deephyper.skopt.space.rv_discrete

deephyper.skopt.space.rv_discrete#

class deephyper.skopt.space.rv_discrete(a=0, b=inf, name=None, badvalue=None, moment_tol=1e-08, values=None, inc=1, longname=None, shapes=None, seed=None)[source]#

Bases: rv_generic

A generic discrete random variable class meant for subclassing.

rv_discrete is a base class to construct specific distribution classes and instances for discrete random variables. It can also be used to construct an arbitrary distribution defined by a list of support points and corresponding probabilities.

Parameters:

a (float, optional) – Lower bound of the support of the distribution, default: 0
b (float, optional) – Upper bound of the support of the distribution, default: plus infinity
moment_tol (float, optional) – The tolerance for the generic calculation of moments.
values (tuple of two array_like, optional) – (xk, pk) where xk are integers and pk are the non-zero probabilities between 0 and 1 with sum(pk) = 1. xk and pk must have the same shape.
inc (integer, optional) – Increment for the support of the distribution. Default is 1. (other values have not been tested)
badvalue (float, optional) – The value in a result arrays that indicates a value that for which some argument restriction is violated, default is np.nan.
name (str, optional) – The name of the instance. This string is used to construct the default example for distributions.
longname (str, optional) – This string is used as part of the first line of the docstring returned when a subclass has no docstring of its own. Note: longname exists for backwards compatibility, do not use for new subclasses.
shapes (str, optional) – The shape of the distribution. For example “m, n” for a distribution that takes two integers as the two shape arguments for all its methods If not provided, shape parameters will be inferred from the signatures of the private methods, _pmf and _cdf of the instance.
seed ({None, int, numpy.random.Generator, numpy.random.RandomState}, optional) – If seed is None (or np.random), the numpy.random.RandomState singleton is used. If seed is an int, a new RandomState instance is used, seeded with seed. If seed is already a Generator or RandomState instance then that instance is used.

rvs()[source]#

pmf()[source]#

logpmf()[source]#

cdf()[source]#

logcdf()[source]#

sf()[source]#

logsf()[source]#

ppf()[source]#

isf()[source]#

moment()#

stats()#

entropy()#

expect()[source]#

median()#

mean()#

std()#

var()#

interval()#

__call__()#

support()#

Notes

This class is similar to rv_continuous. Whether a shape parameter is valid is decided by an _argcheck method (which defaults to checking that its arguments are strictly positive.) The main differences are:

the support of the distribution is a set of integers
instead of the probability density function, pdf (and the corresponding private _pdf), this class defines the probability mass function, pmf (and the corresponding private _pmf.)
scale parameter is not defined.

To create a new discrete distribution, we would do the following:

>>> from scipy.stats import rv_discrete
>>> class poisson_gen(rv_discrete):
...     "Poisson distribution"
...     def _pmf(self, k, mu):
...         return exp(-mu) * mu**k / factorial(k)

and create an instance:

>>> poisson = poisson_gen(name="poisson")

Note that above we defined the Poisson distribution in the standard form. Shifting the distribution can be done by providing the loc parameter to the methods of the instance. For example, poisson.pmf(x, mu, loc) delegates the work to poisson._pmf(x-loc, mu).

Discrete distributions from a list of probabilities

Alternatively, you can construct an arbitrary discrete rv defined on a finite set of values xk with Prob{X=xk} = pk by using the values keyword argument to the rv_discrete constructor.

Deepcopying / Pickling

If a distribution or frozen distribution is deepcopied (pickled/unpickled, etc.), any underlying random number generator is deepcopied with it. An implication is that if a distribution relies on the singleton RandomState before copying, it will rely on a copy of that random state after copying, and np.random.seed will no longer control the state.

Examples

Custom made discrete distribution:

>>> import numpy as np
>>> from scipy import stats
>>> xk = np.arange(7)
>>> pk = (0.1, 0.2, 0.3, 0.1, 0.1, 0.0, 0.2)
>>> custm = stats.rv_discrete(name='custm', values=(xk, pk))
>>>
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)
>>> ax.plot(xk, custm.pmf(xk), 'ro', ms=12, mec='r')
>>> ax.vlines(xk, 0, custm.pmf(xk), colors='r', lw=4)
>>> plt.show()

Random number generation:

>>> R = custm.rvs(size=100)

Methods

`cdf`	Cumulative distribution function of the given RV.
`entropy`	Differential entropy of the RV.
`expect`	Calculate expected value of a function with respect to the distribution for discrete distribution by numerical summation.
`freeze`	Freeze the distribution for the given arguments.
`interval`	Confidence interval with equal areas around the median.
`isf`	Inverse survival function (inverse of sf) at q of the given RV.
`logcdf`	Log of the cumulative distribution function at k of the given RV.
`logpmf`	Log of the probability mass function at k of the given RV.
`logsf`	Log of the survival function of the given RV.
`mean`	Mean of the distribution.
`median`	Median of the distribution.
`moment`	non-central moment of distribution of specified order.
`nnlf`	Negative loglikelihood function.
`pmf`	Probability mass function at k of the given RV.
`ppf`	Percent point function (inverse of cdf) at q of the given RV.
`rvs`	Random variates of given type.
`sf`	Survival function (1 - cdf) at k of the given RV.
`stats`	Some statistics of the given RV.
`std`	Standard deviation of the distribution.
`support`	Support of the distribution.
`var`	Variance of the distribution.

Attributes

random_state

Get or set the generator object for generating random variates.

__call__(*args, **kwds)#

Freeze the distribution for the given arguments.

Parameters:

arg1 (array_like) – The shape parameter(s) for the distribution. Should include all the non-optional arguments, may include loc and scale.
arg2 (array_like) – The shape parameter(s) for the distribution. Should include all the non-optional arguments, may include loc and scale.
arg3 (array_like) – The shape parameter(s) for the distribution. Should include all the non-optional arguments, may include loc and scale.
... (array_like) – The shape parameter(s) for the distribution. Should include all the non-optional arguments, may include loc and scale.

Returns:

rv_frozen – The frozen distribution.

Return type:

rv_frozen instance

cdf(k, *args, **kwds)[source]#

Cumulative distribution function of the given RV.

Parameters:

k (array_like, int) – Quantiles.
arg1 (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
arg2 (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
arg3 (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
... (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
loc (array_like, optional) – Location parameter (default=0).

Returns:

cdf – Cumulative distribution function evaluated at k.

Return type:

ndarray

entropy(*args, **kwds)#

Differential entropy of the RV.

Parameters:

arg1 (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
arg2 (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
arg3 (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
... (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
loc (array_like, optional) – Location parameter (default=0).
scale (array_like, optional (continuous distributions only).) – Scale parameter (default=1).

Notes

Entropy is defined base e:

>>> import numpy as np
>>> drv = rv_discrete(values=((0, 1), (0.5, 0.5)))
>>> np.allclose(drv.entropy(), np.log(2.0))
True

expect(func=None, args=(), loc=0, lb=None, ub=None, conditional=False, maxcount=1000, tolerance=1e-10, chunksize=32)[source]#

Calculate expected value of a function with respect to the distribution for discrete distribution by numerical summation.

Parameters:

func (callable, optional) – Function for which the expectation value is calculated. Takes only one argument. The default is the identity mapping f(k) = k.
args (tuple, optional) – Shape parameters of the distribution.
loc (float, optional) – Location parameter. Default is 0.
lb (int, optional) – Lower and upper bound for the summation, default is set to the support of the distribution, inclusive (lb <= k <= ub).
ub (int, optional) – Lower and upper bound for the summation, default is set to the support of the distribution, inclusive (lb <= k <= ub).
conditional (bool, optional) – If true then the expectation is corrected by the conditional probability of the summation interval. The return value is the expectation of the function, func, conditional on being in the given interval (k such that lb <= k <= ub). Default is False.
maxcount (int, optional) – Maximal number of terms to evaluate (to avoid an endless loop for an infinite sum). Default is 1000.
tolerance (float, optional) – Absolute tolerance for the summation. Default is 1e-10.
chunksize (int, optional) – Iterate over the support of a distributions in chunks of this size. Default is 32.

Returns:

expect – Expected value.

Return type:

float

Notes

For heavy-tailed distributions, the expected value may or may not exist, depending on the function, func. If it does exist, but the sum converges slowly, the accuracy of the result may be rather low. For instance, for zipf(4), accuracy for mean, variance in example is only 1e-5. increasing maxcount and/or chunksize may improve the result, but may also make zipf very slow.

The function is not vectorized.

freeze(*args, **kwds)#

Freeze the distribution for the given arguments.

Parameters:

arg1 (array_like) – The shape parameter(s) for the distribution. Should include all the non-optional arguments, may include loc and scale.
arg2 (array_like) – The shape parameter(s) for the distribution. Should include all the non-optional arguments, may include loc and scale.
arg3 (array_like) – The shape parameter(s) for the distribution. Should include all the non-optional arguments, may include loc and scale.
... (array_like) – The shape parameter(s) for the distribution. Should include all the non-optional arguments, may include loc and scale.

Returns:

rv_frozen – The frozen distribution.

Return type:

rv_frozen instance

interval(confidence, *args, **kwds)#

Confidence interval with equal areas around the median.

Parameters:

confidence (array_like of float) – Probability that an rv will be drawn from the returned range. Each value should be in the range [0, 1].
arg1 (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
arg2 (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
... (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
loc (array_like, optional) – location parameter, Default is 0.
scale (array_like, optional) – scale parameter, Default is 1.

Returns:

a, b – end-points of range that contain 100 * alpha % of the rv’s possible values.

Return type:

ndarray of float

Notes

This is implemented as ppf([p_tail, 1-p_tail]), where ppf is the inverse cumulative distribution function and p_tail = (1-confidence)/2. Suppose [c, d] is the support of a discrete distribution; then ppf([0, 1]) == (c-1, d). Therefore, when confidence=1 and the distribution is discrete, the left end of the interval will be beyond the support of the distribution. For discrete distributions, the interval will limit the probability in each tail to be less than or equal to p_tail (usually strictly less).

isf(q, *args, **kwds)[source]#

Inverse survival function (inverse of sf) at q of the given RV.

Parameters:

q (array_like) – Upper tail probability.
arg1 (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
arg2 (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
arg3 (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
... (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
loc (array_like, optional) – Location parameter (default=0).

Returns:

k – Quantile corresponding to the upper tail probability, q.

Return type:

ndarray or scalar

logcdf(k, *args, **kwds)[source]#

Log of the cumulative distribution function at k of the given RV.

Parameters:

k (array_like, int) – Quantiles.
arg1 (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
arg2 (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
arg3 (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
... (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
loc (array_like, optional) – Location parameter (default=0).

Returns:

logcdf – Log of the cumulative distribution function evaluated at k.

Return type:

array_like

logpmf(k, *args, **kwds)[source]#

Log of the probability mass function at k of the given RV.

Parameters:

k (array_like) – Quantiles.
arg1 (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
arg2 (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
arg3 (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
... (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
loc (array_like, optional) – Location parameter. Default is 0.

Returns:

logpmf – Log of the probability mass function evaluated at k.

Return type:

array_like

logsf(k, *args, **kwds)[source]#

Log of the survival function of the given RV.

Returns the log of the “survival function,” defined as 1 - cdf, evaluated at k.

Parameters:

k (array_like) – Quantiles.
arg1 (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
arg2 (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
arg3 (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
... (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information).
loc (array_like, optional) – Location parameter (default=0).

Returns:

logsf – Log of the survival function evaluated at k.

Return type:

ndarray

mean(*args, **kwds)#

Mean of the distribution.

Parameters:

arg1 (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information)
arg2 (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information)
arg3 (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information)
... (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information)
loc (array_like, optional) – location parameter (default=0)
scale (array_like, optional) – scale parameter (default=1)

Returns:

mean – the mean of the distribution

Return type:

float

median(*args, **kwds)#

Median of the distribution.

Parameters:

arg1 (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information)
arg2 (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information)
arg3 (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information)
... (array_like) – The shape parameter(s) for the distribution (see docstring of the instance object for more information)
loc (array_like, optional) – Location parameter, Default is 0.
scale (array_like, optional) – Scale parameter, Default is 1.

Returns:

median – The median of the distribution.

Return type:

float

deephyper.skopt.space.rv_discrete

Contents

deephyper.skopt.space.rv_discrete#