351 lines
12 KiB
Python
351 lines
12 KiB
Python
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from sympy.concrete.summations import (Sum, summation)
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from sympy.core.basic import Basic
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from sympy.core.cache import cacheit
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from sympy.core.function import Lambda
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from sympy.core.numbers import I
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from sympy.core.relational import (Eq, Ne)
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from sympy.core.singleton import S
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from sympy.core.symbol import (Dummy, symbols)
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from sympy.core.sympify import sympify
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from sympy.functions.combinatorial.factorials import factorial
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from sympy.functions.elementary.exponential import exp
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from sympy.functions.elementary.integers import floor
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from sympy.functions.elementary.piecewise import Piecewise
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from sympy.logic.boolalg import And
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from sympy.polys.polytools import poly
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from sympy.series.series import series
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from sympy.polys.polyerrors import PolynomialError
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from sympy.stats.crv import reduce_rational_inequalities_wrap
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from sympy.stats.rv import (NamedArgsMixin, SinglePSpace, SingleDomain,
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random_symbols, PSpace, ConditionalDomain, RandomDomain,
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ProductDomain, Distribution)
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from sympy.stats.symbolic_probability import Probability
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from sympy.sets.fancysets import Range, FiniteSet
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from sympy.sets.sets import Union
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from sympy.sets.contains import Contains
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from sympy.utilities import filldedent
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from sympy.core.sympify import _sympify
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class DiscreteDistribution(Distribution):
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def __call__(self, *args):
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return self.pdf(*args)
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class SingleDiscreteDistribution(DiscreteDistribution, NamedArgsMixin):
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""" Discrete distribution of a single variable.
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Serves as superclass for PoissonDistribution etc....
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Provides methods for pdf, cdf, and sampling
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See Also:
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sympy.stats.crv_types.*
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"""
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set = S.Integers
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def __new__(cls, *args):
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args = list(map(sympify, args))
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return Basic.__new__(cls, *args)
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@staticmethod
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def check(*args):
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pass
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@cacheit
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def compute_cdf(self, **kwargs):
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""" Compute the CDF from the PDF.
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Returns a Lambda.
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"""
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x = symbols('x', integer=True, cls=Dummy)
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z = symbols('z', real=True, cls=Dummy)
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left_bound = self.set.inf
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# CDF is integral of PDF from left bound to z
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pdf = self.pdf(x)
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cdf = summation(pdf, (x, left_bound, floor(z)), **kwargs)
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# CDF Ensure that CDF left of left_bound is zero
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cdf = Piecewise((cdf, z >= left_bound), (0, True))
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return Lambda(z, cdf)
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def _cdf(self, x):
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return None
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def cdf(self, x, **kwargs):
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""" Cumulative density function """
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if not kwargs:
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cdf = self._cdf(x)
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if cdf is not None:
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return cdf
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return self.compute_cdf(**kwargs)(x)
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@cacheit
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def compute_characteristic_function(self, **kwargs):
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""" Compute the characteristic function from the PDF.
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Returns a Lambda.
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"""
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x, t = symbols('x, t', real=True, cls=Dummy)
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pdf = self.pdf(x)
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cf = summation(exp(I*t*x)*pdf, (x, self.set.inf, self.set.sup))
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return Lambda(t, cf)
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def _characteristic_function(self, t):
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return None
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def characteristic_function(self, t, **kwargs):
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""" Characteristic function """
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if not kwargs:
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cf = self._characteristic_function(t)
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if cf is not None:
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return cf
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return self.compute_characteristic_function(**kwargs)(t)
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@cacheit
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def compute_moment_generating_function(self, **kwargs):
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t = Dummy('t', real=True)
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x = Dummy('x', integer=True)
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pdf = self.pdf(x)
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mgf = summation(exp(t*x)*pdf, (x, self.set.inf, self.set.sup))
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return Lambda(t, mgf)
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def _moment_generating_function(self, t):
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return None
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def moment_generating_function(self, t, **kwargs):
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if not kwargs:
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mgf = self._moment_generating_function(t)
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if mgf is not None:
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return mgf
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return self.compute_moment_generating_function(**kwargs)(t)
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@cacheit
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def compute_quantile(self, **kwargs):
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""" Compute the Quantile from the PDF.
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Returns a Lambda.
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"""
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x = Dummy('x', integer=True)
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p = Dummy('p', real=True)
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left_bound = self.set.inf
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pdf = self.pdf(x)
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cdf = summation(pdf, (x, left_bound, x), **kwargs)
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set = ((x, p <= cdf), )
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return Lambda(p, Piecewise(*set))
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def _quantile(self, x):
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return None
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def quantile(self, x, **kwargs):
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""" Cumulative density function """
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if not kwargs:
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quantile = self._quantile(x)
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if quantile is not None:
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return quantile
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return self.compute_quantile(**kwargs)(x)
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def expectation(self, expr, var, evaluate=True, **kwargs):
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""" Expectation of expression over distribution """
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# TODO: support discrete sets with non integer stepsizes
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if evaluate:
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try:
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p = poly(expr, var)
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t = Dummy('t', real=True)
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mgf = self.moment_generating_function(t)
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deg = p.degree()
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taylor = poly(series(mgf, t, 0, deg + 1).removeO(), t)
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result = 0
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for k in range(deg+1):
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result += p.coeff_monomial(var ** k) * taylor.coeff_monomial(t ** k) * factorial(k)
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return result
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except PolynomialError:
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return summation(expr * self.pdf(var),
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(var, self.set.inf, self.set.sup), **kwargs)
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else:
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return Sum(expr * self.pdf(var),
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(var, self.set.inf, self.set.sup), **kwargs)
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def __call__(self, *args):
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return self.pdf(*args)
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class DiscreteDomain(RandomDomain):
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"""
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A domain with discrete support with step size one.
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Represented using symbols and Range.
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"""
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is_Discrete = True
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class SingleDiscreteDomain(DiscreteDomain, SingleDomain):
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def as_boolean(self):
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return Contains(self.symbol, self.set)
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class ConditionalDiscreteDomain(DiscreteDomain, ConditionalDomain):
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"""
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Domain with discrete support of step size one, that is restricted by
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some condition.
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"""
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@property
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def set(self):
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rv = self.symbols
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if len(self.symbols) > 1:
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raise NotImplementedError(filldedent('''
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Multivariate conditional domains are not yet implemented.'''))
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rv = list(rv)[0]
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return reduce_rational_inequalities_wrap(self.condition,
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rv).intersect(self.fulldomain.set)
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class DiscretePSpace(PSpace):
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is_real = True
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is_Discrete = True
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@property
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def pdf(self):
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return self.density(*self.symbols)
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def where(self, condition):
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rvs = random_symbols(condition)
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assert all(r.symbol in self.symbols for r in rvs)
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if len(rvs) > 1:
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raise NotImplementedError(filldedent('''Multivariate discrete
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random variables are not yet supported.'''))
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conditional_domain = reduce_rational_inequalities_wrap(condition,
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rvs[0])
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conditional_domain = conditional_domain.intersect(self.domain.set)
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return SingleDiscreteDomain(rvs[0].symbol, conditional_domain)
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def probability(self, condition):
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complement = isinstance(condition, Ne)
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if complement:
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condition = Eq(condition.args[0], condition.args[1])
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try:
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_domain = self.where(condition).set
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if condition == False or _domain is S.EmptySet:
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return S.Zero
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if condition == True or _domain == self.domain.set:
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return S.One
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prob = self.eval_prob(_domain)
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except NotImplementedError:
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from sympy.stats.rv import density
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expr = condition.lhs - condition.rhs
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dens = density(expr)
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if not isinstance(dens, DiscreteDistribution):
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from sympy.stats.drv_types import DiscreteDistributionHandmade
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dens = DiscreteDistributionHandmade(dens)
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z = Dummy('z', real=True)
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space = SingleDiscretePSpace(z, dens)
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prob = space.probability(condition.__class__(space.value, 0))
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if prob is None:
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prob = Probability(condition)
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return prob if not complement else S.One - prob
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def eval_prob(self, _domain):
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sym = list(self.symbols)[0]
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if isinstance(_domain, Range):
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n = symbols('n', integer=True)
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inf, sup, step = (r for r in _domain.args)
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summand = ((self.pdf).replace(
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sym, n*step))
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rv = summation(summand,
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(n, inf/step, (sup)/step - 1)).doit()
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return rv
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elif isinstance(_domain, FiniteSet):
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pdf = Lambda(sym, self.pdf)
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rv = sum(pdf(x) for x in _domain)
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return rv
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elif isinstance(_domain, Union):
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rv = sum(self.eval_prob(x) for x in _domain.args)
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return rv
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def conditional_space(self, condition):
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# XXX: Converting from set to tuple. The order matters to Lambda
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# though so we should be starting with a set...
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density = Lambda(tuple(self.symbols), self.pdf/self.probability(condition))
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condition = condition.xreplace({rv: rv.symbol for rv in self.values})
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domain = ConditionalDiscreteDomain(self.domain, condition)
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return DiscretePSpace(domain, density)
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class ProductDiscreteDomain(ProductDomain, DiscreteDomain):
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def as_boolean(self):
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return And(*[domain.as_boolean for domain in self.domains])
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class SingleDiscretePSpace(DiscretePSpace, SinglePSpace):
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""" Discrete probability space over a single univariate variable """
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is_real = True
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@property
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def set(self):
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return self.distribution.set
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@property
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def domain(self):
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return SingleDiscreteDomain(self.symbol, self.set)
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def sample(self, size=(), library='scipy', seed=None):
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"""
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Internal sample method.
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Returns dictionary mapping RandomSymbol to realization value.
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"""
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return {self.value: self.distribution.sample(size, library=library, seed=seed)}
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def compute_expectation(self, expr, rvs=None, evaluate=True, **kwargs):
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rvs = rvs or (self.value,)
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if self.value not in rvs:
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return expr
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expr = _sympify(expr)
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expr = expr.xreplace({rv: rv.symbol for rv in rvs})
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x = self.value.symbol
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try:
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return self.distribution.expectation(expr, x, evaluate=evaluate,
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**kwargs)
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except NotImplementedError:
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return Sum(expr * self.pdf, (x, self.set.inf, self.set.sup),
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**kwargs)
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def compute_cdf(self, expr, **kwargs):
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if expr == self.value:
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x = Dummy("x", real=True)
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return Lambda(x, self.distribution.cdf(x, **kwargs))
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else:
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raise NotImplementedError()
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def compute_density(self, expr, **kwargs):
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if expr == self.value:
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return self.distribution
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raise NotImplementedError()
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def compute_characteristic_function(self, expr, **kwargs):
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if expr == self.value:
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t = Dummy("t", real=True)
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return Lambda(t, self.distribution.characteristic_function(t, **kwargs))
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else:
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raise NotImplementedError()
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def compute_moment_generating_function(self, expr, **kwargs):
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if expr == self.value:
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t = Dummy("t", real=True)
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return Lambda(t, self.distribution.moment_generating_function(t, **kwargs))
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else:
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raise NotImplementedError()
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def compute_quantile(self, expr, **kwargs):
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if expr == self.value:
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p = Dummy("p", real=True)
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return Lambda(p, self.distribution.quantile(p, **kwargs))
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else:
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raise NotImplementedError()
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