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611 lines
21 KiB
611 lines
21 KiB
from math import prod
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from sympy.core.basic import Basic
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from sympy.core.numbers import pi
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from sympy.core.singleton import S
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from sympy.functions.elementary.exponential import exp
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from sympy.functions.special.gamma_functions import multigamma
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from sympy.core.sympify import sympify, _sympify
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from sympy.matrices import (ImmutableMatrix, Inverse, Trace, Determinant,
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MatrixSymbol, MatrixBase, Transpose, MatrixSet,
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matrix2numpy)
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from sympy.stats.rv import (_value_check, RandomMatrixSymbol, NamedArgsMixin, PSpace,
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_symbol_converter, MatrixDomain, Distribution)
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from sympy.external import import_module
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################################################################################
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#------------------------Matrix Probability Space------------------------------#
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################################################################################
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class MatrixPSpace(PSpace):
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"""
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Represents probability space for
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Matrix Distributions.
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"""
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def __new__(cls, sym, distribution, dim_n, dim_m):
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sym = _symbol_converter(sym)
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dim_n, dim_m = _sympify(dim_n), _sympify(dim_m)
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if not (dim_n.is_integer and dim_m.is_integer):
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raise ValueError("Dimensions should be integers")
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return Basic.__new__(cls, sym, distribution, dim_n, dim_m)
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distribution = property(lambda self: self.args[1])
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symbol = property(lambda self: self.args[0])
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@property
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def domain(self):
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return MatrixDomain(self.symbol, self.distribution.set)
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@property
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def value(self):
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return RandomMatrixSymbol(self.symbol, self.args[2], self.args[3], self)
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@property
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def values(self):
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return {self.value}
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def compute_density(self, expr, *args):
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rms = expr.atoms(RandomMatrixSymbol)
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if len(rms) > 1 or (not isinstance(expr, RandomMatrixSymbol)):
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raise NotImplementedError("Currently, no algorithm has been "
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"implemented to handle general expressions containing "
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"multiple matrix distributions.")
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return self.distribution.pdf(expr)
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def sample(self, size=(), library='scipy', seed=None):
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"""
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Internal sample method
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Returns dictionary mapping RandomMatrixSymbol to realization value.
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"""
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return {self.value: self.distribution.sample(size, library=library, seed=seed)}
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def rv(symbol, cls, args):
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args = list(map(sympify, args))
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dist = cls(*args)
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dist.check(*args)
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dim = dist.dimension
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pspace = MatrixPSpace(symbol, dist, dim[0], dim[1])
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return pspace.value
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class SampleMatrixScipy:
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"""Returns the sample from scipy of the given distribution"""
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def __new__(cls, dist, size, seed=None):
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return cls._sample_scipy(dist, size, seed)
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@classmethod
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def _sample_scipy(cls, dist, size, seed):
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"""Sample from SciPy."""
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from scipy import stats as scipy_stats
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import numpy
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scipy_rv_map = {
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'WishartDistribution': lambda dist, size, rand_state: scipy_stats.wishart.rvs(
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df=int(dist.n), scale=matrix2numpy(dist.scale_matrix, float), size=size),
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'MatrixNormalDistribution': lambda dist, size, rand_state: scipy_stats.matrix_normal.rvs(
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mean=matrix2numpy(dist.location_matrix, float),
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rowcov=matrix2numpy(dist.scale_matrix_1, float),
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colcov=matrix2numpy(dist.scale_matrix_2, float), size=size, random_state=rand_state)
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}
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sample_shape = {
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'WishartDistribution': lambda dist: dist.scale_matrix.shape,
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'MatrixNormalDistribution' : lambda dist: dist.location_matrix.shape
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}
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dist_list = scipy_rv_map.keys()
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if dist.__class__.__name__ not in dist_list:
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return None
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if seed is None or isinstance(seed, int):
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rand_state = numpy.random.default_rng(seed=seed)
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else:
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rand_state = seed
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samp = scipy_rv_map[dist.__class__.__name__](dist, prod(size), rand_state)
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return samp.reshape(size + sample_shape[dist.__class__.__name__](dist))
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class SampleMatrixNumpy:
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"""Returns the sample from numpy of the given distribution"""
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### TODO: Add tests after adding matrix distributions in numpy_rv_map
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def __new__(cls, dist, size, seed=None):
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return cls._sample_numpy(dist, size, seed)
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@classmethod
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def _sample_numpy(cls, dist, size, seed):
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"""Sample from NumPy."""
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numpy_rv_map = {
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}
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sample_shape = {
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}
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dist_list = numpy_rv_map.keys()
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if dist.__class__.__name__ not in dist_list:
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return None
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import numpy
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if seed is None or isinstance(seed, int):
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rand_state = numpy.random.default_rng(seed=seed)
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else:
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rand_state = seed
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samp = numpy_rv_map[dist.__class__.__name__](dist, prod(size), rand_state)
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return samp.reshape(size + sample_shape[dist.__class__.__name__](dist))
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class SampleMatrixPymc:
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"""Returns the sample from pymc of the given distribution"""
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def __new__(cls, dist, size, seed=None):
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return cls._sample_pymc(dist, size, seed)
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@classmethod
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def _sample_pymc(cls, dist, size, seed):
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"""Sample from PyMC."""
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try:
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import pymc
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except ImportError:
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import pymc3 as pymc
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pymc_rv_map = {
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'MatrixNormalDistribution': lambda dist: pymc.MatrixNormal('X',
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mu=matrix2numpy(dist.location_matrix, float),
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rowcov=matrix2numpy(dist.scale_matrix_1, float),
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colcov=matrix2numpy(dist.scale_matrix_2, float),
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shape=dist.location_matrix.shape),
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'WishartDistribution': lambda dist: pymc.WishartBartlett('X',
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nu=int(dist.n), S=matrix2numpy(dist.scale_matrix, float))
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}
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sample_shape = {
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'WishartDistribution': lambda dist: dist.scale_matrix.shape,
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'MatrixNormalDistribution' : lambda dist: dist.location_matrix.shape
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}
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dist_list = pymc_rv_map.keys()
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if dist.__class__.__name__ not in dist_list:
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return None
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import logging
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logging.getLogger("pymc").setLevel(logging.ERROR)
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with pymc.Model():
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pymc_rv_map[dist.__class__.__name__](dist)
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samps = pymc.sample(draws=prod(size), chains=1, progressbar=False, random_seed=seed, return_inferencedata=False, compute_convergence_checks=False)['X']
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return samps.reshape(size + sample_shape[dist.__class__.__name__](dist))
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_get_sample_class_matrixrv = {
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'scipy': SampleMatrixScipy,
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'pymc3': SampleMatrixPymc,
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'pymc': SampleMatrixPymc,
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'numpy': SampleMatrixNumpy
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}
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################################################################################
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#-------------------------Matrix Distribution----------------------------------#
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################################################################################
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class MatrixDistribution(Distribution, NamedArgsMixin):
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"""
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Abstract class for Matrix Distribution.
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"""
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def __new__(cls, *args):
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args = [ImmutableMatrix(arg) if isinstance(arg, list)
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else _sympify(arg) for arg in 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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def __call__(self, expr):
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if isinstance(expr, list):
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expr = ImmutableMatrix(expr)
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return self.pdf(expr)
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def sample(self, size=(), library='scipy', seed=None):
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"""
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Internal sample method
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Returns dictionary mapping RandomSymbol to realization value.
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"""
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libraries = ['scipy', 'numpy', 'pymc3', 'pymc']
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if library not in libraries:
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raise NotImplementedError("Sampling from %s is not supported yet."
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% str(library))
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if not import_module(library):
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raise ValueError("Failed to import %s" % library)
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samps = _get_sample_class_matrixrv[library](self, size, seed)
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if samps is not None:
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return samps
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raise NotImplementedError(
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"Sampling for %s is not currently implemented from %s"
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% (self.__class__.__name__, library)
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)
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################################################################################
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#------------------------Matrix Distribution Types-----------------------------#
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################################################################################
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#-------------------------------------------------------------------------------
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# Matrix Gamma distribution ----------------------------------------------------
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class MatrixGammaDistribution(MatrixDistribution):
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_argnames = ('alpha', 'beta', 'scale_matrix')
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@staticmethod
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def check(alpha, beta, scale_matrix):
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if not isinstance(scale_matrix, MatrixSymbol):
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_value_check(scale_matrix.is_positive_definite, "The shape "
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"matrix must be positive definite.")
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_value_check(scale_matrix.is_square, "Should "
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"be square matrix")
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_value_check(alpha.is_positive, "Shape parameter should be positive.")
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_value_check(beta.is_positive, "Scale parameter should be positive.")
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@property
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def set(self):
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k = self.scale_matrix.shape[0]
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return MatrixSet(k, k, S.Reals)
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@property
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def dimension(self):
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return self.scale_matrix.shape
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def pdf(self, x):
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alpha, beta, scale_matrix = self.alpha, self.beta, self.scale_matrix
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p = scale_matrix.shape[0]
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if isinstance(x, list):
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x = ImmutableMatrix(x)
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if not isinstance(x, (MatrixBase, MatrixSymbol)):
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raise ValueError("%s should be an isinstance of Matrix "
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"or MatrixSymbol" % str(x))
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sigma_inv_x = - Inverse(scale_matrix)*x / beta
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term1 = exp(Trace(sigma_inv_x))/((beta**(p*alpha)) * multigamma(alpha, p))
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term2 = (Determinant(scale_matrix))**(-alpha)
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term3 = (Determinant(x))**(alpha - S(p + 1)/2)
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return term1 * term2 * term3
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def MatrixGamma(symbol, alpha, beta, scale_matrix):
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"""
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Creates a random variable with Matrix Gamma Distribution.
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The density of the said distribution can be found at [1].
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Parameters
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==========
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alpha: Positive Real number
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Shape Parameter
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beta: Positive Real number
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Scale Parameter
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scale_matrix: Positive definite real square matrix
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Scale Matrix
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Returns
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=======
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RandomSymbol
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Examples
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========
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>>> from sympy.stats import density, MatrixGamma
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>>> from sympy import MatrixSymbol, symbols
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>>> a, b = symbols('a b', positive=True)
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>>> M = MatrixGamma('M', a, b, [[2, 1], [1, 2]])
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>>> X = MatrixSymbol('X', 2, 2)
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>>> density(M)(X).doit()
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exp(Trace(Matrix([
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[-2/3, 1/3],
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[ 1/3, -2/3]])*X)/b)*Determinant(X)**(a - 3/2)/(3**a*sqrt(pi)*b**(2*a)*gamma(a)*gamma(a - 1/2))
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>>> density(M)([[1, 0], [0, 1]]).doit()
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exp(-4/(3*b))/(3**a*sqrt(pi)*b**(2*a)*gamma(a)*gamma(a - 1/2))
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Matrix_gamma_distribution
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"""
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if isinstance(scale_matrix, list):
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scale_matrix = ImmutableMatrix(scale_matrix)
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return rv(symbol, MatrixGammaDistribution, (alpha, beta, scale_matrix))
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#-------------------------------------------------------------------------------
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# Wishart Distribution ---------------------------------------------------------
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class WishartDistribution(MatrixDistribution):
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_argnames = ('n', 'scale_matrix')
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@staticmethod
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def check(n, scale_matrix):
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if not isinstance(scale_matrix, MatrixSymbol):
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_value_check(scale_matrix.is_positive_definite, "The shape "
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"matrix must be positive definite.")
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_value_check(scale_matrix.is_square, "Should "
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"be square matrix")
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_value_check(n.is_positive, "Shape parameter should be positive.")
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@property
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def set(self):
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k = self.scale_matrix.shape[0]
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return MatrixSet(k, k, S.Reals)
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@property
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def dimension(self):
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return self.scale_matrix.shape
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def pdf(self, x):
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n, scale_matrix = self.n, self.scale_matrix
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p = scale_matrix.shape[0]
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if isinstance(x, list):
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x = ImmutableMatrix(x)
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if not isinstance(x, (MatrixBase, MatrixSymbol)):
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raise ValueError("%s should be an isinstance of Matrix "
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"or MatrixSymbol" % str(x))
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sigma_inv_x = - Inverse(scale_matrix)*x / S(2)
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term1 = exp(Trace(sigma_inv_x))/((2**(p*n/S(2))) * multigamma(n/S(2), p))
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term2 = (Determinant(scale_matrix))**(-n/S(2))
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term3 = (Determinant(x))**(S(n - p - 1)/2)
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return term1 * term2 * term3
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def Wishart(symbol, n, scale_matrix):
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"""
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Creates a random variable with Wishart Distribution.
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The density of the said distribution can be found at [1].
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Parameters
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==========
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n: Positive Real number
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Represents degrees of freedom
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scale_matrix: Positive definite real square matrix
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Scale Matrix
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Returns
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=======
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RandomSymbol
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Examples
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========
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>>> from sympy.stats import density, Wishart
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>>> from sympy import MatrixSymbol, symbols
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>>> n = symbols('n', positive=True)
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>>> W = Wishart('W', n, [[2, 1], [1, 2]])
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>>> X = MatrixSymbol('X', 2, 2)
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>>> density(W)(X).doit()
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exp(Trace(Matrix([
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[-1/3, 1/6],
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[ 1/6, -1/3]])*X))*Determinant(X)**(n/2 - 3/2)/(2**n*3**(n/2)*sqrt(pi)*gamma(n/2)*gamma(n/2 - 1/2))
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>>> density(W)([[1, 0], [0, 1]]).doit()
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exp(-2/3)/(2**n*3**(n/2)*sqrt(pi)*gamma(n/2)*gamma(n/2 - 1/2))
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Wishart_distribution
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"""
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if isinstance(scale_matrix, list):
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scale_matrix = ImmutableMatrix(scale_matrix)
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return rv(symbol, WishartDistribution, (n, scale_matrix))
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#-------------------------------------------------------------------------------
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# Matrix Normal distribution ---------------------------------------------------
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class MatrixNormalDistribution(MatrixDistribution):
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_argnames = ('location_matrix', 'scale_matrix_1', 'scale_matrix_2')
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@staticmethod
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def check(location_matrix, scale_matrix_1, scale_matrix_2):
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if not isinstance(scale_matrix_1, MatrixSymbol):
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_value_check(scale_matrix_1.is_positive_definite, "The shape "
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"matrix must be positive definite.")
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if not isinstance(scale_matrix_2, MatrixSymbol):
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_value_check(scale_matrix_2.is_positive_definite, "The shape "
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"matrix must be positive definite.")
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_value_check(scale_matrix_1.is_square, "Scale matrix 1 should be "
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"be square matrix")
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_value_check(scale_matrix_2.is_square, "Scale matrix 2 should be "
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"be square matrix")
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n = location_matrix.shape[0]
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p = location_matrix.shape[1]
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_value_check(scale_matrix_1.shape[0] == n, "Scale matrix 1 should be"
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" of shape %s x %s"% (str(n), str(n)))
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_value_check(scale_matrix_2.shape[0] == p, "Scale matrix 2 should be"
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" of shape %s x %s"% (str(p), str(p)))
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@property
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def set(self):
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n, p = self.location_matrix.shape
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return MatrixSet(n, p, S.Reals)
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@property
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def dimension(self):
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return self.location_matrix.shape
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def pdf(self, x):
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M, U, V = self.location_matrix, self.scale_matrix_1, self.scale_matrix_2
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n, p = M.shape
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if isinstance(x, list):
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x = ImmutableMatrix(x)
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if not isinstance(x, (MatrixBase, MatrixSymbol)):
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raise ValueError("%s should be an isinstance of Matrix "
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"or MatrixSymbol" % str(x))
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term1 = Inverse(V)*Transpose(x - M)*Inverse(U)*(x - M)
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num = exp(-Trace(term1)/S(2))
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den = (2*pi)**(S(n*p)/2) * Determinant(U)**(S(p)/2) * Determinant(V)**(S(n)/2)
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return num/den
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def MatrixNormal(symbol, location_matrix, scale_matrix_1, scale_matrix_2):
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"""
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Creates a random variable with Matrix Normal Distribution.
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The density of the said distribution can be found at [1].
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Parameters
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==========
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location_matrix: Real ``n x p`` matrix
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Represents degrees of freedom
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scale_matrix_1: Positive definite matrix
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Scale Matrix of shape ``n x n``
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scale_matrix_2: Positive definite matrix
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Scale Matrix of shape ``p x p``
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Returns
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=======
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RandomSymbol
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Examples
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========
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>>> from sympy import MatrixSymbol
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>>> from sympy.stats import density, MatrixNormal
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>>> M = MatrixNormal('M', [[1, 2]], [1], [[1, 0], [0, 1]])
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>>> X = MatrixSymbol('X', 1, 2)
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>>> density(M)(X).doit()
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exp(-Trace((Matrix([
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[-1],
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[-2]]) + X.T)*(Matrix([[-1, -2]]) + X))/2)/(2*pi)
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>>> density(M)([[3, 4]]).doit()
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exp(-4)/(2*pi)
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Matrix_normal_distribution
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"""
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if isinstance(location_matrix, list):
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location_matrix = ImmutableMatrix(location_matrix)
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if isinstance(scale_matrix_1, list):
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scale_matrix_1 = ImmutableMatrix(scale_matrix_1)
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if isinstance(scale_matrix_2, list):
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scale_matrix_2 = ImmutableMatrix(scale_matrix_2)
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args = (location_matrix, scale_matrix_1, scale_matrix_2)
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return rv(symbol, MatrixNormalDistribution, args)
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#-------------------------------------------------------------------------------
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# Matrix Student's T distribution ---------------------------------------------------
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class MatrixStudentTDistribution(MatrixDistribution):
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_argnames = ('nu', 'location_matrix', 'scale_matrix_1', 'scale_matrix_2')
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@staticmethod
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def check(nu, location_matrix, scale_matrix_1, scale_matrix_2):
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if not isinstance(scale_matrix_1, MatrixSymbol):
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_value_check(scale_matrix_1.is_positive_definite != False, "The shape "
|
|
"matrix must be positive definite.")
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if not isinstance(scale_matrix_2, MatrixSymbol):
|
|
_value_check(scale_matrix_2.is_positive_definite != False, "The shape "
|
|
"matrix must be positive definite.")
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|
_value_check(scale_matrix_1.is_square != False, "Scale matrix 1 should be "
|
|
"be square matrix")
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|
_value_check(scale_matrix_2.is_square != False, "Scale matrix 2 should be "
|
|
"be square matrix")
|
|
n = location_matrix.shape[0]
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p = location_matrix.shape[1]
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|
_value_check(scale_matrix_1.shape[0] == p, "Scale matrix 1 should be"
|
|
" of shape %s x %s" % (str(p), str(p)))
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|
_value_check(scale_matrix_2.shape[0] == n, "Scale matrix 2 should be"
|
|
" of shape %s x %s" % (str(n), str(n)))
|
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_value_check(nu.is_positive != False, "Degrees of freedom must be positive")
|
|
|
|
@property
|
|
def set(self):
|
|
n, p = self.location_matrix.shape
|
|
return MatrixSet(n, p, S.Reals)
|
|
|
|
@property
|
|
def dimension(self):
|
|
return self.location_matrix.shape
|
|
|
|
def pdf(self, x):
|
|
from sympy.matrices.dense import eye
|
|
if isinstance(x, list):
|
|
x = ImmutableMatrix(x)
|
|
if not isinstance(x, (MatrixBase, MatrixSymbol)):
|
|
raise ValueError("%s should be an isinstance of Matrix "
|
|
"or MatrixSymbol" % str(x))
|
|
nu, M, Omega, Sigma = self.nu, self.location_matrix, self.scale_matrix_1, self.scale_matrix_2
|
|
n, p = M.shape
|
|
|
|
K = multigamma((nu + n + p - 1)/2, p) * Determinant(Omega)**(-n/2) * Determinant(Sigma)**(-p/2) \
|
|
/ ((pi)**(n*p/2) * multigamma((nu + p - 1)/2, p))
|
|
return K * (Determinant(eye(n) + Inverse(Sigma)*(x - M)*Inverse(Omega)*Transpose(x - M))) \
|
|
**(-(nu + n + p -1)/2)
|
|
|
|
|
|
|
|
def MatrixStudentT(symbol, nu, location_matrix, scale_matrix_1, scale_matrix_2):
|
|
"""
|
|
Creates a random variable with Matrix Gamma Distribution.
|
|
|
|
The density of the said distribution can be found at [1].
|
|
|
|
Parameters
|
|
==========
|
|
|
|
nu: Positive Real number
|
|
degrees of freedom
|
|
location_matrix: Positive definite real square matrix
|
|
Location Matrix of shape ``n x p``
|
|
scale_matrix_1: Positive definite real square matrix
|
|
Scale Matrix of shape ``p x p``
|
|
scale_matrix_2: Positive definite real square matrix
|
|
Scale Matrix of shape ``n x n``
|
|
|
|
Returns
|
|
=======
|
|
|
|
RandomSymbol
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy import MatrixSymbol,symbols
|
|
>>> from sympy.stats import density, MatrixStudentT
|
|
>>> v = symbols('v',positive=True)
|
|
>>> M = MatrixStudentT('M', v, [[1, 2]], [[1, 0], [0, 1]], [1])
|
|
>>> X = MatrixSymbol('X', 1, 2)
|
|
>>> density(M)(X)
|
|
gamma(v/2 + 1)*Determinant((Matrix([[-1, -2]]) + X)*(Matrix([
|
|
[-1],
|
|
[-2]]) + X.T) + Matrix([[1]]))**(-v/2 - 1)/(pi**1.0*gamma(v/2)*Determinant(Matrix([[1]]))**1.0*Determinant(Matrix([
|
|
[1, 0],
|
|
[0, 1]]))**0.5)
|
|
|
|
References
|
|
==========
|
|
|
|
.. [1] https://en.wikipedia.org/wiki/Matrix_t-distribution
|
|
|
|
"""
|
|
if isinstance(location_matrix, list):
|
|
location_matrix = ImmutableMatrix(location_matrix)
|
|
if isinstance(scale_matrix_1, list):
|
|
scale_matrix_1 = ImmutableMatrix(scale_matrix_1)
|
|
if isinstance(scale_matrix_2, list):
|
|
scale_matrix_2 = ImmutableMatrix(scale_matrix_2)
|
|
args = (nu, location_matrix, scale_matrix_1, scale_matrix_2)
|
|
return rv(symbol, MatrixStudentTDistribution, args)
|