520 lines
14 KiB
Python
520 lines
14 KiB
Python
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from sympy.sets import FiniteSet
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from sympy.core.numbers import Rational
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from sympy.core.relational import Eq
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from sympy.core.symbol import Dummy
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from sympy.functions.combinatorial.factorials import FallingFactorial
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from sympy.functions.elementary.exponential import (exp, log)
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from sympy.functions.elementary.miscellaneous import sqrt
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from sympy.functions.elementary.piecewise import piecewise_fold
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from sympy.integrals.integrals import Integral
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from sympy.solvers.solveset import solveset
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from .rv import (probability, expectation, density, where, given, pspace, cdf, PSpace,
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characteristic_function, sample, sample_iter, random_symbols, independent, dependent,
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sampling_density, moment_generating_function, quantile, is_random,
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sample_stochastic_process)
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__all__ = ['P', 'E', 'H', 'density', 'where', 'given', 'sample', 'cdf',
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'characteristic_function', 'pspace', 'sample_iter', 'variance', 'std',
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'skewness', 'kurtosis', 'covariance', 'dependent', 'entropy', 'median',
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'independent', 'random_symbols', 'correlation', 'factorial_moment',
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'moment', 'cmoment', 'sampling_density', 'moment_generating_function',
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'smoment', 'quantile', 'sample_stochastic_process']
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def moment(X, n, c=0, condition=None, *, evaluate=True, **kwargs):
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"""
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Return the nth moment of a random expression about c.
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.. math::
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moment(X, c, n) = E((X-c)^{n})
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Default value of c is 0.
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Examples
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========
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>>> from sympy.stats import Die, moment, E
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>>> X = Die('X', 6)
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>>> moment(X, 1, 6)
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-5/2
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>>> moment(X, 2)
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91/6
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>>> moment(X, 1) == E(X)
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True
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"""
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from sympy.stats.symbolic_probability import Moment
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if evaluate:
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return Moment(X, n, c, condition).doit()
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return Moment(X, n, c, condition).rewrite(Integral)
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def variance(X, condition=None, **kwargs):
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"""
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Variance of a random expression.
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.. math::
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variance(X) = E((X-E(X))^{2})
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Examples
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========
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>>> from sympy.stats import Die, Bernoulli, variance
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>>> from sympy import simplify, Symbol
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>>> X = Die('X', 6)
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>>> p = Symbol('p')
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>>> B = Bernoulli('B', p, 1, 0)
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>>> variance(2*X)
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35/3
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>>> simplify(variance(B))
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p*(1 - p)
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"""
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if is_random(X) and pspace(X) == PSpace():
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from sympy.stats.symbolic_probability import Variance
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return Variance(X, condition)
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return cmoment(X, 2, condition, **kwargs)
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def standard_deviation(X, condition=None, **kwargs):
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r"""
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Standard Deviation of a random expression
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.. math::
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std(X) = \sqrt(E((X-E(X))^{2}))
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Examples
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========
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>>> from sympy.stats import Bernoulli, std
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>>> from sympy import Symbol, simplify
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>>> p = Symbol('p')
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>>> B = Bernoulli('B', p, 1, 0)
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>>> simplify(std(B))
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sqrt(p*(1 - p))
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"""
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return sqrt(variance(X, condition, **kwargs))
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std = standard_deviation
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def entropy(expr, condition=None, **kwargs):
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"""
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Calculuates entropy of a probability distribution.
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Parameters
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==========
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expression : the random expression whose entropy is to be calculated
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condition : optional, to specify conditions on random expression
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b: base of the logarithm, optional
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By default, it is taken as Euler's number
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Returns
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=======
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result : Entropy of the expression, a constant
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Examples
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========
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>>> from sympy.stats import Normal, Die, entropy
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>>> X = Normal('X', 0, 1)
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>>> entropy(X)
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log(2)/2 + 1/2 + log(pi)/2
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>>> D = Die('D', 4)
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>>> entropy(D)
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log(4)
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Entropy_%28information_theory%29
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.. [2] https://www.crmarsh.com/static/pdf/Charles_Marsh_Continuous_Entropy.pdf
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.. [3] https://kconrad.math.uconn.edu/blurbs/analysis/entropypost.pdf
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"""
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pdf = density(expr, condition, **kwargs)
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base = kwargs.get('b', exp(1))
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if isinstance(pdf, dict):
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return sum([-prob*log(prob, base) for prob in pdf.values()])
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return expectation(-log(pdf(expr), base))
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def covariance(X, Y, condition=None, **kwargs):
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"""
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Covariance of two random expressions.
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Explanation
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===========
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The expectation that the two variables will rise and fall together
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.. math::
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covariance(X,Y) = E((X-E(X)) (Y-E(Y)))
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Examples
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========
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>>> from sympy.stats import Exponential, covariance
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>>> from sympy import Symbol
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>>> rate = Symbol('lambda', positive=True, real=True)
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>>> X = Exponential('X', rate)
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>>> Y = Exponential('Y', rate)
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>>> covariance(X, X)
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lambda**(-2)
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>>> covariance(X, Y)
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0
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>>> covariance(X, Y + rate*X)
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1/lambda
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"""
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if (is_random(X) and pspace(X) == PSpace()) or (is_random(Y) and pspace(Y) == PSpace()):
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from sympy.stats.symbolic_probability import Covariance
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return Covariance(X, Y, condition)
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return expectation(
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(X - expectation(X, condition, **kwargs)) *
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(Y - expectation(Y, condition, **kwargs)),
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condition, **kwargs)
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def correlation(X, Y, condition=None, **kwargs):
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r"""
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Correlation of two random expressions, also known as correlation
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coefficient or Pearson's correlation.
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Explanation
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===========
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The normalized expectation that the two variables will rise
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and fall together
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.. math::
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correlation(X,Y) = E((X-E(X))(Y-E(Y)) / (\sigma_x \sigma_y))
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Examples
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========
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>>> from sympy.stats import Exponential, correlation
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>>> from sympy import Symbol
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>>> rate = Symbol('lambda', positive=True, real=True)
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>>> X = Exponential('X', rate)
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>>> Y = Exponential('Y', rate)
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>>> correlation(X, X)
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1
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>>> correlation(X, Y)
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0
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>>> correlation(X, Y + rate*X)
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1/sqrt(1 + lambda**(-2))
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"""
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return covariance(X, Y, condition, **kwargs)/(std(X, condition, **kwargs)
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* std(Y, condition, **kwargs))
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def cmoment(X, n, condition=None, *, evaluate=True, **kwargs):
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"""
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Return the nth central moment of a random expression about its mean.
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.. math::
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cmoment(X, n) = E((X - E(X))^{n})
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Examples
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========
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>>> from sympy.stats import Die, cmoment, variance
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>>> X = Die('X', 6)
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>>> cmoment(X, 3)
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0
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>>> cmoment(X, 2)
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35/12
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>>> cmoment(X, 2) == variance(X)
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True
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"""
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from sympy.stats.symbolic_probability import CentralMoment
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if evaluate:
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return CentralMoment(X, n, condition).doit()
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return CentralMoment(X, n, condition).rewrite(Integral)
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def smoment(X, n, condition=None, **kwargs):
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r"""
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Return the nth Standardized moment of a random expression.
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.. math::
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smoment(X, n) = E(((X - \mu)/\sigma_X)^{n})
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Examples
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========
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>>> from sympy.stats import skewness, Exponential, smoment
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>>> from sympy import Symbol
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>>> rate = Symbol('lambda', positive=True, real=True)
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>>> Y = Exponential('Y', rate)
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>>> smoment(Y, 4)
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9
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>>> smoment(Y, 4) == smoment(3*Y, 4)
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True
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>>> smoment(Y, 3) == skewness(Y)
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True
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"""
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sigma = std(X, condition, **kwargs)
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return (1/sigma)**n*cmoment(X, n, condition, **kwargs)
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def skewness(X, condition=None, **kwargs):
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r"""
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Measure of the asymmetry of the probability distribution.
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Explanation
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===========
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Positive skew indicates that most of the values lie to the right of
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the mean.
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.. math::
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skewness(X) = E(((X - E(X))/\sigma_X)^{3})
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Parameters
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==========
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condition : Expr containing RandomSymbols
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A conditional expression. skewness(X, X>0) is skewness of X given X > 0
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Examples
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========
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>>> from sympy.stats import skewness, Exponential, Normal
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>>> from sympy import Symbol
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>>> X = Normal('X', 0, 1)
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>>> skewness(X)
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0
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>>> skewness(X, X > 0) # find skewness given X > 0
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(-sqrt(2)/sqrt(pi) + 4*sqrt(2)/pi**(3/2))/(1 - 2/pi)**(3/2)
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>>> rate = Symbol('lambda', positive=True, real=True)
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>>> Y = Exponential('Y', rate)
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>>> skewness(Y)
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2
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"""
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return smoment(X, 3, condition=condition, **kwargs)
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def kurtosis(X, condition=None, **kwargs):
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r"""
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Characterizes the tails/outliers of a probability distribution.
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Explanation
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===========
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Kurtosis of any univariate normal distribution is 3. Kurtosis less than
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3 means that the distribution produces fewer and less extreme outliers
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than the normal distribution.
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.. math::
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kurtosis(X) = E(((X - E(X))/\sigma_X)^{4})
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Parameters
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==========
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condition : Expr containing RandomSymbols
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A conditional expression. kurtosis(X, X>0) is kurtosis of X given X > 0
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Examples
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========
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>>> from sympy.stats import kurtosis, Exponential, Normal
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>>> from sympy import Symbol
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>>> X = Normal('X', 0, 1)
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>>> kurtosis(X)
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3
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>>> kurtosis(X, X > 0) # find kurtosis given X > 0
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(-4/pi - 12/pi**2 + 3)/(1 - 2/pi)**2
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>>> rate = Symbol('lamda', positive=True, real=True)
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>>> Y = Exponential('Y', rate)
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>>> kurtosis(Y)
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9
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Kurtosis
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.. [2] https://mathworld.wolfram.com/Kurtosis.html
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"""
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return smoment(X, 4, condition=condition, **kwargs)
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def factorial_moment(X, n, condition=None, **kwargs):
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"""
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The factorial moment is a mathematical quantity defined as the expectation
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or average of the falling factorial of a random variable.
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.. math::
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factorial-moment(X, n) = E(X(X - 1)(X - 2)...(X - n + 1))
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Parameters
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==========
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n: A natural number, n-th factorial moment.
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condition : Expr containing RandomSymbols
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A conditional expression.
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Examples
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========
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>>> from sympy.stats import factorial_moment, Poisson, Binomial
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>>> from sympy import Symbol, S
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>>> lamda = Symbol('lamda')
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>>> X = Poisson('X', lamda)
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>>> factorial_moment(X, 2)
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lamda**2
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>>> Y = Binomial('Y', 2, S.Half)
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>>> factorial_moment(Y, 2)
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1/2
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>>> factorial_moment(Y, 2, Y > 1) # find factorial moment for Y > 1
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2
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Factorial_moment
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.. [2] https://mathworld.wolfram.com/FactorialMoment.html
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"""
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return expectation(FallingFactorial(X, n), condition=condition, **kwargs)
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def median(X, evaluate=True, **kwargs):
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r"""
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Calculuates the median of the probability distribution.
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Explanation
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===========
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Mathematically, median of Probability distribution is defined as all those
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values of `m` for which the following condition is satisfied
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.. math::
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P(X\leq m) \geq \frac{1}{2} \text{ and} \text{ } P(X\geq m)\geq \frac{1}{2}
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Parameters
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||
|
|
==========
|
||
|
|
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|
X: The random expression whose median is to be calculated.
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|
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||
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Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
The FiniteSet or an Interval which contains the median of the
|
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|
|
random expression.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.stats import Normal, Die, median
|
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|
|
>>> N = Normal('N', 3, 1)
|
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|
|
>>> median(N)
|
||
|
|
{3}
|
||
|
|
>>> D = Die('D')
|
||
|
|
>>> median(D)
|
||
|
|
{3, 4}
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
|
||
|
|
.. [1] https://en.wikipedia.org/wiki/Median#Probability_distributions
|
||
|
|
|
||
|
|
"""
|
||
|
|
if not is_random(X):
|
||
|
|
return X
|
||
|
|
|
||
|
|
from sympy.stats.crv import ContinuousPSpace
|
||
|
|
from sympy.stats.drv import DiscretePSpace
|
||
|
|
from sympy.stats.frv import FinitePSpace
|
||
|
|
|
||
|
|
if isinstance(pspace(X), FinitePSpace):
|
||
|
|
cdf = pspace(X).compute_cdf(X)
|
||
|
|
result = []
|
||
|
|
for key, value in cdf.items():
|
||
|
|
if value>= Rational(1, 2) and (1 - value) + \
|
||
|
|
pspace(X).probability(Eq(X, key)) >= Rational(1, 2):
|
||
|
|
result.append(key)
|
||
|
|
return FiniteSet(*result)
|
||
|
|
if isinstance(pspace(X), (ContinuousPSpace, DiscretePSpace)):
|
||
|
|
cdf = pspace(X).compute_cdf(X)
|
||
|
|
x = Dummy('x')
|
||
|
|
result = solveset(piecewise_fold(cdf(x) - Rational(1, 2)), x, pspace(X).set)
|
||
|
|
return result
|
||
|
|
raise NotImplementedError("The median of %s is not implemented."%str(pspace(X)))
|
||
|
|
|
||
|
|
|
||
|
|
def coskewness(X, Y, Z, condition=None, **kwargs):
|
||
|
|
r"""
|
||
|
|
Calculates the co-skewness of three random variables.
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
Mathematically Coskewness is defined as
|
||
|
|
|
||
|
|
.. math::
|
||
|
|
coskewness(X,Y,Z)=\frac{E[(X-E[X]) * (Y-E[Y]) * (Z-E[Z])]} {\sigma_{X}\sigma_{Y}\sigma_{Z}}
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
X : RandomSymbol
|
||
|
|
Random Variable used to calculate coskewness
|
||
|
|
Y : RandomSymbol
|
||
|
|
Random Variable used to calculate coskewness
|
||
|
|
Z : RandomSymbol
|
||
|
|
Random Variable used to calculate coskewness
|
||
|
|
condition : Expr containing RandomSymbols
|
||
|
|
A conditional expression
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.stats import coskewness, Exponential, skewness
|
||
|
|
>>> from sympy import symbols
|
||
|
|
>>> p = symbols('p', positive=True)
|
||
|
|
>>> X = Exponential('X', p)
|
||
|
|
>>> Y = Exponential('Y', 2*p)
|
||
|
|
>>> coskewness(X, Y, Y)
|
||
|
|
0
|
||
|
|
>>> coskewness(X, Y + X, Y + 2*X)
|
||
|
|
16*sqrt(85)/85
|
||
|
|
>>> coskewness(X + 2*Y, Y + X, Y + 2*X, X > 3)
|
||
|
|
9*sqrt(170)/85
|
||
|
|
>>> coskewness(Y, Y, Y) == skewness(Y)
|
||
|
|
True
|
||
|
|
>>> coskewness(X, Y + p*X, Y + 2*p*X)
|
||
|
|
4/(sqrt(1 + 1/(4*p**2))*sqrt(4 + 1/(4*p**2)))
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
coskewness : The coskewness of the three random variables
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
|
||
|
|
.. [1] https://en.wikipedia.org/wiki/Coskewness
|
||
|
|
|
||
|
|
"""
|
||
|
|
num = expectation((X - expectation(X, condition, **kwargs)) \
|
||
|
|
* (Y - expectation(Y, condition, **kwargs)) \
|
||
|
|
* (Z - expectation(Z, condition, **kwargs)), condition, **kwargs)
|
||
|
|
den = std(X, condition, **kwargs) * std(Y, condition, **kwargs) \
|
||
|
|
* std(Z, condition, **kwargs)
|
||
|
|
return num/den
|
||
|
|
|
||
|
|
|
||
|
|
P = probability
|
||
|
|
E = expectation
|
||
|
|
H = entropy
|