2.5 * (np.random.random_integers(5, size=(5,)) - 1) / 4.
array([ 0.625,  1.25 ,  0.625,  0.625,  2.5  ]) # random

Roll two six sided dice 1000 times and sum the results:

>>> d1 = np.random.random_integers(1, 6, 1000)
>>> d2 = np.random.random_integers(1, 6, 1000)
>>> dsums = d1 + d2

Display results as a histogram:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(dsums, 11, density=True)
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.random_sample(size=None)

Return random floats in the half-open interval [0.0, 1.0).

Results are from the “continuous uniform” distribution over the stated interval. To sample \(Unif[a, b), b > a\) multiply the output of random_sample by (b-a) and add a:

(b - a) * random_sample() + a

Note

New code should use the random method of a default_rng() instance instead; see random-quick-start.

size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
out : float or ndarray of floats
Array of random floats of shape size (unless size=None, in which case a single float is returned).

Generator.random: which should be used for new code.

>>> np.random.random_sample()
0.47108547995356098 # random
>>> type(np.random.random_sample())
<class 'float'>
>>> np.random.random_sample((5,))
array([ 0.30220482,  0.86820401,  0.1654503 ,  0.11659149,  0.54323428]) # random

Three-by-two array of random numbers from [-5, 0):

>>> 5 * np.random.random_sample((3, 2)) - 5
array([[-3.99149989, -0.52338984], # random
       [-2.99091858, -0.79479508],
       [-1.23204345, -1.75224494]])
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.rayleigh(scale=1.0, size=None)

Draw samples from a Rayleigh distribution.

The \(\chi\) and Weibull distributions are generalizations of the Rayleigh.

Note

New code should use the rayleigh method of a default_rng() instance instead; see random-quick-start.

scale : float or array_like of floats, optional
Scale, also equals the mode. Must be non-negative. Default is 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if scale is a scalar. Otherwise, np.array(scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Rayleigh distribution.

Generator.rayleigh: which should be used for new code.

The probability density function for the Rayleigh distribution is

\[P(x;scale) = \frac{x}{scale^2}e^{\frac{-x^2}{2 \cdotp scale^2}}\]

The Rayleigh distribution would arise, for example, if the East and North components of the wind velocity had identical zero-mean Gaussian distributions. Then the wind speed would have a Rayleigh distribution.

[1]Brighton Webs Ltd., “Rayleigh Distribution,” https://web.archive.org/web/20090514091424/http://brighton-webs.co.uk:80/distributions/rayleigh.asp
[2]Wikipedia, “Rayleigh distribution” https://en.wikipedia.org/wiki/Rayleigh_distribution

Draw values from the distribution and plot the histogram

>>> from matplotlib.pyplot import hist
>>> values = hist(np.random.rayleigh(3, 100000), bins=200, density=True)

Wave heights tend to follow a Rayleigh distribution. If the mean wave height is 1 meter, what fraction of waves are likely to be larger than 3 meters?

>>> meanvalue = 1
>>> modevalue = np.sqrt(2 / np.pi) * meanvalue
>>> s = np.random.rayleigh(modevalue, 1000000)

The percentage of waves larger than 3 meters is:

>>> 100.*sum(s>3)/1000000.
0.087300000000000003 # random
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.seed(self, seed=None)

Reseed a legacy MT19937 BitGenerator

This is a convenience, legacy function.

The best practice is to not reseed a BitGenerator, rather to recreate a new one. This method is here for legacy reasons. This example demonstrates best practice.

>>> from numpy.random import MT19937
>>> from numpy.random import RandomState, SeedSequence
>>> rs = RandomState(MT19937(SeedSequence(123456789)))
# Later, you want to restart the stream
>>> rs = RandomState(MT19937(SeedSequence(987654321)))
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.set_state(state)

Set the internal state of the generator from a tuple.

For use if one has reason to manually (re-)set the internal state of the bit generator used by the RandomState instance. By default, RandomState uses the “Mersenne Twister”[1]_ pseudo-random number generating algorithm.

state : {tuple(str, ndarray of 624 uints, int, int, float), dict}

The state tuple has the following items:

  1. the string ‘MT19937’, specifying the Mersenne Twister algorithm.
  2. a 1-D array of 624 unsigned integers keys.
  3. an integer pos.
  4. an integer has_gauss.
  5. a float cached_gaussian.

If state is a dictionary, it is directly set using the BitGenerators state property.

out : None
Returns ‘None’ on success.

get_state

set_state and get_state are not needed to work with any of the random distributions in NumPy. If the internal state is manually altered, the user should know exactly what he/she is doing.

For backwards compatibility, the form (str, array of 624 uints, int) is also accepted although it is missing some information about the cached Gaussian value: state = ('MT19937', keys, pos).

[1]M. Matsumoto and T. Nishimura, “Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator,” ACM Trans. on Modeling and Computer Simulation, Vol. 8, No. 1, pp. 3-30, Jan. 1998.
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.shuffle(x)

Modify a sequence in-place by shuffling its contents.

This function only shuffles the array along the first axis of a multi-dimensional array. The order of sub-arrays is changed but their contents remains the same.

Note

New code should use the shuffle method of a default_rng() instance instead; see random-quick-start.

x : array_like
The array or list to be shuffled.

None

Generator.shuffle: which should be used for new code.

>>> arr = np.arange(10)
>>> np.random.shuffle(arr)
>>> arr
[1 7 5 2 9 4 3 6 0 8] # random

Multi-dimensional arrays are only shuffled along the first axis:

>>> arr = np.arange(9).reshape((3, 3))
>>> np.random.shuffle(arr)
>>> arr
array([[3, 4, 5], # random
       [6, 7, 8],
       [0, 1, 2]])
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.standard_cauchy(size=None)

Draw samples from a standard Cauchy distribution with mode = 0.

Also known as the Lorentz distribution.

Note

New code should use the standard_cauchy method of a default_rng() instance instead; see random-quick-start.

size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
samples : ndarray or scalar
The drawn samples.

Generator.standard_cauchy: which should be used for new code.

The probability density function for the full Cauchy distribution is

\[P(x; x_0, \gamma) = \frac{1}{\pi \gamma \bigl[ 1+ (\frac{x-x_0}{\gamma})^2 \bigr] }\]

and the Standard Cauchy distribution just sets \(x_0=0\) and \(\gamma=1\)

The Cauchy distribution arises in the solution to the driven harmonic oscillator problem, and also describes spectral line broadening. It also describes the distribution of values at which a line tilted at a random angle will cut the x axis.

When studying hypothesis tests that assume normality, seeing how the tests perform on data from a Cauchy distribution is a good indicator of their sensitivity to a heavy-tailed distribution, since the Cauchy looks very much like a Gaussian distribution, but with heavier tails.

[1]NIST/SEMATECH e-Handbook of Statistical Methods, “Cauchy Distribution”, https://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm
[2]Weisstein, Eric W. “Cauchy Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/CauchyDistribution.html
[3]Wikipedia, “Cauchy distribution” https://en.wikipedia.org/wiki/Cauchy_distribution

Draw samples and plot the distribution:

>>> import matplotlib.pyplot as plt
>>> s = np.random.standard_cauchy(1000000)
>>> s = s[(s>-25) & (s<25)]  # truncate distribution so it plots well
>>> plt.hist(s, bins=100)
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.standard_exponential(size=None)

Draw samples from the standard exponential distribution.

standard_exponential is identical to the exponential distribution with a scale parameter of 1.

Note

New code should use the standard_exponential method of a default_rng() instance instead; see random-quick-start.

size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
out : float or ndarray
Drawn samples.

Generator.standard_exponential: which should be used for new code.

Output a 3x8000 array:

>>> n = np.random.standard_exponential((3, 8000))
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.standard_gamma(shape, size=None)

Draw samples from a standard Gamma distribution.

Samples are drawn from a Gamma distribution with specified parameters, shape (sometimes designated “k”) and scale=1.

Note

New code should use the standard_gamma method of a default_rng() instance instead; see random-quick-start.

shape : float or array_like of floats
Parameter, must be non-negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if shape is a scalar. Otherwise, np.array(shape).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized standard gamma distribution.
scipy.stats.gamma : probability density function, distribution or
cumulative density function, etc.

Generator.standard_gamma: which should be used for new code.

The probability density for the Gamma distribution is

\[p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},\]

where \(k\) is the shape and \(\theta\) the scale, and \(\Gamma\) is the Gamma function.

The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant.

[1]Weisstein, Eric W. “Gamma Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html
[2]Wikipedia, “Gamma distribution”, https://en.wikipedia.org/wiki/Gamma_distribution

Draw samples from the distribution:

>>> shape, scale = 2., 1. # mean and width
>>> s = np.random.standard_gamma(shape, 1000000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> import scipy.special as sps  
>>> count, bins, ignored = plt.hist(s, 50, density=True)
>>> y = bins**(shape-1) * ((np.exp(-bins/scale))/  
...                       (sps.gamma(shape) * scale**shape))
>>> plt.plot(bins, y, linewidth=2, color='r')  
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.standard_normal(size=None)

Draw samples from a standard Normal distribution (mean=0, stdev=1).

Note

New code should use the standard_normal method of a default_rng() instance instead; see random-quick-start.

size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
out : float or ndarray
A floating-point array of shape size of drawn samples, or a single sample if size was not specified.
normal :
Equivalent function with additional loc and scale arguments for setting the mean and standard deviation.

Generator.standard_normal: which should be used for new code.

For random samples from \(N(\mu, \sigma^2)\), use one of:

mu + sigma * np.random.standard_normal(size=...)
np.random.normal(mu, sigma, size=...)
>>> np.random.standard_normal()
2.1923875335537315 #random
>>> s = np.random.standard_normal(8000)
>>> s
array([ 0.6888893 ,  0.78096262, -0.89086505, ...,  0.49876311,  # random
       -0.38672696, -0.4685006 ])                                # random
>>> s.shape
(8000,)
>>> s = np.random.standard_normal(size=(3, 4, 2))
>>> s.shape
(3, 4, 2)

Two-by-four array of samples from \(N(3, 6.25)\):

>>> 3 + 2.5 * np.random.standard_normal(size=(2, 4))
array([[-4.49401501,  4.00950034, -1.81814867,  7.29718677],   # random
       [ 0.39924804,  4.68456316,  4.99394529,  4.84057254]])  # random
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.standard_t(df, size=None)

Draw samples from a standard Student’s t distribution with df degrees of freedom.

A special case of the hyperbolic distribution. As df gets large, the result resembles that of the standard normal distribution (standard_normal).

Note

New code should use the standard_t method of a default_rng() instance instead; see random-quick-start.

df : float or array_like of floats
Degrees of freedom, must be > 0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if df is a scalar. Otherwise, np.array(df).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized standard Student’s t distribution.

Generator.standard_t: which should be used for new code.

The probability density function for the t distribution is

\[P(x, df) = \frac{\Gamma(\frac{df+1}{2})}{\sqrt{\pi df} \Gamma(\frac{df}{2})}\Bigl( 1+\frac{x^2}{df} \Bigr)^{-(df+1)/2}\]

The t test is based on an assumption that the data come from a Normal distribution. The t test provides a way to test whether the sample mean (that is the mean calculated from the data) is a good estimate of the true mean.

The derivation of the t-distribution was first published in 1908 by William Gosset while working for the Guinness Brewery in Dublin. Due to proprietary issues, he had to publish under a pseudonym, and so he used the name Student.

[1]Dalgaard, Peter, “Introductory Statistics With R”, Springer, 2002.
[2]Wikipedia, “Student’s t-distribution” https://en.wikipedia.org/wiki/Student’s_t-distribution

From Dalgaard page 83 [1]_, suppose the daily energy intake for 11 women in kilojoules (kJ) is:

>>> intake = np.array([5260., 5470, 5640, 6180, 6390, 6515, 6805, 7515, \
...                    7515, 8230, 8770])

Does their energy intake deviate systematically from the recommended value of 7725 kJ?

We have 10 degrees of freedom, so is the sample mean within 95% of the recommended value?

>>> s = np.random.standard_t(10, size=100000)
>>> np.mean(intake)
6753.636363636364
>>> intake.std(ddof=1)
1142.1232221373727

Calculate the t statistic, setting the ddof parameter to the unbiased value so the divisor in the standard deviation will be degrees of freedom, N-1.

>>> t = (np.mean(intake)-7725)/(intake.std(ddof=1)/np.sqrt(len(intake)))
>>> import matplotlib.pyplot as plt
>>> h = plt.hist(s, bins=100, density=True)

For a one-sided t-test, how far out in the distribution does the t statistic appear?

>>> np.sum(s<t) / float(len(s))
0.0090699999999999999  #random

So the p-value is about 0.009, which says the null hypothesis has a probability of about 99% of being true.

tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.triangular(left, mode, right, size=None)

Draw samples from the triangular distribution over the interval [left, right].

The triangular distribution is a continuous probability distribution with lower limit left, peak at mode, and upper limit right. Unlike the other distributions, these parameters directly define the shape of the pdf.

Note

New code should use the triangular method of a default_rng() instance instead; see random-quick-start.

left : float or array_like of floats
Lower limit.
mode : float or array_like of floats
The value where the peak of the distribution occurs. The value must fulfill the condition left <= mode <= right.
right : float or array_like of floats
Upper limit, must be larger than left.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if left, mode, and right are all scalars. Otherwise, np.broadcast(left, mode, right).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized triangular distribution.

Generator.triangular: which should be used for new code.

The probability density function for the triangular distribution is

\[\begin{split}P(x;l, m, r) = \begin{cases} \frac{2(x-l)}{(r-l)(m-l)}& \text{for $l \leq x \leq m$},\\ \frac{2(r-x)}{(r-l)(r-m)}& \text{for $m \leq x \leq r$},\\ 0& \text{otherwise}. \end{cases}\end{split}\]

The triangular distribution is often used in ill-defined problems where the underlying distribution is not known, but some knowledge of the limits and mode exists. Often it is used in simulations.

[1]Wikipedia, “Triangular distribution” https://en.wikipedia.org/wiki/Triangular_distribution

Draw values from the distribution and plot the histogram:

>>> import matplotlib.pyplot as plt
>>> h = plt.hist(np.random.triangular(-3, 0, 8, 100000), bins=200,
...              density=True)
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.uniform(low=0.0, high=1.0, size=None)

Draw samples from a uniform distribution.

Samples are uniformly distributed over the half-open interval [low, high) (includes low, but excludes high). In other words, any value within the given interval is equally likely to be drawn by uniform.

Note

New code should use the uniform method of a default_rng() instance instead; see random-quick-start.

low : float or array_like of floats, optional
Lower boundary of the output interval. All values generated will be greater than or equal to low. The default value is 0.
high : float or array_like of floats
Upper boundary of the output interval. All values generated will be less than or equal to high. The default value is 1.0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if low and high are both scalars. Otherwise, np.broadcast(low, high).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized uniform distribution.

randint : Discrete uniform distribution, yielding integers. random_integers : Discrete uniform distribution over the closed

interval [low, high].

random_sample : Floats uniformly distributed over [0, 1). random : Alias for random_sample. rand : Convenience function that accepts dimensions as input, e.g.,

rand(2,2) would generate a 2-by-2 array of floats, uniformly distributed over [0, 1).

Generator.uniform: which should be used for new code.

The probability density function of the uniform distribution is

\[p(x) = \frac{1}{b - a}\]

anywhere within the interval [a, b), and zero elsewhere.

When high == low, values of low will be returned. If high < low, the results are officially undefined and may eventually raise an error, i.e. do not rely on this function to behave when passed arguments satisfying that inequality condition. The high limit may be included in the returned array of floats due to floating-point rounding in the equation low + (high-low) * random_sample(). For example:

>>> x = np.float32(5*0.99999999)
>>> x
5.0

Draw samples from the distribution:

>>> s = np.random.uniform(-1,0,1000)

All values are within the given interval:

>>> np.all(s >= -1)
True
>>> np.all(s < 0)
True

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 15, density=True)
>>> plt.plot(bins, np.ones_like(bins), linewidth=2, color='r')
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.vonmises(mu, kappa, size=None)

Draw samples from a von Mises distribution.

Samples are drawn from a von Mises distribution with specified mode (mu) and dispersion (kappa), on the interval [-pi, pi].

The von Mises distribution (also known as the circular normal distribution) is a continuous probability distribution on the unit circle. It may be thought of as the circular analogue of the normal distribution.

Note

New code should use the vonmises method of a default_rng() instance instead; see random-quick-start.

mu : float or array_like of floats
Mode (“center”) of the distribution.
kappa : float or array_like of floats
Dispersion of the distribution, has to be >=0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if mu and kappa are both scalars. Otherwise, np.broadcast(mu, kappa).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized von Mises distribution.
scipy.stats.vonmises : probability density function, distribution, or
cumulative density function, etc.

Generator.vonmises: which should be used for new code.

The probability density for the von Mises distribution is

\[p(x) = \frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)},\]

where \(\mu\) is the mode and \(\kappa\) the dispersion, and \(I_0(\kappa)\) is the modified Bessel function of order 0.

The von Mises is named for Richard Edler von Mises, who was born in Austria-Hungary, in what is now the Ukraine. He fled to the United States in 1939 and became a professor at Harvard. He worked in probability theory, aerodynamics, fluid mechanics, and philosophy of science.

[1]Abramowitz, M. and Stegun, I. A. (Eds.). “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing,” New York: Dover, 1972.
[2]von Mises, R., “Mathematical Theory of Probability and Statistics”, New York: Academic Press, 1964.

Draw samples from the distribution:

>>> mu, kappa = 0.0, 4.0 # mean and dispersion
>>> s = np.random.vonmises(mu, kappa, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> from scipy.special import i0  
>>> plt.hist(s, 50, density=True)
>>> x = np.linspace(-np.pi, np.pi, num=51)
>>> y = np.exp(kappa*np.cos(x-mu))/(2*np.pi*i0(kappa))  
>>> plt.plot(x, y, linewidth=2, color='r')  
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.wald(mean, scale, size=None)

Draw samples from a Wald, or inverse Gaussian, distribution.

As the scale approaches infinity, the distribution becomes more like a Gaussian. Some references claim that the Wald is an inverse Gaussian with mean equal to 1, but this is by no means universal.

The inverse Gaussian distribution was first studied in relationship to Brownian motion. In 1956 M.C.K. Tweedie used the name inverse Gaussian because there is an inverse relationship between the time to cover a unit distance and distance covered in unit time.

Note

New code should use the wald method of a default_rng() instance instead; see random-quick-start.

mean : float or array_like of floats
Distribution mean, must be > 0.
scale : float or array_like of floats
Scale parameter, must be > 0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if mean and scale are both scalars. Otherwise, np.broadcast(mean, scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Wald distribution.

Generator.wald: which should be used for new code.

The probability density function for the Wald distribution is

\[P(x;mean,scale) = \sqrt{\frac{scale}{2\pi x^3}}e^ \frac{-scale(x-mean)^2}{2\cdotp mean^2x}\]

As noted above the inverse Gaussian distribution first arise from attempts to model Brownian motion. It is also a competitor to the Weibull for use in reliability modeling and modeling stock returns and interest rate processes.

[1]Brighton Webs Ltd., Wald Distribution, https://web.archive.org/web/20090423014010/http://www.brighton-webs.co.uk:80/distributions/wald.asp
[2]Chhikara, Raj S., and Folks, J. Leroy, “The Inverse Gaussian Distribution: Theory : Methodology, and Applications”, CRC Press, 1988.
[3]Wikipedia, “Inverse Gaussian distribution” https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution

Draw values from the distribution and plot the histogram:

>>> import matplotlib.pyplot as plt
>>> h = plt.hist(np.random.wald(3, 2, 100000), bins=200, density=True)
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.weibull(a, size=None)

Draw samples from a Weibull distribution.

Draw samples from a 1-parameter Weibull distribution with the given shape parameter a.

\[X = (-ln(U))^{1/a}\]

Here, U is drawn from the uniform distribution over (0,1].

The more common 2-parameter Weibull, including a scale parameter \(\lambda\) is just \(X = \lambda(-ln(U))^{1/a}\).

Note

New code should use the weibull method of a default_rng() instance instead; see random-quick-start.

a : float or array_like of floats
Shape parameter of the distribution. Must be nonnegative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if a is a scalar. Otherwise, np.array(a).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Weibull distribution.

scipy.stats.weibull_max scipy.stats.weibull_min scipy.stats.genextreme gumbel Generator.weibull: which should be used for new code.

The Weibull (or Type III asymptotic extreme value distribution for smallest values, SEV Type III, or Rosin-Rammler distribution) is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. This class includes the Gumbel and Frechet distributions.

The probability density for the Weibull distribution is

\[p(x) = \frac{a} {\lambda}(\frac{x}{\lambda})^{a-1}e^{-(x/\lambda)^a},\]

where \(a\) is the shape and \(\lambda\) the scale.

The function has its peak (the mode) at \(\lambda(\frac{a-1}{a})^{1/a}\).

When a = 1, the Weibull distribution reduces to the exponential distribution.

[1]Waloddi Weibull, Royal Technical University, Stockholm, 1939 “A Statistical Theory Of The Strength Of Materials”, Ingeniorsvetenskapsakademiens Handlingar Nr 151, 1939, Generalstabens Litografiska Anstalts Forlag, Stockholm.
[2]Waloddi Weibull, “A Statistical Distribution Function of Wide Applicability”, Journal Of Applied Mechanics ASME Paper 1951.
[3]Wikipedia, “Weibull distribution”, https://en.wikipedia.org/wiki/Weibull_distribution

Draw samples from the distribution:

>>> a = 5. # shape
>>> s = np.random.weibull(a, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> x = np.arange(1,100.)/50.
>>> def weib(x,n,a):
...     return (a / n) * (x / n)**(a - 1) * np.exp(-(x / n)**a)
>>> count, bins, ignored = plt.hist(np.random.weibull(5.,1000))
>>> x = np.arange(1,100.)/50.
>>> scale = count.max()/weib(x, 1., 5.).max()
>>> plt.plot(x, weib(x, 1., 5.)*scale)
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_a.zipf(a, size=None)

Draw samples from a Zipf distribution.

Samples are drawn from a Zipf distribution with specified parameter a > 1.

The Zipf distribution (also known as the zeta distribution) is a continuous probability distribution that satisfies Zipf’s law: the frequency of an item is inversely proportional to its rank in a frequency table.

Note

New code should use the zipf method of a default_rng() instance instead; see random-quick-start.

a : float or array_like of floats
Distribution parameter. Must be greater than 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if a is a scalar. Otherwise, np.array(a).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Zipf distribution.
scipy.stats.zipf : probability density function, distribution, or
cumulative density function, etc.

Generator.zipf: which should be used for new code.

The probability density for the Zipf distribution is

\[p(x) = \frac{x^{-a}}{\zeta(a)},\]

where \(\zeta\) is the Riemann Zeta function.

It is named for the American linguist George Kingsley Zipf, who noted that the frequency of any word in a sample of a language is inversely proportional to its rank in the frequency table.

[1]Zipf, G. K., “Selected Studies of the Principle of Relative Frequency in Language,” Cambridge, MA: Harvard Univ. Press, 1932.

Draw samples from the distribution:

>>> a = 2. # parameter
>>> s = np.random.zipf(a, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> from scipy import special  

Truncate s values at 50 so plot is interesting:

>>> count, bins, ignored = plt.hist(s[s<50], 50, density=True)
>>> x = np.arange(1., 50.)
>>> y = x**(-a) / special.zetac(a)  
>>> plt.plot(x, y/max(y), linewidth=2, color='r')  
>>> plt.show()

region_deterministic_bnm_g2d_b

What:
Reproduces Figure 13, second column, of Sanz-Leon P., Knock, S. A., Spiegler, A. and Jirsa V. Mathematical framework for large-scale brain network modelling in The Virtual Brain. Neuroimage, 2014, (in review)
Needs:
A working installation of tvb
Run:

python region_deterministic_bnm_g2d_b.py -s True -f True

#Subsequent calls can be made with: python region_deterministic_bnm_g2d_b.py -f True

tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.beta(a, b, size=None)

Draw samples from a Beta distribution.

The Beta distribution is a special case of the Dirichlet distribution, and is related to the Gamma distribution. It has the probability distribution function

\[f(x; a,b) = \frac{1}{B(\alpha, \beta)} x^{\alpha - 1} (1 - x)^{\beta - 1},\]

where the normalization, B, is the beta function,

\[B(\alpha, \beta) = \int_0^1 t^{\alpha - 1} (1 - t)^{\beta - 1} dt.\]

It is often seen in Bayesian inference and order statistics.

Note

New code should use the beta method of a default_rng() instance instead; see random-quick-start.

a : float or array_like of floats
Alpha, positive (>0).
b : float or array_like of floats
Beta, positive (>0).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if a and b are both scalars. Otherwise, np.broadcast(a, b).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized beta distribution.

Generator.beta: which should be used for new code.

tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.binomial(n, p, size=None)

Draw samples from a binomial distribution.

Samples are drawn from a binomial distribution with specified parameters, n trials and p probability of success where n an integer >= 0 and p is in the interval [0,1]. (n may be input as a float, but it is truncated to an integer in use)

Note

New code should use the binomial method of a default_rng() instance instead; see random-quick-start.

n : int or array_like of ints
Parameter of the distribution, >= 0. Floats are also accepted, but they will be truncated to integers.
p : float or array_like of floats
Parameter of the distribution, >= 0 and <=1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if n and p are both scalars. Otherwise, np.broadcast(n, p).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized binomial distribution, where each sample is equal to the number of successes over the n trials.
scipy.stats.binom : probability density function, distribution or
cumulative density function, etc.

Generator.binomial: which should be used for new code.

The probability density for the binomial distribution is

\[P(N) = \binom{n}{N}p^N(1-p)^{n-N},\]

where \(n\) is the number of trials, \(p\) is the probability of success, and \(N\) is the number of successes.

When estimating the standard error of a proportion in a population by using a random sample, the normal distribution works well unless the product p*n <=5, where p = population proportion estimate, and n = number of samples, in which case the binomial distribution is used instead. For example, a sample of 15 people shows 4 who are left handed, and 11 who are right handed. Then p = 4/15 = 27%. 0.27*15 = 4, so the binomial distribution should be used in this case.

[1]Dalgaard, Peter, “Introductory Statistics with R”, Springer-Verlag, 2002.
[2]Glantz, Stanton A. “Primer of Biostatistics.”, McGraw-Hill, Fifth Edition, 2002.
[3]Lentner, Marvin, “Elementary Applied Statistics”, Bogden and Quigley, 1972.
[4]Weisstein, Eric W. “Binomial Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/BinomialDistribution.html
[5]Wikipedia, “Binomial distribution”, https://en.wikipedia.org/wiki/Binomial_distribution

Draw samples from the distribution:

>>> n, p = 10, .5  # number of trials, probability of each trial
>>> s = np.random.binomial(n, p, 1000)
# result of flipping a coin 10 times, tested 1000 times.

A real world example. A company drills 9 wild-cat oil exploration wells, each with an estimated probability of success of 0.1. All nine wells fail. What is the probability of that happening?

Let’s do 20,000 trials of the model, and count the number that generate zero positive results.

>>> sum(np.random.binomial(9, 0.1, 20000) == 0)/20000.
# answer = 0.38885, or 38%.
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.chisquare(df, size=None)

Draw samples from a chi-square distribution.

When df independent random variables, each with standard normal distributions (mean 0, variance 1), are squared and summed, the resulting distribution is chi-square (see Notes). This distribution is often used in hypothesis testing.

Note

New code should use the chisquare method of a default_rng() instance instead; see random-quick-start.

df : float or array_like of floats
Number of degrees of freedom, must be > 0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if df is a scalar. Otherwise, np.array(df).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized chi-square distribution.
ValueError
When df <= 0 or when an inappropriate size (e.g. size=-1) is given.

Generator.chisquare: which should be used for new code.

The variable obtained by summing the squares of df independent, standard normally distributed random variables:

\[Q = \sum_{i=0}^{\mathtt{df}} X^2_i\]

is chi-square distributed, denoted

\[Q \sim \chi^2_k.\]

The probability density function of the chi-squared distribution is

\[p(x) = \frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2},\]

where \(\Gamma\) is the gamma function,

\[\Gamma(x) = \int_0^{-\infty} t^{x - 1} e^{-t} dt.\]
[1]NIST “Engineering Statistics Handbook” https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm
>>> np.random.chisquare(2,4)
array([ 1.89920014,  9.00867716,  3.13710533,  5.62318272]) # random
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.choice(a, size=None, replace=True, p=None)

Generates a random sample from a given 1-D array

New in version 1.7.0.

Note

New code should use the choice method of a default_rng() instance instead; see random-quick-start.

a : 1-D array-like or int
If an ndarray, a random sample is generated from its elements. If an int, the random sample is generated as if a were np.arange(a)
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
replace : boolean, optional
Whether the sample is with or without replacement
p : 1-D array-like, optional
The probabilities associated with each entry in a. If not given the sample assumes a uniform distribution over all entries in a.
samples : single item or ndarray
The generated random samples
ValueError
If a is an int and less than zero, if a or p are not 1-dimensional, if a is an array-like of size 0, if p is not a vector of probabilities, if a and p have different lengths, or if replace=False and the sample size is greater than the population size

randint, shuffle, permutation Generator.choice: which should be used in new code

Sampling random rows from a 2-D array is not possible with this function, but is possible with Generator.choice through its axis keyword.

Generate a uniform random sample from np.arange(5) of size 3:

>>> np.random.choice(5, 3)
array([0, 3, 4]) # random
>>> #This is equivalent to np.random.randint(0,5,3)

Generate a non-uniform random sample from np.arange(5) of size 3:

>>> np.random.choice(5, 3, p=[0.1, 0, 0.3, 0.6, 0])
array([3, 3, 0]) # random

Generate a uniform random sample from np.arange(5) of size 3 without replacement:

>>> np.random.choice(5, 3, replace=False)
array([3,1,0]) # random
>>> #This is equivalent to np.random.permutation(np.arange(5))[:3]

Generate a non-uniform random sample from np.arange(5) of size 3 without replacement:

>>> np.random.choice(5, 3, replace=False, p=[0.1, 0, 0.3, 0.6, 0])
array([2, 3, 0]) # random

Any of the above can be repeated with an arbitrary array-like instead of just integers. For instance:

>>> aa_milne_arr = ['pooh', 'rabbit', 'piglet', 'Christopher']
>>> np.random.choice(aa_milne_arr, 5, p=[0.5, 0.1, 0.1, 0.3])
array(['pooh', 'pooh', 'pooh', 'Christopher', 'piglet'], # random
      dtype='<U11')
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.dirichlet(alpha, size=None)

Draw samples from the Dirichlet distribution.

Draw size samples of dimension k from a Dirichlet distribution. A Dirichlet-distributed random variable can be seen as a multivariate generalization of a Beta distribution. The Dirichlet distribution is a conjugate prior of a multinomial distribution in Bayesian inference.

Note

New code should use the dirichlet method of a default_rng() instance instead; see random-quick-start.

alpha : sequence of floats, length k
Parameter of the distribution (length k for sample of length k).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n), then m * n * k samples are drawn. Default is None, in which case a vector of length k is returned.
samples : ndarray,
The drawn samples, of shape (size, k).
ValueError
If any value in alpha is less than or equal to zero

Generator.dirichlet: which should be used for new code.

The Dirichlet distribution is a distribution over vectors \(x\) that fulfil the conditions \(x_i>0\) and \(\sum_{i=1}^k x_i = 1\).

The probability density function \(p\) of a Dirichlet-distributed random vector \(X\) is proportional to

\[p(x) \propto \prod_{i=1}^{k}{x^{\alpha_i-1}_i},\]

where \(\alpha\) is a vector containing the positive concentration parameters.

The method uses the following property for computation: let \(Y\) be a random vector which has components that follow a standard gamma distribution, then \(X = \frac{1}{\sum_{i=1}^k{Y_i}} Y\) is Dirichlet-distributed

[1]David McKay, “Information Theory, Inference and Learning Algorithms,” chapter 23, http://www.inference.org.uk/mackay/itila/
[2]Wikipedia, “Dirichlet distribution”, https://en.wikipedia.org/wiki/Dirichlet_distribution

Taking an example cited in Wikipedia, this distribution can be used if one wanted to cut strings (each of initial length 1.0) into K pieces with different lengths, where each piece had, on average, a designated average length, but allowing some variation in the relative sizes of the pieces.

>>> s = np.random.dirichlet((10, 5, 3), 20).transpose()
>>> import matplotlib.pyplot as plt
>>> plt.barh(range(20), s[0])
>>> plt.barh(range(20), s[1], left=s[0], color='g')
>>> plt.barh(range(20), s[2], left=s[0]+s[1], color='r')
>>> plt.title("Lengths of Strings")
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.exponential(scale=1.0, size=None)

Draw samples from an exponential distribution.

Its probability density function is

\[f(x; \frac{1}{\beta}) = \frac{1}{\beta} \exp(-\frac{x}{\beta}),\]

for x > 0 and 0 elsewhere. \(\beta\) is the scale parameter, which is the inverse of the rate parameter \(\lambda = 1/\beta\). The rate parameter is an alternative, widely used parameterization of the exponential distribution [3]_.

The exponential distribution is a continuous analogue of the geometric distribution. It describes many common situations, such as the size of raindrops measured over many rainstorms [1]_, or the time between page requests to Wikipedia [2]_.

Note

New code should use the exponential method of a default_rng() instance instead; see random-quick-start.

scale : float or array_like of floats
The scale parameter, \(\beta = 1/\lambda\). Must be non-negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if scale is a scalar. Otherwise, np.array(scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized exponential distribution.

Generator.exponential: which should be used for new code.

[1]Peyton Z. Peebles Jr., “Probability, Random Variables and Random Signal Principles”, 4th ed, 2001, p. 57.
[2]Wikipedia, “Poisson process”, https://en.wikipedia.org/wiki/Poisson_process
[3]Wikipedia, “Exponential distribution”, https://en.wikipedia.org/wiki/Exponential_distribution
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.f(dfnum, dfden, size=None)

Draw samples from an F distribution.

Samples are drawn from an F distribution with specified parameters, dfnum (degrees of freedom in numerator) and dfden (degrees of freedom in denominator), where both parameters must be greater than zero.

The random variate of the F distribution (also known as the Fisher distribution) is a continuous probability distribution that arises in ANOVA tests, and is the ratio of two chi-square variates.

Note

New code should use the f method of a default_rng() instance instead; see random-quick-start.

dfnum : float or array_like of floats
Degrees of freedom in numerator, must be > 0.
dfden : float or array_like of float
Degrees of freedom in denominator, must be > 0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if dfnum and dfden are both scalars. Otherwise, np.broadcast(dfnum, dfden).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Fisher distribution.
scipy.stats.f : probability density function, distribution or
cumulative density function, etc.

Generator.f: which should be used for new code.

The F statistic is used to compare in-group variances to between-group variances. Calculating the distribution depends on the sampling, and so it is a function of the respective degrees of freedom in the problem. The variable dfnum is the number of samples minus one, the between-groups degrees of freedom, while dfden is the within-groups degrees of freedom, the sum of the number of samples in each group minus the number of groups.

[1]Glantz, Stanton A. “Primer of Biostatistics.”, McGraw-Hill, Fifth Edition, 2002.
[2]Wikipedia, “F-distribution”, https://en.wikipedia.org/wiki/F-distribution

An example from Glantz[1], pp 47-40:

Two groups, children of diabetics (25 people) and children from people without diabetes (25 controls). Fasting blood glucose was measured, case group had a mean value of 86.1, controls had a mean value of 82.2. Standard deviations were 2.09 and 2.49 respectively. Are these data consistent with the null hypothesis that the parents diabetic status does not affect their children’s blood glucose levels? Calculating the F statistic from the data gives a value of 36.01.

Draw samples from the distribution:

>>> dfnum = 1. # between group degrees of freedom
>>> dfden = 48. # within groups degrees of freedom
>>> s = np.random.f(dfnum, dfden, 1000)

The lower bound for the top 1% of the samples is :

>>> np.sort(s)[-10]
7.61988120985 # random

So there is about a 1% chance that the F statistic will exceed 7.62, the measured value is 36, so the null hypothesis is rejected at the 1% level.

tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.gamma(shape, scale=1.0, size=None)

Draw samples from a Gamma distribution.

Samples are drawn from a Gamma distribution with specified parameters, shape (sometimes designated “k”) and scale (sometimes designated “theta”), where both parameters are > 0.

Note

New code should use the gamma method of a default_rng() instance instead; see random-quick-start.

shape : float or array_like of floats
The shape of the gamma distribution. Must be non-negative.
scale : float or array_like of floats, optional
The scale of the gamma distribution. Must be non-negative. Default is equal to 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if shape and scale are both scalars. Otherwise, np.broadcast(shape, scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized gamma distribution.
scipy.stats.gamma : probability density function, distribution or
cumulative density function, etc.

Generator.gamma: which should be used for new code.

The probability density for the Gamma distribution is

\[p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},\]

where \(k\) is the shape and \(\theta\) the scale, and \(\Gamma\) is the Gamma function.

The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant.

[1]Weisstein, Eric W. “Gamma Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html
[2]Wikipedia, “Gamma distribution”, https://en.wikipedia.org/wiki/Gamma_distribution

Draw samples from the distribution:

>>> shape, scale = 2., 2.  # mean=4, std=2*sqrt(2)
>>> s = np.random.gamma(shape, scale, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> import scipy.special as sps  
>>> count, bins, ignored = plt.hist(s, 50, density=True)
>>> y = bins**(shape-1)*(np.exp(-bins/scale) /  
...                      (sps.gamma(shape)*scale**shape))
>>> plt.plot(bins, y, linewidth=2, color='r')  
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.geometric(p, size=None)

Draw samples from the geometric distribution.

Bernoulli trials are experiments with one of two outcomes: success or failure (an example of such an experiment is flipping a coin). The geometric distribution models the number of trials that must be run in order to achieve success. It is therefore supported on the positive integers, k = 1, 2, ....

The probability mass function of the geometric distribution is

\[f(k) = (1 - p)^{k - 1} p\]

where p is the probability of success of an individual trial.

Note

New code should use the geometric method of a default_rng() instance instead; see random-quick-start.

p : float or array_like of floats
The probability of success of an individual trial.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if p is a scalar. Otherwise, np.array(p).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized geometric distribution.

Generator.geometric: which should be used for new code.

Draw ten thousand values from the geometric distribution, with the probability of an individual success equal to 0.35:

>>> z = np.random.geometric(p=0.35, size=10000)

How many trials succeeded after a single run?

>>> (z == 1).sum() / 10000.
0.34889999999999999 #random
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.get_state()

Return a tuple representing the internal state of the generator.

For more details, see set_state.

legacy : bool, optional
Flag indicating to return a legacy tuple state when the BitGenerator is MT19937, instead of a dict.
out : {tuple(str, ndarray of 624 uints, int, int, float), dict}

The returned tuple has the following items:

  1. the string ‘MT19937’.
  2. a 1-D array of 624 unsigned integer keys.
  3. an integer pos.
  4. an integer has_gauss.
  5. a float cached_gaussian.

If legacy is False, or the BitGenerator is not MT19937, then state is returned as a dictionary.

set_state

set_state and get_state are not needed to work with any of the random distributions in NumPy. If the internal state is manually altered, the user should know exactly what he/she is doing.

tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.gumbel(loc=0.0, scale=1.0, size=None)

Draw samples from a Gumbel distribution.

Draw samples from a Gumbel distribution with specified location and scale. For more information on the Gumbel distribution, see Notes and References below.

Note

New code should use the gumbel method of a default_rng() instance instead; see random-quick-start.

loc : float or array_like of floats, optional
The location of the mode of the distribution. Default is 0.
scale : float or array_like of floats, optional
The scale parameter of the distribution. Default is 1. Must be non- negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if loc and scale are both scalars. Otherwise, np.broadcast(loc, scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Gumbel distribution.

scipy.stats.gumbel_l scipy.stats.gumbel_r scipy.stats.genextreme weibull Generator.gumbel: which should be used for new code.

The Gumbel (or Smallest Extreme Value (SEV) or the Smallest Extreme Value Type I) distribution is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. The Gumbel is a special case of the Extreme Value Type I distribution for maximums from distributions with “exponential-like” tails.

The probability density for the Gumbel distribution is

\[p(x) = \frac{e^{-(x - \mu)/ \beta}}{\beta} e^{ -e^{-(x - \mu)/ \beta}},\]

where \(\mu\) is the mode, a location parameter, and \(\beta\) is the scale parameter.

The Gumbel (named for German mathematician Emil Julius Gumbel) was used very early in the hydrology literature, for modeling the occurrence of flood events. It is also used for modeling maximum wind speed and rainfall rates. It is a “fat-tailed” distribution - the probability of an event in the tail of the distribution is larger than if one used a Gaussian, hence the surprisingly frequent occurrence of 100-year floods. Floods were initially modeled as a Gaussian process, which underestimated the frequency of extreme events.

It is one of a class of extreme value distributions, the Generalized Extreme Value (GEV) distributions, which also includes the Weibull and Frechet.

The function has a mean of \(\mu + 0.57721\beta\) and a variance of \(\frac{\pi^2}{6}\beta^2\).

[1]Gumbel, E. J., “Statistics of Extremes,” New York: Columbia University Press, 1958.
[2]Reiss, R.-D. and Thomas, M., “Statistical Analysis of Extreme Values from Insurance, Finance, Hydrology and Other Fields,” Basel: Birkhauser Verlag, 2001.

Draw samples from the distribution:

>>> mu, beta = 0, 0.1 # location and scale
>>> s = np.random.gumbel(mu, beta, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, density=True)
>>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)
...          * np.exp( -np.exp( -(bins - mu) /beta) ),
...          linewidth=2, color='r')
>>> plt.show()

Show how an extreme value distribution can arise from a Gaussian process and compare to a Gaussian:

>>> means = []
>>> maxima = []
>>> for i in range(0,1000) :
...    a = np.random.normal(mu, beta, 1000)
...    means.append(a.mean())
...    maxima.append(a.max())
>>> count, bins, ignored = plt.hist(maxima, 30, density=True)
>>> beta = np.std(maxima) * np.sqrt(6) / np.pi
>>> mu = np.mean(maxima) - 0.57721*beta
>>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)
...          * np.exp(-np.exp(-(bins - mu)/beta)),
...          linewidth=2, color='r')
>>> plt.plot(bins, 1/(beta * np.sqrt(2 * np.pi))
...          * np.exp(-(bins - mu)**2 / (2 * beta**2)),
...          linewidth=2, color='g')
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.hypergeometric(ngood, nbad, nsample, size=None)

Draw samples from a Hypergeometric distribution.

Samples are drawn from a hypergeometric distribution with specified parameters, ngood (ways to make a good selection), nbad (ways to make a bad selection), and nsample (number of items sampled, which is less than or equal to the sum ngood + nbad).

Note

New code should use the hypergeometric method of a default_rng() instance instead; see random-quick-start.

ngood : int or array_like of ints
Number of ways to make a good selection. Must be nonnegative.
nbad : int or array_like of ints
Number of ways to make a bad selection. Must be nonnegative.
nsample : int or array_like of ints
Number of items sampled. Must be at least 1 and at most ngood + nbad.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if ngood, nbad, and nsample are all scalars. Otherwise, np.broadcast(ngood, nbad, nsample).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized hypergeometric distribution. Each sample is the number of good items within a randomly selected subset of size nsample taken from a set of ngood good items and nbad bad items.
scipy.stats.hypergeom : probability density function, distribution or
cumulative density function, etc.

Generator.hypergeometric: which should be used for new code.

The probability density for the Hypergeometric distribution is

\[P(x) = \frac{\binom{g}{x}\binom{b}{n-x}}{\binom{g+b}{n}},\]

where \(0 \le x \le n\) and \(n-b \le x \le g\)

for P(x) the probability of x good results in the drawn sample, g = ngood, b = nbad, and n = nsample.

Consider an urn with black and white marbles in it, ngood of them are black and nbad are white. If you draw nsample balls without replacement, then the hypergeometric distribution describes the distribution of black balls in the drawn sample.

Note that this distribution is very similar to the binomial distribution, except that in this case, samples are drawn without replacement, whereas in the Binomial case samples are drawn with replacement (or the sample space is infinite). As the sample space becomes large, this distribution approaches the binomial.

[1]Lentner, Marvin, “Elementary Applied Statistics”, Bogden and Quigley, 1972.
[2]Weisstein, Eric W. “Hypergeometric Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/HypergeometricDistribution.html
[3]Wikipedia, “Hypergeometric distribution”, https://en.wikipedia.org/wiki/Hypergeometric_distribution

Draw samples from the distribution:

>>> ngood, nbad, nsamp = 100, 2, 10
# number of good, number of bad, and number of samples
>>> s = np.random.hypergeometric(ngood, nbad, nsamp, 1000)
>>> from matplotlib.pyplot import hist
>>> hist(s)
#   note that it is very unlikely to grab both bad items

Suppose you have an urn with 15 white and 15 black marbles. If you pull 15 marbles at random, how likely is it that 12 or more of them are one color?

>>> s = np.random.hypergeometric(15, 15, 15, 100000)
>>> sum(s>=12)/100000. + sum(s<=3)/100000.
#   answer = 0.003 ... pretty unlikely!
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.laplace(loc=0.0, scale=1.0, size=None)

Draw samples from the Laplace or double exponential distribution with specified location (or mean) and scale (decay).

The Laplace distribution is similar to the Gaussian/normal distribution, but is sharper at the peak and has fatter tails. It represents the difference between two independent, identically distributed exponential random variables.

Note

New code should use the laplace method of a default_rng() instance instead; see random-quick-start.

loc : float or array_like of floats, optional
The position, \(\mu\), of the distribution peak. Default is 0.
scale : float or array_like of floats, optional
\(\lambda\), the exponential decay. Default is 1. Must be non- negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if loc and scale are both scalars. Otherwise, np.broadcast(loc, scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Laplace distribution.

Generator.laplace: which should be used for new code.

It has the probability density function

\[f(x; \mu, \lambda) = \frac{1}{2\lambda} \exp\left(-\frac{|x - \mu|}{\lambda}\right).\]

The first law of Laplace, from 1774, states that the frequency of an error can be expressed as an exponential function of the absolute magnitude of the error, which leads to the Laplace distribution. For many problems in economics and health sciences, this distribution seems to model the data better than the standard Gaussian distribution.

[1]Abramowitz, M. and Stegun, I. A. (Eds.). “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing,” New York: Dover, 1972.
[2]Kotz, Samuel, et. al. “The Laplace Distribution and Generalizations, ” Birkhauser, 2001.
[3]Weisstein, Eric W. “Laplace Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/LaplaceDistribution.html
[4]Wikipedia, “Laplace distribution”, https://en.wikipedia.org/wiki/Laplace_distribution

Draw samples from the distribution

>>> loc, scale = 0., 1.
>>> s = np.random.laplace(loc, scale, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, density=True)
>>> x = np.arange(-8., 8., .01)
>>> pdf = np.exp(-abs(x-loc)/scale)/(2.*scale)
>>> plt.plot(x, pdf)

Plot Gaussian for comparison:

>>> g = (1/(scale * np.sqrt(2 * np.pi)) *
...      np.exp(-(x - loc)**2 / (2 * scale**2)))
>>> plt.plot(x,g)
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.logistic(loc=0.0, scale=1.0, size=None)

Draw samples from a logistic distribution.

Samples are drawn from a logistic distribution with specified parameters, loc (location or mean, also median), and scale (>0).

Note

New code should use the logistic method of a default_rng() instance instead; see random-quick-start.

loc : float or array_like of floats, optional
Parameter of the distribution. Default is 0.
scale : float or array_like of floats, optional
Parameter of the distribution. Must be non-negative. Default is 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if loc and scale are both scalars. Otherwise, np.broadcast(loc, scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized logistic distribution.
scipy.stats.logistic : probability density function, distribution or
cumulative density function, etc.

Generator.logistic: which should be used for new code.

The probability density for the Logistic distribution is

\[P(x) = P(x) = \frac{e^{-(x-\mu)/s}}{s(1+e^{-(x-\mu)/s})^2},\]

where \(\mu\) = location and \(s\) = scale.

The Logistic distribution is used in Extreme Value problems where it can act as a mixture of Gumbel distributions, in Epidemiology, and by the World Chess Federation (FIDE) where it is used in the Elo ranking system, assuming the performance of each player is a logistically distributed random variable.

[1]Reiss, R.-D. and Thomas M. (2001), “Statistical Analysis of Extreme Values, from Insurance, Finance, Hydrology and Other Fields,” Birkhauser Verlag, Basel, pp 132-133.
[2]Weisstein, Eric W. “Logistic Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/LogisticDistribution.html
[3]Wikipedia, “Logistic-distribution”, https://en.wikipedia.org/wiki/Logistic_distribution

Draw samples from the distribution:

>>> loc, scale = 10, 1
>>> s = np.random.logistic(loc, scale, 10000)
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, bins=50)

# plot against distribution

>>> def logist(x, loc, scale):
...     return np.exp((loc-x)/scale)/(scale*(1+np.exp((loc-x)/scale))**2)
>>> lgst_val = logist(bins, loc, scale)
>>> plt.plot(bins, lgst_val * count.max() / lgst_val.max())
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.lognormal(mean=0.0, sigma=1.0, size=None)

Draw samples from a log-normal distribution.

Draw samples from a log-normal distribution with specified mean, standard deviation, and array shape. Note that the mean and standard deviation are not the values for the distribution itself, but of the underlying normal distribution it is derived from.

Note

New code should use the lognormal method of a default_rng() instance instead; see random-quick-start.

mean : float or array_like of floats, optional
Mean value of the underlying normal distribution. Default is 0.
sigma : float or array_like of floats, optional
Standard deviation of the underlying normal distribution. Must be non-negative. Default is 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if mean and sigma are both scalars. Otherwise, np.broadcast(mean, sigma).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized log-normal distribution.
scipy.stats.lognorm : probability density function, distribution,
cumulative density function, etc.

Generator.lognormal: which should be used for new code.

A variable x has a log-normal distribution if log(x) is normally distributed. The probability density function for the log-normal distribution is:

\[p(x) = \frac{1}{\sigma x \sqrt{2\pi}} e^{(-\frac{(ln(x)-\mu)^2}{2\sigma^2})}\]

where \(\mu\) is the mean and \(\sigma\) is the standard deviation of the normally distributed logarithm of the variable. A log-normal distribution results if a random variable is the product of a large number of independent, identically-distributed variables in the same way that a normal distribution results if the variable is the sum of a large number of independent, identically-distributed variables.

[1]Limpert, E., Stahel, W. A., and Abbt, M., “Log-normal Distributions across the Sciences: Keys and Clues,” BioScience, Vol. 51, No. 5, May, 2001. https://stat.ethz.ch/~stahel/lognormal/bioscience.pdf
[2]Reiss, R.D. and Thomas, M., “Statistical Analysis of Extreme Values,” Basel: Birkhauser Verlag, 2001, pp. 31-32.

Draw samples from the distribution:

>>> mu, sigma = 3., 1. # mean and standard deviation
>>> s = np.random.lognormal(mu, sigma, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 100, density=True, align='mid')
>>> x = np.linspace(min(bins), max(bins), 10000)
>>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))
...        / (x * sigma * np.sqrt(2 * np.pi)))
>>> plt.plot(x, pdf, linewidth=2, color='r')
>>> plt.axis('tight')
>>> plt.show()

Demonstrate that taking the products of random samples from a uniform distribution can be fit well by a log-normal probability density function.

>>> # Generate a thousand samples: each is the product of 100 random
>>> # values, drawn from a normal distribution.
>>> b = []
>>> for i in range(1000):
...    a = 10. + np.random.standard_normal(100)
...    b.append(np.product(a))
>>> b = np.array(b) / np.min(b) # scale values to be positive
>>> count, bins, ignored = plt.hist(b, 100, density=True, align='mid')
>>> sigma = np.std(np.log(b))
>>> mu = np.mean(np.log(b))
>>> x = np.linspace(min(bins), max(bins), 10000)
>>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))
...        / (x * sigma * np.sqrt(2 * np.pi)))
>>> plt.plot(x, pdf, color='r', linewidth=2)
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.logseries(p, size=None)

Draw samples from a logarithmic series distribution.

Samples are drawn from a log series distribution with specified shape parameter, 0 < p < 1.

Note

New code should use the logseries method of a default_rng() instance instead; see random-quick-start.

p : float or array_like of floats
Shape parameter for the distribution. Must be in the range (0, 1).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if p is a scalar. Otherwise, np.array(p).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized logarithmic series distribution.
scipy.stats.logser : probability density function, distribution or
cumulative density function, etc.

Generator.logseries: which should be used for new code.

The probability density for the Log Series distribution is

\[P(k) = \frac{-p^k}{k \ln(1-p)},\]

where p = probability.

The log series distribution is frequently used to represent species richness and occurrence, first proposed by Fisher, Corbet, and Williams in 1943 [2]. It may also be used to model the numbers of occupants seen in cars [3].

[1]Buzas, Martin A.; Culver, Stephen J., Understanding regional species diversity through the log series distribution of occurrences: BIODIVERSITY RESEARCH Diversity & Distributions, Volume 5, Number 5, September 1999 , pp. 187-195(9).
[2]Fisher, R.A,, A.S. Corbet, and C.B. Williams. 1943. The relation between the number of species and the number of individuals in a random sample of an animal population. Journal of Animal Ecology, 12:42-58.
[3]D. J. Hand, F. Daly, D. Lunn, E. Ostrowski, A Handbook of Small Data Sets, CRC Press, 1994.
[4]Wikipedia, “Logarithmic distribution”, https://en.wikipedia.org/wiki/Logarithmic_distribution

Draw samples from the distribution:

>>> a = .6
>>> s = np.random.logseries(a, 10000)
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s)

# plot against distribution

>>> def logseries(k, p):
...     return -p**k/(k*np.log(1-p))
>>> plt.plot(bins, logseries(bins, a)*count.max()/
...          logseries(bins, a).max(), 'r')
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.multinomial(n, pvals, size=None)

Draw samples from a multinomial distribution.

The multinomial distribution is a multivariate generalization of the binomial distribution. Take an experiment with one of p possible outcomes. An example of such an experiment is throwing a dice, where the outcome can be 1 through 6. Each sample drawn from the distribution represents n such experiments. Its values, X_i = [X_0, X_1, ..., X_p], represent the number of times the outcome was i.

Note

New code should use the multinomial method of a default_rng() instance instead; see random-quick-start.

n : int
Number of experiments.
pvals : sequence of floats, length p
Probabilities of each of the p different outcomes. These must sum to 1 (however, the last element is always assumed to account for the remaining probability, as long as sum(pvals[:-1]) <= 1).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
out : ndarray

The drawn samples, of shape size, if that was provided. If not, the shape is (N,).

In other words, each entry out[i,j,...,:] is an N-dimensional value drawn from the distribution.

Generator.multinomial: which should be used for new code.

Throw a dice 20 times:

>>> np.random.multinomial(20, [1/6.]*6, size=1)
array([[4, 1, 7, 5, 2, 1]]) # random

It landed 4 times on 1, once on 2, etc.

Now, throw the dice 20 times, and 20 times again:

>>> np.random.multinomial(20, [1/6.]*6, size=2)
array([[3, 4, 3, 3, 4, 3], # random
       [2, 4, 3, 4, 0, 7]])

For the first run, we threw 3 times 1, 4 times 2, etc. For the second, we threw 2 times 1, 4 times 2, etc.

A loaded die is more likely to land on number 6:

>>> np.random.multinomial(100, [1/7.]*5 + [2/7.])
array([11, 16, 14, 17, 16, 26]) # random

The probability inputs should be normalized. As an implementation detail, the value of the last entry is ignored and assumed to take up any leftover probability mass, but this should not be relied on. A biased coin which has twice as much weight on one side as on the other should be sampled like so:

>>> np.random.multinomial(100, [1.0 / 3, 2.0 / 3])  # RIGHT
array([38, 62]) # random

not like:

>>> np.random.multinomial(100, [1.0, 2.0])  # WRONG
Traceback (most recent call last):
ValueError: pvals < 0, pvals > 1 or pvals contains NaNs
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.multivariate_normal(mean, cov, size=None, check_valid='warn', tol=1e-8)

Draw random samples from a multivariate normal distribution.

The multivariate normal, multinormal or Gaussian distribution is a generalization of the one-dimensional normal distribution to higher dimensions. Such a distribution is specified by its mean and covariance matrix. These parameters are analogous to the mean (average or “center”) and variance (standard deviation, or “width,” squared) of the one-dimensional normal distribution.

Note

New code should use the multivariate_normal method of a default_rng() instance instead; see random-quick-start.

mean : 1-D array_like, of length N
Mean of the N-dimensional distribution.
cov : 2-D array_like, of shape (N, N)
Covariance matrix of the distribution. It must be symmetric and positive-semidefinite for proper sampling.
size : int or tuple of ints, optional
Given a shape of, for example, (m,n,k), m*n*k samples are generated, and packed in an m-by-n-by-k arrangement. Because each sample is N-dimensional, the output shape is (m,n,k,N). If no shape is specified, a single (N-D) sample is returned.
check_valid : { ‘warn’, ‘raise’, ‘ignore’ }, optional
Behavior when the covariance matrix is not positive semidefinite.
tol : float, optional
Tolerance when checking the singular values in covariance matrix. cov is cast to double before the check.
out : ndarray

The drawn samples, of shape size, if that was provided. If not, the shape is (N,).

In other words, each entry out[i,j,...,:] is an N-dimensional value drawn from the distribution.

Generator.multivariate_normal: which should be used for new code.

The mean is a coordinate in N-dimensional space, which represents the location where samples are most likely to be generated. This is analogous to the peak of the bell curve for the one-dimensional or univariate normal distribution.

Covariance indicates the level to which two variables vary together. From the multivariate normal distribution, we draw N-dimensional samples, \(X = [x_1, x_2, ... x_N]\). The covariance matrix element \(C_{ij}\) is the covariance of \(x_i\) and \(x_j\). The element \(C_{ii}\) is the variance of \(x_i\) (i.e. its “spread”).

Instead of specifying the full covariance matrix, popular approximations include:

  • Spherical covariance (cov is a multiple of the identity matrix)
  • Diagonal covariance (cov has non-negative elements, and only on the diagonal)

This geometrical property can be seen in two dimensions by plotting generated data-points:

>>> mean = [0, 0]
>>> cov = [[1, 0], [0, 100]]  # diagonal covariance

Diagonal covariance means that points are oriented along x or y-axis:

>>> import matplotlib.pyplot as plt
>>> x, y = np.random.multivariate_normal(mean, cov, 5000).T
>>> plt.plot(x, y, 'x')
>>> plt.axis('equal')
>>> plt.show()

Note that the covariance matrix must be positive semidefinite (a.k.a. nonnegative-definite). Otherwise, the behavior of this method is undefined and backwards compatibility is not guaranteed.

[1]Papoulis, A., “Probability, Random Variables, and Stochastic Processes,” 3rd ed., New York: McGraw-Hill, 1991.
[2]Duda, R. O., Hart, P. E., and Stork, D. G., “Pattern Classification,” 2nd ed., New York: Wiley, 2001.
>>> mean = (1, 2)
>>> cov = [[1, 0], [0, 1]]
>>> x = np.random.multivariate_normal(mean, cov, (3, 3))
>>> x.shape
(3, 3, 2)

The following is probably true, given that 0.6 is roughly twice the standard deviation:

>>> list((x[0,0,:] - mean) < 0.6)
[True, True] # random
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.negative_binomial(n, p, size=None)

Draw samples from a negative binomial distribution.

Samples are drawn from a negative binomial distribution with specified parameters, n successes and p probability of success where n is > 0 and p is in the interval [0, 1].

Note

New code should use the negative_binomial method of a default_rng() instance instead; see random-quick-start.

n : float or array_like of floats
Parameter of the distribution, > 0.
p : float or array_like of floats
Parameter of the distribution, >= 0 and <=1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if n and p are both scalars. Otherwise, np.broadcast(n, p).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized negative binomial distribution, where each sample is equal to N, the number of failures that occurred before a total of n successes was reached.

Generator.negative_binomial: which should be used for new code.

The probability mass function of the negative binomial distribution is

\[P(N;n,p) = \frac{\Gamma(N+n)}{N!\Gamma(n)}p^{n}(1-p)^{N},\]

where \(n\) is the number of successes, \(p\) is the probability of success, \(N+n\) is the number of trials, and \(\Gamma\) is the gamma function. When \(n\) is an integer, \(\frac{\Gamma(N+n)}{N!\Gamma(n)} = \binom{N+n-1}{N}\), which is the more common form of this term in the the pmf. The negative binomial distribution gives the probability of N failures given n successes, with a success on the last trial.

If one throws a die repeatedly until the third time a “1” appears, then the probability distribution of the number of non-“1”s that appear before the third “1” is a negative binomial distribution.

[1]Weisstein, Eric W. “Negative Binomial Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/NegativeBinomialDistribution.html
[2]Wikipedia, “Negative binomial distribution”, https://en.wikipedia.org/wiki/Negative_binomial_distribution

Draw samples from the distribution:

A real world example. A company drills wild-cat oil exploration wells, each with an estimated probability of success of 0.1. What is the probability of having one success for each successive well, that is what is the probability of a single success after drilling 5 wells, after 6 wells, etc.?

>>> s = np.random.negative_binomial(1, 0.1, 100000)
>>> for i in range(1, 11): 
...    probability = sum(s<i) / 100000.
...    print(i, "wells drilled, probability of one success =", probability)
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.noncentral_chisquare(df, nonc, size=None)

Draw samples from a noncentral chi-square distribution.

The noncentral \(\chi^2\) distribution is a generalization of the \(\chi^2\) distribution.

Note

New code should use the noncentral_chisquare method of a default_rng() instance instead; see random-quick-start.

df : float or array_like of floats

Degrees of freedom, must be > 0.

Changed in version 1.10.0: Earlier NumPy versions required dfnum > 1.

nonc : float or array_like of floats
Non-centrality, must be non-negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if df and nonc are both scalars. Otherwise, np.broadcast(df, nonc).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized noncentral chi-square distribution.

Generator.noncentral_chisquare: which should be used for new code.

The probability density function for the noncentral Chi-square distribution is

\[P(x;df,nonc) = \sum^{\infty}_{i=0} \frac{e^{-nonc/2}(nonc/2)^{i}}{i!} P_{Y_{df+2i}}(x),\]

where \(Y_{q}\) is the Chi-square with q degrees of freedom.

[1]Wikipedia, “Noncentral chi-squared distribution” https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution

Draw values from the distribution and plot the histogram

>>> import matplotlib.pyplot as plt
>>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000),
...                   bins=200, density=True)
>>> plt.show()

Draw values from a noncentral chisquare with very small noncentrality, and compare to a chisquare.

>>> plt.figure()
>>> values = plt.hist(np.random.noncentral_chisquare(3, .0000001, 100000),
...                   bins=np.arange(0., 25, .1), density=True)
>>> values2 = plt.hist(np.random.chisquare(3, 100000),
...                    bins=np.arange(0., 25, .1), density=True)
>>> plt.plot(values[1][0:-1], values[0]-values2[0], 'ob')
>>> plt.show()

Demonstrate how large values of non-centrality lead to a more symmetric distribution.

>>> plt.figure()
>>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000),
...                   bins=200, density=True)
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.noncentral_f(dfnum, dfden, nonc, size=None)

Draw samples from the noncentral F distribution.

Samples are drawn from an F distribution with specified parameters, dfnum (degrees of freedom in numerator) and dfden (degrees of freedom in denominator), where both parameters > 1. nonc is the non-centrality parameter.

Note

New code should use the noncentral_f method of a default_rng() instance instead; see random-quick-start.

dfnum : float or array_like of floats

Numerator degrees of freedom, must be > 0.

Changed in version 1.14.0: Earlier NumPy versions required dfnum > 1.

dfden : float or array_like of floats
Denominator degrees of freedom, must be > 0.
nonc : float or array_like of floats
Non-centrality parameter, the sum of the squares of the numerator means, must be >= 0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if dfnum, dfden, and nonc are all scalars. Otherwise, np.broadcast(dfnum, dfden, nonc).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized noncentral Fisher distribution.

Generator.noncentral_f: which should be used for new code.

When calculating the power of an experiment (power = probability of rejecting the null hypothesis when a specific alternative is true) the non-central F statistic becomes important. When the null hypothesis is true, the F statistic follows a central F distribution. When the null hypothesis is not true, then it follows a non-central F statistic.

[1]Weisstein, Eric W. “Noncentral F-Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/NoncentralF-Distribution.html
[2]Wikipedia, “Noncentral F-distribution”, https://en.wikipedia.org/wiki/Noncentral_F-distribution

In a study, testing for a specific alternative to the null hypothesis requires use of the Noncentral F distribution. We need to calculate the area in the tail of the distribution that exceeds the value of the F distribution for the null hypothesis. We’ll plot the two probability distributions for comparison.

>>> dfnum = 3 # between group deg of freedom
>>> dfden = 20 # within groups degrees of freedom
>>> nonc = 3.0
>>> nc_vals = np.random.noncentral_f(dfnum, dfden, nonc, 1000000)
>>> NF = np.histogram(nc_vals, bins=50, density=True)
>>> c_vals = np.random.f(dfnum, dfden, 1000000)
>>> F = np.histogram(c_vals, bins=50, density=True)
>>> import matplotlib.pyplot as plt
>>> plt.plot(F[1][1:], F[0])
>>> plt.plot(NF[1][1:], NF[0])
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.normal(loc=0.0, scale=1.0, size=None)

Draw random samples from a normal (Gaussian) distribution.

The probability density function of the normal distribution, first derived by De Moivre and 200 years later by both Gauss and Laplace independently [2]_, is often called the bell curve because of its characteristic shape (see the example below).

The normal distributions occurs often in nature. For example, it describes the commonly occurring distribution of samples influenced by a large number of tiny, random disturbances, each with its own unique distribution [2]_.

Note

New code should use the normal method of a default_rng() instance instead; see random-quick-start.

loc : float or array_like of floats
Mean (“centre”) of the distribution.
scale : float or array_like of floats
Standard deviation (spread or “width”) of the distribution. Must be non-negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if loc and scale are both scalars. Otherwise, np.broadcast(loc, scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized normal distribution.
scipy.stats.norm : probability density function, distribution or
cumulative density function, etc.

Generator.normal: which should be used for new code.

The probability density for the Gaussian distribution is

\[p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }} e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} },\]

where \(\mu\) is the mean and \(\sigma\) the standard deviation. The square of the standard deviation, \(\sigma^2\), is called the variance.

The function has its peak at the mean, and its “spread” increases with the standard deviation (the function reaches 0.607 times its maximum at \(x + \sigma\) and \(x - \sigma\) [2]_). This implies that normal is more likely to return samples lying close to the mean, rather than those far away.

[1]Wikipedia, “Normal distribution”, https://en.wikipedia.org/wiki/Normal_distribution
[2]P. R. Peebles Jr., “Central Limit Theorem” in “Probability, Random Variables and Random Signal Principles”, 4th ed., 2001, pp. 51, 51, 125.

Draw samples from the distribution:

>>> mu, sigma = 0, 0.1 # mean and standard deviation
>>> s = np.random.normal(mu, sigma, 1000)

Verify the mean and the variance:

>>> abs(mu - np.mean(s))
0.0  # may vary
>>> abs(sigma - np.std(s, ddof=1))
0.1  # may vary

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, density=True)
>>> plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) *
...                np.exp( - (bins - mu)**2 / (2 * sigma**2) ),
...          linewidth=2, color='r')
>>> plt.show()

Two-by-four array of samples from N(3, 6.25):

>>> np.random.normal(3, 2.5, size=(2, 4))
array([[-4.49401501,  4.00950034, -1.81814867,  7.29718677],   # random
       [ 0.39924804,  4.68456316,  4.99394529,  4.84057254]])  # random
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.pareto(a, size=None)

Draw samples from a Pareto II or Lomax distribution with specified shape.

The Lomax or Pareto II distribution is a shifted Pareto distribution. The classical Pareto distribution can be obtained from the Lomax distribution by adding 1 and multiplying by the scale parameter m (see Notes). The smallest value of the Lomax distribution is zero while for the classical Pareto distribution it is mu, where the standard Pareto distribution has location mu = 1. Lomax can also be considered as a simplified version of the Generalized Pareto distribution (available in SciPy), with the scale set to one and the location set to zero.

The Pareto distribution must be greater than zero, and is unbounded above. It is also known as the “80-20 rule”. In this distribution, 80 percent of the weights are in the lowest 20 percent of the range, while the other 20 percent fill the remaining 80 percent of the range.

Note

New code should use the pareto method of a default_rng() instance instead; see random-quick-start.

a : float or array_like of floats
Shape of the distribution. Must be positive.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if a is a scalar. Otherwise, np.array(a).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Pareto distribution.
scipy.stats.lomax : probability density function, distribution or
cumulative density function, etc.
scipy.stats.genpareto : probability density function, distribution or
cumulative density function, etc.

Generator.pareto: which should be used for new code.

The probability density for the Pareto distribution is

\[p(x) = \frac{am^a}{x^{a+1}}\]

where \(a\) is the shape and \(m\) the scale.

The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution useful in many real world problems. Outside the field of economics it is generally referred to as the Bradford distribution. Pareto developed the distribution to describe the distribution of wealth in an economy. It has also found use in insurance, web page access statistics, oil field sizes, and many other problems, including the download frequency for projects in Sourceforge [1]_. It is one of the so-called “fat-tailed” distributions.

[1]Francis Hunt and Paul Johnson, On the Pareto Distribution of Sourceforge projects.
[2]Pareto, V. (1896). Course of Political Economy. Lausanne.
[3]Reiss, R.D., Thomas, M.(2001), Statistical Analysis of Extreme Values, Birkhauser Verlag, Basel, pp 23-30.
[4]Wikipedia, “Pareto distribution”, https://en.wikipedia.org/wiki/Pareto_distribution

Draw samples from the distribution:

>>> a, m = 3., 2.  # shape and mode
>>> s = (np.random.pareto(a, 1000) + 1) * m

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, _ = plt.hist(s, 100, density=True)
>>> fit = a*m**a / bins**(a+1)
>>> plt.plot(bins, max(count)*fit/max(fit), linewidth=2, color='r')
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.permutation(x)

Randomly permute a sequence, or return a permuted range.

If x is a multi-dimensional array, it is only shuffled along its first index.

Note

New code should use the permutation method of a default_rng() instance instead; see random-quick-start.

x : int or array_like
If x is an integer, randomly permute np.arange(x). If x is an array, make a copy and shuffle the elements randomly.
out : ndarray
Permuted sequence or array range.

Generator.permutation: which should be used for new code.

>>> np.random.permutation(10)
array([1, 7, 4, 3, 0, 9, 2, 5, 8, 6]) # random
>>> np.random.permutation([1, 4, 9, 12, 15])
array([15,  1,  9,  4, 12]) # random
>>> arr = np.arange(9).reshape((3, 3))
>>> np.random.permutation(arr)
array([[6, 7, 8], # random
       [0, 1, 2],
       [3, 4, 5]])
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.poisson(lam=1.0, size=None)

Draw samples from a Poisson distribution.

The Poisson distribution is the limit of the binomial distribution for large N.

Note

New code should use the poisson method of a default_rng() instance instead; see random-quick-start.

lam : float or array_like of floats
Expectation of interval, must be >= 0. A sequence of expectation intervals must be broadcastable over the requested size.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if lam is a scalar. Otherwise, np.array(lam).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Poisson distribution.

Generator.poisson: which should be used for new code.

The Poisson distribution

\[f(k; \lambda)=\frac{\lambda^k e^{-\lambda}}{k!}\]

For events with an expected separation \(\lambda\) the Poisson distribution \(f(k; \lambda)\) describes the probability of \(k\) events occurring within the observed interval \(\lambda\).

Because the output is limited to the range of the C int64 type, a ValueError is raised when lam is within 10 sigma of the maximum representable value.

[1]Weisstein, Eric W. “Poisson Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/PoissonDistribution.html
[2]Wikipedia, “Poisson distribution”, https://en.wikipedia.org/wiki/Poisson_distribution

Draw samples from the distribution:

>>> import numpy as np
>>> s = np.random.poisson(5, 10000)

Display histogram of the sample:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 14, density=True)
>>> plt.show()

Draw each 100 values for lambda 100 and 500:

>>> s = np.random.poisson(lam=(100., 500.), size=(100, 2))
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.power(a, size=None)

Draws samples in [0, 1] from a power distribution with positive exponent a - 1.

Also known as the power function distribution.

Note

New code should use the power method of a default_rng() instance instead; see random-quick-start.

a : float or array_like of floats
Parameter of the distribution. Must be non-negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if a is a scalar. Otherwise, np.array(a).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized power distribution.
ValueError
If a < 1.

Generator.power: which should be used for new code.

The probability density function is

\[\begin{split}P(x; a) = ax^{a-1}, 0 \le x \le 1, a>0.\end{split}\]

The power function distribution is just the inverse of the Pareto distribution. It may also be seen as a special case of the Beta distribution.

It is used, for example, in modeling the over-reporting of insurance claims.

[1]Christian Kleiber, Samuel Kotz, “Statistical size distributions in economics and actuarial sciences”, Wiley, 2003.
[2]Heckert, N. A. and Filliben, James J. “NIST Handbook 148: Dataplot Reference Manual, Volume 2: Let Subcommands and Library Functions”, National Institute of Standards and Technology Handbook Series, June 2003. https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/powpdf.pdf

Draw samples from the distribution:

>>> a = 5. # shape
>>> samples = 1000
>>> s = np.random.power(a, samples)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, bins=30)
>>> x = np.linspace(0, 1, 100)
>>> y = a*x**(a-1.)
>>> normed_y = samples*np.diff(bins)[0]*y
>>> plt.plot(x, normed_y)
>>> plt.show()

Compare the power function distribution to the inverse of the Pareto.

>>> from scipy import stats 
>>> rvs = np.random.power(5, 1000000)
>>> rvsp = np.random.pareto(5, 1000000)
>>> xx = np.linspace(0,1,100)
>>> powpdf = stats.powerlaw.pdf(xx,5)  
>>> plt.figure()
>>> plt.hist(rvs, bins=50, density=True)
>>> plt.plot(xx,powpdf,'r-')  
>>> plt.title('np.random.power(5)')
>>> plt.figure()
>>> plt.hist(1./(1.+rvsp), bins=50, density=True)
>>> plt.plot(xx,powpdf,'r-')  
>>> plt.title('inverse of 1 + np.random.pareto(5)')
>>> plt.figure()
>>> plt.hist(1./(1.+rvsp), bins=50, density=True)
>>> plt.plot(xx,powpdf,'r-')  
>>> plt.title('inverse of stats.pareto(5)')
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.rand(d0, d1, ..., dn)

Random values in a given shape.

Note

This is a convenience function for users porting code from Matlab, and wraps random_sample. That function takes a tuple to specify the size of the output, which is consistent with other NumPy functions like numpy.zeros and numpy.ones.

Create an array of the given shape and populate it with random samples from a uniform distribution over [0, 1).

d0, d1, ..., dn : int, optional
The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned.
out : ndarray, shape (d0, d1, ..., dn)
Random values.

random

>>> np.random.rand(3,2)
array([[ 0.14022471,  0.96360618],  #random
       [ 0.37601032,  0.25528411],  #random
       [ 0.49313049,  0.94909878]]) #random
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.randint(low, high=None, size=None, dtype=int)

Return random integers from low (inclusive) to high (exclusive).

Return random integers from the “discrete uniform” distribution of the specified dtype in the “half-open” interval [low, high). If high is None (the default), then results are from [0, low).

Note

New code should use the integers method of a default_rng() instance instead; see random-quick-start.

low : int or array-like of ints
Lowest (signed) integers to be drawn from the distribution (unless high=None, in which case this parameter is one above the highest such integer).
high : int or array-like of ints, optional
If provided, one above the largest (signed) integer to be drawn from the distribution (see above for behavior if high=None). If array-like, must contain integer values
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
dtype : dtype, optional

Desired dtype of the result. Byteorder must be native. The default value is int.

New in version 1.11.0.

out : int or ndarray of ints
size-shaped array of random integers from the appropriate distribution, or a single such random int if size not provided.
random_integers : similar to randint, only for the closed
interval [low, high], and 1 is the lowest value if high is omitted.

Generator.integers: which should be used for new code.

>>> np.random.randint(2, size=10)
array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0]) # random
>>> np.random.randint(1, size=10)
array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0])

Generate a 2 x 4 array of ints between 0 and 4, inclusive:

>>> np.random.randint(5, size=(2, 4))
array([[4, 0, 2, 1], # random
       [3, 2, 2, 0]])

Generate a 1 x 3 array with 3 different upper bounds

>>> np.random.randint(1, [3, 5, 10])
array([2, 2, 9]) # random

Generate a 1 by 3 array with 3 different lower bounds

>>> np.random.randint([1, 5, 7], 10)
array([9, 8, 7]) # random

Generate a 2 by 4 array using broadcasting with dtype of uint8

>>> np.random.randint([1, 3, 5, 7], [[10], [20]], dtype=np.uint8)
array([[ 8,  6,  9,  7], # random
       [ 1, 16,  9, 12]], dtype=uint8)
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.randn(d0, d1, ..., dn)

Return a sample (or samples) from the “standard normal” distribution.

Note

This is a convenience function for users porting code from Matlab, and wraps standard_normal. That function takes a tuple to specify the size of the output, which is consistent with other NumPy functions like numpy.zeros and numpy.ones.

Note

New code should use the standard_normal method of a default_rng() instance instead; see random-quick-start.

If positive int_like arguments are provided, randn generates an array of shape (d0, d1, ..., dn), filled with random floats sampled from a univariate “normal” (Gaussian) distribution of mean 0 and variance 1. A single float randomly sampled from the distribution is returned if no argument is provided.

d0, d1, ..., dn : int, optional
The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned.
Z : ndarray or float
A (d0, d1, ..., dn)-shaped array of floating-point samples from the standard normal distribution, or a single such float if no parameters were supplied.

standard_normal : Similar, but takes a tuple as its argument. normal : Also accepts mu and sigma arguments. Generator.standard_normal: which should be used for new code.

For random samples from \(N(\mu, \sigma^2)\), use:

sigma * np.random.randn(...) + mu

>>> np.random.randn()
2.1923875335537315  # random

Two-by-four array of samples from N(3, 6.25):

>>> 3 + 2.5 * np.random.randn(2, 4)
array([[-4.49401501,  4.00950034, -1.81814867,  7.29718677],   # random
       [ 0.39924804,  4.68456316,  4.99394529,  4.84057254]])  # random
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.random(size=None)

Return random floats in the half-open interval [0.0, 1.0). Alias for random_sample to ease forward-porting to the new random API.

tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.random_integers(low, high=None, size=None)

Random integers of type np.int_ between low and high, inclusive.

Return random integers of type np.int_ from the “discrete uniform” distribution in the closed interval [low, high]. If high is None (the default), then results are from [1, low]. The np.int_ type translates to the C long integer type and its precision is platform dependent.

This function has been deprecated. Use randint instead.

Deprecated since version 1.11.0.

low : int
Lowest (signed) integer to be drawn from the distribution (unless high=None, in which case this parameter is the highest such integer).
high : int, optional
If provided, the largest (signed) integer to be drawn from the distribution (see above for behavior if high=None).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
out : int or ndarray of ints
size-shaped array of random integers from the appropriate distribution, or a single such random int if size not provided.
randint : Similar to random_integers, only for the half-open
interval [low, high), and 0 is the lowest value if high is omitted.

To sample from N evenly spaced floating-point numbers between a and b, use:

a + (b - a) * (np.random.random_integers(N) - 1) / (N - 1.)
>>> np.random.random_integers(5)
4 # random
>>> type(np.random.random_integers(5))
<class 'numpy.int64'>
>>> np.random.random_integers(5, size=(3,2))
array([[5, 4], # random
       [3, 3],
       [4, 5]])

Choose five random numbers from the set of five evenly-spaced numbers between 0 and 2.5, inclusive (i.e., from the set \({0, 5/8, 10/8, 15/8, 20/8}\)):

>>> 2.5 * (np.random.random_integers(5, size=(5,)) - 1) / 4.
array([ 0.625,  1.25 ,  0.625,  0.625,  2.5  ]) # random

Roll two six sided dice 1000 times and sum the results:

>>> d1 = np.random.random_integers(1, 6, 1000)
>>> d2 = np.random.random_integers(1, 6, 1000)
>>> dsums = d1 + d2

Display results as a histogram:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(dsums, 11, density=True)
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.random_sample(size=None)

Return random floats in the half-open interval [0.0, 1.0).

Results are from the “continuous uniform” distribution over the stated interval. To sample \(Unif[a, b), b > a\) multiply the output of random_sample by (b-a) and add a:

(b - a) * random_sample() + a

Note

New code should use the random method of a default_rng() instance instead; see random-quick-start.

size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
out : float or ndarray of floats
Array of random floats of shape size (unless size=None, in which case a single float is returned).

Generator.random: which should be used for new code.

>>> np.random.random_sample()
0.47108547995356098 # random
>>> type(np.random.random_sample())
<class 'float'>
>>> np.random.random_sample((5,))
array([ 0.30220482,  0.86820401,  0.1654503 ,  0.11659149,  0.54323428]) # random

Three-by-two array of random numbers from [-5, 0):

>>> 5 * np.random.random_sample((3, 2)) - 5
array([[-3.99149989, -0.52338984], # random
       [-2.99091858, -0.79479508],
       [-1.23204345, -1.75224494]])
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.rayleigh(scale=1.0, size=None)

Draw samples from a Rayleigh distribution.

The \(\chi\) and Weibull distributions are generalizations of the Rayleigh.

Note

New code should use the rayleigh method of a default_rng() instance instead; see random-quick-start.

scale : float or array_like of floats, optional
Scale, also equals the mode. Must be non-negative. Default is 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if scale is a scalar. Otherwise, np.array(scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Rayleigh distribution.

Generator.rayleigh: which should be used for new code.

The probability density function for the Rayleigh distribution is

\[P(x;scale) = \frac{x}{scale^2}e^{\frac{-x^2}{2 \cdotp scale^2}}\]

The Rayleigh distribution would arise, for example, if the East and North components of the wind velocity had identical zero-mean Gaussian distributions. Then the wind speed would have a Rayleigh distribution.

[1]Brighton Webs Ltd., “Rayleigh Distribution,” https://web.archive.org/web/20090514091424/http://brighton-webs.co.uk:80/distributions/rayleigh.asp
[2]Wikipedia, “Rayleigh distribution” https://en.wikipedia.org/wiki/Rayleigh_distribution

Draw values from the distribution and plot the histogram

>>> from matplotlib.pyplot import hist
>>> values = hist(np.random.rayleigh(3, 100000), bins=200, density=True)

Wave heights tend to follow a Rayleigh distribution. If the mean wave height is 1 meter, what fraction of waves are likely to be larger than 3 meters?

>>> meanvalue = 1
>>> modevalue = np.sqrt(2 / np.pi) * meanvalue
>>> s = np.random.rayleigh(modevalue, 1000000)

The percentage of waves larger than 3 meters is:

>>> 100.*sum(s>3)/1000000.
0.087300000000000003 # random
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.seed(self, seed=None)

Reseed a legacy MT19937 BitGenerator

This is a convenience, legacy function.

The best practice is to not reseed a BitGenerator, rather to recreate a new one. This method is here for legacy reasons. This example demonstrates best practice.

>>> from numpy.random import MT19937
>>> from numpy.random import RandomState, SeedSequence
>>> rs = RandomState(MT19937(SeedSequence(123456789)))
# Later, you want to restart the stream
>>> rs = RandomState(MT19937(SeedSequence(987654321)))
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.set_state(state)

Set the internal state of the generator from a tuple.

For use if one has reason to manually (re-)set the internal state of the bit generator used by the RandomState instance. By default, RandomState uses the “Mersenne Twister”[1]_ pseudo-random number generating algorithm.

state : {tuple(str, ndarray of 624 uints, int, int, float), dict}

The state tuple has the following items:

  1. the string ‘MT19937’, specifying the Mersenne Twister algorithm.
  2. a 1-D array of 624 unsigned integers keys.
  3. an integer pos.
  4. an integer has_gauss.
  5. a float cached_gaussian.

If state is a dictionary, it is directly set using the BitGenerators state property.

out : None
Returns ‘None’ on success.

get_state

set_state and get_state are not needed to work with any of the random distributions in NumPy. If the internal state is manually altered, the user should know exactly what he/she is doing.

For backwards compatibility, the form (str, array of 624 uints, int) is also accepted although it is missing some information about the cached Gaussian value: state = ('MT19937', keys, pos).

[1]M. Matsumoto and T. Nishimura, “Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator,” ACM Trans. on Modeling and Computer Simulation, Vol. 8, No. 1, pp. 3-30, Jan. 1998.
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.shuffle(x)

Modify a sequence in-place by shuffling its contents.

This function only shuffles the array along the first axis of a multi-dimensional array. The order of sub-arrays is changed but their contents remains the same.

Note

New code should use the shuffle method of a default_rng() instance instead; see random-quick-start.

x : array_like
The array or list to be shuffled.

None

Generator.shuffle: which should be used for new code.

>>> arr = np.arange(10)
>>> np.random.shuffle(arr)
>>> arr
[1 7 5 2 9 4 3 6 0 8] # random

Multi-dimensional arrays are only shuffled along the first axis:

>>> arr = np.arange(9).reshape((3, 3))
>>> np.random.shuffle(arr)
>>> arr
array([[3, 4, 5], # random
       [6, 7, 8],
       [0, 1, 2]])
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.standard_cauchy(size=None)

Draw samples from a standard Cauchy distribution with mode = 0.

Also known as the Lorentz distribution.

Note

New code should use the standard_cauchy method of a default_rng() instance instead; see random-quick-start.

size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
samples : ndarray or scalar
The drawn samples.

Generator.standard_cauchy: which should be used for new code.

The probability density function for the full Cauchy distribution is

\[P(x; x_0, \gamma) = \frac{1}{\pi \gamma \bigl[ 1+ (\frac{x-x_0}{\gamma})^2 \bigr] }\]

and the Standard Cauchy distribution just sets \(x_0=0\) and \(\gamma=1\)

The Cauchy distribution arises in the solution to the driven harmonic oscillator problem, and also describes spectral line broadening. It also describes the distribution of values at which a line tilted at a random angle will cut the x axis.

When studying hypothesis tests that assume normality, seeing how the tests perform on data from a Cauchy distribution is a good indicator of their sensitivity to a heavy-tailed distribution, since the Cauchy looks very much like a Gaussian distribution, but with heavier tails.

[1]NIST/SEMATECH e-Handbook of Statistical Methods, “Cauchy Distribution”, https://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm
[2]Weisstein, Eric W. “Cauchy Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/CauchyDistribution.html
[3]Wikipedia, “Cauchy distribution” https://en.wikipedia.org/wiki/Cauchy_distribution

Draw samples and plot the distribution:

>>> import matplotlib.pyplot as plt
>>> s = np.random.standard_cauchy(1000000)
>>> s = s[(s>-25) & (s<25)]  # truncate distribution so it plots well
>>> plt.hist(s, bins=100)
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.standard_exponential(size=None)

Draw samples from the standard exponential distribution.

standard_exponential is identical to the exponential distribution with a scale parameter of 1.

Note

New code should use the standard_exponential method of a default_rng() instance instead; see random-quick-start.

size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
out : float or ndarray
Drawn samples.

Generator.standard_exponential: which should be used for new code.

Output a 3x8000 array:

>>> n = np.random.standard_exponential((3, 8000))
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.standard_gamma(shape, size=None)

Draw samples from a standard Gamma distribution.

Samples are drawn from a Gamma distribution with specified parameters, shape (sometimes designated “k”) and scale=1.

Note

New code should use the standard_gamma method of a default_rng() instance instead; see random-quick-start.

shape : float or array_like of floats
Parameter, must be non-negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if shape is a scalar. Otherwise, np.array(shape).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized standard gamma distribution.
scipy.stats.gamma : probability density function, distribution or
cumulative density function, etc.

Generator.standard_gamma: which should be used for new code.

The probability density for the Gamma distribution is

\[p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},\]

where \(k\) is the shape and \(\theta\) the scale, and \(\Gamma\) is the Gamma function.

The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant.

[1]Weisstein, Eric W. “Gamma Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html
[2]Wikipedia, “Gamma distribution”, https://en.wikipedia.org/wiki/Gamma_distribution

Draw samples from the distribution:

>>> shape, scale = 2., 1. # mean and width
>>> s = np.random.standard_gamma(shape, 1000000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> import scipy.special as sps  
>>> count, bins, ignored = plt.hist(s, 50, density=True)
>>> y = bins**(shape-1) * ((np.exp(-bins/scale))/  
...                       (sps.gamma(shape) * scale**shape))
>>> plt.plot(bins, y, linewidth=2, color='r')  
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.standard_normal(size=None)

Draw samples from a standard Normal distribution (mean=0, stdev=1).

Note

New code should use the standard_normal method of a default_rng() instance instead; see random-quick-start.

size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
out : float or ndarray
A floating-point array of shape size of drawn samples, or a single sample if size was not specified.
normal :
Equivalent function with additional loc and scale arguments for setting the mean and standard deviation.

Generator.standard_normal: which should be used for new code.

For random samples from \(N(\mu, \sigma^2)\), use one of:

mu + sigma * np.random.standard_normal(size=...)
np.random.normal(mu, sigma, size=...)
>>> np.random.standard_normal()
2.1923875335537315 #random
>>> s = np.random.standard_normal(8000)
>>> s
array([ 0.6888893 ,  0.78096262, -0.89086505, ...,  0.49876311,  # random
       -0.38672696, -0.4685006 ])                                # random
>>> s.shape
(8000,)
>>> s = np.random.standard_normal(size=(3, 4, 2))
>>> s.shape
(3, 4, 2)

Two-by-four array of samples from \(N(3, 6.25)\):

>>> 3 + 2.5 * np.random.standard_normal(size=(2, 4))
array([[-4.49401501,  4.00950034, -1.81814867,  7.29718677],   # random
       [ 0.39924804,  4.68456316,  4.99394529,  4.84057254]])  # random
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.standard_t(df, size=None)

Draw samples from a standard Student’s t distribution with df degrees of freedom.

A special case of the hyperbolic distribution. As df gets large, the result resembles that of the standard normal distribution (standard_normal).

Note

New code should use the standard_t method of a default_rng() instance instead; see random-quick-start.

df : float or array_like of floats
Degrees of freedom, must be > 0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if df is a scalar. Otherwise, np.array(df).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized standard Student’s t distribution.

Generator.standard_t: which should be used for new code.

The probability density function for the t distribution is

\[P(x, df) = \frac{\Gamma(\frac{df+1}{2})}{\sqrt{\pi df} \Gamma(\frac{df}{2})}\Bigl( 1+\frac{x^2}{df} \Bigr)^{-(df+1)/2}\]

The t test is based on an assumption that the data come from a Normal distribution. The t test provides a way to test whether the sample mean (that is the mean calculated from the data) is a good estimate of the true mean.

The derivation of the t-distribution was first published in 1908 by William Gosset while working for the Guinness Brewery in Dublin. Due to proprietary issues, he had to publish under a pseudonym, and so he used the name Student.

[1]Dalgaard, Peter, “Introductory Statistics With R”, Springer, 2002.
[2]Wikipedia, “Student’s t-distribution” https://en.wikipedia.org/wiki/Student’s_t-distribution

From Dalgaard page 83 [1]_, suppose the daily energy intake for 11 women in kilojoules (kJ) is:

>>> intake = np.array([5260., 5470, 5640, 6180, 6390, 6515, 6805, 7515, \
...                    7515, 8230, 8770])

Does their energy intake deviate systematically from the recommended value of 7725 kJ?

We have 10 degrees of freedom, so is the sample mean within 95% of the recommended value?

>>> s = np.random.standard_t(10, size=100000)
>>> np.mean(intake)
6753.636363636364
>>> intake.std(ddof=1)
1142.1232221373727

Calculate the t statistic, setting the ddof parameter to the unbiased value so the divisor in the standard deviation will be degrees of freedom, N-1.

>>> t = (np.mean(intake)-7725)/(intake.std(ddof=1)/np.sqrt(len(intake)))
>>> import matplotlib.pyplot as plt
>>> h = plt.hist(s, bins=100, density=True)

For a one-sided t-test, how far out in the distribution does the t statistic appear?

>>> np.sum(s<t) / float(len(s))
0.0090699999999999999  #random

So the p-value is about 0.009, which says the null hypothesis has a probability of about 99% of being true.

tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.triangular(left, mode, right, size=None)

Draw samples from the triangular distribution over the interval [left, right].

The triangular distribution is a continuous probability distribution with lower limit left, peak at mode, and upper limit right. Unlike the other distributions, these parameters directly define the shape of the pdf.

Note

New code should use the triangular method of a default_rng() instance instead; see random-quick-start.

left : float or array_like of floats
Lower limit.
mode : float or array_like of floats
The value where the peak of the distribution occurs. The value must fulfill the condition left <= mode <= right.
right : float or array_like of floats
Upper limit, must be larger than left.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if left, mode, and right are all scalars. Otherwise, np.broadcast(left, mode, right).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized triangular distribution.

Generator.triangular: which should be used for new code.

The probability density function for the triangular distribution is

\[\begin{split}P(x;l, m, r) = \begin{cases} \frac{2(x-l)}{(r-l)(m-l)}& \text{for $l \leq x \leq m$},\\ \frac{2(r-x)}{(r-l)(r-m)}& \text{for $m \leq x \leq r$},\\ 0& \text{otherwise}. \end{cases}\end{split}\]

The triangular distribution is often used in ill-defined problems where the underlying distribution is not known, but some knowledge of the limits and mode exists. Often it is used in simulations.

[1]Wikipedia, “Triangular distribution” https://en.wikipedia.org/wiki/Triangular_distribution

Draw values from the distribution and plot the histogram:

>>> import matplotlib.pyplot as plt
>>> h = plt.hist(np.random.triangular(-3, 0, 8, 100000), bins=200,
...              density=True)
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.uniform(low=0.0, high=1.0, size=None)

Draw samples from a uniform distribution.

Samples are uniformly distributed over the half-open interval [low, high) (includes low, but excludes high). In other words, any value within the given interval is equally likely to be drawn by uniform.

Note

New code should use the uniform method of a default_rng() instance instead; see random-quick-start.

low : float or array_like of floats, optional
Lower boundary of the output interval. All values generated will be greater than or equal to low. The default value is 0.
high : float or array_like of floats
Upper boundary of the output interval. All values generated will be less than or equal to high. The default value is 1.0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if low and high are both scalars. Otherwise, np.broadcast(low, high).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized uniform distribution.

randint : Discrete uniform distribution, yielding integers. random_integers : Discrete uniform distribution over the closed

interval [low, high].

random_sample : Floats uniformly distributed over [0, 1). random : Alias for random_sample. rand : Convenience function that accepts dimensions as input, e.g.,

rand(2,2) would generate a 2-by-2 array of floats, uniformly distributed over [0, 1).

Generator.uniform: which should be used for new code.

The probability density function of the uniform distribution is

\[p(x) = \frac{1}{b - a}\]

anywhere within the interval [a, b), and zero elsewhere.

When high == low, values of low will be returned. If high < low, the results are officially undefined and may eventually raise an error, i.e. do not rely on this function to behave when passed arguments satisfying that inequality condition. The high limit may be included in the returned array of floats due to floating-point rounding in the equation low + (high-low) * random_sample(). For example:

>>> x = np.float32(5*0.99999999)
>>> x
5.0

Draw samples from the distribution:

>>> s = np.random.uniform(-1,0,1000)

All values are within the given interval:

>>> np.all(s >= -1)
True
>>> np.all(s < 0)
True

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 15, density=True)
>>> plt.plot(bins, np.ones_like(bins), linewidth=2, color='r')
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.vonmises(mu, kappa, size=None)

Draw samples from a von Mises distribution.

Samples are drawn from a von Mises distribution with specified mode (mu) and dispersion (kappa), on the interval [-pi, pi].

The von Mises distribution (also known as the circular normal distribution) is a continuous probability distribution on the unit circle. It may be thought of as the circular analogue of the normal distribution.

Note

New code should use the vonmises method of a default_rng() instance instead; see random-quick-start.

mu : float or array_like of floats
Mode (“center”) of the distribution.
kappa : float or array_like of floats
Dispersion of the distribution, has to be >=0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if mu and kappa are both scalars. Otherwise, np.broadcast(mu, kappa).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized von Mises distribution.
scipy.stats.vonmises : probability density function, distribution, or
cumulative density function, etc.

Generator.vonmises: which should be used for new code.

The probability density for the von Mises distribution is

\[p(x) = \frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)},\]

where \(\mu\) is the mode and \(\kappa\) the dispersion, and \(I_0(\kappa)\) is the modified Bessel function of order 0.

The von Mises is named for Richard Edler von Mises, who was born in Austria-Hungary, in what is now the Ukraine. He fled to the United States in 1939 and became a professor at Harvard. He worked in probability theory, aerodynamics, fluid mechanics, and philosophy of science.

[1]Abramowitz, M. and Stegun, I. A. (Eds.). “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing,” New York: Dover, 1972.
[2]von Mises, R., “Mathematical Theory of Probability and Statistics”, New York: Academic Press, 1964.

Draw samples from the distribution:

>>> mu, kappa = 0.0, 4.0 # mean and dispersion
>>> s = np.random.vonmises(mu, kappa, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> from scipy.special import i0  
>>> plt.hist(s, 50, density=True)
>>> x = np.linspace(-np.pi, np.pi, num=51)
>>> y = np.exp(kappa*np.cos(x-mu))/(2*np.pi*i0(kappa))  
>>> plt.plot(x, y, linewidth=2, color='r')  
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.wald(mean, scale, size=None)

Draw samples from a Wald, or inverse Gaussian, distribution.

As the scale approaches infinity, the distribution becomes more like a Gaussian. Some references claim that the Wald is an inverse Gaussian with mean equal to 1, but this is by no means universal.

The inverse Gaussian distribution was first studied in relationship to Brownian motion. In 1956 M.C.K. Tweedie used the name inverse Gaussian because there is an inverse relationship between the time to cover a unit distance and distance covered in unit time.

Note

New code should use the wald method of a default_rng() instance instead; see random-quick-start.

mean : float or array_like of floats
Distribution mean, must be > 0.
scale : float or array_like of floats
Scale parameter, must be > 0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if mean and scale are both scalars. Otherwise, np.broadcast(mean, scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Wald distribution.

Generator.wald: which should be used for new code.

The probability density function for the Wald distribution is

\[P(x;mean,scale) = \sqrt{\frac{scale}{2\pi x^3}}e^ \frac{-scale(x-mean)^2}{2\cdotp mean^2x}\]

As noted above the inverse Gaussian distribution first arise from attempts to model Brownian motion. It is also a competitor to the Weibull for use in reliability modeling and modeling stock returns and interest rate processes.

[1]Brighton Webs Ltd., Wald Distribution, https://web.archive.org/web/20090423014010/http://www.brighton-webs.co.uk:80/distributions/wald.asp
[2]Chhikara, Raj S., and Folks, J. Leroy, “The Inverse Gaussian Distribution: Theory : Methodology, and Applications”, CRC Press, 1988.
[3]Wikipedia, “Inverse Gaussian distribution” https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution

Draw values from the distribution and plot the histogram:

>>> import matplotlib.pyplot as plt
>>> h = plt.hist(np.random.wald(3, 2, 100000), bins=200, density=True)
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.weibull(a, size=None)

Draw samples from a Weibull distribution.

Draw samples from a 1-parameter Weibull distribution with the given shape parameter a.

\[X = (-ln(U))^{1/a}\]

Here, U is drawn from the uniform distribution over (0,1].

The more common 2-parameter Weibull, including a scale parameter \(\lambda\) is just \(X = \lambda(-ln(U))^{1/a}\).

Note

New code should use the weibull method of a default_rng() instance instead; see random-quick-start.

a : float or array_like of floats
Shape parameter of the distribution. Must be nonnegative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if a is a scalar. Otherwise, np.array(a).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Weibull distribution.

scipy.stats.weibull_max scipy.stats.weibull_min scipy.stats.genextreme gumbel Generator.weibull: which should be used for new code.

The Weibull (or Type III asymptotic extreme value distribution for smallest values, SEV Type III, or Rosin-Rammler distribution) is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. This class includes the Gumbel and Frechet distributions.

The probability density for the Weibull distribution is

\[p(x) = \frac{a} {\lambda}(\frac{x}{\lambda})^{a-1}e^{-(x/\lambda)^a},\]

where \(a\) is the shape and \(\lambda\) the scale.

The function has its peak (the mode) at \(\lambda(\frac{a-1}{a})^{1/a}\).

When a = 1, the Weibull distribution reduces to the exponential distribution.

[1]Waloddi Weibull, Royal Technical University, Stockholm, 1939 “A Statistical Theory Of The Strength Of Materials”, Ingeniorsvetenskapsakademiens Handlingar Nr 151, 1939, Generalstabens Litografiska Anstalts Forlag, Stockholm.
[2]Waloddi Weibull, “A Statistical Distribution Function of Wide Applicability”, Journal Of Applied Mechanics ASME Paper 1951.
[3]Wikipedia, “Weibull distribution”, https://en.wikipedia.org/wiki/Weibull_distribution

Draw samples from the distribution:

>>> a = 5. # shape
>>> s = np.random.weibull(a, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> x = np.arange(1,100.)/50.
>>> def weib(x,n,a):
...     return (a / n) * (x / n)**(a - 1) * np.exp(-(x / n)**a)
>>> count, bins, ignored = plt.hist(np.random.weibull(5.,1000))
>>> x = np.arange(1,100.)/50.
>>> scale = count.max()/weib(x, 1., 5.).max()
>>> plt.plot(x, weib(x, 1., 5.)*scale)
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_b.zipf(a, size=None)

Draw samples from a Zipf distribution.

Samples are drawn from a Zipf distribution with specified parameter a > 1.

The Zipf distribution (also known as the zeta distribution) is a continuous probability distribution that satisfies Zipf’s law: the frequency of an item is inversely proportional to its rank in a frequency table.

Note

New code should use the zipf method of a default_rng() instance instead; see random-quick-start.

a : float or array_like of floats
Distribution parameter. Must be greater than 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if a is a scalar. Otherwise, np.array(a).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Zipf distribution.
scipy.stats.zipf : probability density function, distribution, or
cumulative density function, etc.

Generator.zipf: which should be used for new code.

The probability density for the Zipf distribution is

\[p(x) = \frac{x^{-a}}{\zeta(a)},\]

where \(\zeta\) is the Riemann Zeta function.

It is named for the American linguist George Kingsley Zipf, who noted that the frequency of any word in a sample of a language is inversely proportional to its rank in the frequency table.

[1]Zipf, G. K., “Selected Studies of the Principle of Relative Frequency in Language,” Cambridge, MA: Harvard Univ. Press, 1932.

Draw samples from the distribution:

>>> a = 2. # parameter
>>> s = np.random.zipf(a, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> from scipy import special  

Truncate s values at 50 so plot is interesting:

>>> count, bins, ignored = plt.hist(s[s<50], 50, density=True)
>>> x = np.arange(1., 50.)
>>> y = x**(-a) / special.zetac(a)  
>>> plt.plot(x, y/max(y), linewidth=2, color='r')  
>>> plt.show()

region_deterministic_bnm_g2d_c

What:
Reproduces Figure 13, third column, of Sanz-Leon P., Knock, S. A., Spiegler, A. and Jirsa V. Mathematical framework for large-scale brain network modelling in The Virtual Brain. Neuroimage, 2014, (in review)
Needs:
A working installation of tvb
Run:

python region_deterministic_bnm_g2d_c.py -s True -f True

#Subsequent calls can be made with: python region_deterministic_bnm_g2d_c.py -f True

tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.beta(a, b, size=None)

Draw samples from a Beta distribution.

The Beta distribution is a special case of the Dirichlet distribution, and is related to the Gamma distribution. It has the probability distribution function

\[f(x; a,b) = \frac{1}{B(\alpha, \beta)} x^{\alpha - 1} (1 - x)^{\beta - 1},\]

where the normalization, B, is the beta function,

\[B(\alpha, \beta) = \int_0^1 t^{\alpha - 1} (1 - t)^{\beta - 1} dt.\]

It is often seen in Bayesian inference and order statistics.

Note

New code should use the beta method of a default_rng() instance instead; see random-quick-start.

a : float or array_like of floats
Alpha, positive (>0).
b : float or array_like of floats
Beta, positive (>0).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if a and b are both scalars. Otherwise, np.broadcast(a, b).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized beta distribution.

Generator.beta: which should be used for new code.

tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.binomial(n, p, size=None)

Draw samples from a binomial distribution.

Samples are drawn from a binomial distribution with specified parameters, n trials and p probability of success where n an integer >= 0 and p is in the interval [0,1]. (n may be input as a float, but it is truncated to an integer in use)

Note

New code should use the binomial method of a default_rng() instance instead; see random-quick-start.

n : int or array_like of ints
Parameter of the distribution, >= 0. Floats are also accepted, but they will be truncated to integers.
p : float or array_like of floats
Parameter of the distribution, >= 0 and <=1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if n and p are both scalars. Otherwise, np.broadcast(n, p).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized binomial distribution, where each sample is equal to the number of successes over the n trials.
scipy.stats.binom : probability density function, distribution or
cumulative density function, etc.

Generator.binomial: which should be used for new code.

The probability density for the binomial distribution is

\[P(N) = \binom{n}{N}p^N(1-p)^{n-N},\]

where \(n\) is the number of trials, \(p\) is the probability of success, and \(N\) is the number of successes.

When estimating the standard error of a proportion in a population by using a random sample, the normal distribution works well unless the product p*n <=5, where p = population proportion estimate, and n = number of samples, in which case the binomial distribution is used instead. For example, a sample of 15 people shows 4 who are left handed, and 11 who are right handed. Then p = 4/15 = 27%. 0.27*15 = 4, so the binomial distribution should be used in this case.

[1]Dalgaard, Peter, “Introductory Statistics with R”, Springer-Verlag, 2002.
[2]Glantz, Stanton A. “Primer of Biostatistics.”, McGraw-Hill, Fifth Edition, 2002.
[3]Lentner, Marvin, “Elementary Applied Statistics”, Bogden and Quigley, 1972.
[4]Weisstein, Eric W. “Binomial Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/BinomialDistribution.html
[5]Wikipedia, “Binomial distribution”, https://en.wikipedia.org/wiki/Binomial_distribution

Draw samples from the distribution:

>>> n, p = 10, .5  # number of trials, probability of each trial
>>> s = np.random.binomial(n, p, 1000)
# result of flipping a coin 10 times, tested 1000 times.

A real world example. A company drills 9 wild-cat oil exploration wells, each with an estimated probability of success of 0.1. All nine wells fail. What is the probability of that happening?

Let’s do 20,000 trials of the model, and count the number that generate zero positive results.

>>> sum(np.random.binomial(9, 0.1, 20000) == 0)/20000.
# answer = 0.38885, or 38%.
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.chisquare(df, size=None)

Draw samples from a chi-square distribution.

When df independent random variables, each with standard normal distributions (mean 0, variance 1), are squared and summed, the resulting distribution is chi-square (see Notes). This distribution is often used in hypothesis testing.

Note

New code should use the chisquare method of a default_rng() instance instead; see random-quick-start.

df : float or array_like of floats
Number of degrees of freedom, must be > 0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if df is a scalar. Otherwise, np.array(df).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized chi-square distribution.
ValueError
When df <= 0 or when an inappropriate size (e.g. size=-1) is given.

Generator.chisquare: which should be used for new code.

The variable obtained by summing the squares of df independent, standard normally distributed random variables:

\[Q = \sum_{i=0}^{\mathtt{df}} X^2_i\]

is chi-square distributed, denoted

\[Q \sim \chi^2_k.\]

The probability density function of the chi-squared distribution is

\[p(x) = \frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2},\]

where \(\Gamma\) is the gamma function,

\[\Gamma(x) = \int_0^{-\infty} t^{x - 1} e^{-t} dt.\]
[1]NIST “Engineering Statistics Handbook” https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm
>>> np.random.chisquare(2,4)
array([ 1.89920014,  9.00867716,  3.13710533,  5.62318272]) # random
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.choice(a, size=None, replace=True, p=None)

Generates a random sample from a given 1-D array

New in version 1.7.0.

Note

New code should use the choice method of a default_rng() instance instead; see random-quick-start.

a : 1-D array-like or int
If an ndarray, a random sample is generated from its elements. If an int, the random sample is generated as if a were np.arange(a)
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
replace : boolean, optional
Whether the sample is with or without replacement
p : 1-D array-like, optional
The probabilities associated with each entry in a. If not given the sample assumes a uniform distribution over all entries in a.
samples : single item or ndarray
The generated random samples
ValueError
If a is an int and less than zero, if a or p are not 1-dimensional, if a is an array-like of size 0, if p is not a vector of probabilities, if a and p have different lengths, or if replace=False and the sample size is greater than the population size

randint, shuffle, permutation Generator.choice: which should be used in new code

Sampling random rows from a 2-D array is not possible with this function, but is possible with Generator.choice through its axis keyword.

Generate a uniform random sample from np.arange(5) of size 3:

>>> np.random.choice(5, 3)
array([0, 3, 4]) # random
>>> #This is equivalent to np.random.randint(0,5,3)

Generate a non-uniform random sample from np.arange(5) of size 3:

>>> np.random.choice(5, 3, p=[0.1, 0, 0.3, 0.6, 0])
array([3, 3, 0]) # random

Generate a uniform random sample from np.arange(5) of size 3 without replacement:

>>> np.random.choice(5, 3, replace=False)
array([3,1,0]) # random
>>> #This is equivalent to np.random.permutation(np.arange(5))[:3]

Generate a non-uniform random sample from np.arange(5) of size 3 without replacement:

>>> np.random.choice(5, 3, replace=False, p=[0.1, 0, 0.3, 0.6, 0])
array([2, 3, 0]) # random

Any of the above can be repeated with an arbitrary array-like instead of just integers. For instance:

>>> aa_milne_arr = ['pooh', 'rabbit', 'piglet', 'Christopher']
>>> np.random.choice(aa_milne_arr, 5, p=[0.5, 0.1, 0.1, 0.3])
array(['pooh', 'pooh', 'pooh', 'Christopher', 'piglet'], # random
      dtype='<U11')
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.dirichlet(alpha, size=None)

Draw samples from the Dirichlet distribution.

Draw size samples of dimension k from a Dirichlet distribution. A Dirichlet-distributed random variable can be seen as a multivariate generalization of a Beta distribution. The Dirichlet distribution is a conjugate prior of a multinomial distribution in Bayesian inference.

Note

New code should use the dirichlet method of a default_rng() instance instead; see random-quick-start.

alpha : sequence of floats, length k
Parameter of the distribution (length k for sample of length k).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n), then m * n * k samples are drawn. Default is None, in which case a vector of length k is returned.
samples : ndarray,
The drawn samples, of shape (size, k).
ValueError
If any value in alpha is less than or equal to zero

Generator.dirichlet: which should be used for new code.

The Dirichlet distribution is a distribution over vectors \(x\) that fulfil the conditions \(x_i>0\) and \(\sum_{i=1}^k x_i = 1\).

The probability density function \(p\) of a Dirichlet-distributed random vector \(X\) is proportional to

\[p(x) \propto \prod_{i=1}^{k}{x^{\alpha_i-1}_i},\]

where \(\alpha\) is a vector containing the positive concentration parameters.

The method uses the following property for computation: let \(Y\) be a random vector which has components that follow a standard gamma distribution, then \(X = \frac{1}{\sum_{i=1}^k{Y_i}} Y\) is Dirichlet-distributed

[1]David McKay, “Information Theory, Inference and Learning Algorithms,” chapter 23, http://www.inference.org.uk/mackay/itila/
[2]Wikipedia, “Dirichlet distribution”, https://en.wikipedia.org/wiki/Dirichlet_distribution

Taking an example cited in Wikipedia, this distribution can be used if one wanted to cut strings (each of initial length 1.0) into K pieces with different lengths, where each piece had, on average, a designated average length, but allowing some variation in the relative sizes of the pieces.

>>> s = np.random.dirichlet((10, 5, 3), 20).transpose()
>>> import matplotlib.pyplot as plt
>>> plt.barh(range(20), s[0])
>>> plt.barh(range(20), s[1], left=s[0], color='g')
>>> plt.barh(range(20), s[2], left=s[0]+s[1], color='r')
>>> plt.title("Lengths of Strings")
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.exponential(scale=1.0, size=None)

Draw samples from an exponential distribution.

Its probability density function is

\[f(x; \frac{1}{\beta}) = \frac{1}{\beta} \exp(-\frac{x}{\beta}),\]

for x > 0 and 0 elsewhere. \(\beta\) is the scale parameter, which is the inverse of the rate parameter \(\lambda = 1/\beta\). The rate parameter is an alternative, widely used parameterization of the exponential distribution [3]_.

The exponential distribution is a continuous analogue of the geometric distribution. It describes many common situations, such as the size of raindrops measured over many rainstorms [1]_, or the time between page requests to Wikipedia [2]_.

Note

New code should use the exponential method of a default_rng() instance instead; see random-quick-start.

scale : float or array_like of floats
The scale parameter, \(\beta = 1/\lambda\). Must be non-negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if scale is a scalar. Otherwise, np.array(scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized exponential distribution.

Generator.exponential: which should be used for new code.

[1]Peyton Z. Peebles Jr., “Probability, Random Variables and Random Signal Principles”, 4th ed, 2001, p. 57.
[2]Wikipedia, “Poisson process”, https://en.wikipedia.org/wiki/Poisson_process
[3]Wikipedia, “Exponential distribution”, https://en.wikipedia.org/wiki/Exponential_distribution
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.f(dfnum, dfden, size=None)

Draw samples from an F distribution.

Samples are drawn from an F distribution with specified parameters, dfnum (degrees of freedom in numerator) and dfden (degrees of freedom in denominator), where both parameters must be greater than zero.

The random variate of the F distribution (also known as the Fisher distribution) is a continuous probability distribution that arises in ANOVA tests, and is the ratio of two chi-square variates.

Note

New code should use the f method of a default_rng() instance instead; see random-quick-start.

dfnum : float or array_like of floats
Degrees of freedom in numerator, must be > 0.
dfden : float or array_like of float
Degrees of freedom in denominator, must be > 0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if dfnum and dfden are both scalars. Otherwise, np.broadcast(dfnum, dfden).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Fisher distribution.
scipy.stats.f : probability density function, distribution or
cumulative density function, etc.

Generator.f: which should be used for new code.

The F statistic is used to compare in-group variances to between-group variances. Calculating the distribution depends on the sampling, and so it is a function of the respective degrees of freedom in the problem. The variable dfnum is the number of samples minus one, the between-groups degrees of freedom, while dfden is the within-groups degrees of freedom, the sum of the number of samples in each group minus the number of groups.

[1]Glantz, Stanton A. “Primer of Biostatistics.”, McGraw-Hill, Fifth Edition, 2002.
[2]Wikipedia, “F-distribution”, https://en.wikipedia.org/wiki/F-distribution

An example from Glantz[1], pp 47-40:

Two groups, children of diabetics (25 people) and children from people without diabetes (25 controls). Fasting blood glucose was measured, case group had a mean value of 86.1, controls had a mean value of 82.2. Standard deviations were 2.09 and 2.49 respectively. Are these data consistent with the null hypothesis that the parents diabetic status does not affect their children’s blood glucose levels? Calculating the F statistic from the data gives a value of 36.01.

Draw samples from the distribution:

>>> dfnum = 1. # between group degrees of freedom
>>> dfden = 48. # within groups degrees of freedom
>>> s = np.random.f(dfnum, dfden, 1000)

The lower bound for the top 1% of the samples is :

>>> np.sort(s)[-10]
7.61988120985 # random

So there is about a 1% chance that the F statistic will exceed 7.62, the measured value is 36, so the null hypothesis is rejected at the 1% level.

tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.gamma(shape, scale=1.0, size=None)

Draw samples from a Gamma distribution.

Samples are drawn from a Gamma distribution with specified parameters, shape (sometimes designated “k”) and scale (sometimes designated “theta”), where both parameters are > 0.

Note

New code should use the gamma method of a default_rng() instance instead; see random-quick-start.

shape : float or array_like of floats
The shape of the gamma distribution. Must be non-negative.
scale : float or array_like of floats, optional
The scale of the gamma distribution. Must be non-negative. Default is equal to 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if shape and scale are both scalars. Otherwise, np.broadcast(shape, scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized gamma distribution.
scipy.stats.gamma : probability density function, distribution or
cumulative density function, etc.

Generator.gamma: which should be used for new code.

The probability density for the Gamma distribution is

\[p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},\]

where \(k\) is the shape and \(\theta\) the scale, and \(\Gamma\) is the Gamma function.

The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant.

[1]Weisstein, Eric W. “Gamma Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html
[2]Wikipedia, “Gamma distribution”, https://en.wikipedia.org/wiki/Gamma_distribution

Draw samples from the distribution:

>>> shape, scale = 2., 2.  # mean=4, std=2*sqrt(2)
>>> s = np.random.gamma(shape, scale, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> import scipy.special as sps  
>>> count, bins, ignored = plt.hist(s, 50, density=True)
>>> y = bins**(shape-1)*(np.exp(-bins/scale) /  
...                      (sps.gamma(shape)*scale**shape))
>>> plt.plot(bins, y, linewidth=2, color='r')  
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.geometric(p, size=None)

Draw samples from the geometric distribution.

Bernoulli trials are experiments with one of two outcomes: success or failure (an example of such an experiment is flipping a coin). The geometric distribution models the number of trials that must be run in order to achieve success. It is therefore supported on the positive integers, k = 1, 2, ....

The probability mass function of the geometric distribution is

\[f(k) = (1 - p)^{k - 1} p\]

where p is the probability of success of an individual trial.

Note

New code should use the geometric method of a default_rng() instance instead; see random-quick-start.

p : float or array_like of floats
The probability of success of an individual trial.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if p is a scalar. Otherwise, np.array(p).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized geometric distribution.

Generator.geometric: which should be used for new code.

Draw ten thousand values from the geometric distribution, with the probability of an individual success equal to 0.35:

>>> z = np.random.geometric(p=0.35, size=10000)

How many trials succeeded after a single run?

>>> (z == 1).sum() / 10000.
0.34889999999999999 #random
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.get_state()

Return a tuple representing the internal state of the generator.

For more details, see set_state.

legacy : bool, optional
Flag indicating to return a legacy tuple state when the BitGenerator is MT19937, instead of a dict.
out : {tuple(str, ndarray of 624 uints, int, int, float), dict}

The returned tuple has the following items:

  1. the string ‘MT19937’.
  2. a 1-D array of 624 unsigned integer keys.
  3. an integer pos.
  4. an integer has_gauss.
  5. a float cached_gaussian.

If legacy is False, or the BitGenerator is not MT19937, then state is returned as a dictionary.

set_state

set_state and get_state are not needed to work with any of the random distributions in NumPy. If the internal state is manually altered, the user should know exactly what he/she is doing.

tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.gumbel(loc=0.0, scale=1.0, size=None)

Draw samples from a Gumbel distribution.

Draw samples from a Gumbel distribution with specified location and scale. For more information on the Gumbel distribution, see Notes and References below.

Note

New code should use the gumbel method of a default_rng() instance instead; see random-quick-start.

loc : float or array_like of floats, optional
The location of the mode of the distribution. Default is 0.
scale : float or array_like of floats, optional
The scale parameter of the distribution. Default is 1. Must be non- negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if loc and scale are both scalars. Otherwise, np.broadcast(loc, scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Gumbel distribution.

scipy.stats.gumbel_l scipy.stats.gumbel_r scipy.stats.genextreme weibull Generator.gumbel: which should be used for new code.

The Gumbel (or Smallest Extreme Value (SEV) or the Smallest Extreme Value Type I) distribution is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. The Gumbel is a special case of the Extreme Value Type I distribution for maximums from distributions with “exponential-like” tails.

The probability density for the Gumbel distribution is

\[p(x) = \frac{e^{-(x - \mu)/ \beta}}{\beta} e^{ -e^{-(x - \mu)/ \beta}},\]

where \(\mu\) is the mode, a location parameter, and \(\beta\) is the scale parameter.

The Gumbel (named for German mathematician Emil Julius Gumbel) was used very early in the hydrology literature, for modeling the occurrence of flood events. It is also used for modeling maximum wind speed and rainfall rates. It is a “fat-tailed” distribution - the probability of an event in the tail of the distribution is larger than if one used a Gaussian, hence the surprisingly frequent occurrence of 100-year floods. Floods were initially modeled as a Gaussian process, which underestimated the frequency of extreme events.

It is one of a class of extreme value distributions, the Generalized Extreme Value (GEV) distributions, which also includes the Weibull and Frechet.

The function has a mean of \(\mu + 0.57721\beta\) and a variance of \(\frac{\pi^2}{6}\beta^2\).

[1]Gumbel, E. J., “Statistics of Extremes,” New York: Columbia University Press, 1958.
[2]Reiss, R.-D. and Thomas, M., “Statistical Analysis of Extreme Values from Insurance, Finance, Hydrology and Other Fields,” Basel: Birkhauser Verlag, 2001.

Draw samples from the distribution:

>>> mu, beta = 0, 0.1 # location and scale
>>> s = np.random.gumbel(mu, beta, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, density=True)
>>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)
...          * np.exp( -np.exp( -(bins - mu) /beta) ),
...          linewidth=2, color='r')
>>> plt.show()

Show how an extreme value distribution can arise from a Gaussian process and compare to a Gaussian:

>>> means = []
>>> maxima = []
>>> for i in range(0,1000) :
...    a = np.random.normal(mu, beta, 1000)
...    means.append(a.mean())
...    maxima.append(a.max())
>>> count, bins, ignored = plt.hist(maxima, 30, density=True)
>>> beta = np.std(maxima) * np.sqrt(6) / np.pi
>>> mu = np.mean(maxima) - 0.57721*beta
>>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)
...          * np.exp(-np.exp(-(bins - mu)/beta)),
...          linewidth=2, color='r')
>>> plt.plot(bins, 1/(beta * np.sqrt(2 * np.pi))
...          * np.exp(-(bins - mu)**2 / (2 * beta**2)),
...          linewidth=2, color='g')
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.hypergeometric(ngood, nbad, nsample, size=None)

Draw samples from a Hypergeometric distribution.

Samples are drawn from a hypergeometric distribution with specified parameters, ngood (ways to make a good selection), nbad (ways to make a bad selection), and nsample (number of items sampled, which is less than or equal to the sum ngood + nbad).

Note

New code should use the hypergeometric method of a default_rng() instance instead; see random-quick-start.

ngood : int or array_like of ints
Number of ways to make a good selection. Must be nonnegative.
nbad : int or array_like of ints
Number of ways to make a bad selection. Must be nonnegative.
nsample : int or array_like of ints
Number of items sampled. Must be at least 1 and at most ngood + nbad.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if ngood, nbad, and nsample are all scalars. Otherwise, np.broadcast(ngood, nbad, nsample).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized hypergeometric distribution. Each sample is the number of good items within a randomly selected subset of size nsample taken from a set of ngood good items and nbad bad items.
scipy.stats.hypergeom : probability density function, distribution or
cumulative density function, etc.

Generator.hypergeometric: which should be used for new code.

The probability density for the Hypergeometric distribution is

\[P(x) = \frac{\binom{g}{x}\binom{b}{n-x}}{\binom{g+b}{n}},\]

where \(0 \le x \le n\) and \(n-b \le x \le g\)

for P(x) the probability of x good results in the drawn sample, g = ngood, b = nbad, and n = nsample.

Consider an urn with black and white marbles in it, ngood of them are black and nbad are white. If you draw nsample balls without replacement, then the hypergeometric distribution describes the distribution of black balls in the drawn sample.

Note that this distribution is very similar to the binomial distribution, except that in this case, samples are drawn without replacement, whereas in the Binomial case samples are drawn with replacement (or the sample space is infinite). As the sample space becomes large, this distribution approaches the binomial.

[1]Lentner, Marvin, “Elementary Applied Statistics”, Bogden and Quigley, 1972.
[2]Weisstein, Eric W. “Hypergeometric Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/HypergeometricDistribution.html
[3]Wikipedia, “Hypergeometric distribution”, https://en.wikipedia.org/wiki/Hypergeometric_distribution

Draw samples from the distribution:

>>> ngood, nbad, nsamp = 100, 2, 10
# number of good, number of bad, and number of samples
>>> s = np.random.hypergeometric(ngood, nbad, nsamp, 1000)
>>> from matplotlib.pyplot import hist
>>> hist(s)
#   note that it is very unlikely to grab both bad items

Suppose you have an urn with 15 white and 15 black marbles. If you pull 15 marbles at random, how likely is it that 12 or more of them are one color?

>>> s = np.random.hypergeometric(15, 15, 15, 100000)
>>> sum(s>=12)/100000. + sum(s<=3)/100000.
#   answer = 0.003 ... pretty unlikely!
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.laplace(loc=0.0, scale=1.0, size=None)

Draw samples from the Laplace or double exponential distribution with specified location (or mean) and scale (decay).

The Laplace distribution is similar to the Gaussian/normal distribution, but is sharper at the peak and has fatter tails. It represents the difference between two independent, identically distributed exponential random variables.

Note

New code should use the laplace method of a default_rng() instance instead; see random-quick-start.

loc : float or array_like of floats, optional
The position, \(\mu\), of the distribution peak. Default is 0.
scale : float or array_like of floats, optional
\(\lambda\), the exponential decay. Default is 1. Must be non- negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if loc and scale are both scalars. Otherwise, np.broadcast(loc, scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Laplace distribution.

Generator.laplace: which should be used for new code.

It has the probability density function

\[f(x; \mu, \lambda) = \frac{1}{2\lambda} \exp\left(-\frac{|x - \mu|}{\lambda}\right).\]

The first law of Laplace, from 1774, states that the frequency of an error can be expressed as an exponential function of the absolute magnitude of the error, which leads to the Laplace distribution. For many problems in economics and health sciences, this distribution seems to model the data better than the standard Gaussian distribution.

[1]Abramowitz, M. and Stegun, I. A. (Eds.). “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing,” New York: Dover, 1972.
[2]Kotz, Samuel, et. al. “The Laplace Distribution and Generalizations, ” Birkhauser, 2001.
[3]Weisstein, Eric W. “Laplace Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/LaplaceDistribution.html
[4]Wikipedia, “Laplace distribution”, https://en.wikipedia.org/wiki/Laplace_distribution

Draw samples from the distribution

>>> loc, scale = 0., 1.
>>> s = np.random.laplace(loc, scale, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, density=True)
>>> x = np.arange(-8., 8., .01)
>>> pdf = np.exp(-abs(x-loc)/scale)/(2.*scale)
>>> plt.plot(x, pdf)

Plot Gaussian for comparison:

>>> g = (1/(scale * np.sqrt(2 * np.pi)) *
...      np.exp(-(x - loc)**2 / (2 * scale**2)))
>>> plt.plot(x,g)
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.logistic(loc=0.0, scale=1.0, size=None)

Draw samples from a logistic distribution.

Samples are drawn from a logistic distribution with specified parameters, loc (location or mean, also median), and scale (>0).

Note

New code should use the logistic method of a default_rng() instance instead; see random-quick-start.

loc : float or array_like of floats, optional
Parameter of the distribution. Default is 0.
scale : float or array_like of floats, optional
Parameter of the distribution. Must be non-negative. Default is 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if loc and scale are both scalars. Otherwise, np.broadcast(loc, scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized logistic distribution.
scipy.stats.logistic : probability density function, distribution or
cumulative density function, etc.

Generator.logistic: which should be used for new code.

The probability density for the Logistic distribution is

\[P(x) = P(x) = \frac{e^{-(x-\mu)/s}}{s(1+e^{-(x-\mu)/s})^2},\]

where \(\mu\) = location and \(s\) = scale.

The Logistic distribution is used in Extreme Value problems where it can act as a mixture of Gumbel distributions, in Epidemiology, and by the World Chess Federation (FIDE) where it is used in the Elo ranking system, assuming the performance of each player is a logistically distributed random variable.

[1]Reiss, R.-D. and Thomas M. (2001), “Statistical Analysis of Extreme Values, from Insurance, Finance, Hydrology and Other Fields,” Birkhauser Verlag, Basel, pp 132-133.
[2]Weisstein, Eric W. “Logistic Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/LogisticDistribution.html
[3]Wikipedia, “Logistic-distribution”, https://en.wikipedia.org/wiki/Logistic_distribution

Draw samples from the distribution:

>>> loc, scale = 10, 1
>>> s = np.random.logistic(loc, scale, 10000)
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, bins=50)

# plot against distribution

>>> def logist(x, loc, scale):
...     return np.exp((loc-x)/scale)/(scale*(1+np.exp((loc-x)/scale))**2)
>>> lgst_val = logist(bins, loc, scale)
>>> plt.plot(bins, lgst_val * count.max() / lgst_val.max())
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.lognormal(mean=0.0, sigma=1.0, size=None)

Draw samples from a log-normal distribution.

Draw samples from a log-normal distribution with specified mean, standard deviation, and array shape. Note that the mean and standard deviation are not the values for the distribution itself, but of the underlying normal distribution it is derived from.

Note

New code should use the lognormal method of a default_rng() instance instead; see random-quick-start.

mean : float or array_like of floats, optional
Mean value of the underlying normal distribution. Default is 0.
sigma : float or array_like of floats, optional
Standard deviation of the underlying normal distribution. Must be non-negative. Default is 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if mean and sigma are both scalars. Otherwise, np.broadcast(mean, sigma).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized log-normal distribution.
scipy.stats.lognorm : probability density function, distribution,
cumulative density function, etc.

Generator.lognormal: which should be used for new code.

A variable x has a log-normal distribution if log(x) is normally distributed. The probability density function for the log-normal distribution is:

\[p(x) = \frac{1}{\sigma x \sqrt{2\pi}} e^{(-\frac{(ln(x)-\mu)^2}{2\sigma^2})}\]

where \(\mu\) is the mean and \(\sigma\) is the standard deviation of the normally distributed logarithm of the variable. A log-normal distribution results if a random variable is the product of a large number of independent, identically-distributed variables in the same way that a normal distribution results if the variable is the sum of a large number of independent, identically-distributed variables.

[1]Limpert, E., Stahel, W. A., and Abbt, M., “Log-normal Distributions across the Sciences: Keys and Clues,” BioScience, Vol. 51, No. 5, May, 2001. https://stat.ethz.ch/~stahel/lognormal/bioscience.pdf
[2]Reiss, R.D. and Thomas, M., “Statistical Analysis of Extreme Values,” Basel: Birkhauser Verlag, 2001, pp. 31-32.

Draw samples from the distribution:

>>> mu, sigma = 3., 1. # mean and standard deviation
>>> s = np.random.lognormal(mu, sigma, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 100, density=True, align='mid')
>>> x = np.linspace(min(bins), max(bins), 10000)
>>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))
...        / (x * sigma * np.sqrt(2 * np.pi)))
>>> plt.plot(x, pdf, linewidth=2, color='r')
>>> plt.axis('tight')
>>> plt.show()

Demonstrate that taking the products of random samples from a uniform distribution can be fit well by a log-normal probability density function.

>>> # Generate a thousand samples: each is the product of 100 random
>>> # values, drawn from a normal distribution.
>>> b = []
>>> for i in range(1000):
...    a = 10. + np.random.standard_normal(100)
...    b.append(np.product(a))
>>> b = np.array(b) / np.min(b) # scale values to be positive
>>> count, bins, ignored = plt.hist(b, 100, density=True, align='mid')
>>> sigma = np.std(np.log(b))
>>> mu = np.mean(np.log(b))
>>> x = np.linspace(min(bins), max(bins), 10000)
>>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))
...        / (x * sigma * np.sqrt(2 * np.pi)))
>>> plt.plot(x, pdf, color='r', linewidth=2)
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.logseries(p, size=None)

Draw samples from a logarithmic series distribution.

Samples are drawn from a log series distribution with specified shape parameter, 0 < p < 1.

Note

New code should use the logseries method of a default_rng() instance instead; see random-quick-start.

p : float or array_like of floats
Shape parameter for the distribution. Must be in the range (0, 1).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if p is a scalar. Otherwise, np.array(p).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized logarithmic series distribution.
scipy.stats.logser : probability density function, distribution or
cumulative density function, etc.

Generator.logseries: which should be used for new code.

The probability density for the Log Series distribution is

\[P(k) = \frac{-p^k}{k \ln(1-p)},\]

where p = probability.

The log series distribution is frequently used to represent species richness and occurrence, first proposed by Fisher, Corbet, and Williams in 1943 [2]. It may also be used to model the numbers of occupants seen in cars [3].

[1]Buzas, Martin A.; Culver, Stephen J., Understanding regional species diversity through the log series distribution of occurrences: BIODIVERSITY RESEARCH Diversity & Distributions, Volume 5, Number 5, September 1999 , pp. 187-195(9).
[2]Fisher, R.A,, A.S. Corbet, and C.B. Williams. 1943. The relation between the number of species and the number of individuals in a random sample of an animal population. Journal of Animal Ecology, 12:42-58.
[3]D. J. Hand, F. Daly, D. Lunn, E. Ostrowski, A Handbook of Small Data Sets, CRC Press, 1994.
[4]Wikipedia, “Logarithmic distribution”, https://en.wikipedia.org/wiki/Logarithmic_distribution

Draw samples from the distribution:

>>> a = .6
>>> s = np.random.logseries(a, 10000)
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s)

# plot against distribution

>>> def logseries(k, p):
...     return -p**k/(k*np.log(1-p))
>>> plt.plot(bins, logseries(bins, a)*count.max()/
...          logseries(bins, a).max(), 'r')
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.multinomial(n, pvals, size=None)

Draw samples from a multinomial distribution.

The multinomial distribution is a multivariate generalization of the binomial distribution. Take an experiment with one of p possible outcomes. An example of such an experiment is throwing a dice, where the outcome can be 1 through 6. Each sample drawn from the distribution represents n such experiments. Its values, X_i = [X_0, X_1, ..., X_p], represent the number of times the outcome was i.

Note

New code should use the multinomial method of a default_rng() instance instead; see random-quick-start.

n : int
Number of experiments.
pvals : sequence of floats, length p
Probabilities of each of the p different outcomes. These must sum to 1 (however, the last element is always assumed to account for the remaining probability, as long as sum(pvals[:-1]) <= 1).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
out : ndarray

The drawn samples, of shape size, if that was provided. If not, the shape is (N,).

In other words, each entry out[i,j,...,:] is an N-dimensional value drawn from the distribution.

Generator.multinomial: which should be used for new code.

Throw a dice 20 times:

>>> np.random.multinomial(20, [1/6.]*6, size=1)
array([[4, 1, 7, 5, 2, 1]]) # random

It landed 4 times on 1, once on 2, etc.

Now, throw the dice 20 times, and 20 times again:

>>> np.random.multinomial(20, [1/6.]*6, size=2)
array([[3, 4, 3, 3, 4, 3], # random
       [2, 4, 3, 4, 0, 7]])

For the first run, we threw 3 times 1, 4 times 2, etc. For the second, we threw 2 times 1, 4 times 2, etc.

A loaded die is more likely to land on number 6:

>>> np.random.multinomial(100, [1/7.]*5 + [2/7.])
array([11, 16, 14, 17, 16, 26]) # random

The probability inputs should be normalized. As an implementation detail, the value of the last entry is ignored and assumed to take up any leftover probability mass, but this should not be relied on. A biased coin which has twice as much weight on one side as on the other should be sampled like so:

>>> np.random.multinomial(100, [1.0 / 3, 2.0 / 3])  # RIGHT
array([38, 62]) # random

not like:

>>> np.random.multinomial(100, [1.0, 2.0])  # WRONG
Traceback (most recent call last):
ValueError: pvals < 0, pvals > 1 or pvals contains NaNs
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.multivariate_normal(mean, cov, size=None, check_valid='warn', tol=1e-8)

Draw random samples from a multivariate normal distribution.

The multivariate normal, multinormal or Gaussian distribution is a generalization of the one-dimensional normal distribution to higher dimensions. Such a distribution is specified by its mean and covariance matrix. These parameters are analogous to the mean (average or “center”) and variance (standard deviation, or “width,” squared) of the one-dimensional normal distribution.

Note

New code should use the multivariate_normal method of a default_rng() instance instead; see random-quick-start.

mean : 1-D array_like, of length N
Mean of the N-dimensional distribution.
cov : 2-D array_like, of shape (N, N)
Covariance matrix of the distribution. It must be symmetric and positive-semidefinite for proper sampling.
size : int or tuple of ints, optional
Given a shape of, for example, (m,n,k), m*n*k samples are generated, and packed in an m-by-n-by-k arrangement. Because each sample is N-dimensional, the output shape is (m,n,k,N). If no shape is specified, a single (N-D) sample is returned.
check_valid : { ‘warn’, ‘raise’, ‘ignore’ }, optional
Behavior when the covariance matrix is not positive semidefinite.
tol : float, optional
Tolerance when checking the singular values in covariance matrix. cov is cast to double before the check.
out : ndarray

The drawn samples, of shape size, if that was provided. If not, the shape is (N,).

In other words, each entry out[i,j,...,:] is an N-dimensional value drawn from the distribution.

Generator.multivariate_normal: which should be used for new code.

The mean is a coordinate in N-dimensional space, which represents the location where samples are most likely to be generated. This is analogous to the peak of the bell curve for the one-dimensional or univariate normal distribution.

Covariance indicates the level to which two variables vary together. From the multivariate normal distribution, we draw N-dimensional samples, \(X = [x_1, x_2, ... x_N]\). The covariance matrix element \(C_{ij}\) is the covariance of \(x_i\) and \(x_j\). The element \(C_{ii}\) is the variance of \(x_i\) (i.e. its “spread”).

Instead of specifying the full covariance matrix, popular approximations include:

  • Spherical covariance (cov is a multiple of the identity matrix)
  • Diagonal covariance (cov has non-negative elements, and only on the diagonal)

This geometrical property can be seen in two dimensions by plotting generated data-points:

>>> mean = [0, 0]
>>> cov = [[1, 0], [0, 100]]  # diagonal covariance

Diagonal covariance means that points are oriented along x or y-axis:

>>> import matplotlib.pyplot as plt
>>> x, y = np.random.multivariate_normal(mean, cov, 5000).T
>>> plt.plot(x, y, 'x')
>>> plt.axis('equal')
>>> plt.show()

Note that the covariance matrix must be positive semidefinite (a.k.a. nonnegative-definite). Otherwise, the behavior of this method is undefined and backwards compatibility is not guaranteed.

[1]Papoulis, A., “Probability, Random Variables, and Stochastic Processes,” 3rd ed., New York: McGraw-Hill, 1991.
[2]Duda, R. O., Hart, P. E., and Stork, D. G., “Pattern Classification,” 2nd ed., New York: Wiley, 2001.
>>> mean = (1, 2)
>>> cov = [[1, 0], [0, 1]]
>>> x = np.random.multivariate_normal(mean, cov, (3, 3))
>>> x.shape
(3, 3, 2)

The following is probably true, given that 0.6 is roughly twice the standard deviation:

>>> list((x[0,0,:] - mean) < 0.6)
[True, True] # random
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.negative_binomial(n, p, size=None)

Draw samples from a negative binomial distribution.

Samples are drawn from a negative binomial distribution with specified parameters, n successes and p probability of success where n is > 0 and p is in the interval [0, 1].

Note

New code should use the negative_binomial method of a default_rng() instance instead; see random-quick-start.

n : float or array_like of floats
Parameter of the distribution, > 0.
p : float or array_like of floats
Parameter of the distribution, >= 0 and <=1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if n and p are both scalars. Otherwise, np.broadcast(n, p).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized negative binomial distribution, where each sample is equal to N, the number of failures that occurred before a total of n successes was reached.

Generator.negative_binomial: which should be used for new code.

The probability mass function of the negative binomial distribution is

\[P(N;n,p) = \frac{\Gamma(N+n)}{N!\Gamma(n)}p^{n}(1-p)^{N},\]

where \(n\) is the number of successes, \(p\) is the probability of success, \(N+n\) is the number of trials, and \(\Gamma\) is the gamma function. When \(n\) is an integer, \(\frac{\Gamma(N+n)}{N!\Gamma(n)} = \binom{N+n-1}{N}\), which is the more common form of this term in the the pmf. The negative binomial distribution gives the probability of N failures given n successes, with a success on the last trial.

If one throws a die repeatedly until the third time a “1” appears, then the probability distribution of the number of non-“1”s that appear before the third “1” is a negative binomial distribution.

[1]Weisstein, Eric W. “Negative Binomial Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/NegativeBinomialDistribution.html
[2]Wikipedia, “Negative binomial distribution”, https://en.wikipedia.org/wiki/Negative_binomial_distribution

Draw samples from the distribution:

A real world example. A company drills wild-cat oil exploration wells, each with an estimated probability of success of 0.1. What is the probability of having one success for each successive well, that is what is the probability of a single success after drilling 5 wells, after 6 wells, etc.?

>>> s = np.random.negative_binomial(1, 0.1, 100000)
>>> for i in range(1, 11): 
...    probability = sum(s<i) / 100000.
...    print(i, "wells drilled, probability of one success =", probability)
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.noncentral_chisquare(df, nonc, size=None)

Draw samples from a noncentral chi-square distribution.

The noncentral \(\chi^2\) distribution is a generalization of the \(\chi^2\) distribution.

Note

New code should use the noncentral_chisquare method of a default_rng() instance instead; see random-quick-start.

df : float or array_like of floats

Degrees of freedom, must be > 0.

Changed in version 1.10.0: Earlier NumPy versions required dfnum > 1.

nonc : float or array_like of floats
Non-centrality, must be non-negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if df and nonc are both scalars. Otherwise, np.broadcast(df, nonc).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized noncentral chi-square distribution.

Generator.noncentral_chisquare: which should be used for new code.

The probability density function for the noncentral Chi-square distribution is

\[P(x;df,nonc) = \sum^{\infty}_{i=0} \frac{e^{-nonc/2}(nonc/2)^{i}}{i!} P_{Y_{df+2i}}(x),\]

where \(Y_{q}\) is the Chi-square with q degrees of freedom.

[1]Wikipedia, “Noncentral chi-squared distribution” https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution

Draw values from the distribution and plot the histogram

>>> import matplotlib.pyplot as plt
>>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000),
...                   bins=200, density=True)
>>> plt.show()

Draw values from a noncentral chisquare with very small noncentrality, and compare to a chisquare.

>>> plt.figure()
>>> values = plt.hist(np.random.noncentral_chisquare(3, .0000001, 100000),
...                   bins=np.arange(0., 25, .1), density=True)
>>> values2 = plt.hist(np.random.chisquare(3, 100000),
...                    bins=np.arange(0., 25, .1), density=True)
>>> plt.plot(values[1][0:-1], values[0]-values2[0], 'ob')
>>> plt.show()

Demonstrate how large values of non-centrality lead to a more symmetric distribution.

>>> plt.figure()
>>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000),
...                   bins=200, density=True)
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.noncentral_f(dfnum, dfden, nonc, size=None)

Draw samples from the noncentral F distribution.

Samples are drawn from an F distribution with specified parameters, dfnum (degrees of freedom in numerator) and dfden (degrees of freedom in denominator), where both parameters > 1. nonc is the non-centrality parameter.

Note

New code should use the noncentral_f method of a default_rng() instance instead; see random-quick-start.

dfnum : float or array_like of floats

Numerator degrees of freedom, must be > 0.

Changed in version 1.14.0: Earlier NumPy versions required dfnum > 1.

dfden : float or array_like of floats
Denominator degrees of freedom, must be > 0.
nonc : float or array_like of floats
Non-centrality parameter, the sum of the squares of the numerator means, must be >= 0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if dfnum, dfden, and nonc are all scalars. Otherwise, np.broadcast(dfnum, dfden, nonc).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized noncentral Fisher distribution.

Generator.noncentral_f: which should be used for new code.

When calculating the power of an experiment (power = probability of rejecting the null hypothesis when a specific alternative is true) the non-central F statistic becomes important. When the null hypothesis is true, the F statistic follows a central F distribution. When the null hypothesis is not true, then it follows a non-central F statistic.

[1]Weisstein, Eric W. “Noncentral F-Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/NoncentralF-Distribution.html
[2]Wikipedia, “Noncentral F-distribution”, https://en.wikipedia.org/wiki/Noncentral_F-distribution

In a study, testing for a specific alternative to the null hypothesis requires use of the Noncentral F distribution. We need to calculate the area in the tail of the distribution that exceeds the value of the F distribution for the null hypothesis. We’ll plot the two probability distributions for comparison.

>>> dfnum = 3 # between group deg of freedom
>>> dfden = 20 # within groups degrees of freedom
>>> nonc = 3.0
>>> nc_vals = np.random.noncentral_f(dfnum, dfden, nonc, 1000000)
>>> NF = np.histogram(nc_vals, bins=50, density=True)
>>> c_vals = np.random.f(dfnum, dfden, 1000000)
>>> F = np.histogram(c_vals, bins=50, density=True)
>>> import matplotlib.pyplot as plt
>>> plt.plot(F[1][1:], F[0])
>>> plt.plot(NF[1][1:], NF[0])
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.normal(loc=0.0, scale=1.0, size=None)

Draw random samples from a normal (Gaussian) distribution.

The probability density function of the normal distribution, first derived by De Moivre and 200 years later by both Gauss and Laplace independently [2]_, is often called the bell curve because of its characteristic shape (see the example below).

The normal distributions occurs often in nature. For example, it describes the commonly occurring distribution of samples influenced by a large number of tiny, random disturbances, each with its own unique distribution [2]_.

Note

New code should use the normal method of a default_rng() instance instead; see random-quick-start.

loc : float or array_like of floats
Mean (“centre”) of the distribution.
scale : float or array_like of floats
Standard deviation (spread or “width”) of the distribution. Must be non-negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if loc and scale are both scalars. Otherwise, np.broadcast(loc, scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized normal distribution.
scipy.stats.norm : probability density function, distribution or
cumulative density function, etc.

Generator.normal: which should be used for new code.

The probability density for the Gaussian distribution is

\[p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }} e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} },\]

where \(\mu\) is the mean and \(\sigma\) the standard deviation. The square of the standard deviation, \(\sigma^2\), is called the variance.

The function has its peak at the mean, and its “spread” increases with the standard deviation (the function reaches 0.607 times its maximum at \(x + \sigma\) and \(x - \sigma\) [2]_). This implies that normal is more likely to return samples lying close to the mean, rather than those far away.

[1]Wikipedia, “Normal distribution”, https://en.wikipedia.org/wiki/Normal_distribution
[2]P. R. Peebles Jr., “Central Limit Theorem” in “Probability, Random Variables and Random Signal Principles”, 4th ed., 2001, pp. 51, 51, 125.

Draw samples from the distribution:

>>> mu, sigma = 0, 0.1 # mean and standard deviation
>>> s = np.random.normal(mu, sigma, 1000)

Verify the mean and the variance:

>>> abs(mu - np.mean(s))
0.0  # may vary
>>> abs(sigma - np.std(s, ddof=1))
0.1  # may vary

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, density=True)
>>> plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) *
...                np.exp( - (bins - mu)**2 / (2 * sigma**2) ),
...          linewidth=2, color='r')
>>> plt.show()

Two-by-four array of samples from N(3, 6.25):

>>> np.random.normal(3, 2.5, size=(2, 4))
array([[-4.49401501,  4.00950034, -1.81814867,  7.29718677],   # random
       [ 0.39924804,  4.68456316,  4.99394529,  4.84057254]])  # random
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.pareto(a, size=None)

Draw samples from a Pareto II or Lomax distribution with specified shape.

The Lomax or Pareto II distribution is a shifted Pareto distribution. The classical Pareto distribution can be obtained from the Lomax distribution by adding 1 and multiplying by the scale parameter m (see Notes). The smallest value of the Lomax distribution is zero while for the classical Pareto distribution it is mu, where the standard Pareto distribution has location mu = 1. Lomax can also be considered as a simplified version of the Generalized Pareto distribution (available in SciPy), with the scale set to one and the location set to zero.

The Pareto distribution must be greater than zero, and is unbounded above. It is also known as the “80-20 rule”. In this distribution, 80 percent of the weights are in the lowest 20 percent of the range, while the other 20 percent fill the remaining 80 percent of the range.

Note

New code should use the pareto method of a default_rng() instance instead; see random-quick-start.

a : float or array_like of floats
Shape of the distribution. Must be positive.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if a is a scalar. Otherwise, np.array(a).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Pareto distribution.
scipy.stats.lomax : probability density function, distribution or
cumulative density function, etc.
scipy.stats.genpareto : probability density function, distribution or
cumulative density function, etc.

Generator.pareto: which should be used for new code.

The probability density for the Pareto distribution is

\[p(x) = \frac{am^a}{x^{a+1}}\]

where \(a\) is the shape and \(m\) the scale.

The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution useful in many real world problems. Outside the field of economics it is generally referred to as the Bradford distribution. Pareto developed the distribution to describe the distribution of wealth in an economy. It has also found use in insurance, web page access statistics, oil field sizes, and many other problems, including the download frequency for projects in Sourceforge [1]_. It is one of the so-called “fat-tailed” distributions.

[1]Francis Hunt and Paul Johnson, On the Pareto Distribution of Sourceforge projects.
[2]Pareto, V. (1896). Course of Political Economy. Lausanne.
[3]Reiss, R.D., Thomas, M.(2001), Statistical Analysis of Extreme Values, Birkhauser Verlag, Basel, pp 23-30.
[4]Wikipedia, “Pareto distribution”, https://en.wikipedia.org/wiki/Pareto_distribution

Draw samples from the distribution:

>>> a, m = 3., 2.  # shape and mode
>>> s = (np.random.pareto(a, 1000) + 1) * m

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, _ = plt.hist(s, 100, density=True)
>>> fit = a*m**a / bins**(a+1)
>>> plt.plot(bins, max(count)*fit/max(fit), linewidth=2, color='r')
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.permutation(x)

Randomly permute a sequence, or return a permuted range.

If x is a multi-dimensional array, it is only shuffled along its first index.

Note

New code should use the permutation method of a default_rng() instance instead; see random-quick-start.

x : int or array_like
If x is an integer, randomly permute np.arange(x). If x is an array, make a copy and shuffle the elements randomly.
out : ndarray
Permuted sequence or array range.

Generator.permutation: which should be used for new code.

>>> np.random.permutation(10)
array([1, 7, 4, 3, 0, 9, 2, 5, 8, 6]) # random
>>> np.random.permutation([1, 4, 9, 12, 15])
array([15,  1,  9,  4, 12]) # random
>>> arr = np.arange(9).reshape((3, 3))
>>> np.random.permutation(arr)
array([[6, 7, 8], # random
       [0, 1, 2],
       [3, 4, 5]])
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.poisson(lam=1.0, size=None)

Draw samples from a Poisson distribution.

The Poisson distribution is the limit of the binomial distribution for large N.

Note

New code should use the poisson method of a default_rng() instance instead; see random-quick-start.

lam : float or array_like of floats
Expectation of interval, must be >= 0. A sequence of expectation intervals must be broadcastable over the requested size.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if lam is a scalar. Otherwise, np.array(lam).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Poisson distribution.

Generator.poisson: which should be used for new code.

The Poisson distribution

\[f(k; \lambda)=\frac{\lambda^k e^{-\lambda}}{k!}\]

For events with an expected separation \(\lambda\) the Poisson distribution \(f(k; \lambda)\) describes the probability of \(k\) events occurring within the observed interval \(\lambda\).

Because the output is limited to the range of the C int64 type, a ValueError is raised when lam is within 10 sigma of the maximum representable value.

[1]Weisstein, Eric W. “Poisson Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/PoissonDistribution.html
[2]Wikipedia, “Poisson distribution”, https://en.wikipedia.org/wiki/Poisson_distribution

Draw samples from the distribution:

>>> import numpy as np
>>> s = np.random.poisson(5, 10000)

Display histogram of the sample:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 14, density=True)
>>> plt.show()

Draw each 100 values for lambda 100 and 500:

>>> s = np.random.poisson(lam=(100., 500.), size=(100, 2))
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.power(a, size=None)

Draws samples in [0, 1] from a power distribution with positive exponent a - 1.

Also known as the power function distribution.

Note

New code should use the power method of a default_rng() instance instead; see random-quick-start.

a : float or array_like of floats
Parameter of the distribution. Must be non-negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if a is a scalar. Otherwise, np.array(a).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized power distribution.
ValueError
If a < 1.

Generator.power: which should be used for new code.

The probability density function is

\[\begin{split}P(x; a) = ax^{a-1}, 0 \le x \le 1, a>0.\end{split}\]

The power function distribution is just the inverse of the Pareto distribution. It may also be seen as a special case of the Beta distribution.

It is used, for example, in modeling the over-reporting of insurance claims.

[1]Christian Kleiber, Samuel Kotz, “Statistical size distributions in economics and actuarial sciences”, Wiley, 2003.
[2]Heckert, N. A. and Filliben, James J. “NIST Handbook 148: Dataplot Reference Manual, Volume 2: Let Subcommands and Library Functions”, National Institute of Standards and Technology Handbook Series, June 2003. https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/powpdf.pdf

Draw samples from the distribution:

>>> a = 5. # shape
>>> samples = 1000
>>> s = np.random.power(a, samples)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, bins=30)
>>> x = np.linspace(0, 1, 100)
>>> y = a*x**(a-1.)
>>> normed_y = samples*np.diff(bins)[0]*y
>>> plt.plot(x, normed_y)
>>> plt.show()

Compare the power function distribution to the inverse of the Pareto.

>>> from scipy import stats 
>>> rvs = np.random.power(5, 1000000)
>>> rvsp = np.random.pareto(5, 1000000)
>>> xx = np.linspace(0,1,100)
>>> powpdf = stats.powerlaw.pdf(xx,5)  
>>> plt.figure()
>>> plt.hist(rvs, bins=50, density=True)
>>> plt.plot(xx,powpdf,'r-')  
>>> plt.title('np.random.power(5)')
>>> plt.figure()
>>> plt.hist(1./(1.+rvsp), bins=50, density=True)
>>> plt.plot(xx,powpdf,'r-')  
>>> plt.title('inverse of 1 + np.random.pareto(5)')
>>> plt.figure()
>>> plt.hist(1./(1.+rvsp), bins=50, density=True)
>>> plt.plot(xx,powpdf,'r-')  
>>> plt.title('inverse of stats.pareto(5)')
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.rand(d0, d1, ..., dn)

Random values in a given shape.

Note

This is a convenience function for users porting code from Matlab, and wraps random_sample. That function takes a tuple to specify the size of the output, which is consistent with other NumPy functions like numpy.zeros and numpy.ones.

Create an array of the given shape and populate it with random samples from a uniform distribution over [0, 1).

d0, d1, ..., dn : int, optional
The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned.
out : ndarray, shape (d0, d1, ..., dn)
Random values.

random

>>> np.random.rand(3,2)
array([[ 0.14022471,  0.96360618],  #random
       [ 0.37601032,  0.25528411],  #random
       [ 0.49313049,  0.94909878]]) #random
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.randint(low, high=None, size=None, dtype=int)

Return random integers from low (inclusive) to high (exclusive).

Return random integers from the “discrete uniform” distribution of the specified dtype in the “half-open” interval [low, high). If high is None (the default), then results are from [0, low).

Note

New code should use the integers method of a default_rng() instance instead; see random-quick-start.

low : int or array-like of ints
Lowest (signed) integers to be drawn from the distribution (unless high=None, in which case this parameter is one above the highest such integer).
high : int or array-like of ints, optional
If provided, one above the largest (signed) integer to be drawn from the distribution (see above for behavior if high=None). If array-like, must contain integer values
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
dtype : dtype, optional

Desired dtype of the result. Byteorder must be native. The default value is int.

New in version 1.11.0.

out : int or ndarray of ints
size-shaped array of random integers from the appropriate distribution, or a single such random int if size not provided.
random_integers : similar to randint, only for the closed
interval [low, high], and 1 is the lowest value if high is omitted.

Generator.integers: which should be used for new code.

>>> np.random.randint(2, size=10)
array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0]) # random
>>> np.random.randint(1, size=10)
array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0])

Generate a 2 x 4 array of ints between 0 and 4, inclusive:

>>> np.random.randint(5, size=(2, 4))
array([[4, 0, 2, 1], # random
       [3, 2, 2, 0]])

Generate a 1 x 3 array with 3 different upper bounds

>>> np.random.randint(1, [3, 5, 10])
array([2, 2, 9]) # random

Generate a 1 by 3 array with 3 different lower bounds

>>> np.random.randint([1, 5, 7], 10)
array([9, 8, 7]) # random

Generate a 2 by 4 array using broadcasting with dtype of uint8

>>> np.random.randint([1, 3, 5, 7], [[10], [20]], dtype=np.uint8)
array([[ 8,  6,  9,  7], # random
       [ 1, 16,  9, 12]], dtype=uint8)
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.randn(d0, d1, ..., dn)

Return a sample (or samples) from the “standard normal” distribution.

Note

This is a convenience function for users porting code from Matlab, and wraps standard_normal. That function takes a tuple to specify the size of the output, which is consistent with other NumPy functions like numpy.zeros and numpy.ones.

Note

New code should use the standard_normal method of a default_rng() instance instead; see random-quick-start.

If positive int_like arguments are provided, randn generates an array of shape (d0, d1, ..., dn), filled with random floats sampled from a univariate “normal” (Gaussian) distribution of mean 0 and variance 1. A single float randomly sampled from the distribution is returned if no argument is provided.

d0, d1, ..., dn : int, optional
The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned.
Z : ndarray or float
A (d0, d1, ..., dn)-shaped array of floating-point samples from the standard normal distribution, or a single such float if no parameters were supplied.

standard_normal : Similar, but takes a tuple as its argument. normal : Also accepts mu and sigma arguments. Generator.standard_normal: which should be used for new code.

For random samples from \(N(\mu, \sigma^2)\), use:

sigma * np.random.randn(...) + mu

>>> np.random.randn()
2.1923875335537315  # random

Two-by-four array of samples from N(3, 6.25):

>>> 3 + 2.5 * np.random.randn(2, 4)
array([[-4.49401501,  4.00950034, -1.81814867,  7.29718677],   # random
       [ 0.39924804,  4.68456316,  4.99394529,  4.84057254]])  # random
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.random(size=None)

Return random floats in the half-open interval [0.0, 1.0). Alias for random_sample to ease forward-porting to the new random API.

tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.random_integers(low, high=None, size=None)

Random integers of type np.int_ between low and high, inclusive.

Return random integers of type np.int_ from the “discrete uniform” distribution in the closed interval [low, high]. If high is None (the default), then results are from [1, low]. The np.int_ type translates to the C long integer type and its precision is platform dependent.

This function has been deprecated. Use randint instead.

Deprecated since version 1.11.0.

low : int
Lowest (signed) integer to be drawn from the distribution (unless high=None, in which case this parameter is the highest such integer).
high : int, optional
If provided, the largest (signed) integer to be drawn from the distribution (see above for behavior if high=None).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
out : int or ndarray of ints
size-shaped array of random integers from the appropriate distribution, or a single such random int if size not provided.
randint : Similar to random_integers, only for the half-open
interval [low, high), and 0 is the lowest value if high is omitted.

To sample from N evenly spaced floating-point numbers between a and b, use:

a + (b - a) * (np.random.random_integers(N) - 1) / (N - 1.)
>>> np.random.random_integers(5)
4 # random
>>> type(np.random.random_integers(5))
<class 'numpy.int64'>
>>> np.random.random_integers(5, size=(3,2))
array([[5, 4], # random
       [3, 3],
       [4, 5]])

Choose five random numbers from the set of five evenly-spaced numbers between 0 and 2.5, inclusive (i.e., from the set \({0, 5/8, 10/8, 15/8, 20/8}\)):

>>> 2.5 * (np.random.random_integers(5, size=(5,)) - 1) / 4.
array([ 0.625,  1.25 ,  0.625,  0.625,  2.5  ]) # random

Roll two six sided dice 1000 times and sum the results:

>>> d1 = np.random.random_integers(1, 6, 1000)
>>> d2 = np.random.random_integers(1, 6, 1000)
>>> dsums = d1 + d2

Display results as a histogram:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(dsums, 11, density=True)
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.random_sample(size=None)

Return random floats in the half-open interval [0.0, 1.0).

Results are from the “continuous uniform” distribution over the stated interval. To sample \(Unif[a, b), b > a\) multiply the output of random_sample by (b-a) and add a:

(b - a) * random_sample() + a

Note

New code should use the random method of a default_rng() instance instead; see random-quick-start.

size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
out : float or ndarray of floats
Array of random floats of shape size (unless size=None, in which case a single float is returned).

Generator.random: which should be used for new code.

>>> np.random.random_sample()
0.47108547995356098 # random
>>> type(np.random.random_sample())
<class 'float'>
>>> np.random.random_sample((5,))
array([ 0.30220482,  0.86820401,  0.1654503 ,  0.11659149,  0.54323428]) # random

Three-by-two array of random numbers from [-5, 0):

>>> 5 * np.random.random_sample((3, 2)) - 5
array([[-3.99149989, -0.52338984], # random
       [-2.99091858, -0.79479508],
       [-1.23204345, -1.75224494]])
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.rayleigh(scale=1.0, size=None)

Draw samples from a Rayleigh distribution.

The \(\chi\) and Weibull distributions are generalizations of the Rayleigh.

Note

New code should use the rayleigh method of a default_rng() instance instead; see random-quick-start.

scale : float or array_like of floats, optional
Scale, also equals the mode. Must be non-negative. Default is 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if scale is a scalar. Otherwise, np.array(scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Rayleigh distribution.

Generator.rayleigh: which should be used for new code.

The probability density function for the Rayleigh distribution is

\[P(x;scale) = \frac{x}{scale^2}e^{\frac{-x^2}{2 \cdotp scale^2}}\]

The Rayleigh distribution would arise, for example, if the East and North components of the wind velocity had identical zero-mean Gaussian distributions. Then the wind speed would have a Rayleigh distribution.

[1]Brighton Webs Ltd., “Rayleigh Distribution,” https://web.archive.org/web/20090514091424/http://brighton-webs.co.uk:80/distributions/rayleigh.asp
[2]Wikipedia, “Rayleigh distribution” https://en.wikipedia.org/wiki/Rayleigh_distribution

Draw values from the distribution and plot the histogram

>>> from matplotlib.pyplot import hist
>>> values = hist(np.random.rayleigh(3, 100000), bins=200, density=True)

Wave heights tend to follow a Rayleigh distribution. If the mean wave height is 1 meter, what fraction of waves are likely to be larger than 3 meters?

>>> meanvalue = 1
>>> modevalue = np.sqrt(2 / np.pi) * meanvalue
>>> s = np.random.rayleigh(modevalue, 1000000)

The percentage of waves larger than 3 meters is:

>>> 100.*sum(s>3)/1000000.
0.087300000000000003 # random
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.seed(self, seed=None)

Reseed a legacy MT19937 BitGenerator

This is a convenience, legacy function.

The best practice is to not reseed a BitGenerator, rather to recreate a new one. This method is here for legacy reasons. This example demonstrates best practice.

>>> from numpy.random import MT19937
>>> from numpy.random import RandomState, SeedSequence
>>> rs = RandomState(MT19937(SeedSequence(123456789)))
# Later, you want to restart the stream
>>> rs = RandomState(MT19937(SeedSequence(987654321)))
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.set_state(state)

Set the internal state of the generator from a tuple.

For use if one has reason to manually (re-)set the internal state of the bit generator used by the RandomState instance. By default, RandomState uses the “Mersenne Twister”[1]_ pseudo-random number generating algorithm.

state : {tuple(str, ndarray of 624 uints, int, int, float), dict}

The state tuple has the following items:

  1. the string ‘MT19937’, specifying the Mersenne Twister algorithm.
  2. a 1-D array of 624 unsigned integers keys.
  3. an integer pos.
  4. an integer has_gauss.
  5. a float cached_gaussian.

If state is a dictionary, it is directly set using the BitGenerators state property.

out : None
Returns ‘None’ on success.

get_state

set_state and get_state are not needed to work with any of the random distributions in NumPy. If the internal state is manually altered, the user should know exactly what he/she is doing.

For backwards compatibility, the form (str, array of 624 uints, int) is also accepted although it is missing some information about the cached Gaussian value: state = ('MT19937', keys, pos).

[1]M. Matsumoto and T. Nishimura, “Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator,” ACM Trans. on Modeling and Computer Simulation, Vol. 8, No. 1, pp. 3-30, Jan. 1998.
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.shuffle(x)

Modify a sequence in-place by shuffling its contents.

This function only shuffles the array along the first axis of a multi-dimensional array. The order of sub-arrays is changed but their contents remains the same.

Note

New code should use the shuffle method of a default_rng() instance instead; see random-quick-start.

x : array_like
The array or list to be shuffled.

None

Generator.shuffle: which should be used for new code.

>>> arr = np.arange(10)
>>> np.random.shuffle(arr)
>>> arr
[1 7 5 2 9 4 3 6 0 8] # random

Multi-dimensional arrays are only shuffled along the first axis:

>>> arr = np.arange(9).reshape((3, 3))
>>> np.random.shuffle(arr)
>>> arr
array([[3, 4, 5], # random
       [6, 7, 8],
       [0, 1, 2]])
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.standard_cauchy(size=None)

Draw samples from a standard Cauchy distribution with mode = 0.

Also known as the Lorentz distribution.

Note

New code should use the standard_cauchy method of a default_rng() instance instead; see random-quick-start.

size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
samples : ndarray or scalar
The drawn samples.

Generator.standard_cauchy: which should be used for new code.

The probability density function for the full Cauchy distribution is

\[P(x; x_0, \gamma) = \frac{1}{\pi \gamma \bigl[ 1+ (\frac{x-x_0}{\gamma})^2 \bigr] }\]

and the Standard Cauchy distribution just sets \(x_0=0\) and \(\gamma=1\)

The Cauchy distribution arises in the solution to the driven harmonic oscillator problem, and also describes spectral line broadening. It also describes the distribution of values at which a line tilted at a random angle will cut the x axis.

When studying hypothesis tests that assume normality, seeing how the tests perform on data from a Cauchy distribution is a good indicator of their sensitivity to a heavy-tailed distribution, since the Cauchy looks very much like a Gaussian distribution, but with heavier tails.

[1]NIST/SEMATECH e-Handbook of Statistical Methods, “Cauchy Distribution”, https://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm
[2]Weisstein, Eric W. “Cauchy Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/CauchyDistribution.html
[3]Wikipedia, “Cauchy distribution” https://en.wikipedia.org/wiki/Cauchy_distribution

Draw samples and plot the distribution:

>>> import matplotlib.pyplot as plt
>>> s = np.random.standard_cauchy(1000000)
>>> s = s[(s>-25) & (s<25)]  # truncate distribution so it plots well
>>> plt.hist(s, bins=100)
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.standard_exponential(size=None)

Draw samples from the standard exponential distribution.

standard_exponential is identical to the exponential distribution with a scale parameter of 1.

Note

New code should use the standard_exponential method of a default_rng() instance instead; see random-quick-start.

size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
out : float or ndarray
Drawn samples.

Generator.standard_exponential: which should be used for new code.

Output a 3x8000 array:

>>> n = np.random.standard_exponential((3, 8000))
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.standard_gamma(shape, size=None)

Draw samples from a standard Gamma distribution.

Samples are drawn from a Gamma distribution with specified parameters, shape (sometimes designated “k”) and scale=1.

Note

New code should use the standard_gamma method of a default_rng() instance instead; see random-quick-start.

shape : float or array_like of floats
Parameter, must be non-negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if shape is a scalar. Otherwise, np.array(shape).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized standard gamma distribution.
scipy.stats.gamma : probability density function, distribution or
cumulative density function, etc.

Generator.standard_gamma: which should be used for new code.

The probability density for the Gamma distribution is

\[p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},\]

where \(k\) is the shape and \(\theta\) the scale, and \(\Gamma\) is the Gamma function.

The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant.

[1]Weisstein, Eric W. “Gamma Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html
[2]Wikipedia, “Gamma distribution”, https://en.wikipedia.org/wiki/Gamma_distribution

Draw samples from the distribution:

>>> shape, scale = 2., 1. # mean and width
>>> s = np.random.standard_gamma(shape, 1000000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> import scipy.special as sps  
>>> count, bins, ignored = plt.hist(s, 50, density=True)
>>> y = bins**(shape-1) * ((np.exp(-bins/scale))/  
...                       (sps.gamma(shape) * scale**shape))
>>> plt.plot(bins, y, linewidth=2, color='r')  
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.standard_normal(size=None)

Draw samples from a standard Normal distribution (mean=0, stdev=1).

Note

New code should use the standard_normal method of a default_rng() instance instead; see random-quick-start.

size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
out : float or ndarray
A floating-point array of shape size of drawn samples, or a single sample if size was not specified.
normal :
Equivalent function with additional loc and scale arguments for setting the mean and standard deviation.

Generator.standard_normal: which should be used for new code.

For random samples from \(N(\mu, \sigma^2)\), use one of:

mu + sigma * np.random.standard_normal(size=...)
np.random.normal(mu, sigma, size=...)
>>> np.random.standard_normal()
2.1923875335537315 #random
>>> s = np.random.standard_normal(8000)
>>> s
array([ 0.6888893 ,  0.78096262, -0.89086505, ...,  0.49876311,  # random
       -0.38672696, -0.4685006 ])                                # random
>>> s.shape
(8000,)
>>> s = np.random.standard_normal(size=(3, 4, 2))
>>> s.shape
(3, 4, 2)

Two-by-four array of samples from \(N(3, 6.25)\):

>>> 3 + 2.5 * np.random.standard_normal(size=(2, 4))
array([[-4.49401501,  4.00950034, -1.81814867,  7.29718677],   # random
       [ 0.39924804,  4.68456316,  4.99394529,  4.84057254]])  # random
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.standard_t(df, size=None)

Draw samples from a standard Student’s t distribution with df degrees of freedom.

A special case of the hyperbolic distribution. As df gets large, the result resembles that of the standard normal distribution (standard_normal).

Note

New code should use the standard_t method of a default_rng() instance instead; see random-quick-start.

df : float or array_like of floats
Degrees of freedom, must be > 0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if df is a scalar. Otherwise, np.array(df).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized standard Student’s t distribution.

Generator.standard_t: which should be used for new code.

The probability density function for the t distribution is

\[P(x, df) = \frac{\Gamma(\frac{df+1}{2})}{\sqrt{\pi df} \Gamma(\frac{df}{2})}\Bigl( 1+\frac{x^2}{df} \Bigr)^{-(df+1)/2}\]

The t test is based on an assumption that the data come from a Normal distribution. The t test provides a way to test whether the sample mean (that is the mean calculated from the data) is a good estimate of the true mean.

The derivation of the t-distribution was first published in 1908 by William Gosset while working for the Guinness Brewery in Dublin. Due to proprietary issues, he had to publish under a pseudonym, and so he used the name Student.

[1]Dalgaard, Peter, “Introductory Statistics With R”, Springer, 2002.
[2]Wikipedia, “Student’s t-distribution” https://en.wikipedia.org/wiki/Student’s_t-distribution

From Dalgaard page 83 [1]_, suppose the daily energy intake for 11 women in kilojoules (kJ) is:

>>> intake = np.array([5260., 5470, 5640, 6180, 6390, 6515, 6805, 7515, \
...                    7515, 8230, 8770])

Does their energy intake deviate systematically from the recommended value of 7725 kJ?

We have 10 degrees of freedom, so is the sample mean within 95% of the recommended value?

>>> s = np.random.standard_t(10, size=100000)
>>> np.mean(intake)
6753.636363636364
>>> intake.std(ddof=1)
1142.1232221373727

Calculate the t statistic, setting the ddof parameter to the unbiased value so the divisor in the standard deviation will be degrees of freedom, N-1.

>>> t = (np.mean(intake)-7725)/(intake.std(ddof=1)/np.sqrt(len(intake)))
>>> import matplotlib.pyplot as plt
>>> h = plt.hist(s, bins=100, density=True)

For a one-sided t-test, how far out in the distribution does the t statistic appear?

>>> np.sum(s<t) / float(len(s))
0.0090699999999999999  #random

So the p-value is about 0.009, which says the null hypothesis has a probability of about 99% of being true.

tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.triangular(left, mode, right, size=None)

Draw samples from the triangular distribution over the interval [left, right].

The triangular distribution is a continuous probability distribution with lower limit left, peak at mode, and upper limit right. Unlike the other distributions, these parameters directly define the shape of the pdf.

Note

New code should use the triangular method of a default_rng() instance instead; see random-quick-start.

left : float or array_like of floats
Lower limit.
mode : float or array_like of floats
The value where the peak of the distribution occurs. The value must fulfill the condition left <= mode <= right.
right : float or array_like of floats
Upper limit, must be larger than left.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if left, mode, and right are all scalars. Otherwise, np.broadcast(left, mode, right).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized triangular distribution.

Generator.triangular: which should be used for new code.

The probability density function for the triangular distribution is

\[\begin{split}P(x;l, m, r) = \begin{cases} \frac{2(x-l)}{(r-l)(m-l)}& \text{for $l \leq x \leq m$},\\ \frac{2(r-x)}{(r-l)(r-m)}& \text{for $m \leq x \leq r$},\\ 0& \text{otherwise}. \end{cases}\end{split}\]

The triangular distribution is often used in ill-defined problems where the underlying distribution is not known, but some knowledge of the limits and mode exists. Often it is used in simulations.

[1]Wikipedia, “Triangular distribution” https://en.wikipedia.org/wiki/Triangular_distribution

Draw values from the distribution and plot the histogram:

>>> import matplotlib.pyplot as plt
>>> h = plt.hist(np.random.triangular(-3, 0, 8, 100000), bins=200,
...              density=True)
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.uniform(low=0.0, high=1.0, size=None)

Draw samples from a uniform distribution.

Samples are uniformly distributed over the half-open interval [low, high) (includes low, but excludes high). In other words, any value within the given interval is equally likely to be drawn by uniform.

Note

New code should use the uniform method of a default_rng() instance instead; see random-quick-start.

low : float or array_like of floats, optional
Lower boundary of the output interval. All values generated will be greater than or equal to low. The default value is 0.
high : float or array_like of floats
Upper boundary of the output interval. All values generated will be less than or equal to high. The default value is 1.0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if low and high are both scalars. Otherwise, np.broadcast(low, high).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized uniform distribution.

randint : Discrete uniform distribution, yielding integers. random_integers : Discrete uniform distribution over the closed

interval [low, high].

random_sample : Floats uniformly distributed over [0, 1). random : Alias for random_sample. rand : Convenience function that accepts dimensions as input, e.g.,

rand(2,2) would generate a 2-by-2 array of floats, uniformly distributed over [0, 1).

Generator.uniform: which should be used for new code.

The probability density function of the uniform distribution is

\[p(x) = \frac{1}{b - a}\]

anywhere within the interval [a, b), and zero elsewhere.

When high == low, values of low will be returned. If high < low, the results are officially undefined and may eventually raise an error, i.e. do not rely on this function to behave when passed arguments satisfying that inequality condition. The high limit may be included in the returned array of floats due to floating-point rounding in the equation low + (high-low) * random_sample(). For example:

>>> x = np.float32(5*0.99999999)
>>> x
5.0

Draw samples from the distribution:

>>> s = np.random.uniform(-1,0,1000)

All values are within the given interval:

>>> np.all(s >= -1)
True
>>> np.all(s < 0)
True

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 15, density=True)
>>> plt.plot(bins, np.ones_like(bins), linewidth=2, color='r')
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.vonmises(mu, kappa, size=None)

Draw samples from a von Mises distribution.

Samples are drawn from a von Mises distribution with specified mode (mu) and dispersion (kappa), on the interval [-pi, pi].

The von Mises distribution (also known as the circular normal distribution) is a continuous probability distribution on the unit circle. It may be thought of as the circular analogue of the normal distribution.

Note

New code should use the vonmises method of a default_rng() instance instead; see random-quick-start.

mu : float or array_like of floats
Mode (“center”) of the distribution.
kappa : float or array_like of floats
Dispersion of the distribution, has to be >=0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if mu and kappa are both scalars. Otherwise, np.broadcast(mu, kappa).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized von Mises distribution.
scipy.stats.vonmises : probability density function, distribution, or
cumulative density function, etc.

Generator.vonmises: which should be used for new code.

The probability density for the von Mises distribution is

\[p(x) = \frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)},\]

where \(\mu\) is the mode and \(\kappa\) the dispersion, and \(I_0(\kappa)\) is the modified Bessel function of order 0.

The von Mises is named for Richard Edler von Mises, who was born in Austria-Hungary, in what is now the Ukraine. He fled to the United States in 1939 and became a professor at Harvard. He worked in probability theory, aerodynamics, fluid mechanics, and philosophy of science.

[1]Abramowitz, M. and Stegun, I. A. (Eds.). “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing,” New York: Dover, 1972.
[2]von Mises, R., “Mathematical Theory of Probability and Statistics”, New York: Academic Press, 1964.

Draw samples from the distribution:

>>> mu, kappa = 0.0, 4.0 # mean and dispersion
>>> s = np.random.vonmises(mu, kappa, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> from scipy.special import i0  
>>> plt.hist(s, 50, density=True)
>>> x = np.linspace(-np.pi, np.pi, num=51)
>>> y = np.exp(kappa*np.cos(x-mu))/(2*np.pi*i0(kappa))  
>>> plt.plot(x, y, linewidth=2, color='r')  
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.wald(mean, scale, size=None)

Draw samples from a Wald, or inverse Gaussian, distribution.

As the scale approaches infinity, the distribution becomes more like a Gaussian. Some references claim that the Wald is an inverse Gaussian with mean equal to 1, but this is by no means universal.

The inverse Gaussian distribution was first studied in relationship to Brownian motion. In 1956 M.C.K. Tweedie used the name inverse Gaussian because there is an inverse relationship between the time to cover a unit distance and distance covered in unit time.

Note

New code should use the wald method of a default_rng() instance instead; see random-quick-start.

mean : float or array_like of floats
Distribution mean, must be > 0.
scale : float or array_like of floats
Scale parameter, must be > 0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if mean and scale are both scalars. Otherwise, np.broadcast(mean, scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Wald distribution.

Generator.wald: which should be used for new code.

The probability density function for the Wald distribution is

\[P(x;mean,scale) = \sqrt{\frac{scale}{2\pi x^3}}e^ \frac{-scale(x-mean)^2}{2\cdotp mean^2x}\]

As noted above the inverse Gaussian distribution first arise from attempts to model Brownian motion. It is also a competitor to the Weibull for use in reliability modeling and modeling stock returns and interest rate processes.

[1]Brighton Webs Ltd., Wald Distribution, https://web.archive.org/web/20090423014010/http://www.brighton-webs.co.uk:80/distributions/wald.asp
[2]Chhikara, Raj S., and Folks, J. Leroy, “The Inverse Gaussian Distribution: Theory : Methodology, and Applications”, CRC Press, 1988.
[3]Wikipedia, “Inverse Gaussian distribution” https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution

Draw values from the distribution and plot the histogram:

>>> import matplotlib.pyplot as plt
>>> h = plt.hist(np.random.wald(3, 2, 100000), bins=200, density=True)
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.weibull(a, size=None)

Draw samples from a Weibull distribution.

Draw samples from a 1-parameter Weibull distribution with the given shape parameter a.

\[X = (-ln(U))^{1/a}\]

Here, U is drawn from the uniform distribution over (0,1].

The more common 2-parameter Weibull, including a scale parameter \(\lambda\) is just \(X = \lambda(-ln(U))^{1/a}\).

Note

New code should use the weibull method of a default_rng() instance instead; see random-quick-start.

a : float or array_like of floats
Shape parameter of the distribution. Must be nonnegative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if a is a scalar. Otherwise, np.array(a).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Weibull distribution.

scipy.stats.weibull_max scipy.stats.weibull_min scipy.stats.genextreme gumbel Generator.weibull: which should be used for new code.

The Weibull (or Type III asymptotic extreme value distribution for smallest values, SEV Type III, or Rosin-Rammler distribution) is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. This class includes the Gumbel and Frechet distributions.

The probability density for the Weibull distribution is

\[p(x) = \frac{a} {\lambda}(\frac{x}{\lambda})^{a-1}e^{-(x/\lambda)^a},\]

where \(a\) is the shape and \(\lambda\) the scale.

The function has its peak (the mode) at \(\lambda(\frac{a-1}{a})^{1/a}\).

When a = 1, the Weibull distribution reduces to the exponential distribution.

[1]Waloddi Weibull, Royal Technical University, Stockholm, 1939 “A Statistical Theory Of The Strength Of Materials”, Ingeniorsvetenskapsakademiens Handlingar Nr 151, 1939, Generalstabens Litografiska Anstalts Forlag, Stockholm.
[2]Waloddi Weibull, “A Statistical Distribution Function of Wide Applicability”, Journal Of Applied Mechanics ASME Paper 1951.
[3]Wikipedia, “Weibull distribution”, https://en.wikipedia.org/wiki/Weibull_distribution

Draw samples from the distribution:

>>> a = 5. # shape
>>> s = np.random.weibull(a, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> x = np.arange(1,100.)/50.
>>> def weib(x,n,a):
...     return (a / n) * (x / n)**(a - 1) * np.exp(-(x / n)**a)
>>> count, bins, ignored = plt.hist(np.random.weibull(5.,1000))
>>> x = np.arange(1,100.)/50.
>>> scale = count.max()/weib(x, 1., 5.).max()
>>> plt.plot(x, weib(x, 1., 5.)*scale)
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_c.zipf(a, size=None)

Draw samples from a Zipf distribution.

Samples are drawn from a Zipf distribution with specified parameter a > 1.

The Zipf distribution (also known as the zeta distribution) is a continuous probability distribution that satisfies Zipf’s law: the frequency of an item is inversely proportional to its rank in a frequency table.

Note

New code should use the zipf method of a default_rng() instance instead; see random-quick-start.

a : float or array_like of floats
Distribution parameter. Must be greater than 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if a is a scalar. Otherwise, np.array(a).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Zipf distribution.
scipy.stats.zipf : probability density function, distribution, or
cumulative density function, etc.

Generator.zipf: which should be used for new code.

The probability density for the Zipf distribution is

\[p(x) = \frac{x^{-a}}{\zeta(a)},\]

where \(\zeta\) is the Riemann Zeta function.

It is named for the American linguist George Kingsley Zipf, who noted that the frequency of any word in a sample of a language is inversely proportional to its rank in the frequency table.

[1]Zipf, G. K., “Selected Studies of the Principle of Relative Frequency in Language,” Cambridge, MA: Harvard Univ. Press, 1932.

Draw samples from the distribution:

>>> a = 2. # parameter
>>> s = np.random.zipf(a, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> from scipy import special  

Truncate s values at 50 so plot is interesting:

>>> count, bins, ignored = plt.hist(s[s<50], 50, density=True)
>>> x = np.arange(1., 50.)
>>> y = x**(-a) / special.zetac(a)  
>>> plt.plot(x, y/max(y), linewidth=2, color='r')  
>>> plt.show()

region_deterministic_bnm_g2d_d

What:
Reproduces Figure 11, of Sanz-Leon P., Knock, S. A., Spiegler, A. and Jirsa V. Mathematical framework for large-scale brain network modelling in The Virtual Brain. Neuroimage, 2014, (in review)
Needs:
A working installation of tvb
Run:

python region_deterministic_bnm_g2d_d.py -s True -f True

#Subsequent calls can be made with: python region_deterministic_bnm_g2d_d.py -f True

tvb.contrib.from_articles.region_deterministic_bnm_g2d_d.binomial(n, p, size=None)

Draw samples from a binomial distribution.

Samples are drawn from a binomial distribution with specified parameters, n trials and p probability of success where n an integer >= 0 and p is in the interval [0,1]. (n may be input as a float, but it is truncated to an integer in use)

Note

New code should use the binomial method of a default_rng() instance instead; see random-quick-start.

n : int or array_like of ints
Parameter of the distribution, >= 0. Floats are also accepted, but they will be truncated to integers.
p : float or array_like of floats
Parameter of the distribution, >= 0 and <=1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if n and p are both scalars. Otherwise, np.broadcast(n, p).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized binomial distribution, where each sample is equal to the number of successes over the n trials.
scipy.stats.binom : probability density function, distribution or
cumulative density function, etc.

Generator.binomial: which should be used for new code.

The probability density for the binomial distribution is

\[P(N) = \binom{n}{N}p^N(1-p)^{n-N},\]

where \(n\) is the number of trials, \(p\) is the probability of success, and \(N\) is the number of successes.

When estimating the standard error of a proportion in a population by using a random sample, the normal distribution works well unless the product p*n <=5, where p = population proportion estimate, and n = number of samples, in which case the binomial distribution is used instead. For example, a sample of 15 people shows 4 who are left handed, and 11 who are right handed. Then p = 4/15 = 27%. 0.27*15 = 4, so the binomial distribution should be used in this case.

[1]Dalgaard, Peter, “Introductory Statistics with R”, Springer-Verlag, 2002.
[2]Glantz, Stanton A. “Primer of Biostatistics.”, McGraw-Hill, Fifth Edition, 2002.
[3]Lentner, Marvin, “Elementary Applied Statistics”, Bogden and Quigley, 1972.
[4]Weisstein, Eric W. “Binomial Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/BinomialDistribution.html
[5]Wikipedia, “Binomial distribution”, https://en.wikipedia.org/wiki/Binomial_distribution

Draw samples from the distribution:

>>> n, p = 10, .5  # number of trials, probability of each trial
>>> s = np.random.binomial(n, p, 1000)
# result of flipping a coin 10 times, tested 1000 times.

A real world example. A company drills 9 wild-cat oil exploration wells, each with an estimated probability of success of 0.1. All nine wells fail. What is the probability of that happening?

Let’s do 20,000 trials of the model, and count the number that generate zero positive results.

>>> sum(np.random.binomial(9, 0.1, 20000) == 0)/20000.
# answer = 0.38885, or 38%.
tvb.contrib.from_articles.region_deterministic_bnm_g2d_d.chisquare(df, size=None)

Draw samples from a chi-square distribution.

When df independent random variables, each with standard normal distributions (mean 0, variance 1), are squared and summed, the resulting distribution is chi-square (see Notes). This distribution is often used in hypothesis testing.

Note

New code should use the chisquare method of a default_rng() instance instead; see random-quick-start.

df : float or array_like of floats
Number of degrees of freedom, must be > 0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if df is a scalar. Otherwise, np.array(df).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized chi-square distribution.
ValueError
When df <= 0 or when an inappropriate size (e.g. size=-1) is given.

Generator.chisquare: which should be used for new code.

The variable obtained by summing the squares of df independent, standard normally distributed random variables:

\[Q = \sum_{i=0}^{\mathtt{df}} X^2_i\]

is chi-square distributed, denoted

\[Q \sim \chi^2_k.\]

The probability density function of the chi-squared distribution is

\[p(x) = \frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2},\]

where \(\Gamma\) is the gamma function,

\[\Gamma(x) = \int_0^{-\infty} t^{x - 1} e^{-t} dt.\]
[1]NIST “Engineering Statistics Handbook” https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm
>>> np.random.chisquare(2,4)
array([ 1.89920014,  9.00867716,  3.13710533,  5.62318272]) # random
tvb.contrib.from_articles.region_deterministic_bnm_g2d_d.choice(a, size=None, replace=True, p=None)

Generates a random sample from a given 1-D array

New in version 1.7.0.

Note

New code should use the choice method of a default_rng() instance instead; see random-quick-start.

a : 1-D array-like or int
If an ndarray, a random sample is generated from its elements. If an int, the random sample is generated as if a were np.arange(a)
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
replace : boolean, optional
Whether the sample is with or without replacement
p : 1-D array-like, optional
The probabilities associated with each entry in a. If not given the sample assumes a uniform distribution over all entries in a.
samples : single item or ndarray
The generated random samples
ValueError
If a is an int and less than zero, if a or p are not 1-dimensional, if a is an array-like of size 0, if p is not a vector of probabilities, if a and p have different lengths, or if replace=False and the sample size is greater than the population size

randint, shuffle, permutation Generator.choice: which should be used in new code

Sampling random rows from a 2-D array is not possible with this function, but is possible with Generator.choice through its axis keyword.

Generate a uniform random sample from np.arange(5) of size 3:

>>> np.random.choice(5, 3)
array([0, 3, 4]) # random
>>> #This is equivalent to np.random.randint(0,5,3)

Generate a non-uniform random sample from np.arange(5) of size 3:

>>> np.random.choice(5, 3, p=[0.1, 0, 0.3, 0.6, 0])
array([3, 3, 0]) # random

Generate a uniform random sample from np.arange(5) of size 3 without replacement:

>>> np.random.choice(5, 3, replace=False)
array([3,1,0]) # random
>>> #This is equivalent to np.random.permutation(np.arange(5))[:3]

Generate a non-uniform random sample from np.arange(5) of size 3 without replacement:

>>> np.random.choice(5, 3, replace=False, p=[0.1, 0, 0.3, 0.6, 0])
array([2, 3, 0]) # random

Any of the above can be repeated with an arbitrary array-like instead of just integers. For instance:

>>> aa_milne_arr = ['pooh', 'rabbit', 'piglet', 'Christopher']
>>> np.random.choice(aa_milne_arr, 5, p=[0.5, 0.1, 0.1, 0.3])
array(['pooh', 'pooh', 'pooh', 'Christopher', 'piglet'], # random
      dtype='<U11')
tvb.contrib.from_articles.region_deterministic_bnm_g2d_d.dirichlet(alpha, size=None)

Draw samples from the Dirichlet distribution.

Draw size samples of dimension k from a Dirichlet distribution. A Dirichlet-distributed random variable can be seen as a multivariate generalization of a Beta distribution. The Dirichlet distribution is a conjugate prior of a multinomial distribution in Bayesian inference.

Note

New code should use the dirichlet method of a default_rng() instance instead; see random-quick-start.

alpha : sequence of floats, length k
Parameter of the distribution (length k for sample of length k).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n), then m * n * k samples are drawn. Default is None, in which case a vector of length k is returned.
samples : ndarray,
The drawn samples, of shape (size, k).
ValueError
If any value in alpha is less than or equal to zero

Generator.dirichlet: which should be used for new code.

The Dirichlet distribution is a distribution over vectors \(x\) that fulfil the conditions \(x_i>0\) and \(\sum_{i=1}^k x_i = 1\).

The probability density function \(p\) of a Dirichlet-distributed random vector \(X\) is proportional to

\[p(x) \propto \prod_{i=1}^{k}{x^{\alpha_i-1}_i},\]

where \(\alpha\) is a vector containing the positive concentration parameters.

The method uses the following property for computation: let \(Y\) be a random vector which has components that follow a standard gamma distribution, then \(X = \frac{1}{\sum_{i=1}^k{Y_i}} Y\) is Dirichlet-distributed

[1]David McKay, “Information Theory, Inference and Learning Algorithms,” chapter 23, http://www.inference.org.uk/mackay/itila/
[2]Wikipedia, “Dirichlet distribution”, https://en.wikipedia.org/wiki/Dirichlet_distribution

Taking an example cited in Wikipedia, this distribution can be used if one wanted to cut strings (each of initial length 1.0) into K pieces with different lengths, where each piece had, on average, a designated average length, but allowing some variation in the relative sizes of the pieces.

>>> s = np.random.dirichlet((10, 5, 3), 20).transpose()
>>> import matplotlib.pyplot as plt
>>> plt.barh(range(20), s[0])
>>> plt.barh(range(20), s[1], left=s[0], color='g')
>>> plt.barh(range(20), s[2], left=s[0]+s[1], color='r')
>>> plt.title("Lengths of Strings")
tvb.contrib.from_articles.region_deterministic_bnm_g2d_d.exponential(scale=1.0, size=None)

Draw samples from an exponential distribution.

Its probability density function is

\[f(x; \frac{1}{\beta}) = \frac{1}{\beta} \exp(-\frac{x}{\beta}),\]

for x > 0 and 0 elsewhere. \(\beta\) is the scale parameter, which is the inverse of the rate parameter \(\lambda = 1/\beta\). The rate parameter is an alternative, widely used parameterization of the exponential distribution [3]_.

The exponential distribution is a continuous analogue of the geometric distribution. It describes many common situations, such as the size of raindrops measured over many rainstorms [1]_, or the time between page requests to Wikipedia [2]_.

Note

New code should use the exponential method of a default_rng() instance instead; see random-quick-start.

scale : float or array_like of floats
The scale parameter, \(\beta = 1/\lambda\). Must be non-negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if scale is a scalar. Otherwise, np.array(scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized exponential distribution.

Generator.exponential: which should be used for new code.

[1]Peyton Z. Peebles Jr., “Probability, Random Variables and Random Signal Principles”, 4th ed, 2001, p. 57.
[2]Wikipedia, “Poisson process”, https://en.wikipedia.org/wiki/Poisson_process
[3]Wikipedia, “Exponential distribution”, https://en.wikipedia.org/wiki/Exponential_distribution
tvb.contrib.from_articles.region_deterministic_bnm_g2d_d.f(dfnum, dfden, size=None)

Draw samples from an F distribution.

Samples are drawn from an F distribution with specified parameters, dfnum (degrees of freedom in numerator) and dfden (degrees of freedom in denominator), where both parameters must be greater than zero.

The random variate of the F distribution (also known as the Fisher distribution) is a continuous probability distribution that arises in ANOVA tests, and is the ratio of two chi-square variates.

Note

New code should use the f method of a default_rng() instance instead; see random-quick-start.

dfnum : float or array_like of floats
Degrees of freedom in numerator, must be > 0.
dfden : float or array_like of float
Degrees of freedom in denominator, must be > 0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if dfnum and dfden are both scalars. Otherwise, np.broadcast(dfnum, dfden).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Fisher distribution.
scipy.stats.f : probability density function, distribution or
cumulative density function, etc.

Generator.f: which should be used for new code.

The F statistic is used to compare in-group variances to between-group variances. Calculating the distribution depends on the sampling, and so it is a function of the respective degrees of freedom in the problem. The variable dfnum is the number of samples minus one, the between-groups degrees of freedom, while dfden is the within-groups degrees of freedom, the sum of the number of samples in each group minus the number of groups.

[1]Glantz, Stanton A. “Primer of Biostatistics.”, McGraw-Hill, Fifth Edition, 2002.
[2]Wikipedia, “F-distribution”, https://en.wikipedia.org/wiki/F-distribution

An example from Glantz[1], pp 47-40:

Two groups, children of diabetics (25 people) and children from people without diabetes (25 controls). Fasting blood glucose was measured, case group had a mean value of 86.1, controls had a mean value of 82.2. Standard deviations were 2.09 and 2.49 respectively. Are these data consistent with the null hypothesis that the parents diabetic status does not affect their children’s blood glucose levels? Calculating the F statistic from the data gives a value of 36.01.

Draw samples from the distribution:

>>> dfnum = 1. # between group degrees of freedom
>>> dfden = 48. # within groups degrees of freedom
>>> s = np.random.f(dfnum, dfden, 1000)

The lower bound for the top 1% of the samples is :

>>> np.sort(s)[-10]
7.61988120985 # random

So there is about a 1% chance that the F statistic will exceed 7.62, the measured value is 36, so the null hypothesis is rejected at the 1% level.

tvb.contrib.from_articles.region_deterministic_bnm_g2d_d.gamma(shape, scale=1.0, size=None)

Draw samples from a Gamma distribution.

Samples are drawn from a Gamma distribution with specified parameters, shape (sometimes designated “k”) and scale (sometimes designated “theta”), where both parameters are > 0.

Note

New code should use the gamma method of a default_rng() instance instead; see random-quick-start.

shape : float or array_like of floats
The shape of the gamma distribution. Must be non-negative.
scale : float or array_like of floats, optional
The scale of the gamma distribution. Must be non-negative. Default is equal to 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if shape and scale are both scalars. Otherwise, np.broadcast(shape, scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized gamma distribution.
scipy.stats.gamma : probability density function, distribution or
cumulative density function, etc.

Generator.gamma: which should be used for new code.

The probability density for the Gamma distribution is

\[p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},\]

where \(k\) is the shape and \(\theta\) the scale, and \(\Gamma\) is the Gamma function.

The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant.

[1]Weisstein, Eric W. “Gamma Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html
[2]Wikipedia, “Gamma distribution”, https://en.wikipedia.org/wiki/Gamma_distribution

Draw samples from the distribution:

>>> shape, scale = 2., 2.  # mean=4, std=2*sqrt(2)
>>> s = np.random.gamma(shape, scale, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> import scipy.special as sps  
>>> count, bins, ignored = plt.hist(s, 50, density=True)
>>> y = bins**(shape-1)*(np.exp(-bins/scale) /  
...                      (sps.gamma(shape)*scale**shape))
>>> plt.plot(bins, y, linewidth=2, color='r')  
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_d.geometric(p, size=None)

Draw samples from the geometric distribution.

Bernoulli trials are experiments with one of two outcomes: success or failure (an example of such an experiment is flipping a coin). The geometric distribution models the number of trials that must be run in order to achieve success. It is therefore supported on the positive integers, k = 1, 2, ....

The probability mass function of the geometric distribution is

\[f(k) = (1 - p)^{k - 1} p\]

where p is the probability of success of an individual trial.

Note

New code should use the geometric method of a default_rng() instance instead; see random-quick-start.

p : float or array_like of floats
The probability of success of an individual trial.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if p is a scalar. Otherwise, np.array(p).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized geometric distribution.

Generator.geometric: which should be used for new code.

Draw ten thousand values from the geometric distribution, with the probability of an individual success equal to 0.35:

>>> z = np.random.geometric(p=0.35, size=10000)

How many trials succeeded after a single run?

>>> (z == 1).sum() / 10000.
0.34889999999999999 #random
tvb.contrib.from_articles.region_deterministic_bnm_g2d_d.get_state()

Return a tuple representing the internal state of the generator.

For more details, see set_state.

legacy : bool, optional
Flag indicating to return a legacy tuple state when the BitGenerator is MT19937, instead of a dict.
out : {tuple(str, ndarray of 624 uints, int, int, float), dict}

The returned tuple has the following items:

  1. the string ‘MT19937’.
  2. a 1-D array of 624 unsigned integer keys.
  3. an integer pos.
  4. an integer has_gauss.
  5. a float cached_gaussian.

If legacy is False, or the BitGenerator is not MT19937, then state is returned as a dictionary.

set_state

set_state and get_state are not needed to work with any of the random distributions in NumPy. If the internal state is manually altered, the user should know exactly what he/she is doing.

tvb.contrib.from_articles.region_deterministic_bnm_g2d_d.gumbel(loc=0.0, scale=1.0, size=None)

Draw samples from a Gumbel distribution.

Draw samples from a Gumbel distribution with specified location and scale. For more information on the Gumbel distribution, see Notes and References below.

Note

New code should use the gumbel method of a default_rng() instance instead; see random-quick-start.

loc : float or array_like of floats, optional
The location of the mode of the distribution. Default is 0.
scale : float or array_like of floats, optional
The scale parameter of the distribution. Default is 1. Must be non- negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if loc and scale are both scalars. Otherwise, np.broadcast(loc, scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Gumbel distribution.

scipy.stats.gumbel_l scipy.stats.gumbel_r scipy.stats.genextreme weibull Generator.gumbel: which should be used for new code.

The Gumbel (or Smallest Extreme Value (SEV) or the Smallest Extreme Value Type I) distribution is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. The Gumbel is a special case of the Extreme Value Type I distribution for maximums from distributions with “exponential-like” tails.

The probability density for the Gumbel distribution is

\[p(x) = \frac{e^{-(x - \mu)/ \beta}}{\beta} e^{ -e^{-(x - \mu)/ \beta}},\]

where \(\mu\) is the mode, a location parameter, and \(\beta\) is the scale parameter.

The Gumbel (named for German mathematician Emil Julius Gumbel) was used very early in the hydrology literature, for modeling the occurrence of flood events. It is also used for modeling maximum wind speed and rainfall rates. It is a “fat-tailed” distribution - the probability of an event in the tail of the distribution is larger than if one used a Gaussian, hence the surprisingly frequent occurrence of 100-year floods. Floods were initially modeled as a Gaussian process, which underestimated the frequency of extreme events.

It is one of a class of extreme value distributions, the Generalized Extreme Value (GEV) distributions, which also includes the Weibull and Frechet.

The function has a mean of \(\mu + 0.57721\beta\) and a variance of \(\frac{\pi^2}{6}\beta^2\).

[1]Gumbel, E. J., “Statistics of Extremes,” New York: Columbia University Press, 1958.
[2]Reiss, R.-D. and Thomas, M., “Statistical Analysis of Extreme Values from Insurance, Finance, Hydrology and Other Fields,” Basel: Birkhauser Verlag, 2001.

Draw samples from the distribution:

>>> mu, beta = 0, 0.1 # location and scale
>>> s = np.random.gumbel(mu, beta, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, density=True)
>>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)
...          * np.exp( -np.exp( -(bins - mu) /beta) ),
...          linewidth=2, color='r')
>>> plt.show()

Show how an extreme value distribution can arise from a Gaussian process and compare to a Gaussian:

>>> means = []
>>> maxima = []
>>> for i in range(0,1000) :
...    a = np.random.normal(mu, beta, 1000)
...    means.append(a.mean())
...    maxima.append(a.max())
>>> count, bins, ignored = plt.hist(maxima, 30, density=True)
>>> beta = np.std(maxima) * np.sqrt(6) / np.pi
>>> mu = np.mean(maxima) - 0.57721*beta
>>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)
...          * np.exp(-np.exp(-(bins - mu)/beta)),
...          linewidth=2, color='r')
>>> plt.plot(bins, 1/(beta * np.sqrt(2 * np.pi))
...          * np.exp(-(bins - mu)**2 / (2 * beta**2)),
...          linewidth=2, color='g')
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_d.hypergeometric(ngood, nbad, nsample, size=None)

Draw samples from a Hypergeometric distribution.

Samples are drawn from a hypergeometric distribution with specified parameters, ngood (ways to make a good selection), nbad (ways to make a bad selection), and nsample (number of items sampled, which is less than or equal to the sum ngood + nbad).

Note

New code should use the hypergeometric method of a default_rng() instance instead; see random-quick-start.

ngood : int or array_like of ints
Number of ways to make a good selection. Must be nonnegative.
nbad : int or array_like of ints
Number of ways to make a bad selection. Must be nonnegative.
nsample : int or array_like of ints
Number of items sampled. Must be at least 1 and at most ngood + nbad.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if ngood, nbad, and nsample are all scalars. Otherwise, np.broadcast(ngood, nbad, nsample).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized hypergeometric distribution. Each sample is the number of good items within a randomly selected subset of size nsample taken from a set of ngood good items and nbad bad items.
scipy.stats.hypergeom : probability density function, distribution or
cumulative density function, etc.

Generator.hypergeometric: which should be used for new code.

The probability density for the Hypergeometric distribution is

\[P(x) = \frac{\binom{g}{x}\binom{b}{n-x}}{\binom{g+b}{n}},\]

where \(0 \le x \le n\) and \(n-b \le x \le g\)

for P(x) the probability of x good results in the drawn sample, g = ngood, b = nbad, and n = nsample.

Consider an urn with black and white marbles in it, ngood of them are black and nbad are white. If you draw nsample balls without replacement, then the hypergeometric distribution describes the distribution of black balls in the drawn sample.

Note that this distribution is very similar to the binomial distribution, except that in this case, samples are drawn without replacement, whereas in the Binomial case samples are drawn with replacement (or the sample space is infinite). As the sample space becomes large, this distribution approaches the binomial.

[1]Lentner, Marvin, “Elementary Applied Statistics”, Bogden and Quigley, 1972.
[2]Weisstein, Eric W. “Hypergeometric Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/HypergeometricDistribution.html
[3]Wikipedia, “Hypergeometric distribution”, https://en.wikipedia.org/wiki/Hypergeometric_distribution

Draw samples from the distribution:

>>> ngood, nbad, nsamp = 100, 2, 10
# number of good, number of bad, and number of samples
>>> s = np.random.hypergeometric(ngood, nbad, nsamp, 1000)
>>> from matplotlib.pyplot import hist
>>> hist(s)
#   note that it is very unlikely to grab both bad items

Suppose you have an urn with 15 white and 15 black marbles. If you pull 15 marbles at random, how likely is it that 12 or more of them are one color?

>>> s = np.random.hypergeometric(15, 15, 15, 100000)
>>> sum(s>=12)/100000. + sum(s<=3)/100000.
#   answer = 0.003 ... pretty unlikely!
tvb.contrib.from_articles.region_deterministic_bnm_g2d_d.laplace(loc=0.0, scale=1.0, size=None)

Draw samples from the Laplace or double exponential distribution with specified location (or mean) and scale (decay).

The Laplace distribution is similar to the Gaussian/normal distribution, but is sharper at the peak and has fatter tails. It represents the difference between two independent, identically distributed exponential random variables.

Note

New code should use the laplace method of a default_rng() instance instead; see random-quick-start.

loc : float or array_like of floats, optional
The position, \(\mu\), of the distribution peak. Default is 0.
scale : float or array_like of floats, optional
\(\lambda\), the exponential decay. Default is 1. Must be non- negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if loc and scale are both scalars. Otherwise, np.broadcast(loc, scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized Laplace distribution.

Generator.laplace: which should be used for new code.

It has the probability density function

\[f(x; \mu, \lambda) = \frac{1}{2\lambda} \exp\left(-\frac{|x - \mu|}{\lambda}\right).\]

The first law of Laplace, from 1774, states that the frequency of an error can be expressed as an exponential function of the absolute magnitude of the error, which leads to the Laplace distribution. For many problems in economics and health sciences, this distribution seems to model the data better than the standard Gaussian distribution.

[1]Abramowitz, M. and Stegun, I. A. (Eds.). “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing,” New York: Dover, 1972.
[2]Kotz, Samuel, et. al. “The Laplace Distribution and Generalizations, ” Birkhauser, 2001.
[3]Weisstein, Eric W. “Laplace Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/LaplaceDistribution.html
[4]Wikipedia, “Laplace distribution”, https://en.wikipedia.org/wiki/Laplace_distribution

Draw samples from the distribution

>>> loc, scale = 0., 1.
>>> s = np.random.laplace(loc, scale, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, density=True)
>>> x = np.arange(-8., 8., .01)
>>> pdf = np.exp(-abs(x-loc)/scale)/(2.*scale)
>>> plt.plot(x, pdf)

Plot Gaussian for comparison:

>>> g = (1/(scale * np.sqrt(2 * np.pi)) *
...      np.exp(-(x - loc)**2 / (2 * scale**2)))
>>> plt.plot(x,g)
tvb.contrib.from_articles.region_deterministic_bnm_g2d_d.logistic(loc=0.0, scale=1.0, size=None)

Draw samples from a logistic distribution.

Samples are drawn from a logistic distribution with specified parameters, loc (location or mean, also median), and scale (>0).

Note

New code should use the logistic method of a default_rng() instance instead; see random-quick-start.

loc : float or array_like of floats, optional
Parameter of the distribution. Default is 0.
scale : float or array_like of floats, optional
Parameter of the distribution. Must be non-negative. Default is 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if loc and scale are both scalars. Otherwise, np.broadcast(loc, scale).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized logistic distribution.
scipy.stats.logistic : probability density function, distribution or
cumulative density function, etc.

Generator.logistic: which should be used for new code.

The probability density for the Logistic distribution is

\[P(x) = P(x) = \frac{e^{-(x-\mu)/s}}{s(1+e^{-(x-\mu)/s})^2},\]

where \(\mu\) = location and \(s\) = scale.

The Logistic distribution is used in Extreme Value problems where it can act as a mixture of Gumbel distributions, in Epidemiology, and by the World Chess Federation (FIDE) where it is used in the Elo ranking system, assuming the performance of each player is a logistically distributed random variable.

[1]Reiss, R.-D. and Thomas M. (2001), “Statistical Analysis of Extreme Values, from Insurance, Finance, Hydrology and Other Fields,” Birkhauser Verlag, Basel, pp 132-133.
[2]Weisstein, Eric W. “Logistic Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/LogisticDistribution.html
[3]Wikipedia, “Logistic-distribution”, https://en.wikipedia.org/wiki/Logistic_distribution

Draw samples from the distribution:

>>> loc, scale = 10, 1
>>> s = np.random.logistic(loc, scale, 10000)
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, bins=50)

# plot against distribution

>>> def logist(x, loc, scale):
...     return np.exp((loc-x)/scale)/(scale*(1+np.exp((loc-x)/scale))**2)
>>> lgst_val = logist(bins, loc, scale)
>>> plt.plot(bins, lgst_val * count.max() / lgst_val.max())
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_d.lognormal(mean=0.0, sigma=1.0, size=None)

Draw samples from a log-normal distribution.

Draw samples from a log-normal distribution with specified mean, standard deviation, and array shape. Note that the mean and standard deviation are not the values for the distribution itself, but of the underlying normal distribution it is derived from.

Note

New code should use the lognormal method of a default_rng() instance instead; see random-quick-start.

mean : float or array_like of floats, optional
Mean value of the underlying normal distribution. Default is 0.
sigma : float or array_like of floats, optional
Standard deviation of the underlying normal distribution. Must be non-negative. Default is 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if mean and sigma are both scalars. Otherwise, np.broadcast(mean, sigma).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized log-normal distribution.
scipy.stats.lognorm : probability density function, distribution,
cumulative density function, etc.

Generator.lognormal: which should be used for new code.

A variable x has a log-normal distribution if log(x) is normally distributed. The probability density function for the log-normal distribution is:

\[p(x) = \frac{1}{\sigma x \sqrt{2\pi}} e^{(-\frac{(ln(x)-\mu)^2}{2\sigma^2})}\]

where \(\mu\) is the mean and \(\sigma\) is the standard deviation of the normally distributed logarithm of the variable. A log-normal distribution results if a random variable is the product of a large number of independent, identically-distributed variables in the same way that a normal distribution results if the variable is the sum of a large number of independent, identically-distributed variables.

[1]Limpert, E., Stahel, W. A., and Abbt, M., “Log-normal Distributions across the Sciences: Keys and Clues,” BioScience, Vol. 51, No. 5, May, 2001. https://stat.ethz.ch/~stahel/lognormal/bioscience.pdf
[2]Reiss, R.D. and Thomas, M., “Statistical Analysis of Extreme Values,” Basel: Birkhauser Verlag, 2001, pp. 31-32.

Draw samples from the distribution:

>>> mu, sigma = 3., 1. # mean and standard deviation
>>> s = np.random.lognormal(mu, sigma, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 100, density=True, align='mid')
>>> x = np.linspace(min(bins), max(bins), 10000)
>>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))
...        / (x * sigma * np.sqrt(2 * np.pi)))
>>> plt.plot(x, pdf, linewidth=2, color='r')
>>> plt.axis('tight')
>>> plt.show()

Demonstrate that taking the products of random samples from a uniform distribution can be fit well by a log-normal probability density function.

>>> # Generate a thousand samples: each is the product of 100 random
>>> # values, drawn from a normal distribution.
>>> b = []
>>> for i in range(1000):
...    a = 10. + np.random.standard_normal(100)
...    b.append(np.product(a))
>>> b = np.array(b) / np.min(b) # scale values to be positive
>>> count, bins, ignored = plt.hist(b, 100, density=True, align='mid')
>>> sigma = np.std(np.log(b))
>>> mu = np.mean(np.log(b))
>>> x = np.linspace(min(bins), max(bins), 10000)
>>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))
...        / (x * sigma * np.sqrt(2 * np.pi)))
>>> plt.plot(x, pdf, color='r', linewidth=2)
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_d.logseries(p, size=None)

Draw samples from a logarithmic series distribution.

Samples are drawn from a log series distribution with specified shape parameter, 0 < p < 1.

Note

New code should use the logseries method of a default_rng() instance instead; see random-quick-start.

p : float or array_like of floats
Shape parameter for the distribution. Must be in the range (0, 1).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if p is a scalar. Otherwise, np.array(p).size samples are drawn.
out : ndarray or scalar
Drawn samples from the parameterized logarithmic series distribution.
scipy.stats.logser : probability density function, distribution or
cumulative density function, etc.

Generator.logseries: which should be used for new code.

The probability density for the Log Series distribution is

\[P(k) = \frac{-p^k}{k \ln(1-p)},\]

where p = probability.

The log series distribution is frequently used to represent species richness and occurrence, first proposed by Fisher, Corbet, and Williams in 1943 [2]. It may also be used to model the numbers of occupants seen in cars [3].

[1]Buzas, Martin A.; Culver, Stephen J., Understanding regional species diversity through the log series distribution of occurrences: BIODIVERSITY RESEARCH Diversity & Distributions, Volume 5, Number 5, September 1999 , pp. 187-195(9).
[2]Fisher, R.A,, A.S. Corbet, and C.B. Williams. 1943. The relation between the number of species and the number of individuals in a random sample of an animal population. Journal of Animal Ecology, 12:42-58.
[3]D. J. Hand, F. Daly, D. Lunn, E. Ostrowski, A Handbook of Small Data Sets, CRC Press, 1994.
[4]Wikipedia, “Logarithmic distribution”, https://en.wikipedia.org/wiki/Logarithmic_distribution

Draw samples from the distribution:

>>> a = .6
>>> s = np.random.logseries(a, 10000)
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s)

# plot against distribution

>>> def logseries(k, p):
...     return -p**k/(k*np.log(1-p))
>>> plt.plot(bins, logseries(bins, a)*count.max()/
...          logseries(bins, a).max(), 'r')
>>> plt.show()
tvb.contrib.from_articles.region_deterministic_bnm_g2d_d.multinomial(n, pvals, size=None)

Draw samples from a multinomial distribution.

The multinomial distribution is a multivariate generalization of the binomial distribution. Take an experiment with one of p possible outcomes. An example of such an experiment is throwing a dice, where the outcome can be 1 through 6. Each sample drawn from the distribution represents n such experiments. Its values, X_i = [X_0, X_1, ..., X_p], represent the number of times the outcome was i.

Note

New code should use the multinomial method of a default_rng() instance instead; see random-quick-start.

n : int
Number of experiments.
pvals : sequence of floats, length p
Probabilities of each of the p different outcomes. These must sum to 1 (however, the last element is always assumed to account for the remaining probability, as long as sum(pvals[:-1]) <= 1).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
out : ndarray

The drawn samples, of shape size, if that was provided. If not, the shape is (N,).

In other words, each entry out[i,j,...,:] is an N-dimensional value drawn from the distribution.

Generator.multinomial: which should be used for new code.

Throw a dice 20 times:

>>> np.random.multinomial(20, [1/6.]*6, size=1)
array([[4, 1, 7, 5, 2, 1]]) # random

It landed 4 times on 1, once on 2, etc.

Now, throw the dice 20 times, and 20 times again:

>>> np.random.multinomial(20, [1/6.]*6, size=2)
array([[3, 4, 3, 3, 4, 3], # random
       [2, 4, 3, 4, 0, 7]])

For the first run, we threw 3 times 1, 4 times 2, etc. For the second, we threw 2 times 1, 4 times 2, etc.

A loaded die is more likely to land on number 6:

>>> np.random.multinomial(100, [1/7.]*5 + [2/7.])
array([11, 16, 14, 17, 16, 26]) # random

The probability inputs should be normalized. As an implementation detail, the value of the last entry is ign