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1793 lines
53 KiB
1793 lines
53 KiB
"""
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Main Random Variables Module
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Defines abstract random variable type.
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Contains interfaces for probability space object (PSpace) as well as standard
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operators, P, E, sample, density, where, quantile
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See Also
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========
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sympy.stats.crv
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sympy.stats.frv
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sympy.stats.rv_interface
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"""
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from __future__ import annotations
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from functools import singledispatch
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from math import prod
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from sympy.core.add import Add
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from sympy.core.basic import Basic
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from sympy.core.containers import Tuple
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from sympy.core.expr import Expr
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from sympy.core.function import (Function, Lambda)
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from sympy.core.logic import fuzzy_and
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from sympy.core.mul import Mul
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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, Symbol)
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from sympy.core.sympify import sympify
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from sympy.functions.special.delta_functions import DiracDelta
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from sympy.functions.special.tensor_functions import KroneckerDelta
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from sympy.logic.boolalg import (And, Or)
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from sympy.matrices.expressions.matexpr import MatrixSymbol
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from sympy.tensor.indexed import Indexed
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from sympy.utilities.lambdify import lambdify
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from sympy.core.relational import Relational
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from sympy.core.sympify import _sympify
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from sympy.sets.sets import FiniteSet, ProductSet, Intersection
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from sympy.solvers.solveset import solveset
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from sympy.external import import_module
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from sympy.utilities.decorator import doctest_depends_on
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from sympy.utilities.exceptions import sympy_deprecation_warning
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from sympy.utilities.iterables import iterable
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x = Symbol('x')
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@singledispatch
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def is_random(x):
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return False
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@is_random.register(Basic)
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def _(x):
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atoms = x.free_symbols
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return any(is_random(i) for i in atoms)
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class RandomDomain(Basic):
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"""
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Represents a set of variables and the values which they can take.
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See Also
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========
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sympy.stats.crv.ContinuousDomain
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sympy.stats.frv.FiniteDomain
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"""
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is_ProductDomain = False
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is_Finite = False
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is_Continuous = False
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is_Discrete = False
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def __new__(cls, symbols, *args):
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symbols = FiniteSet(*symbols)
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return Basic.__new__(cls, symbols, *args)
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@property
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def symbols(self):
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return self.args[0]
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@property
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def set(self):
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return self.args[1]
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def __contains__(self, other):
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raise NotImplementedError()
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def compute_expectation(self, expr):
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raise NotImplementedError()
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class SingleDomain(RandomDomain):
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"""
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A single variable and its domain.
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See Also
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========
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sympy.stats.crv.SingleContinuousDomain
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sympy.stats.frv.SingleFiniteDomain
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"""
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def __new__(cls, symbol, set):
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assert symbol.is_Symbol
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return Basic.__new__(cls, symbol, set)
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@property
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def symbol(self):
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return self.args[0]
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@property
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def symbols(self):
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return FiniteSet(self.symbol)
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def __contains__(self, other):
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if len(other) != 1:
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return False
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sym, val = tuple(other)[0]
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return self.symbol == sym and val in self.set
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class MatrixDomain(RandomDomain):
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"""
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A Random Matrix variable and its domain.
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"""
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def __new__(cls, symbol, set):
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symbol, set = _symbol_converter(symbol), _sympify(set)
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return Basic.__new__(cls, symbol, set)
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@property
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def symbol(self):
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return self.args[0]
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@property
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def symbols(self):
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return FiniteSet(self.symbol)
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class ConditionalDomain(RandomDomain):
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"""
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A RandomDomain with an attached condition.
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See Also
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========
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sympy.stats.crv.ConditionalContinuousDomain
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sympy.stats.frv.ConditionalFiniteDomain
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"""
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def __new__(cls, fulldomain, condition):
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condition = condition.xreplace({rs: rs.symbol
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for rs in random_symbols(condition)})
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return Basic.__new__(cls, fulldomain, condition)
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@property
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def symbols(self):
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return self.fulldomain.symbols
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@property
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def fulldomain(self):
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return self.args[0]
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@property
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def condition(self):
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return self.args[1]
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@property
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def set(self):
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raise NotImplementedError("Set of Conditional Domain not Implemented")
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def as_boolean(self):
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return And(self.fulldomain.as_boolean(), self.condition)
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class PSpace(Basic):
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"""
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A Probability Space.
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Explanation
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===========
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Probability Spaces encode processes that equal different values
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probabilistically. These underly Random Symbols which occur in SymPy
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expressions and contain the mechanics to evaluate statistical statements.
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See Also
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========
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sympy.stats.crv.ContinuousPSpace
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sympy.stats.frv.FinitePSpace
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"""
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is_Finite = None # type: bool
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is_Continuous = None # type: bool
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is_Discrete = None # type: bool
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is_real = None # type: bool
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@property
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def domain(self):
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return self.args[0]
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@property
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def density(self):
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return self.args[1]
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@property
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def values(self):
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return frozenset(RandomSymbol(sym, self) for sym in self.symbols)
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@property
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def symbols(self):
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return self.domain.symbols
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def where(self, condition):
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raise NotImplementedError()
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def compute_density(self, expr):
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raise NotImplementedError()
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def sample(self, size=(), library='scipy', seed=None):
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raise NotImplementedError()
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def probability(self, condition):
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raise NotImplementedError()
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def compute_expectation(self, expr):
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raise NotImplementedError()
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class SinglePSpace(PSpace):
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"""
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Represents the probabilities of a set of random events that can be
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attributed to a single variable/symbol.
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"""
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def __new__(cls, s, distribution):
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s = _symbol_converter(s)
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return Basic.__new__(cls, s, distribution)
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@property
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def value(self):
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return RandomSymbol(self.symbol, self)
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@property
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def symbol(self):
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return self.args[0]
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@property
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def distribution(self):
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return self.args[1]
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@property
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def pdf(self):
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return self.distribution.pdf(self.symbol)
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class RandomSymbol(Expr):
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"""
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Random Symbols represent ProbabilitySpaces in SymPy Expressions.
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In principle they can take on any value that their symbol can take on
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within the associated PSpace with probability determined by the PSpace
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Density.
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Explanation
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===========
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Random Symbols contain pspace and symbol properties.
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The pspace property points to the represented Probability Space
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The symbol is a standard SymPy Symbol that is used in that probability space
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for example in defining a density.
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You can form normal SymPy expressions using RandomSymbols and operate on
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those expressions with the Functions
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E - Expectation of a random expression
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P - Probability of a condition
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density - Probability Density of an expression
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given - A new random expression (with new random symbols) given a condition
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An object of the RandomSymbol type should almost never be created by the
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user. They tend to be created instead by the PSpace class's value method.
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Traditionally a user does not even do this but instead calls one of the
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convenience functions Normal, Exponential, Coin, Die, FiniteRV, etc....
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"""
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def __new__(cls, symbol, pspace=None):
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from sympy.stats.joint_rv import JointRandomSymbol
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if pspace is None:
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# Allow single arg, representing pspace == PSpace()
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pspace = PSpace()
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symbol = _symbol_converter(symbol)
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if not isinstance(pspace, PSpace):
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raise TypeError("pspace variable should be of type PSpace")
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if cls == JointRandomSymbol and isinstance(pspace, SinglePSpace):
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cls = RandomSymbol
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return Basic.__new__(cls, symbol, pspace)
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is_finite = True
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is_symbol = True
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is_Atom = True
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_diff_wrt = True
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pspace = property(lambda self: self.args[1])
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symbol = property(lambda self: self.args[0])
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name = property(lambda self: self.symbol.name)
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def _eval_is_positive(self):
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return self.symbol.is_positive
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def _eval_is_integer(self):
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return self.symbol.is_integer
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def _eval_is_real(self):
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return self.symbol.is_real or self.pspace.is_real
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@property
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def is_commutative(self):
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return self.symbol.is_commutative
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@property
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def free_symbols(self):
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return {self}
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class RandomIndexedSymbol(RandomSymbol):
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def __new__(cls, idx_obj, pspace=None):
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if pspace is None:
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# Allow single arg, representing pspace == PSpace()
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pspace = PSpace()
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if not isinstance(idx_obj, (Indexed, Function)):
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raise TypeError("An Function or Indexed object is expected not %s"%(idx_obj))
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return Basic.__new__(cls, idx_obj, pspace)
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symbol = property(lambda self: self.args[0])
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name = property(lambda self: str(self.args[0]))
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@property
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def key(self):
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if isinstance(self.symbol, Indexed):
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return self.symbol.args[1]
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elif isinstance(self.symbol, Function):
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return self.symbol.args[0]
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@property
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def free_symbols(self):
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if self.key.free_symbols:
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free_syms = self.key.free_symbols
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free_syms.add(self)
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return free_syms
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return {self}
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@property
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def pspace(self):
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return self.args[1]
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class RandomMatrixSymbol(RandomSymbol, MatrixSymbol): # type: ignore
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def __new__(cls, symbol, n, m, pspace=None):
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n, m = _sympify(n), _sympify(m)
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symbol = _symbol_converter(symbol)
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if pspace is None:
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# Allow single arg, representing pspace == PSpace()
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pspace = PSpace()
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return Basic.__new__(cls, symbol, n, m, pspace)
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symbol = property(lambda self: self.args[0])
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pspace = property(lambda self: self.args[3])
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class ProductPSpace(PSpace):
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"""
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Abstract class for representing probability spaces with multiple random
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variables.
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See Also
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========
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sympy.stats.rv.IndependentProductPSpace
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sympy.stats.joint_rv.JointPSpace
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"""
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pass
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class IndependentProductPSpace(ProductPSpace):
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"""
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A probability space resulting from the merger of two independent probability
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spaces.
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Often created using the function, pspace.
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"""
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def __new__(cls, *spaces):
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rs_space_dict = {}
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for space in spaces:
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for value in space.values:
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rs_space_dict[value] = space
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symbols = FiniteSet(*[val.symbol for val in rs_space_dict.keys()])
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# Overlapping symbols
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from sympy.stats.joint_rv import MarginalDistribution
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from sympy.stats.compound_rv import CompoundDistribution
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if len(symbols) < sum(len(space.symbols) for space in spaces if not
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isinstance(space.distribution, (
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CompoundDistribution, MarginalDistribution))):
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raise ValueError("Overlapping Random Variables")
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if all(space.is_Finite for space in spaces):
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from sympy.stats.frv import ProductFinitePSpace
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cls = ProductFinitePSpace
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obj = Basic.__new__(cls, *FiniteSet(*spaces))
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return obj
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@property
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def pdf(self):
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p = Mul(*[space.pdf for space in self.spaces])
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return p.subs({rv: rv.symbol for rv in self.values})
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@property
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def rs_space_dict(self):
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d = {}
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for space in self.spaces:
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for value in space.values:
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d[value] = space
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return d
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|
@property
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|
def symbols(self):
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return FiniteSet(*[val.symbol for val in self.rs_space_dict.keys()])
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|
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@property
|
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def spaces(self):
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return FiniteSet(*self.args)
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@property
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def values(self):
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return sumsets(space.values for space in self.spaces)
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|
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def compute_expectation(self, expr, rvs=None, evaluate=False, **kwargs):
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rvs = rvs or self.values
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rvs = frozenset(rvs)
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for space in self.spaces:
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expr = space.compute_expectation(expr, rvs & space.values, evaluate=False, **kwargs)
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if evaluate and hasattr(expr, 'doit'):
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return expr.doit(**kwargs)
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return expr
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|
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@property
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def domain(self):
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return ProductDomain(*[space.domain for space in self.spaces])
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|
@property
|
|
def density(self):
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raise NotImplementedError("Density not available for ProductSpaces")
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def sample(self, size=(), library='scipy', seed=None):
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return {k: v for space in self.spaces
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for k, v in space.sample(size=size, library=library, seed=seed).items()}
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def probability(self, condition, **kwargs):
|
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cond_inv = False
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if isinstance(condition, Ne):
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condition = Eq(condition.args[0], condition.args[1])
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cond_inv = True
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elif isinstance(condition, And): # they are independent
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return Mul(*[self.probability(arg) for arg in condition.args])
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elif isinstance(condition, Or): # they are independent
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return Add(*[self.probability(arg) for arg in condition.args])
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expr = condition.lhs - condition.rhs
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rvs = random_symbols(expr)
|
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dens = self.compute_density(expr)
|
|
if any(pspace(rv).is_Continuous for rv in rvs):
|
|
from sympy.stats.crv import SingleContinuousPSpace
|
|
from sympy.stats.crv_types import ContinuousDistributionHandmade
|
|
if expr in self.values:
|
|
# Marginalize all other random symbols out of the density
|
|
randomsymbols = tuple(set(self.values) - frozenset([expr]))
|
|
symbols = tuple(rs.symbol for rs in randomsymbols)
|
|
pdf = self.domain.integrate(self.pdf, symbols, **kwargs)
|
|
return Lambda(expr.symbol, pdf)
|
|
dens = ContinuousDistributionHandmade(dens)
|
|
z = Dummy('z', real=True)
|
|
space = SingleContinuousPSpace(z, dens)
|
|
result = space.probability(condition.__class__(space.value, 0))
|
|
else:
|
|
from sympy.stats.drv import SingleDiscretePSpace
|
|
from sympy.stats.drv_types import DiscreteDistributionHandmade
|
|
dens = DiscreteDistributionHandmade(dens)
|
|
z = Dummy('z', integer=True)
|
|
space = SingleDiscretePSpace(z, dens)
|
|
result = space.probability(condition.__class__(space.value, 0))
|
|
return result if not cond_inv else S.One - result
|
|
|
|
def compute_density(self, expr, **kwargs):
|
|
rvs = random_symbols(expr)
|
|
if any(pspace(rv).is_Continuous for rv in rvs):
|
|
z = Dummy('z', real=True)
|
|
expr = self.compute_expectation(DiracDelta(expr - z),
|
|
**kwargs)
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|
else:
|
|
z = Dummy('z', integer=True)
|
|
expr = self.compute_expectation(KroneckerDelta(expr, z),
|
|
**kwargs)
|
|
return Lambda(z, expr)
|
|
|
|
def compute_cdf(self, expr, **kwargs):
|
|
raise ValueError("CDF not well defined on multivariate expressions")
|
|
|
|
def conditional_space(self, condition, normalize=True, **kwargs):
|
|
rvs = random_symbols(condition)
|
|
condition = condition.xreplace({rv: rv.symbol for rv in self.values})
|
|
pspaces = [pspace(rv) for rv in rvs]
|
|
if any(ps.is_Continuous for ps in pspaces):
|
|
from sympy.stats.crv import (ConditionalContinuousDomain,
|
|
ContinuousPSpace)
|
|
space = ContinuousPSpace
|
|
domain = ConditionalContinuousDomain(self.domain, condition)
|
|
elif any(ps.is_Discrete for ps in pspaces):
|
|
from sympy.stats.drv import (ConditionalDiscreteDomain,
|
|
DiscretePSpace)
|
|
space = DiscretePSpace
|
|
domain = ConditionalDiscreteDomain(self.domain, condition)
|
|
elif all(ps.is_Finite for ps in pspaces):
|
|
from sympy.stats.frv import FinitePSpace
|
|
return FinitePSpace.conditional_space(self, condition)
|
|
if normalize:
|
|
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 symbols from 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 space(domain, density)
|
|
|
|
class ProductDomain(RandomDomain):
|
|
"""
|
|
A domain resulting from the merger of two independent domains.
|
|
|
|
See Also
|
|
========
|
|
sympy.stats.crv.ProductContinuousDomain
|
|
sympy.stats.frv.ProductFiniteDomain
|
|
"""
|
|
is_ProductDomain = True
|
|
|
|
def __new__(cls, *domains):
|
|
# Flatten any product of products
|
|
domains2 = []
|
|
for domain in domains:
|
|
if not domain.is_ProductDomain:
|
|
domains2.append(domain)
|
|
else:
|
|
domains2.extend(domain.domains)
|
|
domains2 = FiniteSet(*domains2)
|
|
|
|
if all(domain.is_Finite for domain in domains2):
|
|
from sympy.stats.frv import ProductFiniteDomain
|
|
cls = ProductFiniteDomain
|
|
if all(domain.is_Continuous for domain in domains2):
|
|
from sympy.stats.crv import ProductContinuousDomain
|
|
cls = ProductContinuousDomain
|
|
if all(domain.is_Discrete for domain in domains2):
|
|
from sympy.stats.drv import ProductDiscreteDomain
|
|
cls = ProductDiscreteDomain
|
|
|
|
return Basic.__new__(cls, *domains2)
|
|
|
|
@property
|
|
def sym_domain_dict(self):
|
|
return {symbol: domain for domain in self.domains
|
|
for symbol in domain.symbols}
|
|
|
|
@property
|
|
def symbols(self):
|
|
return FiniteSet(*[sym for domain in self.domains
|
|
for sym in domain.symbols])
|
|
|
|
@property
|
|
def domains(self):
|
|
return self.args
|
|
|
|
@property
|
|
def set(self):
|
|
return ProductSet(*(domain.set for domain in self.domains))
|
|
|
|
def __contains__(self, other):
|
|
# Split event into each subdomain
|
|
for domain in self.domains:
|
|
# Collect the parts of this event which associate to this domain
|
|
elem = frozenset([item for item in other
|
|
if sympify(domain.symbols.contains(item[0]))
|
|
is S.true])
|
|
# Test this sub-event
|
|
if elem not in domain:
|
|
return False
|
|
# All subevents passed
|
|
return True
|
|
|
|
def as_boolean(self):
|
|
return And(*[domain.as_boolean() for domain in self.domains])
|
|
|
|
|
|
def random_symbols(expr):
|
|
"""
|
|
Returns all RandomSymbols within a SymPy Expression.
|
|
"""
|
|
atoms = getattr(expr, 'atoms', None)
|
|
if atoms is not None:
|
|
comp = lambda rv: rv.symbol.name
|
|
l = list(atoms(RandomSymbol))
|
|
return sorted(l, key=comp)
|
|
else:
|
|
return []
|
|
|
|
|
|
def pspace(expr):
|
|
"""
|
|
Returns the underlying Probability Space of a random expression.
|
|
|
|
For internal use.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.stats import pspace, Normal
|
|
>>> X = Normal('X', 0, 1)
|
|
>>> pspace(2*X + 1) == X.pspace
|
|
True
|
|
"""
|
|
expr = sympify(expr)
|
|
if isinstance(expr, RandomSymbol) and expr.pspace is not None:
|
|
return expr.pspace
|
|
if expr.has(RandomMatrixSymbol):
|
|
rm = list(expr.atoms(RandomMatrixSymbol))[0]
|
|
return rm.pspace
|
|
|
|
rvs = random_symbols(expr)
|
|
if not rvs:
|
|
raise ValueError("Expression containing Random Variable expected, not %s" % (expr))
|
|
# If only one space present
|
|
if all(rv.pspace == rvs[0].pspace for rv in rvs):
|
|
return rvs[0].pspace
|
|
from sympy.stats.compound_rv import CompoundPSpace
|
|
from sympy.stats.stochastic_process import StochasticPSpace
|
|
for rv in rvs:
|
|
if isinstance(rv.pspace, (CompoundPSpace, StochasticPSpace)):
|
|
return rv.pspace
|
|
# Otherwise make a product space
|
|
return IndependentProductPSpace(*[rv.pspace for rv in rvs])
|
|
|
|
|
|
def sumsets(sets):
|
|
"""
|
|
Union of sets
|
|
"""
|
|
return frozenset().union(*sets)
|
|
|
|
|
|
def rs_swap(a, b):
|
|
"""
|
|
Build a dictionary to swap RandomSymbols based on their underlying symbol.
|
|
|
|
i.e.
|
|
if ``X = ('x', pspace1)``
|
|
and ``Y = ('x', pspace2)``
|
|
then ``X`` and ``Y`` match and the key, value pair
|
|
``{X:Y}`` will appear in the result
|
|
|
|
Inputs: collections a and b of random variables which share common symbols
|
|
Output: dict mapping RVs in a to RVs in b
|
|
"""
|
|
d = {}
|
|
for rsa in a:
|
|
d[rsa] = [rsb for rsb in b if rsa.symbol == rsb.symbol][0]
|
|
return d
|
|
|
|
|
|
def given(expr, condition=None, **kwargs):
|
|
r""" Conditional Random Expression.
|
|
|
|
Explanation
|
|
===========
|
|
|
|
From a random expression and a condition on that expression creates a new
|
|
probability space from the condition and returns the same expression on that
|
|
conditional probability space.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.stats import given, density, Die
|
|
>>> X = Die('X', 6)
|
|
>>> Y = given(X, X > 3)
|
|
>>> density(Y).dict
|
|
{4: 1/3, 5: 1/3, 6: 1/3}
|
|
|
|
Following convention, if the condition is a random symbol then that symbol
|
|
is considered fixed.
|
|
|
|
>>> from sympy.stats import Normal
|
|
>>> from sympy import pprint
|
|
>>> from sympy.abc import z
|
|
|
|
>>> X = Normal('X', 0, 1)
|
|
>>> Y = Normal('Y', 0, 1)
|
|
>>> pprint(density(X + Y, Y)(z), use_unicode=False)
|
|
2
|
|
-(-Y + z)
|
|
-----------
|
|
___ 2
|
|
\/ 2 *e
|
|
------------------
|
|
____
|
|
2*\/ pi
|
|
"""
|
|
|
|
if not is_random(condition) or pspace_independent(expr, condition):
|
|
return expr
|
|
|
|
if isinstance(condition, RandomSymbol):
|
|
condition = Eq(condition, condition.symbol)
|
|
|
|
condsymbols = random_symbols(condition)
|
|
if (isinstance(condition, Eq) and len(condsymbols) == 1 and
|
|
not isinstance(pspace(expr).domain, ConditionalDomain)):
|
|
rv = tuple(condsymbols)[0]
|
|
|
|
results = solveset(condition, rv)
|
|
if isinstance(results, Intersection) and S.Reals in results.args:
|
|
results = list(results.args[1])
|
|
|
|
sums = 0
|
|
for res in results:
|
|
temp = expr.subs(rv, res)
|
|
if temp == True:
|
|
return True
|
|
if temp != False:
|
|
# XXX: This seems nonsensical but preserves existing behaviour
|
|
# after the change that Relational is no longer a subclass of
|
|
# Expr. Here expr is sometimes Relational and sometimes Expr
|
|
# but we are trying to add them with +=. This needs to be
|
|
# fixed somehow.
|
|
if sums == 0 and isinstance(expr, Relational):
|
|
sums = expr.subs(rv, res)
|
|
else:
|
|
sums += expr.subs(rv, res)
|
|
if sums == 0:
|
|
return False
|
|
return sums
|
|
|
|
# Get full probability space of both the expression and the condition
|
|
fullspace = pspace(Tuple(expr, condition))
|
|
# Build new space given the condition
|
|
space = fullspace.conditional_space(condition, **kwargs)
|
|
# Dictionary to swap out RandomSymbols in expr with new RandomSymbols
|
|
# That point to the new conditional space
|
|
swapdict = rs_swap(fullspace.values, space.values)
|
|
# Swap random variables in the expression
|
|
expr = expr.xreplace(swapdict)
|
|
return expr
|
|
|
|
|
|
def expectation(expr, condition=None, numsamples=None, evaluate=True, **kwargs):
|
|
"""
|
|
Returns the expected value of a random expression.
|
|
|
|
Parameters
|
|
==========
|
|
|
|
expr : Expr containing RandomSymbols
|
|
The expression of which you want to compute the expectation value
|
|
given : Expr containing RandomSymbols
|
|
A conditional expression. E(X, X>0) is expectation of X given X > 0
|
|
numsamples : int
|
|
Enables sampling and approximates the expectation with this many samples
|
|
evalf : Bool (defaults to True)
|
|
If sampling return a number rather than a complex expression
|
|
evaluate : Bool (defaults to True)
|
|
In case of continuous systems return unevaluated integral
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.stats import E, Die
|
|
>>> X = Die('X', 6)
|
|
>>> E(X)
|
|
7/2
|
|
>>> E(2*X + 1)
|
|
8
|
|
|
|
>>> E(X, X > 3) # Expectation of X given that it is above 3
|
|
5
|
|
"""
|
|
|
|
if not is_random(expr): # expr isn't random?
|
|
return expr
|
|
kwargs['numsamples'] = numsamples
|
|
from sympy.stats.symbolic_probability import Expectation
|
|
if evaluate:
|
|
return Expectation(expr, condition).doit(**kwargs)
|
|
return Expectation(expr, condition)
|
|
|
|
|
|
def probability(condition, given_condition=None, numsamples=None,
|
|
evaluate=True, **kwargs):
|
|
"""
|
|
Probability that a condition is true, optionally given a second condition.
|
|
|
|
Parameters
|
|
==========
|
|
|
|
condition : Combination of Relationals containing RandomSymbols
|
|
The condition of which you want to compute the probability
|
|
given_condition : Combination of Relationals containing RandomSymbols
|
|
A conditional expression. P(X > 1, X > 0) is expectation of X > 1
|
|
given X > 0
|
|
numsamples : int
|
|
Enables sampling and approximates the probability with this many samples
|
|
evaluate : Bool (defaults to True)
|
|
In case of continuous systems return unevaluated integral
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.stats import P, Die
|
|
>>> from sympy import Eq
|
|
>>> X, Y = Die('X', 6), Die('Y', 6)
|
|
>>> P(X > 3)
|
|
1/2
|
|
>>> P(Eq(X, 5), X > 2) # Probability that X == 5 given that X > 2
|
|
1/4
|
|
>>> P(X > Y)
|
|
5/12
|
|
"""
|
|
|
|
kwargs['numsamples'] = numsamples
|
|
from sympy.stats.symbolic_probability import Probability
|
|
if evaluate:
|
|
return Probability(condition, given_condition).doit(**kwargs)
|
|
return Probability(condition, given_condition)
|
|
|
|
|
|
class Density(Basic):
|
|
expr = property(lambda self: self.args[0])
|
|
|
|
def __new__(cls, expr, condition = None):
|
|
expr = _sympify(expr)
|
|
if condition is None:
|
|
obj = Basic.__new__(cls, expr)
|
|
else:
|
|
condition = _sympify(condition)
|
|
obj = Basic.__new__(cls, expr, condition)
|
|
return obj
|
|
|
|
@property
|
|
def condition(self):
|
|
if len(self.args) > 1:
|
|
return self.args[1]
|
|
else:
|
|
return None
|
|
|
|
def doit(self, evaluate=True, **kwargs):
|
|
from sympy.stats.random_matrix import RandomMatrixPSpace
|
|
from sympy.stats.joint_rv import JointPSpace
|
|
from sympy.stats.matrix_distributions import MatrixPSpace
|
|
from sympy.stats.compound_rv import CompoundPSpace
|
|
from sympy.stats.frv import SingleFiniteDistribution
|
|
expr, condition = self.expr, self.condition
|
|
|
|
if isinstance(expr, SingleFiniteDistribution):
|
|
return expr.dict
|
|
if condition is not None:
|
|
# Recompute on new conditional expr
|
|
expr = given(expr, condition, **kwargs)
|
|
if not random_symbols(expr):
|
|
return Lambda(x, DiracDelta(x - expr))
|
|
if isinstance(expr, RandomSymbol):
|
|
if isinstance(expr.pspace, (SinglePSpace, JointPSpace, MatrixPSpace)) and \
|
|
hasattr(expr.pspace, 'distribution'):
|
|
return expr.pspace.distribution
|
|
elif isinstance(expr.pspace, RandomMatrixPSpace):
|
|
return expr.pspace.model
|
|
if isinstance(pspace(expr), CompoundPSpace):
|
|
kwargs['compound_evaluate'] = evaluate
|
|
result = pspace(expr).compute_density(expr, **kwargs)
|
|
|
|
if evaluate and hasattr(result, 'doit'):
|
|
return result.doit()
|
|
else:
|
|
return result
|
|
|
|
|
|
def density(expr, condition=None, evaluate=True, numsamples=None, **kwargs):
|
|
"""
|
|
Probability density of a random expression, optionally given a second
|
|
condition.
|
|
|
|
Explanation
|
|
===========
|
|
|
|
This density will take on different forms for different types of
|
|
probability spaces. Discrete variables produce Dicts. Continuous
|
|
variables produce Lambdas.
|
|
|
|
Parameters
|
|
==========
|
|
|
|
expr : Expr containing RandomSymbols
|
|
The expression of which you want to compute the density value
|
|
condition : Relational containing RandomSymbols
|
|
A conditional expression. density(X > 1, X > 0) is density of X > 1
|
|
given X > 0
|
|
numsamples : int
|
|
Enables sampling and approximates the density with this many samples
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.stats import density, Die, Normal
|
|
>>> from sympy import Symbol
|
|
|
|
>>> x = Symbol('x')
|
|
>>> D = Die('D', 6)
|
|
>>> X = Normal(x, 0, 1)
|
|
|
|
>>> density(D).dict
|
|
{1: 1/6, 2: 1/6, 3: 1/6, 4: 1/6, 5: 1/6, 6: 1/6}
|
|
>>> density(2*D).dict
|
|
{2: 1/6, 4: 1/6, 6: 1/6, 8: 1/6, 10: 1/6, 12: 1/6}
|
|
>>> density(X)(x)
|
|
sqrt(2)*exp(-x**2/2)/(2*sqrt(pi))
|
|
"""
|
|
|
|
if numsamples:
|
|
return sampling_density(expr, condition, numsamples=numsamples,
|
|
**kwargs)
|
|
|
|
return Density(expr, condition).doit(evaluate=evaluate, **kwargs)
|
|
|
|
|
|
def cdf(expr, condition=None, evaluate=True, **kwargs):
|
|
"""
|
|
Cumulative Distribution Function of a random expression.
|
|
|
|
optionally given a second condition.
|
|
|
|
Explanation
|
|
===========
|
|
|
|
This density will take on different forms for different types of
|
|
probability spaces.
|
|
Discrete variables produce Dicts.
|
|
Continuous variables produce Lambdas.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.stats import density, Die, Normal, cdf
|
|
|
|
>>> D = Die('D', 6)
|
|
>>> X = Normal('X', 0, 1)
|
|
|
|
>>> density(D).dict
|
|
{1: 1/6, 2: 1/6, 3: 1/6, 4: 1/6, 5: 1/6, 6: 1/6}
|
|
>>> cdf(D)
|
|
{1: 1/6, 2: 1/3, 3: 1/2, 4: 2/3, 5: 5/6, 6: 1}
|
|
>>> cdf(3*D, D > 2)
|
|
{9: 1/4, 12: 1/2, 15: 3/4, 18: 1}
|
|
|
|
>>> cdf(X)
|
|
Lambda(_z, erf(sqrt(2)*_z/2)/2 + 1/2)
|
|
"""
|
|
if condition is not None: # If there is a condition
|
|
# Recompute on new conditional expr
|
|
return cdf(given(expr, condition, **kwargs), **kwargs)
|
|
|
|
# Otherwise pass work off to the ProbabilitySpace
|
|
result = pspace(expr).compute_cdf(expr, **kwargs)
|
|
|
|
if evaluate and hasattr(result, 'doit'):
|
|
return result.doit()
|
|
else:
|
|
return result
|
|
|
|
|
|
def characteristic_function(expr, condition=None, evaluate=True, **kwargs):
|
|
"""
|
|
Characteristic function of a random expression, optionally given a second condition.
|
|
|
|
Returns a Lambda.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.stats import Normal, DiscreteUniform, Poisson, characteristic_function
|
|
|
|
>>> X = Normal('X', 0, 1)
|
|
>>> characteristic_function(X)
|
|
Lambda(_t, exp(-_t**2/2))
|
|
|
|
>>> Y = DiscreteUniform('Y', [1, 2, 7])
|
|
>>> characteristic_function(Y)
|
|
Lambda(_t, exp(7*_t*I)/3 + exp(2*_t*I)/3 + exp(_t*I)/3)
|
|
|
|
>>> Z = Poisson('Z', 2)
|
|
>>> characteristic_function(Z)
|
|
Lambda(_t, exp(2*exp(_t*I) - 2))
|
|
"""
|
|
if condition is not None:
|
|
return characteristic_function(given(expr, condition, **kwargs), **kwargs)
|
|
|
|
result = pspace(expr).compute_characteristic_function(expr, **kwargs)
|
|
|
|
if evaluate and hasattr(result, 'doit'):
|
|
return result.doit()
|
|
else:
|
|
return result
|
|
|
|
def moment_generating_function(expr, condition=None, evaluate=True, **kwargs):
|
|
if condition is not None:
|
|
return moment_generating_function(given(expr, condition, **kwargs), **kwargs)
|
|
|
|
result = pspace(expr).compute_moment_generating_function(expr, **kwargs)
|
|
|
|
if evaluate and hasattr(result, 'doit'):
|
|
return result.doit()
|
|
else:
|
|
return result
|
|
|
|
def where(condition, given_condition=None, **kwargs):
|
|
"""
|
|
Returns the domain where a condition is True.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.stats import where, Die, Normal
|
|
>>> from sympy import And
|
|
|
|
>>> D1, D2 = Die('a', 6), Die('b', 6)
|
|
>>> a, b = D1.symbol, D2.symbol
|
|
>>> X = Normal('x', 0, 1)
|
|
|
|
>>> where(X**2<1)
|
|
Domain: (-1 < x) & (x < 1)
|
|
|
|
>>> where(X**2<1).set
|
|
Interval.open(-1, 1)
|
|
|
|
>>> where(And(D1<=D2, D2<3))
|
|
Domain: (Eq(a, 1) & Eq(b, 1)) | (Eq(a, 1) & Eq(b, 2)) | (Eq(a, 2) & Eq(b, 2))
|
|
"""
|
|
if given_condition is not None: # If there is a condition
|
|
# Recompute on new conditional expr
|
|
return where(given(condition, given_condition, **kwargs), **kwargs)
|
|
|
|
# Otherwise pass work off to the ProbabilitySpace
|
|
return pspace(condition).where(condition, **kwargs)
|
|
|
|
|
|
@doctest_depends_on(modules=('scipy',))
|
|
def sample(expr, condition=None, size=(), library='scipy',
|
|
numsamples=1, seed=None, **kwargs):
|
|
"""
|
|
A realization of the random expression.
|
|
|
|
Parameters
|
|
==========
|
|
|
|
expr : Expression of random variables
|
|
Expression from which sample is extracted
|
|
condition : Expr containing RandomSymbols
|
|
A conditional expression
|
|
size : int, tuple
|
|
Represents size of each sample in numsamples
|
|
library : str
|
|
- 'scipy' : Sample using scipy
|
|
- 'numpy' : Sample using numpy
|
|
- 'pymc' : Sample using PyMC
|
|
|
|
Choose any of the available options to sample from as string,
|
|
by default is 'scipy'
|
|
numsamples : int
|
|
Number of samples, each with size as ``size``.
|
|
|
|
.. deprecated:: 1.9
|
|
|
|
The ``numsamples`` parameter is deprecated and is only provided for
|
|
compatibility with v1.8. Use a list comprehension or an additional
|
|
dimension in ``size`` instead. See
|
|
:ref:`deprecated-sympy-stats-numsamples` for details.
|
|
|
|
seed :
|
|
An object to be used as seed by the given external library for sampling `expr`.
|
|
Following is the list of possible types of object for the supported libraries,
|
|
|
|
- 'scipy': int, numpy.random.RandomState, numpy.random.Generator
|
|
- 'numpy': int, numpy.random.RandomState, numpy.random.Generator
|
|
- 'pymc': int
|
|
|
|
Optional, by default None, in which case seed settings
|
|
related to the given library will be used.
|
|
No modifications to environment's global seed settings
|
|
are done by this argument.
|
|
|
|
Returns
|
|
=======
|
|
|
|
sample: float/list/numpy.ndarray
|
|
one sample or a collection of samples of the random expression.
|
|
|
|
- sample(X) returns float/numpy.float64/numpy.int64 object.
|
|
- sample(X, size=int/tuple) returns numpy.ndarray object.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.stats import Die, sample, Normal, Geometric
|
|
>>> X, Y, Z = Die('X', 6), Die('Y', 6), Die('Z', 6) # Finite Random Variable
|
|
>>> die_roll = sample(X + Y + Z)
|
|
>>> die_roll # doctest: +SKIP
|
|
3
|
|
>>> N = Normal('N', 3, 4) # Continuous Random Variable
|
|
>>> samp = sample(N)
|
|
>>> samp in N.pspace.domain.set
|
|
True
|
|
>>> samp = sample(N, N>0)
|
|
>>> samp > 0
|
|
True
|
|
>>> samp_list = sample(N, size=4)
|
|
>>> [sam in N.pspace.domain.set for sam in samp_list]
|
|
[True, True, True, True]
|
|
>>> sample(N, size = (2,3)) # doctest: +SKIP
|
|
array([[5.42519758, 6.40207856, 4.94991743],
|
|
[1.85819627, 6.83403519, 1.9412172 ]])
|
|
>>> G = Geometric('G', 0.5) # Discrete Random Variable
|
|
>>> samp_list = sample(G, size=3)
|
|
>>> samp_list # doctest: +SKIP
|
|
[1, 3, 2]
|
|
>>> [sam in G.pspace.domain.set for sam in samp_list]
|
|
[True, True, True]
|
|
>>> MN = Normal("MN", [3, 4], [[2, 1], [1, 2]]) # Joint Random Variable
|
|
>>> samp_list = sample(MN, size=4)
|
|
>>> samp_list # doctest: +SKIP
|
|
[array([2.85768055, 3.38954165]),
|
|
array([4.11163337, 4.3176591 ]),
|
|
array([0.79115232, 1.63232916]),
|
|
array([4.01747268, 3.96716083])]
|
|
>>> [tuple(sam) in MN.pspace.domain.set for sam in samp_list]
|
|
[True, True, True, True]
|
|
|
|
.. versionchanged:: 1.7.0
|
|
sample used to return an iterator containing the samples instead of value.
|
|
|
|
.. versionchanged:: 1.9.0
|
|
sample returns values or array of values instead of an iterator and numsamples is deprecated.
|
|
|
|
"""
|
|
|
|
iterator = sample_iter(expr, condition, size=size, library=library,
|
|
numsamples=numsamples, seed=seed)
|
|
|
|
if numsamples != 1:
|
|
sympy_deprecation_warning(
|
|
f"""
|
|
The numsamples parameter to sympy.stats.sample() is deprecated.
|
|
Either use a list comprehension, like
|
|
|
|
[sample(...) for i in range({numsamples})]
|
|
|
|
or add a dimension to size, like
|
|
|
|
sample(..., size={(numsamples,) + size})
|
|
""",
|
|
deprecated_since_version="1.9",
|
|
active_deprecations_target="deprecated-sympy-stats-numsamples",
|
|
)
|
|
return [next(iterator) for i in range(numsamples)]
|
|
|
|
return next(iterator)
|
|
|
|
|
|
def quantile(expr, evaluate=True, **kwargs):
|
|
r"""
|
|
Return the :math:`p^{th}` order quantile of a probability distribution.
|
|
|
|
Explanation
|
|
===========
|
|
|
|
Quantile is defined as the value at which the probability of the random
|
|
variable is less than or equal to the given probability.
|
|
|
|
.. math::
|
|
Q(p) = \inf\{x \in (-\infty, \infty) : p \le F(x)\}
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.stats import quantile, Die, Exponential
|
|
>>> from sympy import Symbol, pprint
|
|
>>> p = Symbol("p")
|
|
|
|
>>> l = Symbol("lambda", positive=True)
|
|
>>> X = Exponential("x", l)
|
|
>>> quantile(X)(p)
|
|
-log(1 - p)/lambda
|
|
|
|
>>> D = Die("d", 6)
|
|
>>> pprint(quantile(D)(p), use_unicode=False)
|
|
/nan for Or(p > 1, p < 0)
|
|
|
|
|
| 1 for p <= 1/6
|
|
|
|
|
| 2 for p <= 1/3
|
|
|
|
|
< 3 for p <= 1/2
|
|
|
|
|
| 4 for p <= 2/3
|
|
|
|
|
| 5 for p <= 5/6
|
|
|
|
|
\ 6 for p <= 1
|
|
|
|
"""
|
|
result = pspace(expr).compute_quantile(expr, **kwargs)
|
|
|
|
if evaluate and hasattr(result, 'doit'):
|
|
return result.doit()
|
|
else:
|
|
return result
|
|
|
|
def sample_iter(expr, condition=None, size=(), library='scipy',
|
|
numsamples=S.Infinity, seed=None, **kwargs):
|
|
|
|
"""
|
|
Returns an iterator of realizations from the expression given a condition.
|
|
|
|
Parameters
|
|
==========
|
|
|
|
expr: Expr
|
|
Random expression to be realized
|
|
condition: Expr, optional
|
|
A conditional expression
|
|
size : int, tuple
|
|
Represents size of each sample in numsamples
|
|
numsamples: integer, optional
|
|
Length of the iterator (defaults to infinity)
|
|
seed :
|
|
An object to be used as seed by the given external library for sampling `expr`.
|
|
Following is the list of possible types of object for the supported libraries,
|
|
|
|
- 'scipy': int, numpy.random.RandomState, numpy.random.Generator
|
|
- 'numpy': int, numpy.random.RandomState, numpy.random.Generator
|
|
- 'pymc': int
|
|
|
|
Optional, by default None, in which case seed settings
|
|
related to the given library will be used.
|
|
No modifications to environment's global seed settings
|
|
are done by this argument.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.stats import Normal, sample_iter
|
|
>>> X = Normal('X', 0, 1)
|
|
>>> expr = X*X + 3
|
|
>>> iterator = sample_iter(expr, numsamples=3) # doctest: +SKIP
|
|
>>> list(iterator) # doctest: +SKIP
|
|
[12, 4, 7]
|
|
|
|
Returns
|
|
=======
|
|
|
|
sample_iter: iterator object
|
|
iterator object containing the sample/samples of given expr
|
|
|
|
See Also
|
|
========
|
|
|
|
sample
|
|
sampling_P
|
|
sampling_E
|
|
|
|
"""
|
|
from sympy.stats.joint_rv import JointRandomSymbol
|
|
if not import_module(library):
|
|
raise ValueError("Failed to import %s" % library)
|
|
|
|
if condition is not None:
|
|
ps = pspace(Tuple(expr, condition))
|
|
else:
|
|
ps = pspace(expr)
|
|
|
|
rvs = list(ps.values)
|
|
if isinstance(expr, JointRandomSymbol):
|
|
expr = expr.subs({expr: RandomSymbol(expr.symbol, expr.pspace)})
|
|
else:
|
|
sub = {}
|
|
for arg in expr.args:
|
|
if isinstance(arg, JointRandomSymbol):
|
|
sub[arg] = RandomSymbol(arg.symbol, arg.pspace)
|
|
expr = expr.subs(sub)
|
|
|
|
def fn_subs(*args):
|
|
return expr.subs({rv: arg for rv, arg in zip(rvs, args)})
|
|
|
|
def given_fn_subs(*args):
|
|
if condition is not None:
|
|
return condition.subs({rv: arg for rv, arg in zip(rvs, args)})
|
|
return False
|
|
|
|
if library in ('pymc', 'pymc3'):
|
|
# Currently unable to lambdify in pymc
|
|
# TODO : Remove when lambdify accepts 'pymc' as module
|
|
fn = lambdify(rvs, expr, **kwargs)
|
|
else:
|
|
fn = lambdify(rvs, expr, modules=library, **kwargs)
|
|
|
|
|
|
if condition is not None:
|
|
given_fn = lambdify(rvs, condition, **kwargs)
|
|
|
|
def return_generator_infinite():
|
|
count = 0
|
|
_size = (1,)+((size,) if isinstance(size, int) else size)
|
|
while count < numsamples:
|
|
d = ps.sample(size=_size, library=library, seed=seed) # a dictionary that maps RVs to values
|
|
args = [d[rv][0] for rv in rvs]
|
|
|
|
if condition is not None: # Check that these values satisfy the condition
|
|
# TODO: Replace the try-except block with only given_fn(*args)
|
|
# once lambdify works with unevaluated SymPy objects.
|
|
try:
|
|
gd = given_fn(*args)
|
|
except (NameError, TypeError):
|
|
gd = given_fn_subs(*args)
|
|
if gd != True and gd != False:
|
|
raise ValueError(
|
|
"Conditions must not contain free symbols")
|
|
if not gd: # If the values don't satisfy then try again
|
|
continue
|
|
|
|
yield fn(*args)
|
|
count += 1
|
|
|
|
def return_generator_finite():
|
|
faulty = True
|
|
while faulty:
|
|
d = ps.sample(size=(numsamples,) + ((size,) if isinstance(size, int) else size),
|
|
library=library, seed=seed) # a dictionary that maps RVs to values
|
|
|
|
faulty = False
|
|
count = 0
|
|
while count < numsamples and not faulty:
|
|
args = [d[rv][count] for rv in rvs]
|
|
if condition is not None: # Check that these values satisfy the condition
|
|
# TODO: Replace the try-except block with only given_fn(*args)
|
|
# once lambdify works with unevaluated SymPy objects.
|
|
try:
|
|
gd = given_fn(*args)
|
|
except (NameError, TypeError):
|
|
gd = given_fn_subs(*args)
|
|
if gd != True and gd != False:
|
|
raise ValueError(
|
|
"Conditions must not contain free symbols")
|
|
if not gd: # If the values don't satisfy then try again
|
|
faulty = True
|
|
|
|
count += 1
|
|
|
|
count = 0
|
|
while count < numsamples:
|
|
args = [d[rv][count] for rv in rvs]
|
|
# TODO: Replace the try-except block with only fn(*args)
|
|
# once lambdify works with unevaluated SymPy objects.
|
|
try:
|
|
yield fn(*args)
|
|
except (NameError, TypeError):
|
|
yield fn_subs(*args)
|
|
count += 1
|
|
|
|
if numsamples is S.Infinity:
|
|
return return_generator_infinite()
|
|
|
|
return return_generator_finite()
|
|
|
|
def sample_iter_lambdify(expr, condition=None, size=(),
|
|
numsamples=S.Infinity, seed=None, **kwargs):
|
|
|
|
return sample_iter(expr, condition=condition, size=size,
|
|
numsamples=numsamples, seed=seed, **kwargs)
|
|
|
|
def sample_iter_subs(expr, condition=None, size=(),
|
|
numsamples=S.Infinity, seed=None, **kwargs):
|
|
|
|
return sample_iter(expr, condition=condition, size=size,
|
|
numsamples=numsamples, seed=seed, **kwargs)
|
|
|
|
|
|
def sampling_P(condition, given_condition=None, library='scipy', numsamples=1,
|
|
evalf=True, seed=None, **kwargs):
|
|
"""
|
|
Sampling version of P.
|
|
|
|
See Also
|
|
========
|
|
|
|
P
|
|
sampling_E
|
|
sampling_density
|
|
|
|
"""
|
|
|
|
count_true = 0
|
|
count_false = 0
|
|
samples = sample_iter(condition, given_condition, library=library,
|
|
numsamples=numsamples, seed=seed, **kwargs)
|
|
|
|
for sample in samples:
|
|
if sample:
|
|
count_true += 1
|
|
else:
|
|
count_false += 1
|
|
|
|
result = S(count_true) / numsamples
|
|
if evalf:
|
|
return result.evalf()
|
|
else:
|
|
return result
|
|
|
|
|
|
def sampling_E(expr, given_condition=None, library='scipy', numsamples=1,
|
|
evalf=True, seed=None, **kwargs):
|
|
"""
|
|
Sampling version of E.
|
|
|
|
See Also
|
|
========
|
|
|
|
P
|
|
sampling_P
|
|
sampling_density
|
|
"""
|
|
samples = list(sample_iter(expr, given_condition, library=library,
|
|
numsamples=numsamples, seed=seed, **kwargs))
|
|
result = Add(*samples) / numsamples
|
|
|
|
if evalf:
|
|
return result.evalf()
|
|
else:
|
|
return result
|
|
|
|
def sampling_density(expr, given_condition=None, library='scipy',
|
|
numsamples=1, seed=None, **kwargs):
|
|
"""
|
|
Sampling version of density.
|
|
|
|
See Also
|
|
========
|
|
density
|
|
sampling_P
|
|
sampling_E
|
|
"""
|
|
|
|
results = {}
|
|
for result in sample_iter(expr, given_condition, library=library,
|
|
numsamples=numsamples, seed=seed, **kwargs):
|
|
results[result] = results.get(result, 0) + 1
|
|
|
|
return results
|
|
|
|
|
|
def dependent(a, b):
|
|
"""
|
|
Dependence of two random expressions.
|
|
|
|
Two expressions are independent if knowledge of one does not change
|
|
computations on the other.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.stats import Normal, dependent, given
|
|
>>> from sympy import Tuple, Eq
|
|
|
|
>>> X, Y = Normal('X', 0, 1), Normal('Y', 0, 1)
|
|
>>> dependent(X, Y)
|
|
False
|
|
>>> dependent(2*X + Y, -Y)
|
|
True
|
|
>>> X, Y = given(Tuple(X, Y), Eq(X + Y, 3))
|
|
>>> dependent(X, Y)
|
|
True
|
|
|
|
See Also
|
|
========
|
|
|
|
independent
|
|
"""
|
|
if pspace_independent(a, b):
|
|
return False
|
|
|
|
z = Symbol('z', real=True)
|
|
# Dependent if density is unchanged when one is given information about
|
|
# the other
|
|
return (density(a, Eq(b, z)) != density(a) or
|
|
density(b, Eq(a, z)) != density(b))
|
|
|
|
|
|
def independent(a, b):
|
|
"""
|
|
Independence of two random expressions.
|
|
|
|
Two expressions are independent if knowledge of one does not change
|
|
computations on the other.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.stats import Normal, independent, given
|
|
>>> from sympy import Tuple, Eq
|
|
|
|
>>> X, Y = Normal('X', 0, 1), Normal('Y', 0, 1)
|
|
>>> independent(X, Y)
|
|
True
|
|
>>> independent(2*X + Y, -Y)
|
|
False
|
|
>>> X, Y = given(Tuple(X, Y), Eq(X + Y, 3))
|
|
>>> independent(X, Y)
|
|
False
|
|
|
|
See Also
|
|
========
|
|
|
|
dependent
|
|
"""
|
|
return not dependent(a, b)
|
|
|
|
|
|
def pspace_independent(a, b):
|
|
"""
|
|
Tests for independence between a and b by checking if their PSpaces have
|
|
overlapping symbols. This is a sufficient but not necessary condition for
|
|
independence and is intended to be used internally.
|
|
|
|
Notes
|
|
=====
|
|
|
|
pspace_independent(a, b) implies independent(a, b)
|
|
independent(a, b) does not imply pspace_independent(a, b)
|
|
"""
|
|
a_symbols = set(pspace(b).symbols)
|
|
b_symbols = set(pspace(a).symbols)
|
|
|
|
if len(set(random_symbols(a)).intersection(random_symbols(b))) != 0:
|
|
return False
|
|
|
|
if len(a_symbols.intersection(b_symbols)) == 0:
|
|
return True
|
|
return None
|
|
|
|
|
|
def rv_subs(expr, symbols=None):
|
|
"""
|
|
Given a random expression replace all random variables with their symbols.
|
|
|
|
If symbols keyword is given restrict the swap to only the symbols listed.
|
|
"""
|
|
if symbols is None:
|
|
symbols = random_symbols(expr)
|
|
if not symbols:
|
|
return expr
|
|
swapdict = {rv: rv.symbol for rv in symbols}
|
|
return expr.subs(swapdict)
|
|
|
|
|
|
class NamedArgsMixin:
|
|
_argnames: tuple[str, ...] = ()
|
|
|
|
def __getattr__(self, attr):
|
|
try:
|
|
return self.args[self._argnames.index(attr)]
|
|
except ValueError:
|
|
raise AttributeError("'%s' object has no attribute '%s'" % (
|
|
type(self).__name__, attr))
|
|
|
|
|
|
class Distribution(Basic):
|
|
|
|
def sample(self, size=(), library='scipy', seed=None):
|
|
""" A random realization from the distribution """
|
|
|
|
module = import_module(library)
|
|
if library in {'scipy', 'numpy', 'pymc3', 'pymc'} and module is None:
|
|
raise ValueError("Failed to import %s" % library)
|
|
|
|
if library == 'scipy':
|
|
# scipy does not require map as it can handle using custom distributions.
|
|
# However, we will still use a map where we can.
|
|
|
|
# TODO: do this for drv.py and frv.py if necessary.
|
|
# TODO: add more distributions here if there are more
|
|
# See links below referring to sections beginning with "A common parametrization..."
|
|
# I will remove all these comments if everything is ok.
|
|
|
|
from sympy.stats.sampling.sample_scipy import do_sample_scipy
|
|
import numpy
|
|
if seed is None or isinstance(seed, int):
|
|
rand_state = numpy.random.default_rng(seed=seed)
|
|
else:
|
|
rand_state = seed
|
|
samps = do_sample_scipy(self, size, rand_state)
|
|
|
|
elif library == 'numpy':
|
|
from sympy.stats.sampling.sample_numpy import do_sample_numpy
|
|
import numpy
|
|
if seed is None or isinstance(seed, int):
|
|
rand_state = numpy.random.default_rng(seed=seed)
|
|
else:
|
|
rand_state = seed
|
|
_size = None if size == () else size
|
|
samps = do_sample_numpy(self, _size, rand_state)
|
|
elif library in ('pymc', 'pymc3'):
|
|
from sympy.stats.sampling.sample_pymc import do_sample_pymc
|
|
import logging
|
|
logging.getLogger("pymc").setLevel(logging.ERROR)
|
|
try:
|
|
import pymc
|
|
except ImportError:
|
|
import pymc3 as pymc
|
|
|
|
with pymc.Model():
|
|
if do_sample_pymc(self):
|
|
samps = pymc.sample(draws=prod(size), chains=1, compute_convergence_checks=False,
|
|
progressbar=False, random_seed=seed, return_inferencedata=False)[:]['X']
|
|
samps = samps.reshape(size)
|
|
else:
|
|
samps = None
|
|
|
|
else:
|
|
raise NotImplementedError("Sampling from %s is not supported yet."
|
|
% str(library))
|
|
|
|
if samps is not None:
|
|
return samps
|
|
raise NotImplementedError(
|
|
"Sampling for %s is not currently implemented from %s"
|
|
% (self, library))
|
|
|
|
|
|
def _value_check(condition, message):
|
|
"""
|
|
Raise a ValueError with message if condition is False, else
|
|
return True if all conditions were True, else False.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.stats.rv import _value_check
|
|
>>> from sympy.abc import a, b, c
|
|
>>> from sympy import And, Dummy
|
|
|
|
>>> _value_check(2 < 3, '')
|
|
True
|
|
|
|
Here, the condition is not False, but it does not evaluate to True
|
|
so False is returned (but no error is raised). So checking if the
|
|
return value is True or False will tell you if all conditions were
|
|
evaluated.
|
|
|
|
>>> _value_check(a < b, '')
|
|
False
|
|
|
|
In this case the condition is False so an error is raised:
|
|
|
|
>>> r = Dummy(real=True)
|
|
>>> _value_check(r < r - 1, 'condition is not true')
|
|
Traceback (most recent call last):
|
|
...
|
|
ValueError: condition is not true
|
|
|
|
If no condition of many conditions must be False, they can be
|
|
checked by passing them as an iterable:
|
|
|
|
>>> _value_check((a < 0, b < 0, c < 0), '')
|
|
False
|
|
|
|
The iterable can be a generator, too:
|
|
|
|
>>> _value_check((i < 0 for i in (a, b, c)), '')
|
|
False
|
|
|
|
The following are equivalent to the above but do not pass
|
|
an iterable:
|
|
|
|
>>> all(_value_check(i < 0, '') for i in (a, b, c))
|
|
False
|
|
>>> _value_check(And(a < 0, b < 0, c < 0), '')
|
|
False
|
|
"""
|
|
if not iterable(condition):
|
|
condition = [condition]
|
|
truth = fuzzy_and(condition)
|
|
if truth == False:
|
|
raise ValueError(message)
|
|
return truth == True
|
|
|
|
def _symbol_converter(sym):
|
|
"""
|
|
Casts the parameter to Symbol if it is 'str'
|
|
otherwise no operation is performed on it.
|
|
|
|
Parameters
|
|
==========
|
|
|
|
sym
|
|
The parameter to be converted.
|
|
|
|
Returns
|
|
=======
|
|
|
|
Symbol
|
|
the parameter converted to Symbol.
|
|
|
|
Raises
|
|
======
|
|
|
|
TypeError
|
|
If the parameter is not an instance of both str and
|
|
Symbol.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy import Symbol
|
|
>>> from sympy.stats.rv import _symbol_converter
|
|
>>> s = _symbol_converter('s')
|
|
>>> isinstance(s, Symbol)
|
|
True
|
|
>>> _symbol_converter(1)
|
|
Traceback (most recent call last):
|
|
...
|
|
TypeError: 1 is neither a Symbol nor a string
|
|
>>> r = Symbol('r')
|
|
>>> isinstance(r, Symbol)
|
|
True
|
|
"""
|
|
if isinstance(sym, str):
|
|
sym = Symbol(sym)
|
|
if not isinstance(sym, Symbol):
|
|
raise TypeError("%s is neither a Symbol nor a string"%(sym))
|
|
return sym
|
|
|
|
def sample_stochastic_process(process):
|
|
"""
|
|
This function is used to sample from stochastic process.
|
|
|
|
Parameters
|
|
==========
|
|
|
|
process: StochasticProcess
|
|
Process used to extract the samples. It must be an instance of
|
|
StochasticProcess
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.stats import sample_stochastic_process, DiscreteMarkovChain
|
|
>>> from sympy import Matrix
|
|
>>> T = Matrix([[0.5, 0.2, 0.3],[0.2, 0.5, 0.3],[0.2, 0.3, 0.5]])
|
|
>>> Y = DiscreteMarkovChain("Y", [0, 1, 2], T)
|
|
>>> next(sample_stochastic_process(Y)) in Y.state_space
|
|
True
|
|
>>> next(sample_stochastic_process(Y)) # doctest: +SKIP
|
|
0
|
|
>>> next(sample_stochastic_process(Y)) # doctest: +SKIP
|
|
2
|
|
|
|
Returns
|
|
=======
|
|
|
|
sample: iterator object
|
|
iterator object containing the sample of given process
|
|
|
|
"""
|
|
from sympy.stats.stochastic_process_types import StochasticProcess
|
|
if not isinstance(process, StochasticProcess):
|
|
raise ValueError("Process must be an instance of Stochastic Process")
|
|
return process.sample()
|