Sample Bernoulli random values with given shape and mean.
Parameters:
p (float, array_like, optional) – A float or array of floats for the mean of the random
variables. Must be broadcast-compatible with shape and the values
should be within [0, 1]. Default 0.5.
size (optional, tuple of int, int) – A tuple of nonnegative integers representing the result
shape. Must be broadcast-compatible with p.shape. The default (None)
produces a result shape equal to p.shape.
Returns:
out – A random array with boolean dtype and shape given by shape if shape
is not None, or else p.shape.
The Beta distribution is a special case of the Dirichlet distribution,
and is related to the Gamma distribution. It has the probability
distribution function
It is often seen in Bayesian inference and order statistics.
Parameters:
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.
Returns:
out – Drawn samples from the parameterized beta 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)
Parameters:
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.
Returns:
out – Drawn samples from the parameterized binomial distribution, where
each sample is equal to the number of successes over the n trials.
Return type:
ndarray or scalar
See also
scipy.stats.binom
probability density function, distribution or cumulative density function, etc.
Notes
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.
References
Examples
Draw samples from the distribution:
>>> n,p=10,.5# number of trials, probability of each trial>>> s=bm.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(bm.random.binomial(9,0.1,20000)==0)/20000.# answer = 0.38885, or 38%.
Sample random values from categorical distributions.
Parameters:
logits – Unnormalized log probabilities of the categorical distribution(s) to sample from,
so that softmax(logits, axis) gives the corresponding probabilities.
axis (int) – Axis along which logits belong to the same categorical distribution.
shape – Optional, a tuple of nonnegative integers representing the result shape.
Must be broadcast-compatible with np.delete(logits.shape,axis).
The default (None) produces a result shape equal to np.delete(logits.shape,axis).
key (Union[int, Array, None]) – a PRNG key used as the random key.
Returns:
A random array with int dtype and shape given by shape if shape
is not None, or else np.delete(logits.shape,axis).
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.
Parameters:
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.
Returns:
out – Drawn samples from the parameterized chi-square distribution.
Return type:
ndarray or scalar
Raises:
ValueError – When df <= 0 or when an inappropriate size (e.g. size=-1)
is given.
Notes
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
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 it 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. Default is True,
meaning that a value of a can be selected multiple times.
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.
Returns:
samples – The generated random samples
Return type:
single item or ndarray
Raises:
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
Setting user-specified probabilities through p uses a more general but less
efficient sampler than the default. The general sampler produces a different sample
than the optimized sampler even if each element of p is 1 / len(a).
Sampling random rows from a 2-D array is not possible with this function,
but is possible with Generator.choice through its axis keyword.
Examples
Generate a uniform random sample from np.arange(5) of size 3:
>>> importbrainpy.mathasbm>>> bm.random.choice(5,3)array([0, 3, 4]) # random>>> #This is equivalent to brainpy.math.random.randint(0,5,3)
Generate a non-uniform random sample from np.arange(5) of size 3:
>>> bm.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:
>>> bm.random.choice(5,3,replace=False)array([3,1,0]) # random>>> #This is equivalent to brainpy.math.random.permutation(np.arange(5))[:3]
Generate a non-uniform random sample from np.arange(5) of size
3 without replacement:
>>> bm.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']>>> bm.random.choice(aa_milne_arr,5,p=[0.5,0.1,0.1,0.3])array(['pooh', 'pooh', 'pooh', 'Christopher', 'piglet'], # random dtype='<U11')
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.
Parameters:
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.
Returns:
samples – The drawn samples, of shape (size,k).
Return type:
ndarray,
Raises:
ValueError – If any value in alpha is less than or equal to zero
Notes
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
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
References
Examples
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.
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]_.
Parameters:
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.
Returns:
out – Drawn samples from the parameterized exponential 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.
Parameters:
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.
Returns:
out – Drawn samples from the parameterized Fisher distribution.
Return type:
ndarray or scalar
See also
scipy.stats.f
probability density function, distribution or cumulative density function, etc.
Notes
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.
References
Examples
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=bm.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.
Samples are drawn from a Gamma distribution with specified parameters,
shape (sometimes designated “k”) and scale (sometimes designated
“theta”), where both parameters are > 0.
Parameters:
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.
Returns:
out – Drawn samples from the parameterized gamma distribution.
Return type:
ndarray or scalar
Notes
The probability density for the Gamma distribution is
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.
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.
Parameters:
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.
Returns:
out – Drawn samples from the parameterized geometric distribution.
Return type:
ndarray or scalar
Examples
Draw ten thousand values from the geometric distribution,
with the probability of an individual success equal to 0.35:
Draw samples from a Gumbel distribution with specified location and
scale. For more information on the Gumbel distribution, see
Notes and References below.
Parameters:
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.
Returns:
out – Drawn samples from the parameterized Gumbel distribution.
Return type:
ndarray or scalar
Notes
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
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 \(\frac{\pi^2}{6}\beta^2\).
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).
Parameters:
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.
Returns:
out – 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.
Return type:
ndarray or scalar
See also
scipy.stats.hypergeom
probability density function, distribution or cumulative density function, etc.
Notes
The probability density for the Hypergeometric distribution is
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.
References
Examples
Draw samples from the distribution:
>>> ngood,nbad,nsamp=100,2,10# number of good, number of bad, and number of samples>>> s=bm.random.hypergeometric(ngood,nbad,nsamp,1000)>>> frommatplotlib.pyplotimporthist>>> 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?
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.
Parameters:
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.
Returns:
out – Drawn samples from the parameterized Laplace distribution.
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.
a (float, array_like) – A float or array of floats broadcast-compatible with shape representing the parameter of the distribution.
size (optional, int, tuple of int) – A tuple of nonnegative integers specifying the result shape.
Must be broadcast-compatible with a. The default (None) produces a result shape equal to a.shape.
Samples are drawn from a logistic distribution with specified
parameters, loc (location or mean, also median), and scale (>0).
Parameters:
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.
Returns:
out – Drawn samples from the parameterized logistic distribution.
Return type:
ndarray or scalar
Notes
The probability density for the Logistic distribution is
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.
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.
Parameters:
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.
Returns:
out – Drawn samples from the parameterized log-normal distribution.
Return type:
ndarray or scalar
See also
scipy.stats.lognorm
probability density function, distribution, cumulative density function, etc.
Notes
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.
References
Examples
Draw samples from the distribution:
>>> mu,sigma=3.,1.# mean and standard deviation>>> s=bm.random.lognormal(mu,sigma,1000)
Display the histogram of the samples, along with
the probability density function:
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=[]>>> foriinrange(1000):... a=10.+bm.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))
Draw samples from a logarithmic series distribution.
Samples are drawn from a log series distribution with specified
shape parameter, 0 <= p < 1.
Parameters:
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.
Returns:
out – Drawn samples from the parameterized logarithmic series distribution.
Return type:
ndarray or scalar
See also
scipy.stats.logser
probability density function, distribution or cumulative density function, etc.
Notes
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].
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.
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.
Returns:
out – 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.
Return type:
ndarray
Examples
Throw a dice 20 times:
>>> bm.random.multinomial(20,[1/6.]*6,size=1)array([[4, 1, 7, 5, 2, 1]]) # random
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:
>>> bm.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:
>>> bm.random.multinomial(100,[1.0/3,2.0/3])# RIGHTarray([38, 62]) # random
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.
Parameters:
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.
Returns:
out – 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.
Return type:
ndarray
Notes
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:
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.
Here we generate 800 samples from the bivariate normal distribution
with mean [0, 0] and covariance matrix [[6, -3], [-3, 3.5]]. The
expected variances of the first and second components of the sample
are 6 and 3.5, respectively, and the expected correlation
coefficient is -3/sqrt(6*3.5) ≈ -0.65465.
Check that the mean, covariance, and correlation coefficient of the
sample are close to the expected values:
>>> pts.mean(axis=0)array([ 0.0326911 , -0.01280782]) # may vary>>> np.cov(pts.T)array([[ 5.96202397, -2.85602287], [-2.85602287, 3.47613949]]) # may vary>>> np.corrcoef(pts.T)[0,1]-0.6273591314603949 # may vary
We can visualize this data with a scatter plot. The orientation
of the point cloud illustrates the negative correlation of the
components of this sample.
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].
Parameters:
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.
Returns:
out – 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.
Return type:
ndarray or scalar
Notes
The probability mass function of the negative binomial distribution is
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 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.
References
Examples
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=bm.random.negative_binomial(1,0.1,100000)>>> foriinrange(1,11):... probability=sum(s<i)/100000.... print(i,"wells drilled, probability of one success =",probability)
Draw samples from a noncentral chi-square distribution.
The noncentral \(\chi^2\) distribution is a generalization of
the \(\chi^2\) distribution.
Parameters:
df (float or array_like of floats) – Degrees of freedom, must be > 0.
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.
Returns:
out – Drawn samples from the parameterized noncentral chi-square distribution.
Return type:
ndarray or scalar
Notes
The probability density function for the noncentral Chi-square
distribution is
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.
Parameters:
dfnum (float or array_like of floats) – Numerator degrees of freedom, must be > 0.
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.
Returns:
out – Drawn samples from the parameterized noncentral Fisher distribution.
Return type:
ndarray or scalar
Notes
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.
References
Examples
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=bm.random.noncentral_f(dfnum,dfden,nonc,1000000)>>> NF=np.histogram(nc_vals,bins=50,density=True)>>> c_vals=bm.random.f(dfnum,dfden,1000000)>>> F=np.histogram(c_vals,bins=50,density=True)>>> importmatplotlib.pyplotasplt>>> plt.plot(F[1][1:],F[0])>>> plt.plot(NF[1][1:],NF[0])>>> plt.show()
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]_.
Parameters:
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.
Returns:
out – Drawn samples from the parameterized normal distribution.
Return type:
ndarray or scalar
Notes
The probability density for the Gaussian distribution is
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.
References
Examples
Draw samples from the distribution:
>>> mu,sigma=0,0.1# mean and standard deviation>>> s=bm.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:
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.
Parameters:
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.
Returns:
out – Drawn samples from the parameterized Pareto distribution.
Return type:
ndarray or scalar
See also
scipy.stats.lomax
probability density function, distribution or cumulative density function, etc.
scipy.stats.genpareto
probability density function, distribution or cumulative density function, etc.
Notes
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.
References
Examples
Draw samples from the distribution:
>>> a,m=3.,2.# shape and mode>>> s=(bm.random.pareto(a,1000)+1)*m
Display the histogram of the samples, along with the probability
density function:
The Poisson distribution is the limit of the binomial distribution
for large N.
Parameters:
lam (float or array_like of floats) – Expected number of events occurring in a fixed-time interval,
must be >= 0. A sequence 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.
Returns:
out – Drawn samples from the parameterized Poisson distribution.
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.
Draws samples in [0, 1] from a power distribution with positive
exponent a - 1.
Also known as the power function distribution.
Parameters:
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.
Returns:
out – Drawn samples from the parameterized power distribution.
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).
Parameters:
d0 (int, optional) – The dimensions of the returned array, must be non-negative.
If no argument is given a single Python float is returned.
d1 (int, optional) – The dimensions of the returned array, must be non-negative.
If no argument is given a single Python float is returned.
... (int, optional) – The dimensions of the returned array, must be non-negative.
If no argument is given a single Python float is returned.
dn (int, optional) – The dimensions of the returned array, must be non-negative.
If no argument is given a single Python float is returned.
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).
Parameters:
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.
Returns:
out – size-shaped array of random integers from the appropriate
distribution, or a single such random int if size not provided.
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; please see the 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.
Parameters:
d0 (int, optional) – The dimensions of the returned array, must be non-negative.
If no argument is given a single Python float is returned.
d1 (int, optional) – The dimensions of the returned array, must be non-negative.
If no argument is given a single Python float is returned.
... (int, optional) – The dimensions of the returned array, must be non-negative.
If no argument is given a single Python float is returned.
dn (int, optional) – The dimensions of the returned array, must be non-negative.
If no argument is given a single Python float is returned.
Returns:
Z – 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.
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.
Parameters:
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.
Returns:
out – size-shaped array of random integers from the appropriate
distribution, or a single such random int if size not provided.
Choose five random numbers from the set of five evenly-spaced
numbers between 0 and 2.5, inclusive (i.e., from the set
\({0, 5/8, 10/8, 15/8, 20/8}\)):
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; please see the random-quick-start.
Parameters:
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.
Returns:
out – Array of random floats of shape size (unless size=None, in which
case a single float is returned).
The \(\chi\) and Weibull distributions are generalizations of the
Rayleigh.
Parameters:
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.
Returns:
out – Drawn samples from the parameterized Rayleigh distribution.
Return type:
ndarray or scalar
Notes
The probability density function for the Rayleigh distribution is
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.
References
Examples
Draw values from the distribution and plot the histogram
It can be an integer for initial seed of the random number generator,
or it can be a JAX’s PRNKey, which is an array with two elements and uint32 dtype.
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.
Parameters:
x (ndarray or MutableSequence) – The array, list or mutable sequence to be shuffled.
Draw samples from a standard Cauchy distribution with mode = 0.
Also known as the Lorentz distribution.
Parameters:
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.
Returns:
samples – The drawn samples.
Return type:
ndarray or scalar
Notes
The probability density function for the full Cauchy distribution is
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.
References
Examples
Draw samples and plot the distribution:
>>> importmatplotlib.pyplotasplt>>> s=bm.random.standard_cauchy(1000000)>>> s=s[(s>-25)&(s<25)]# truncate distribution so it plots well>>> plt.hist(s,bins=100)>>> plt.show()
Draw samples from the standard exponential distribution.
standard_exponential is identical to the exponential distribution
with a scale parameter of 1.
Parameters:
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 are drawn from a Gamma distribution with specified parameters,
shape (sometimes designated “k”) and scale=1.
Parameters:
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.
Returns:
out – Drawn samples from the parameterized standard gamma distribution.
Return type:
ndarray or scalar
See also
scipy.stats.gamma
probability density function, distribution or cumulative density function, etc.
Notes
The probability density for the Gamma distribution is
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.
References
Examples
Draw samples from the distribution:
>>> shape,scale=2.,1.# mean and width>>> s=bm.random.standard_gamma(shape,1000000)
Display the histogram of the samples, along with
the probability density function:
Draw samples from a standard Normal distribution (mean=0, stdev=1).
Parameters:
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.
Returns:
out – A floating-point array of shape size of drawn samples, or a
single sample if size was not specified.
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).
Parameters:
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.
Returns:
out – Drawn samples from the parameterized standard Student’s t distribution.
Return type:
ndarray or scalar
Notes
The probability density function for the t distribution is
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.
References
Examples
From Dalgaard page 83 [1]_, suppose the daily energy intake for 11
women in kilojoules (kJ) is:
Does their energy intake deviate systematically from the recommended
value of 7725 kJ? Our null hypothesis will be the absence of deviation,
and the alternate hypothesis will be the presence of an effect that could be
either positive or negative, hence making our test 2-tailed.
Because we are estimating the mean and we have N=11 values in our sample,
we have N-1=10 degrees of freedom. We set our significance level to 95% and
compute the t statistic using the empirical mean and empirical standard
deviation of our intake. We use a ddof of 1 to base the computation of our
empirical standard deviation on an unbiased estimate of the variance (note:
the final estimate is not unbiased due to the concave nature of the square
root).
Does our t statistic land in one of the two critical regions found at
both tails of the distribution?
>>> np.sum(np.abs(t)<np.abs(s))/float(len(s))0.018318 #random < 0.05, statistic is in critical region
The probability value for this 2-tailed test is about 1.83%, which is
lower than the 5% pre-determined significance threshold.
Therefore, the probability of observing values as extreme as our intake
conditionally on the null hypothesis being true is too low, and we reject
the null hypothesis of no deviation.
df (float, array_like) – A float or array of floats broadcast-compatible with shape representing the parameter of the distribution.
size (optional, int, tuple of int) – A tuple of non-negative integers specifying the result shape.
Must be broadcast-compatible with df. The default (None) produces a result shape equal to df.shape.
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.
Parameters:
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.
Returns:
out – Drawn samples from the parameterized triangular distribution.
Return type:
ndarray or scalar
Notes
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.
References
Examples
Draw values from the distribution and plot the histogram:
This distribution is the normal distribution centered on loc (default
0), with standard deviation scale (default 1), and clipped at a,
b standard deviations to the left, right (respectively) from loc.
If myclip_a and myclip_b are clip values in the sample space (as
opposed to the number of standard deviations) then they can be converted
to the required form according to:
a,b=(myclip_a-loc)/scale,(myclip_b-loc)/scale
Parameters:
lower (float, ndarray) – A float or array of floats representing the lower bound for
truncation. Must be broadcast-compatible with upper.
upper (float, ndarray) – A float or array of floats representing the upper bound for
truncation. Must be broadcast-compatible with lower.
loc (optional, float, ndarray) – Mean (“centre”) of the distribution before truncating. Note that
the mean of the truncated distribution will not be exactly equal
to loc.
size (optional, list of int, tuple of int) – A tuple of nonnegative integers specifying the result
shape. Must be broadcast-compatible with lower and upper. The
default (None) produces a result shape by broadcasting lower and
upper.
loc – A float or array of floats representing the mean of the
distribution. Default is 0.
scale (float, ndarray) – Standard deviation (spread or “width”) of the distribution. Must be
non-negative. Default is 1.
dtype (optional) – The float dtype for the returned values (default float64 if
jax_enable_x64 is true, otherwise float32).
key (jax.Array) – The key for random generator. Consistent with the jax’s random
paradigm.
Returns:
out – A random array with the specified dtype and shape given by shape if
shape is not None, or else by broadcasting lower and upper.
Returns values in the open interval (lower,upper).
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.
Parameters:
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 high limit may be included in the
returned array of floats due to floating-point rounding in the
equation low+(high-low)*random_sample(). 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.
Returns:
out – Drawn samples from the parameterized uniform distribution.
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).
Notes
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)>>> x5.0
Examples
Draw samples from the distribution:
>>> s=bm.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:
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.
Parameters:
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.
Returns:
out – Drawn samples from the parameterized von Mises distribution.
Return type:
ndarray or scalar
See also
scipy.stats.vonmises
probability density function, distribution, or cumulative density function, etc.
Notes
The probability density for the von Mises distribution is
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.
References
Examples
Draw samples from the distribution:
>>> mu,kappa=0.0,4.0# mean and dispersion>>> s=bm.random.vonmises(mu,kappa,1000)
Display the histogram of the samples, along with
the probability density function:
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.
Parameters:
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.
Returns:
out – Drawn samples from the parameterized Wald distribution.
Return type:
ndarray or scalar
Notes
The probability density function for the Wald distribution is
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.
References
Examples
Draw values from the distribution and plot the histogram:
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; please see the random-quick-start.
Parameters:
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.
Returns:
out – Drawn samples from the parameterized Weibull distribution.
Return type:
ndarray or scalar
Notes
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
Samples are drawn from a Zipf distribution with specified parameter
a > 1.
The Zipf distribution (also known as the zeta distribution) is a
discrete 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; please see the random-quick-start.
Parameters:
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.
Returns:
out – Drawn samples from the parameterized Zipf distribution.
Return type:
ndarray or scalar
See also
scipy.stats.zipf
probability density function, distribution, or cumulative density function, etc.
Notes
The probability density for the Zipf distribution is
\[p(k) = \frac{k^{-a}}{\zeta(a)},\]
for integers \(k \geq 1\), 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.