571 lines
20 KiB
Python
571 lines
20 KiB
Python
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"""
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Continuous Random Variables Module
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See Also
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========
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sympy.stats.crv_types
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sympy.stats.rv
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sympy.stats.frv
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"""
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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, PoleError
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from sympy.core.numbers import (I, nan, oo)
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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, 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.piecewise import Piecewise
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from sympy.functions.special.delta_functions import DiracDelta
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from sympy.integrals.integrals import (Integral, integrate)
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from sympy.logic.boolalg import (And, Or)
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from sympy.polys.polyerrors import PolynomialError
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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.sets.sets import (FiniteSet, Intersection, Interval, Union)
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from sympy.solvers.solveset import solveset
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from sympy.solvers.inequalities import reduce_rational_inequalities
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from sympy.stats.rv import (RandomDomain, SingleDomain, ConditionalDomain, is_random,
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ProductDomain, PSpace, SinglePSpace, random_symbols, NamedArgsMixin, Distribution)
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class ContinuousDomain(RandomDomain):
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"""
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A domain with continuous support
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Represented using symbols and Intervals.
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"""
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is_Continuous = True
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def as_boolean(self):
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raise NotImplementedError("Not Implemented for generic Domains")
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class SingleContinuousDomain(ContinuousDomain, SingleDomain):
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"""
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A univariate domain with continuous support
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Represented using a single symbol and interval.
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"""
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def compute_expectation(self, expr, variables=None, **kwargs):
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if variables is None:
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variables = self.symbols
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if not variables:
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return expr
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if frozenset(variables) != frozenset(self.symbols):
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raise ValueError("Values should be equal")
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# assumes only intervals
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return Integral(expr, (self.symbol, self.set), **kwargs)
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def as_boolean(self):
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return self.set.as_relational(self.symbol)
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class ProductContinuousDomain(ProductDomain, ContinuousDomain):
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"""
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A collection of independent domains with continuous support
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"""
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def compute_expectation(self, expr, variables=None, **kwargs):
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if variables is None:
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variables = self.symbols
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for domain in self.domains:
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domain_vars = frozenset(variables) & frozenset(domain.symbols)
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if domain_vars:
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expr = domain.compute_expectation(expr, domain_vars, **kwargs)
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return expr
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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 ConditionalContinuousDomain(ContinuousDomain, ConditionalDomain):
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"""
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A domain with continuous support that has been further restricted by a
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condition such as $x > 3$.
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"""
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def compute_expectation(self, expr, variables=None, **kwargs):
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if variables is None:
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variables = self.symbols
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if not variables:
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return expr
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# Extract the full integral
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fullintgrl = self.fulldomain.compute_expectation(expr, variables)
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# separate into integrand and limits
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integrand, limits = fullintgrl.function, list(fullintgrl.limits)
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conditions = [self.condition]
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while conditions:
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cond = conditions.pop()
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if cond.is_Boolean:
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if isinstance(cond, And):
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conditions.extend(cond.args)
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elif isinstance(cond, Or):
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raise NotImplementedError("Or not implemented here")
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elif cond.is_Relational:
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if cond.is_Equality:
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# Add the appropriate Delta to the integrand
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integrand *= DiracDelta(cond.lhs - cond.rhs)
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else:
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symbols = cond.free_symbols & set(self.symbols)
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if len(symbols) != 1: # Can't handle x > y
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raise NotImplementedError(
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"Multivariate Inequalities not yet implemented")
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# Can handle x > 0
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symbol = symbols.pop()
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# Find the limit with x, such as (x, -oo, oo)
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for i, limit in enumerate(limits):
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if limit[0] == symbol:
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# Make condition into an Interval like [0, oo]
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cintvl = reduce_rational_inequalities_wrap(
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cond, symbol)
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# Make limit into an Interval like [-oo, oo]
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lintvl = Interval(limit[1], limit[2])
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# Intersect them to get [0, oo]
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intvl = cintvl.intersect(lintvl)
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# Put back into limits list
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limits[i] = (symbol, intvl.left, intvl.right)
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else:
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raise TypeError(
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"Condition %s is not a relational or Boolean" % cond)
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return Integral(integrand, *limits, **kwargs)
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def as_boolean(self):
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return And(self.fulldomain.as_boolean(), self.condition)
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@property
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def set(self):
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if len(self.symbols) == 1:
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return (self.fulldomain.set & reduce_rational_inequalities_wrap(
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self.condition, tuple(self.symbols)[0]))
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else:
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raise NotImplementedError(
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"Set of Conditional Domain not Implemented")
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class ContinuousDistribution(Distribution):
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def __call__(self, *args):
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return self.pdf(*args)
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class SingleContinuousDistribution(ContinuousDistribution, NamedArgsMixin):
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""" Continuous distribution of a single variable.
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Explanation
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===========
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Serves as superclass for Normal/Exponential/UniformDistribution etc....
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Represented by parameters for each of the specific classes. E.g
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NormalDistribution is represented by a mean and standard deviation.
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Provides methods for pdf, cdf, and sampling.
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See Also
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========
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sympy.stats.crv_types.*
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"""
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set = Interval(-oo, oo)
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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, z = symbols('x, z', real=True, cls=Dummy)
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left_bound = self.set.start
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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 = integrate(pdf.doit(), (x, left_bound, 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 len(kwargs) == 0:
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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 = integrate(exp(I*t*x)*pdf, (x, self.set))
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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 len(kwargs) == 0:
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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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""" Compute the moment generating 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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mgf = integrate(exp(t * x) * pdf, (x, self.set))
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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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""" Moment generating function """
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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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def expectation(self, expr, var, evaluate=True, **kwargs):
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""" Expectation of expression over distribution """
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if evaluate:
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try:
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p = poly(expr, var)
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if p.is_zero:
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return S.Zero
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t = Dummy('t', real=True)
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mgf = self._moment_generating_function(t)
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if mgf is None:
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return integrate(expr * self.pdf(var), (var, self.set), **kwargs)
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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 integrate(expr * self.pdf(var), (var, self.set), **kwargs)
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else:
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return Integral(expr * self.pdf(var), (var, self.set), **kwargs)
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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, p = symbols('x, p', real=True, cls=Dummy)
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left_bound = self.set.start
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pdf = self.pdf(x)
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cdf = integrate(pdf, (x, left_bound, x), **kwargs)
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quantile = solveset(cdf - p, x, self.set)
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return Lambda(p, Piecewise((quantile, (p >= 0) & (p <= 1) ), (nan, True)))
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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 len(kwargs) == 0:
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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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class ContinuousPSpace(PSpace):
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""" Continuous Probability Space
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Represents the likelihood of an event space defined over a continuum.
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Represented with a ContinuousDomain and a PDF (Lambda-Like)
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"""
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is_Continuous = True
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is_real = True
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@property
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def pdf(self):
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return self.density(*self.domain.symbols)
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def compute_expectation(self, expr, rvs=None, evaluate=False, **kwargs):
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if rvs is None:
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rvs = self.values
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else:
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rvs = frozenset(rvs)
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expr = expr.xreplace({rv: rv.symbol for rv in rvs})
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domain_symbols = frozenset(rv.symbol for rv in rvs)
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return self.domain.compute_expectation(self.pdf * expr,
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domain_symbols, **kwargs)
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def compute_density(self, expr, **kwargs):
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# Common case Density(X) where X in self.values
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if expr in self.values:
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# Marginalize all other random symbols out of the density
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randomsymbols = tuple(set(self.values) - frozenset([expr]))
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symbols = tuple(rs.symbol for rs in randomsymbols)
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pdf = self.domain.compute_expectation(self.pdf, symbols, **kwargs)
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return Lambda(expr.symbol, pdf)
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z = Dummy('z', real=True)
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return Lambda(z, self.compute_expectation(DiracDelta(expr - z), **kwargs))
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@cacheit
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def compute_cdf(self, expr, **kwargs):
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if not self.domain.set.is_Interval:
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raise ValueError(
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"CDF not well defined on multivariate expressions")
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d = self.compute_density(expr, **kwargs)
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x, z = symbols('x, z', real=True, cls=Dummy)
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left_bound = self.domain.set.start
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# CDF is integral of PDF from left bound to z
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cdf = integrate(d(x), (x, left_bound, 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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@cacheit
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def compute_characteristic_function(self, expr, **kwargs):
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if not self.domain.set.is_Interval:
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raise NotImplementedError("Characteristic function of multivariate expressions not implemented")
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d = self.compute_density(expr, **kwargs)
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x, t = symbols('x, t', real=True, cls=Dummy)
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cf = integrate(exp(I*t*x)*d(x), (x, -oo, oo), **kwargs)
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return Lambda(t, cf)
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@cacheit
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def compute_moment_generating_function(self, expr, **kwargs):
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if not self.domain.set.is_Interval:
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raise NotImplementedError("Moment generating function of multivariate expressions not implemented")
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d = self.compute_density(expr, **kwargs)
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x, t = symbols('x, t', real=True, cls=Dummy)
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|
|
mgf = integrate(exp(t * x) * d(x), (x, -oo, oo), **kwargs)
|
||
|
|
return Lambda(t, mgf)
|
||
|
|
|
||
|
|
@cacheit
|
||
|
|
def compute_quantile(self, expr, **kwargs):
|
||
|
|
if not self.domain.set.is_Interval:
|
||
|
|
raise ValueError(
|
||
|
|
"Quantile not well defined on multivariate expressions")
|
||
|
|
|
||
|
|
d = self.compute_cdf(expr, **kwargs)
|
||
|
|
x = Dummy('x', real=True)
|
||
|
|
p = Dummy('p', positive=True)
|
||
|
|
|
||
|
|
quantile = solveset(d(x) - p, x, self.set)
|
||
|
|
|
||
|
|
return Lambda(p, quantile)
|
||
|
|
|
||
|
|
def probability(self, condition, **kwargs):
|
||
|
|
z = Dummy('z', real=True)
|
||
|
|
cond_inv = False
|
||
|
|
if isinstance(condition, Ne):
|
||
|
|
condition = Eq(condition.args[0], condition.args[1])
|
||
|
|
cond_inv = True
|
||
|
|
# Univariate case can be handled by where
|
||
|
|
try:
|
||
|
|
domain = self.where(condition)
|
||
|
|
rv = [rv for rv in self.values if rv.symbol == domain.symbol][0]
|
||
|
|
# Integrate out all other random variables
|
||
|
|
pdf = self.compute_density(rv, **kwargs)
|
||
|
|
# return S.Zero if `domain` is empty set
|
||
|
|
if domain.set is S.EmptySet or isinstance(domain.set, FiniteSet):
|
||
|
|
return S.Zero if not cond_inv else S.One
|
||
|
|
if isinstance(domain.set, Union):
|
||
|
|
return sum(
|
||
|
|
Integral(pdf(z), (z, subset), **kwargs) for subset in
|
||
|
|
domain.set.args if isinstance(subset, Interval))
|
||
|
|
# Integrate out the last variable over the special domain
|
||
|
|
return Integral(pdf(z), (z, domain.set), **kwargs)
|
||
|
|
|
||
|
|
# Other cases can be turned into univariate case
|
||
|
|
# by computing a density handled by density computation
|
||
|
|
except NotImplementedError:
|
||
|
|
from sympy.stats.rv import density
|
||
|
|
expr = condition.lhs - condition.rhs
|
||
|
|
if not is_random(expr):
|
||
|
|
dens = self.density
|
||
|
|
comp = condition.rhs
|
||
|
|
else:
|
||
|
|
dens = density(expr, **kwargs)
|
||
|
|
comp = 0
|
||
|
|
if not isinstance(dens, ContinuousDistribution):
|
||
|
|
from sympy.stats.crv_types import ContinuousDistributionHandmade
|
||
|
|
dens = ContinuousDistributionHandmade(dens, set=self.domain.set)
|
||
|
|
# Turn problem into univariate case
|
||
|
|
space = SingleContinuousPSpace(z, dens)
|
||
|
|
result = space.probability(condition.__class__(space.value, comp))
|
||
|
|
return result if not cond_inv else S.One - result
|
||
|
|
|
||
|
|
def where(self, condition):
|
||
|
|
rvs = frozenset(random_symbols(condition))
|
||
|
|
if not (len(rvs) == 1 and rvs.issubset(self.values)):
|
||
|
|
raise NotImplementedError(
|
||
|
|
"Multiple continuous random variables not supported")
|
||
|
|
rv = tuple(rvs)[0]
|
||
|
|
interval = reduce_rational_inequalities_wrap(condition, rv)
|
||
|
|
interval = interval.intersect(self.domain.set)
|
||
|
|
return SingleContinuousDomain(rv.symbol, interval)
|
||
|
|
|
||
|
|
def conditional_space(self, condition, normalize=True, **kwargs):
|
||
|
|
condition = condition.xreplace({rv: rv.symbol for rv in self.values})
|
||
|
|
domain = ConditionalContinuousDomain(self.domain, condition)
|
||
|
|
if normalize:
|
||
|
|
# create a clone of the variable to
|
||
|
|
# make sure that variables in nested integrals are different
|
||
|
|
# from the variables outside the integral
|
||
|
|
# this makes sure that they are evaluated separately
|
||
|
|
# and in the correct order
|
||
|
|
replacement = {rv: Dummy(str(rv)) for rv in self.symbols}
|
||
|
|
norm = domain.compute_expectation(self.pdf, **kwargs)
|
||
|
|
pdf = self.pdf / norm.xreplace(replacement)
|
||
|
|
# XXX: Converting set to tuple. The order matters to Lambda though
|
||
|
|
# so we shouldn't be starting with a set here...
|
||
|
|
density = Lambda(tuple(domain.symbols), pdf)
|
||
|
|
|
||
|
|
return ContinuousPSpace(domain, density)
|
||
|
|
|
||
|
|
|
||
|
|
class SingleContinuousPSpace(ContinuousPSpace, SinglePSpace):
|
||
|
|
"""
|
||
|
|
A continuous probability space over a single univariate variable.
|
||
|
|
|
||
|
|
These consist of a Symbol and a SingleContinuousDistribution
|
||
|
|
|
||
|
|
This class is normally accessed through the various random variable
|
||
|
|
functions, Normal, Exponential, Uniform, etc....
|
||
|
|
"""
|
||
|
|
|
||
|
|
@property
|
||
|
|
def set(self):
|
||
|
|
return self.distribution.set
|
||
|
|
|
||
|
|
@property
|
||
|
|
def domain(self):
|
||
|
|
return SingleContinuousDomain(sympify(self.symbol), self.set)
|
||
|
|
|
||
|
|
def sample(self, size=(), library='scipy', seed=None):
|
||
|
|
"""
|
||
|
|
Internal sample method.
|
||
|
|
|
||
|
|
Returns dictionary mapping RandomSymbol to realization value.
|
||
|
|
"""
|
||
|
|
return {self.value: self.distribution.sample(size, library=library, seed=seed)}
|
||
|
|
|
||
|
|
def compute_expectation(self, expr, rvs=None, evaluate=False, **kwargs):
|
||
|
|
rvs = rvs or (self.value,)
|
||
|
|
if self.value not in rvs:
|
||
|
|
return expr
|
||
|
|
|
||
|
|
expr = _sympify(expr)
|
||
|
|
expr = expr.xreplace({rv: rv.symbol for rv in rvs})
|
||
|
|
|
||
|
|
x = self.value.symbol
|
||
|
|
try:
|
||
|
|
return self.distribution.expectation(expr, x, evaluate=evaluate, **kwargs)
|
||
|
|
except PoleError:
|
||
|
|
return Integral(expr * self.pdf, (x, self.set), **kwargs)
|
||
|
|
|
||
|
|
def compute_cdf(self, expr, **kwargs):
|
||
|
|
if expr == self.value:
|
||
|
|
z = Dummy("z", real=True)
|
||
|
|
return Lambda(z, self.distribution.cdf(z, **kwargs))
|
||
|
|
else:
|
||
|
|
return ContinuousPSpace.compute_cdf(self, expr, **kwargs)
|
||
|
|
|
||
|
|
def compute_characteristic_function(self, expr, **kwargs):
|
||
|
|
if expr == self.value:
|
||
|
|
t = Dummy("t", real=True)
|
||
|
|
return Lambda(t, self.distribution.characteristic_function(t, **kwargs))
|
||
|
|
else:
|
||
|
|
return ContinuousPSpace.compute_characteristic_function(self, expr, **kwargs)
|
||
|
|
|
||
|
|
def compute_moment_generating_function(self, expr, **kwargs):
|
||
|
|
if expr == self.value:
|
||
|
|
t = Dummy("t", real=True)
|
||
|
|
return Lambda(t, self.distribution.moment_generating_function(t, **kwargs))
|
||
|
|
else:
|
||
|
|
return ContinuousPSpace.compute_moment_generating_function(self, expr, **kwargs)
|
||
|
|
|
||
|
|
def compute_density(self, expr, **kwargs):
|
||
|
|
# https://en.wikipedia.org/wiki/Random_variable#Functions_of_random_variables
|
||
|
|
if expr == self.value:
|
||
|
|
return self.density
|
||
|
|
y = Dummy('y', real=True)
|
||
|
|
|
||
|
|
gs = solveset(expr - y, self.value, S.Reals)
|
||
|
|
|
||
|
|
if isinstance(gs, Intersection) and S.Reals in gs.args:
|
||
|
|
gs = list(gs.args[1])
|
||
|
|
|
||
|
|
if not gs:
|
||
|
|
raise ValueError("Can not solve %s for %s"%(expr, self.value))
|
||
|
|
fx = self.compute_density(self.value)
|
||
|
|
fy = sum(fx(g) * abs(g.diff(y)) for g in gs)
|
||
|
|
return Lambda(y, fy)
|
||
|
|
|
||
|
|
def compute_quantile(self, expr, **kwargs):
|
||
|
|
|
||
|
|
if expr == self.value:
|
||
|
|
p = Dummy("p", real=True)
|
||
|
|
return Lambda(p, self.distribution.quantile(p, **kwargs))
|
||
|
|
else:
|
||
|
|
return ContinuousPSpace.compute_quantile(self, expr, **kwargs)
|
||
|
|
|
||
|
|
def _reduce_inequalities(conditions, var, **kwargs):
|
||
|
|
try:
|
||
|
|
return reduce_rational_inequalities(conditions, var, **kwargs)
|
||
|
|
except PolynomialError:
|
||
|
|
raise ValueError("Reduction of condition failed %s\n" % conditions[0])
|
||
|
|
|
||
|
|
|
||
|
|
def reduce_rational_inequalities_wrap(condition, var):
|
||
|
|
if condition.is_Relational:
|
||
|
|
return _reduce_inequalities([[condition]], var, relational=False)
|
||
|
|
if isinstance(condition, Or):
|
||
|
|
return Union(*[_reduce_inequalities([[arg]], var, relational=False)
|
||
|
|
for arg in condition.args])
|
||
|
|
if isinstance(condition, And):
|
||
|
|
intervals = [_reduce_inequalities([[arg]], var, relational=False)
|
||
|
|
for arg in condition.args]
|
||
|
|
I = intervals[0]
|
||
|
|
for i in intervals:
|
||
|
|
I = I.intersect(i)
|
||
|
|
return I
|