7282 lines
243 KiB
Python
7282 lines
243 KiB
Python
#
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# Author: Joris Vankerschaver 2013
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#
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import math
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import warnings
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import threading
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import numpy as np
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import scipy.linalg
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from scipy._lib import doccer
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from scipy.special import (gammaln, psi, multigammaln, xlogy, entr, betaln,
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ive, loggamma)
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from scipy import special
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import scipy._lib.array_api_extra as xpx
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from scipy._lib._util import check_random_state
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from scipy.linalg.blas import drot, get_blas_funcs
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from ._continuous_distns import norm, invgamma
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from ._discrete_distns import binom
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from . import _covariance, _rcont
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from ._qmvnt import _qmvt, _qmvn, _qauto
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from ._morestats import directional_stats
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from scipy.optimize import root_scalar
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__all__ = ['multivariate_normal',
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'matrix_normal',
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'dirichlet',
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'dirichlet_multinomial',
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'wishart',
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'invwishart',
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'multinomial',
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'special_ortho_group',
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'ortho_group',
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'random_correlation',
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'unitary_group',
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'multivariate_t',
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'multivariate_hypergeom',
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'random_table',
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'uniform_direction',
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'vonmises_fisher',
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'normal_inverse_gamma']
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_LOG_2PI = np.log(2 * np.pi)
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_LOG_2 = np.log(2)
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_LOG_PI = np.log(np.pi)
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MVN_LOCK = threading.Lock()
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_doc_random_state = """\
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seed : {None, int, np.random.RandomState, np.random.Generator}, optional
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Used for drawing random variates.
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If `seed` is `None`, the `~np.random.RandomState` singleton is used.
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If `seed` is an int, a new ``RandomState`` instance is used, seeded
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with seed.
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If `seed` is already a ``RandomState`` or ``Generator`` instance,
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then that object is used.
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Default is `None`.
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"""
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def _squeeze_output(out):
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"""
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Remove single-dimensional entries from array and convert to scalar,
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if necessary.
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"""
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out = out.squeeze()
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if out.ndim == 0:
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out = out[()]
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return out
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def _eigvalsh_to_eps(spectrum, cond=None, rcond=None):
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"""Determine which eigenvalues are "small" given the spectrum.
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This is for compatibility across various linear algebra functions
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that should agree about whether or not a Hermitian matrix is numerically
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singular and what is its numerical matrix rank.
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This is designed to be compatible with scipy.linalg.pinvh.
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Parameters
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----------
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spectrum : 1d ndarray
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Array of eigenvalues of a Hermitian matrix.
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cond, rcond : float, optional
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Cutoff for small eigenvalues.
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Singular values smaller than rcond * largest_eigenvalue are
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considered zero.
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If None or -1, suitable machine precision is used.
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Returns
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-------
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eps : float
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Magnitude cutoff for numerical negligibility.
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"""
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if rcond is not None:
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cond = rcond
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if cond in [None, -1]:
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t = spectrum.dtype.char.lower()
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factor = {'f': 1E3, 'd': 1E6}
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cond = factor[t] * np.finfo(t).eps
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eps = cond * np.max(abs(spectrum))
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return eps
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def _pinv_1d(v, eps=1e-5):
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"""A helper function for computing the pseudoinverse.
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Parameters
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----------
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v : iterable of numbers
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This may be thought of as a vector of eigenvalues or singular values.
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eps : float
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Values with magnitude no greater than eps are considered negligible.
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Returns
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-------
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v_pinv : 1d float ndarray
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A vector of pseudo-inverted numbers.
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"""
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return np.array([0 if abs(x) <= eps else 1/x for x in v], dtype=float)
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class _PSD:
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"""
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Compute coordinated functions of a symmetric positive semidefinite matrix.
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This class addresses two issues. Firstly it allows the pseudoinverse,
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the logarithm of the pseudo-determinant, and the rank of the matrix
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to be computed using one call to eigh instead of three.
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Secondly it allows these functions to be computed in a way
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that gives mutually compatible results.
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All of the functions are computed with a common understanding as to
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which of the eigenvalues are to be considered negligibly small.
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The functions are designed to coordinate with scipy.linalg.pinvh()
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but not necessarily with np.linalg.det() or with np.linalg.matrix_rank().
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Parameters
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----------
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M : array_like
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Symmetric positive semidefinite matrix (2-D).
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cond, rcond : float, optional
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Cutoff for small eigenvalues.
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Singular values smaller than rcond * largest_eigenvalue are
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considered zero.
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If None or -1, suitable machine precision is used.
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lower : bool, optional
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Whether the pertinent array data is taken from the lower
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or upper triangle of M. (Default: lower)
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check_finite : bool, optional
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Whether to check that the input matrices contain only finite
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numbers. Disabling may give a performance gain, but may result
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in problems (crashes, non-termination) if the inputs do contain
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infinities or NaNs.
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allow_singular : bool, optional
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Whether to allow a singular matrix. (Default: True)
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Notes
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-----
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The arguments are similar to those of scipy.linalg.pinvh().
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"""
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def __init__(self, M, cond=None, rcond=None, lower=True,
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check_finite=True, allow_singular=True):
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self._M = np.asarray(M)
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# Compute the symmetric eigendecomposition.
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# Note that eigh takes care of array conversion, chkfinite,
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# and assertion that the matrix is square.
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s, u = scipy.linalg.eigh(M, lower=lower, check_finite=check_finite)
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eps = _eigvalsh_to_eps(s, cond, rcond)
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if np.min(s) < -eps:
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msg = "The input matrix must be symmetric positive semidefinite."
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raise ValueError(msg)
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d = s[s > eps]
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if len(d) < len(s) and not allow_singular:
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msg = ("When `allow_singular is False`, the input matrix must be "
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"symmetric positive definite.")
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raise np.linalg.LinAlgError(msg)
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s_pinv = _pinv_1d(s, eps)
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U = np.multiply(u, np.sqrt(s_pinv))
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# Save the eigenvector basis, and tolerance for testing support
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self.eps = 1e3*eps
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self.V = u[:, s <= eps]
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# Initialize the eagerly precomputed attributes.
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self.rank = len(d)
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self.U = U
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self.log_pdet = np.sum(np.log(d))
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# Initialize attributes to be lazily computed.
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self._pinv = None
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def _support_mask(self, x):
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"""
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Check whether x lies in the support of the distribution.
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"""
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residual = np.linalg.norm(x @ self.V, axis=-1)
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in_support = residual < self.eps
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return in_support
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@property
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def pinv(self):
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if self._pinv is None:
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self._pinv = np.dot(self.U, self.U.T)
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return self._pinv
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class multi_rv_generic:
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"""
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Class which encapsulates common functionality between all multivariate
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distributions.
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"""
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def __init__(self, seed=None):
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super().__init__()
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self._random_state = check_random_state(seed)
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@property
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def random_state(self):
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""" Get or set the Generator object for generating random variates.
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If `seed` is None (or `np.random`), the `numpy.random.RandomState`
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singleton is used.
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If `seed` is an int, a new ``RandomState`` instance is used,
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seeded with `seed`.
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If `seed` is already a ``Generator`` or ``RandomState`` instance then
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that instance is used.
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"""
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return self._random_state
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@random_state.setter
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def random_state(self, seed):
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self._random_state = check_random_state(seed)
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def _get_random_state(self, random_state):
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if random_state is not None:
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return check_random_state(random_state)
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else:
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return self._random_state
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class multi_rv_frozen:
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"""
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Class which encapsulates common functionality between all frozen
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multivariate distributions.
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"""
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@property
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def random_state(self):
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return self._dist._random_state
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@random_state.setter
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def random_state(self, seed):
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self._dist._random_state = check_random_state(seed)
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_mvn_doc_default_callparams = """\
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mean : array_like, default: ``[0]``
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Mean of the distribution.
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cov : array_like or `Covariance`, default: ``[1]``
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Symmetric positive (semi)definite covariance matrix of the distribution.
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allow_singular : bool, default: ``False``
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Whether to allow a singular covariance matrix. This is ignored if `cov` is
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a `Covariance` object.
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"""
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_mvn_doc_callparams_note = """\
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Setting the parameter `mean` to `None` is equivalent to having `mean`
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be the zero-vector. The parameter `cov` can be a scalar, in which case
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the covariance matrix is the identity times that value, a vector of
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diagonal entries for the covariance matrix, a two-dimensional array_like,
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or a `Covariance` object.
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"""
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_mvn_doc_frozen_callparams = ""
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_mvn_doc_frozen_callparams_note = """\
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See class definition for a detailed description of parameters."""
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mvn_docdict_params = {
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'_mvn_doc_default_callparams': _mvn_doc_default_callparams,
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'_mvn_doc_callparams_note': _mvn_doc_callparams_note,
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'_doc_random_state': _doc_random_state
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}
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mvn_docdict_noparams = {
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'_mvn_doc_default_callparams': _mvn_doc_frozen_callparams,
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'_mvn_doc_callparams_note': _mvn_doc_frozen_callparams_note,
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'_doc_random_state': _doc_random_state
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}
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class multivariate_normal_gen(multi_rv_generic):
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r"""A multivariate normal random variable.
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The `mean` keyword specifies the mean. The `cov` keyword specifies the
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covariance matrix.
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Methods
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-------
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pdf(x, mean=None, cov=1, allow_singular=False)
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Probability density function.
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logpdf(x, mean=None, cov=1, allow_singular=False)
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Log of the probability density function.
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cdf(x, mean=None, cov=1, allow_singular=False, maxpts=1000000*dim, abseps=1e-5, releps=1e-5, lower_limit=None)
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Cumulative distribution function.
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logcdf(x, mean=None, cov=1, allow_singular=False, maxpts=1000000*dim, abseps=1e-5, releps=1e-5)
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Log of the cumulative distribution function.
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rvs(mean=None, cov=1, size=1, random_state=None)
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Draw random samples from a multivariate normal distribution.
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entropy(mean=None, cov=1)
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Compute the differential entropy of the multivariate normal.
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fit(x, fix_mean=None, fix_cov=None)
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Fit a multivariate normal distribution to data.
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Parameters
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----------
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%(_mvn_doc_default_callparams)s
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%(_doc_random_state)s
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Notes
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-----
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%(_mvn_doc_callparams_note)s
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The covariance matrix `cov` may be an instance of a subclass of
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`Covariance`, e.g. `scipy.stats.CovViaPrecision`. If so, `allow_singular`
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is ignored.
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Otherwise, `cov` must be a symmetric positive semidefinite
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matrix when `allow_singular` is True; it must be (strictly) positive
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definite when `allow_singular` is False.
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Symmetry is not checked; only the lower triangular portion is used.
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The determinant and inverse of `cov` are computed
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as the pseudo-determinant and pseudo-inverse, respectively, so
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that `cov` does not need to have full rank.
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The probability density function for `multivariate_normal` is
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.. math::
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f(x) = \frac{1}{\sqrt{(2 \pi)^k \det \Sigma}}
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\exp\left( -\frac{1}{2} (x - \mu)^T \Sigma^{-1} (x - \mu) \right),
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where :math:`\mu` is the mean, :math:`\Sigma` the covariance matrix,
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:math:`k` the rank of :math:`\Sigma`. In case of singular :math:`\Sigma`,
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SciPy extends this definition according to [1]_.
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.. versionadded:: 0.14.0
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References
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----------
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.. [1] Multivariate Normal Distribution - Degenerate Case, Wikipedia,
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https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Degenerate_case
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Examples
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--------
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>>> import numpy as np
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>>> import matplotlib.pyplot as plt
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>>> from scipy.stats import multivariate_normal
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>>> x = np.linspace(0, 5, 10, endpoint=False)
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>>> y = multivariate_normal.pdf(x, mean=2.5, cov=0.5); y
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array([ 0.00108914, 0.01033349, 0.05946514, 0.20755375, 0.43939129,
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0.56418958, 0.43939129, 0.20755375, 0.05946514, 0.01033349])
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>>> fig1 = plt.figure()
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>>> ax = fig1.add_subplot(111)
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>>> ax.plot(x, y)
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>>> plt.show()
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Alternatively, the object may be called (as a function) to fix the mean
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and covariance parameters, returning a "frozen" multivariate normal
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random variable:
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>>> rv = multivariate_normal(mean=None, cov=1, allow_singular=False)
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>>> # Frozen object with the same methods but holding the given
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>>> # mean and covariance fixed.
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The input quantiles can be any shape of array, as long as the last
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axis labels the components. This allows us for instance to
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display the frozen pdf for a non-isotropic random variable in 2D as
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follows:
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>>> x, y = np.mgrid[-1:1:.01, -1:1:.01]
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>>> pos = np.dstack((x, y))
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>>> rv = multivariate_normal([0.5, -0.2], [[2.0, 0.3], [0.3, 0.5]])
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>>> fig2 = plt.figure()
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>>> ax2 = fig2.add_subplot(111)
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>>> ax2.contourf(x, y, rv.pdf(pos))
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""" # noqa: E501
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def __init__(self, seed=None):
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super().__init__(seed)
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self.__doc__ = doccer.docformat(self.__doc__, mvn_docdict_params)
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def __call__(self, mean=None, cov=1, allow_singular=False, seed=None, **kwds):
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"""Create a frozen multivariate normal distribution.
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See `multivariate_normal_frozen` for more information.
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"""
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return multivariate_normal_frozen(mean, cov,
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allow_singular=allow_singular,
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seed=seed, **kwds)
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def _process_parameters(self, mean, cov, allow_singular=True):
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"""
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Infer dimensionality from mean or covariance matrix, ensure that
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mean and covariance are full vector resp. matrix.
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"""
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if isinstance(cov, _covariance.Covariance):
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return self._process_parameters_Covariance(mean, cov)
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else:
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# Before `Covariance` classes were introduced,
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# `multivariate_normal` accepted plain arrays as `cov` and used the
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# following input validation. To avoid disturbing the behavior of
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# `multivariate_normal` when plain arrays are used, we use the
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# original input validation here.
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dim, mean, cov = self._process_parameters_psd(None, mean, cov)
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# After input validation, some methods then processed the arrays
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# with a `_PSD` object and used that to perform computation.
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# To avoid branching statements in each method depending on whether
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# `cov` is an array or `Covariance` object, we always process the
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# array with `_PSD`, and then use wrapper that satisfies the
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# `Covariance` interface, `CovViaPSD`.
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psd = _PSD(cov, allow_singular=allow_singular)
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cov_object = _covariance.CovViaPSD(psd)
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return dim, mean, cov_object
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def _process_parameters_Covariance(self, mean, cov):
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dim = cov.shape[-1]
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mean = np.array([0.]) if mean is None else mean
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message = (f"`cov` represents a covariance matrix in {dim} dimensions,"
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f"and so `mean` must be broadcastable to shape {(dim,)}")
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try:
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mean = np.broadcast_to(mean, dim)
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except ValueError as e:
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raise ValueError(message) from e
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return dim, mean, cov
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def _process_parameters_psd(self, dim, mean, cov):
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# Try to infer dimensionality
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if dim is None:
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if mean is None:
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if cov is None:
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dim = 1
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else:
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cov = np.asarray(cov, dtype=float)
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if cov.ndim < 2:
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dim = 1
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else:
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dim = cov.shape[0]
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else:
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mean = np.asarray(mean, dtype=float)
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dim = mean.size
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else:
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if not np.isscalar(dim):
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raise ValueError("Dimension of random variable must be "
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"a scalar.")
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# Check input sizes and return full arrays for mean and cov if
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# necessary
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if mean is None:
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mean = np.zeros(dim)
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mean = np.asarray(mean, dtype=float)
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if cov is None:
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cov = 1.0
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cov = np.asarray(cov, dtype=float)
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if dim == 1:
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mean = mean.reshape(1)
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cov = cov.reshape(1, 1)
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if mean.ndim != 1 or mean.shape[0] != dim:
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raise ValueError(f"Array 'mean' must be a vector of length {dim}.")
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if cov.ndim == 0:
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cov = cov * np.eye(dim)
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elif cov.ndim == 1:
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cov = np.diag(cov)
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elif cov.ndim == 2 and cov.shape != (dim, dim):
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rows, cols = cov.shape
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if rows != cols:
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msg = ("Array 'cov' must be square if it is two dimensional,"
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f" but cov.shape = {str(cov.shape)}.")
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else:
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msg = (f"Dimension mismatch: array 'cov' is of shape {cov.shape}, "
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f"but 'mean' is a vector of length {len(mean)}.")
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raise ValueError(msg)
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elif cov.ndim > 2:
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raise ValueError(f"Array 'cov' must be at most two-dimensional, "
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f"but cov.ndim = {cov.ndim}")
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return dim, mean, cov
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def _process_quantiles(self, x, dim):
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"""
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Adjust quantiles array so that last axis labels the components of
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|
each data point.
|
|
"""
|
|
x = np.asarray(x, dtype=float)
|
|
|
|
if x.ndim == 0:
|
|
x = x[np.newaxis]
|
|
elif x.ndim == 1:
|
|
if dim == 1:
|
|
x = x[:, np.newaxis]
|
|
else:
|
|
x = x[np.newaxis, :]
|
|
|
|
return x
|
|
|
|
def _logpdf(self, x, mean, cov_object):
|
|
"""Log of the multivariate normal probability density function.
|
|
|
|
Parameters
|
|
----------
|
|
x : ndarray
|
|
Points at which to evaluate the log of the probability
|
|
density function
|
|
mean : ndarray
|
|
Mean of the distribution
|
|
cov_object : Covariance
|
|
An object representing the Covariance matrix
|
|
|
|
Notes
|
|
-----
|
|
As this function does no argument checking, it should not be
|
|
called directly; use 'logpdf' instead.
|
|
|
|
"""
|
|
log_det_cov, rank = cov_object.log_pdet, cov_object.rank
|
|
dev = x - mean
|
|
if dev.ndim > 1:
|
|
log_det_cov = log_det_cov[..., np.newaxis]
|
|
rank = rank[..., np.newaxis]
|
|
maha = np.sum(np.square(cov_object.whiten(dev)), axis=-1)
|
|
return -0.5 * (rank * _LOG_2PI + log_det_cov + maha)
|
|
|
|
def logpdf(self, x, mean=None, cov=1, allow_singular=False):
|
|
"""Log of the multivariate normal probability density function.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
Quantiles, with the last axis of `x` denoting the components.
|
|
%(_mvn_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
pdf : ndarray or scalar
|
|
Log of the probability density function evaluated at `x`
|
|
|
|
Notes
|
|
-----
|
|
%(_mvn_doc_callparams_note)s
|
|
|
|
"""
|
|
params = self._process_parameters(mean, cov, allow_singular)
|
|
dim, mean, cov_object = params
|
|
x = self._process_quantiles(x, dim)
|
|
out = self._logpdf(x, mean, cov_object)
|
|
if np.any(cov_object.rank < dim):
|
|
out_of_bounds = ~cov_object._support_mask(x-mean)
|
|
out[out_of_bounds] = -np.inf
|
|
return _squeeze_output(out)
|
|
|
|
def pdf(self, x, mean=None, cov=1, allow_singular=False):
|
|
"""Multivariate normal probability density function.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
Quantiles, with the last axis of `x` denoting the components.
|
|
%(_mvn_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
pdf : ndarray or scalar
|
|
Probability density function evaluated at `x`
|
|
|
|
Notes
|
|
-----
|
|
%(_mvn_doc_callparams_note)s
|
|
|
|
"""
|
|
params = self._process_parameters(mean, cov, allow_singular)
|
|
dim, mean, cov_object = params
|
|
x = self._process_quantiles(x, dim)
|
|
out = np.exp(self._logpdf(x, mean, cov_object))
|
|
if np.any(cov_object.rank < dim):
|
|
out_of_bounds = ~cov_object._support_mask(x-mean)
|
|
out[out_of_bounds] = 0.0
|
|
return _squeeze_output(out)
|
|
|
|
def _cdf(self, x, mean, cov, maxpts, abseps, releps, lower_limit, rng):
|
|
"""Multivariate normal cumulative distribution function.
|
|
|
|
Parameters
|
|
----------
|
|
x : ndarray
|
|
Points at which to evaluate the cumulative distribution function.
|
|
mean : ndarray
|
|
Mean of the distribution
|
|
cov : array_like
|
|
Covariance matrix of the distribution
|
|
maxpts : integer
|
|
The maximum number of points to use for integration
|
|
abseps : float
|
|
Absolute error tolerance
|
|
releps : float
|
|
Relative error tolerance
|
|
lower_limit : array_like, optional
|
|
Lower limit of integration of the cumulative distribution function.
|
|
Default is negative infinity. Must be broadcastable with `x`.
|
|
rng : Generator
|
|
an instance of ``np.random.Generator``, which is used internally
|
|
for QMC integration.
|
|
|
|
Notes
|
|
-----
|
|
As this function does no argument checking, it should not be
|
|
called directly; use 'cdf' instead.
|
|
|
|
|
|
.. versionadded:: 1.0.0
|
|
|
|
"""
|
|
lower = (np.full(mean.shape, -np.inf)
|
|
if lower_limit is None else lower_limit)
|
|
# In 2d, _mvn.mvnun accepts input in which `lower` bound elements
|
|
# are greater than `x`. Not so in other dimensions. Fix this by
|
|
# ensuring that lower bounds are indeed lower when passed, then
|
|
# set signs of resulting CDF manually.
|
|
b, a = np.broadcast_arrays(x, lower)
|
|
b, a = b - mean, a - mean # _qmvn only accepts zero mean
|
|
i_swap = b < a
|
|
signs = (-1)**(i_swap.sum(axis=-1)) # odd # of swaps -> negative
|
|
a, b = a.copy(), b.copy()
|
|
a[i_swap], b[i_swap] = b[i_swap], a[i_swap]
|
|
n = x.shape[-1]
|
|
limits = np.concatenate((a, b), axis=-1)
|
|
|
|
# qmvn expects 1-d arguments, so process points sequentially
|
|
# XXX: if cov.ndim == 2 and limits.ndim == 1, can avoid apply_along_axis
|
|
def func1d(limits):
|
|
# res0 = _qmvn(maxpts, cov, limits[:n], limits[n:], rng)[0]
|
|
res = _qauto(_qmvn, cov, limits[:n], limits[n:],
|
|
rng, error=abseps, limit=maxpts, n_batches=10)
|
|
return np.squeeze(res[0])
|
|
|
|
out = np.apply_along_axis(func1d, -1, limits) * signs
|
|
return _squeeze_output(out)
|
|
|
|
def logcdf(self, x, mean=None, cov=1, allow_singular=False, maxpts=None,
|
|
abseps=1e-5, releps=1e-5, *, lower_limit=None, rng=None):
|
|
"""Log of the multivariate normal cumulative distribution function.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
Quantiles, with the last axis of `x` denoting the components.
|
|
%(_mvn_doc_default_callparams)s
|
|
maxpts : integer, optional
|
|
The maximum number of points to use for integration
|
|
(default ``1000000*dim``)
|
|
abseps : float, optional
|
|
Absolute error tolerance (default 1e-5)
|
|
releps : float, optional
|
|
Relative error tolerance (default 1e-5)
|
|
lower_limit : array_like, optional
|
|
Lower limit of integration of the cumulative distribution function.
|
|
Default is negative infinity. Must be broadcastable with `x`.
|
|
rng : Generator, optional
|
|
an instance of ``np.random.Generator``, which is used internally
|
|
for QMC integration.
|
|
|
|
Returns
|
|
-------
|
|
cdf : ndarray or scalar
|
|
Log of the cumulative distribution function evaluated at `x`
|
|
|
|
Notes
|
|
-----
|
|
%(_mvn_doc_callparams_note)s
|
|
|
|
.. versionadded:: 1.0.0
|
|
|
|
"""
|
|
params = self._process_parameters(mean, cov, allow_singular)
|
|
dim, mean, cov_object = params
|
|
cov = cov_object.covariance
|
|
x = self._process_quantiles(x, dim)
|
|
if not maxpts:
|
|
maxpts = 1000000 * dim
|
|
|
|
rng = self._get_random_state(rng)
|
|
cdf = self._cdf(x, mean, cov, maxpts, abseps, releps, lower_limit, rng)
|
|
# the log of a negative real is complex, and cdf can be negative
|
|
# if lower limit is greater than upper limit
|
|
cdf = cdf + 0j if np.any(cdf < 0) else cdf
|
|
out = np.log(cdf)
|
|
return out
|
|
|
|
def cdf(self, x, mean=None, cov=1, allow_singular=False, maxpts=None,
|
|
abseps=1e-5, releps=1e-5, *, lower_limit=None, rng=None):
|
|
"""Multivariate normal cumulative distribution function.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
Quantiles, with the last axis of `x` denoting the components.
|
|
%(_mvn_doc_default_callparams)s
|
|
maxpts : integer, optional
|
|
The maximum number of points to use for integration
|
|
(default ``1000000*dim``)
|
|
abseps : float, optional
|
|
Absolute error tolerance (default 1e-5)
|
|
releps : float, optional
|
|
Relative error tolerance (default 1e-5)
|
|
lower_limit : array_like, optional
|
|
Lower limit of integration of the cumulative distribution function.
|
|
Default is negative infinity. Must be broadcastable with `x`.
|
|
rng : Generator, optional
|
|
an instance of ``np.random.Generator``, which is used internally
|
|
for QMC integration.
|
|
|
|
Returns
|
|
-------
|
|
cdf : ndarray or scalar
|
|
Cumulative distribution function evaluated at `x`
|
|
|
|
Notes
|
|
-----
|
|
%(_mvn_doc_callparams_note)s
|
|
|
|
.. versionadded:: 1.0.0
|
|
|
|
"""
|
|
params = self._process_parameters(mean, cov, allow_singular)
|
|
dim, mean, cov_object = params
|
|
cov = cov_object.covariance
|
|
x = self._process_quantiles(x, dim)
|
|
if not maxpts:
|
|
maxpts = 1000000 * dim
|
|
rng = self._get_random_state(rng)
|
|
out = self._cdf(x, mean, cov, maxpts, abseps, releps, lower_limit, rng)
|
|
return out
|
|
|
|
def rvs(self, mean=None, cov=1, size=1, random_state=None):
|
|
"""Draw random samples from a multivariate normal distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_mvn_doc_default_callparams)s
|
|
size : integer, optional
|
|
Number of samples to draw (default 1).
|
|
%(_doc_random_state)s
|
|
|
|
Returns
|
|
-------
|
|
rvs : ndarray or scalar
|
|
Random variates of size (`size`, `N`), where `N` is the
|
|
dimension of the random variable.
|
|
|
|
Notes
|
|
-----
|
|
%(_mvn_doc_callparams_note)s
|
|
|
|
"""
|
|
dim, mean, cov_object = self._process_parameters(mean, cov)
|
|
random_state = self._get_random_state(random_state)
|
|
|
|
if isinstance(cov_object, _covariance.CovViaPSD):
|
|
cov = cov_object.covariance
|
|
out = random_state.multivariate_normal(mean, cov, size)
|
|
out = _squeeze_output(out)
|
|
else:
|
|
size = size or tuple()
|
|
if not np.iterable(size):
|
|
size = (size,)
|
|
shape = tuple(size) + (cov_object.shape[-1],)
|
|
x = random_state.normal(size=shape)
|
|
out = mean + cov_object.colorize(x)
|
|
return out
|
|
|
|
def entropy(self, mean=None, cov=1):
|
|
"""Compute the differential entropy of the multivariate normal.
|
|
|
|
Parameters
|
|
----------
|
|
%(_mvn_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
h : scalar
|
|
Entropy of the multivariate normal distribution
|
|
|
|
Notes
|
|
-----
|
|
%(_mvn_doc_callparams_note)s
|
|
|
|
"""
|
|
dim, mean, cov_object = self._process_parameters(mean, cov)
|
|
return 0.5 * (cov_object.rank * (_LOG_2PI + 1) + cov_object.log_pdet)
|
|
|
|
def fit(self, x, fix_mean=None, fix_cov=None):
|
|
"""Fit a multivariate normal distribution to data.
|
|
|
|
Parameters
|
|
----------
|
|
x : ndarray (m, n)
|
|
Data the distribution is fitted to. Must have two axes.
|
|
The first axis of length `m` represents the number of vectors
|
|
the distribution is fitted to. The second axis of length `n`
|
|
determines the dimensionality of the fitted distribution.
|
|
fix_mean : ndarray(n, )
|
|
Fixed mean vector. Must have length `n`.
|
|
fix_cov: ndarray (n, n)
|
|
Fixed covariance matrix. Must have shape ``(n, n)``.
|
|
|
|
Returns
|
|
-------
|
|
mean : ndarray (n, )
|
|
Maximum likelihood estimate of the mean vector
|
|
cov : ndarray (n, n)
|
|
Maximum likelihood estimate of the covariance matrix
|
|
|
|
"""
|
|
# input validation for data to be fitted
|
|
x = np.asarray(x)
|
|
if x.ndim != 2:
|
|
raise ValueError("`x` must be two-dimensional.")
|
|
|
|
n_vectors, dim = x.shape
|
|
|
|
# parameter estimation
|
|
# reference: https://home.ttic.edu/~shubhendu/Slides/Estimation.pdf
|
|
if fix_mean is not None:
|
|
# input validation for `fix_mean`
|
|
fix_mean = np.atleast_1d(fix_mean)
|
|
if fix_mean.shape != (dim, ):
|
|
msg = ("`fix_mean` must be a one-dimensional array the same "
|
|
"length as the dimensionality of the vectors `x`.")
|
|
raise ValueError(msg)
|
|
mean = fix_mean
|
|
else:
|
|
mean = x.mean(axis=0)
|
|
|
|
if fix_cov is not None:
|
|
# input validation for `fix_cov`
|
|
fix_cov = np.atleast_2d(fix_cov)
|
|
# validate shape
|
|
if fix_cov.shape != (dim, dim):
|
|
msg = ("`fix_cov` must be a two-dimensional square array "
|
|
"of same side length as the dimensionality of the "
|
|
"vectors `x`.")
|
|
raise ValueError(msg)
|
|
# validate positive semidefiniteness
|
|
# a trimmed down copy from _PSD
|
|
s, u = scipy.linalg.eigh(fix_cov, lower=True, check_finite=True)
|
|
eps = _eigvalsh_to_eps(s)
|
|
if np.min(s) < -eps:
|
|
msg = "`fix_cov` must be symmetric positive semidefinite."
|
|
raise ValueError(msg)
|
|
cov = fix_cov
|
|
else:
|
|
centered_data = x - mean
|
|
cov = centered_data.T @ centered_data / n_vectors
|
|
return mean, cov
|
|
|
|
|
|
multivariate_normal = multivariate_normal_gen()
|
|
|
|
|
|
class multivariate_normal_frozen(multi_rv_frozen):
|
|
def __init__(self, mean=None, cov=1, allow_singular=False, seed=None,
|
|
maxpts=None, abseps=1e-5, releps=1e-5):
|
|
"""Create a frozen multivariate normal distribution.
|
|
|
|
Parameters
|
|
----------
|
|
mean : array_like, default: ``[0]``
|
|
Mean of the distribution.
|
|
cov : array_like, default: ``[1]``
|
|
Symmetric positive (semi)definite covariance matrix of the
|
|
distribution.
|
|
allow_singular : bool, default: ``False``
|
|
Whether to allow a singular covariance matrix.
|
|
seed : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional
|
|
If `seed` is None (or `np.random`), the `numpy.random.RandomState`
|
|
singleton is used.
|
|
If `seed` is an int, a new ``RandomState`` instance is used,
|
|
seeded with `seed`.
|
|
If `seed` is already a ``Generator`` or ``RandomState`` instance
|
|
then that instance is used.
|
|
maxpts : integer, optional
|
|
The maximum number of points to use for integration of the
|
|
cumulative distribution function (default ``1000000*dim``)
|
|
abseps : float, optional
|
|
Absolute error tolerance for the cumulative distribution function
|
|
(default 1e-5)
|
|
releps : float, optional
|
|
Relative error tolerance for the cumulative distribution function
|
|
(default 1e-5)
|
|
|
|
Examples
|
|
--------
|
|
When called with the default parameters, this will create a 1D random
|
|
variable with mean 0 and covariance 1:
|
|
|
|
>>> from scipy.stats import multivariate_normal
|
|
>>> r = multivariate_normal()
|
|
>>> r.mean
|
|
array([ 0.])
|
|
>>> r.cov
|
|
array([[1.]])
|
|
|
|
""" # numpy/numpydoc#87 # noqa: E501
|
|
self._dist = multivariate_normal_gen(seed)
|
|
self.dim, self.mean, self.cov_object = (
|
|
self._dist._process_parameters(mean, cov, allow_singular))
|
|
self.allow_singular = allow_singular or self.cov_object._allow_singular
|
|
if not maxpts:
|
|
maxpts = 1000000 * self.dim
|
|
self.maxpts = maxpts
|
|
self.abseps = abseps
|
|
self.releps = releps
|
|
|
|
@property
|
|
def cov(self):
|
|
return self.cov_object.covariance
|
|
|
|
def logpdf(self, x):
|
|
x = self._dist._process_quantiles(x, self.dim)
|
|
out = self._dist._logpdf(x, self.mean, self.cov_object)
|
|
if np.any(self.cov_object.rank < self.dim):
|
|
out_of_bounds = ~self.cov_object._support_mask(x-self.mean)
|
|
out[out_of_bounds] = -np.inf
|
|
return _squeeze_output(out)
|
|
|
|
def pdf(self, x):
|
|
return np.exp(self.logpdf(x))
|
|
|
|
def logcdf(self, x, *, lower_limit=None, rng=None):
|
|
cdf = self.cdf(x, lower_limit=lower_limit, rng=rng)
|
|
# the log of a negative real is complex, and cdf can be negative
|
|
# if lower limit is greater than upper limit
|
|
cdf = cdf + 0j if np.any(cdf < 0) else cdf
|
|
out = np.log(cdf)
|
|
return out
|
|
|
|
def cdf(self, x, *, lower_limit=None, rng=None):
|
|
x = self._dist._process_quantiles(x, self.dim)
|
|
rng = self._dist._get_random_state(rng)
|
|
out = self._dist._cdf(x, self.mean, self.cov_object.covariance,
|
|
self.maxpts, self.abseps, self.releps,
|
|
lower_limit, rng)
|
|
return _squeeze_output(out)
|
|
|
|
def rvs(self, size=1, random_state=None):
|
|
return self._dist.rvs(self.mean, self.cov_object, size, random_state)
|
|
|
|
def entropy(self):
|
|
"""Computes the differential entropy of the multivariate normal.
|
|
|
|
Returns
|
|
-------
|
|
h : scalar
|
|
Entropy of the multivariate normal distribution
|
|
|
|
"""
|
|
log_pdet = self.cov_object.log_pdet
|
|
rank = self.cov_object.rank
|
|
return 0.5 * (rank * (_LOG_2PI + 1) + log_pdet)
|
|
|
|
|
|
# Set frozen generator docstrings from corresponding docstrings in
|
|
# multivariate_normal_gen and fill in default strings in class docstrings
|
|
for name in ['logpdf', 'pdf', 'logcdf', 'cdf', 'rvs']:
|
|
method = multivariate_normal_gen.__dict__[name]
|
|
method_frozen = multivariate_normal_frozen.__dict__[name]
|
|
method_frozen.__doc__ = doccer.docformat(method.__doc__,
|
|
mvn_docdict_noparams)
|
|
method.__doc__ = doccer.docformat(method.__doc__, mvn_docdict_params)
|
|
|
|
_matnorm_doc_default_callparams = """\
|
|
mean : array_like, optional
|
|
Mean of the distribution (default: `None`)
|
|
rowcov : array_like, optional
|
|
Among-row covariance matrix of the distribution (default: ``1``)
|
|
colcov : array_like, optional
|
|
Among-column covariance matrix of the distribution (default: ``1``)
|
|
"""
|
|
|
|
_matnorm_doc_callparams_note = """\
|
|
If `mean` is set to `None` then a matrix of zeros is used for the mean.
|
|
The dimensions of this matrix are inferred from the shape of `rowcov` and
|
|
`colcov`, if these are provided, or set to ``1`` if ambiguous.
|
|
|
|
`rowcov` and `colcov` can be two-dimensional array_likes specifying the
|
|
covariance matrices directly. Alternatively, a one-dimensional array will
|
|
be be interpreted as the entries of a diagonal matrix, and a scalar or
|
|
zero-dimensional array will be interpreted as this value times the
|
|
identity matrix.
|
|
"""
|
|
|
|
_matnorm_doc_frozen_callparams = ""
|
|
|
|
_matnorm_doc_frozen_callparams_note = """\
|
|
See class definition for a detailed description of parameters."""
|
|
|
|
matnorm_docdict_params = {
|
|
'_matnorm_doc_default_callparams': _matnorm_doc_default_callparams,
|
|
'_matnorm_doc_callparams_note': _matnorm_doc_callparams_note,
|
|
'_doc_random_state': _doc_random_state
|
|
}
|
|
|
|
matnorm_docdict_noparams = {
|
|
'_matnorm_doc_default_callparams': _matnorm_doc_frozen_callparams,
|
|
'_matnorm_doc_callparams_note': _matnorm_doc_frozen_callparams_note,
|
|
'_doc_random_state': _doc_random_state
|
|
}
|
|
|
|
|
|
class matrix_normal_gen(multi_rv_generic):
|
|
r"""A matrix normal random variable.
|
|
|
|
The `mean` keyword specifies the mean. The `rowcov` keyword specifies the
|
|
among-row covariance matrix. The 'colcov' keyword specifies the
|
|
among-column covariance matrix.
|
|
|
|
Methods
|
|
-------
|
|
pdf(X, mean=None, rowcov=1, colcov=1)
|
|
Probability density function.
|
|
logpdf(X, mean=None, rowcov=1, colcov=1)
|
|
Log of the probability density function.
|
|
rvs(mean=None, rowcov=1, colcov=1, size=1, random_state=None)
|
|
Draw random samples.
|
|
entropy(rowcol=1, colcov=1)
|
|
Differential entropy.
|
|
|
|
Parameters
|
|
----------
|
|
%(_matnorm_doc_default_callparams)s
|
|
%(_doc_random_state)s
|
|
|
|
Notes
|
|
-----
|
|
%(_matnorm_doc_callparams_note)s
|
|
|
|
The covariance matrices specified by `rowcov` and `colcov` must be
|
|
(symmetric) positive definite. If the samples in `X` are
|
|
:math:`m \times n`, then `rowcov` must be :math:`m \times m` and
|
|
`colcov` must be :math:`n \times n`. `mean` must be the same shape as `X`.
|
|
|
|
The probability density function for `matrix_normal` is
|
|
|
|
.. math::
|
|
|
|
f(X) = (2 \pi)^{-\frac{mn}{2}}|U|^{-\frac{n}{2}} |V|^{-\frac{m}{2}}
|
|
\exp\left( -\frac{1}{2} \mathrm{Tr}\left[ U^{-1} (X-M) V^{-1}
|
|
(X-M)^T \right] \right),
|
|
|
|
where :math:`M` is the mean, :math:`U` the among-row covariance matrix,
|
|
:math:`V` the among-column covariance matrix.
|
|
|
|
The `allow_singular` behaviour of the `multivariate_normal`
|
|
distribution is not currently supported. Covariance matrices must be
|
|
full rank.
|
|
|
|
The `matrix_normal` distribution is closely related to the
|
|
`multivariate_normal` distribution. Specifically, :math:`\mathrm{Vec}(X)`
|
|
(the vector formed by concatenating the columns of :math:`X`) has a
|
|
multivariate normal distribution with mean :math:`\mathrm{Vec}(M)`
|
|
and covariance :math:`V \otimes U` (where :math:`\otimes` is the Kronecker
|
|
product). Sampling and pdf evaluation are
|
|
:math:`\mathcal{O}(m^3 + n^3 + m^2 n + m n^2)` for the matrix normal, but
|
|
:math:`\mathcal{O}(m^3 n^3)` for the equivalent multivariate normal,
|
|
making this equivalent form algorithmically inefficient.
|
|
|
|
.. versionadded:: 0.17.0
|
|
|
|
Examples
|
|
--------
|
|
|
|
>>> import numpy as np
|
|
>>> from scipy.stats import matrix_normal
|
|
|
|
>>> M = np.arange(6).reshape(3,2); M
|
|
array([[0, 1],
|
|
[2, 3],
|
|
[4, 5]])
|
|
>>> U = np.diag([1,2,3]); U
|
|
array([[1, 0, 0],
|
|
[0, 2, 0],
|
|
[0, 0, 3]])
|
|
>>> V = 0.3*np.identity(2); V
|
|
array([[ 0.3, 0. ],
|
|
[ 0. , 0.3]])
|
|
>>> X = M + 0.1; X
|
|
array([[ 0.1, 1.1],
|
|
[ 2.1, 3.1],
|
|
[ 4.1, 5.1]])
|
|
>>> matrix_normal.pdf(X, mean=M, rowcov=U, colcov=V)
|
|
0.023410202050005054
|
|
|
|
>>> # Equivalent multivariate normal
|
|
>>> from scipy.stats import multivariate_normal
|
|
>>> vectorised_X = X.T.flatten()
|
|
>>> equiv_mean = M.T.flatten()
|
|
>>> equiv_cov = np.kron(V,U)
|
|
>>> multivariate_normal.pdf(vectorised_X, mean=equiv_mean, cov=equiv_cov)
|
|
0.023410202050005054
|
|
|
|
Alternatively, the object may be called (as a function) to fix the mean
|
|
and covariance parameters, returning a "frozen" matrix normal
|
|
random variable:
|
|
|
|
>>> rv = matrix_normal(mean=None, rowcov=1, colcov=1)
|
|
>>> # Frozen object with the same methods but holding the given
|
|
>>> # mean and covariance fixed.
|
|
|
|
"""
|
|
|
|
def __init__(self, seed=None):
|
|
super().__init__(seed)
|
|
self.__doc__ = doccer.docformat(self.__doc__, matnorm_docdict_params)
|
|
|
|
def __call__(self, mean=None, rowcov=1, colcov=1, seed=None):
|
|
"""Create a frozen matrix normal distribution.
|
|
|
|
See `matrix_normal_frozen` for more information.
|
|
|
|
"""
|
|
return matrix_normal_frozen(mean, rowcov, colcov, seed=seed)
|
|
|
|
def _process_parameters(self, mean, rowcov, colcov):
|
|
"""
|
|
Infer dimensionality from mean or covariance matrices. Handle
|
|
defaults. Ensure compatible dimensions.
|
|
"""
|
|
|
|
# Process mean
|
|
if mean is not None:
|
|
mean = np.asarray(mean, dtype=float)
|
|
meanshape = mean.shape
|
|
if len(meanshape) != 2:
|
|
raise ValueError("Array `mean` must be two dimensional.")
|
|
if np.any(meanshape == 0):
|
|
raise ValueError("Array `mean` has invalid shape.")
|
|
|
|
# Process among-row covariance
|
|
rowcov = np.asarray(rowcov, dtype=float)
|
|
if rowcov.ndim == 0:
|
|
if mean is not None:
|
|
rowcov = rowcov * np.identity(meanshape[0])
|
|
else:
|
|
rowcov = rowcov * np.identity(1)
|
|
elif rowcov.ndim == 1:
|
|
rowcov = np.diag(rowcov)
|
|
rowshape = rowcov.shape
|
|
if len(rowshape) != 2:
|
|
raise ValueError("`rowcov` must be a scalar or a 2D array.")
|
|
if rowshape[0] != rowshape[1]:
|
|
raise ValueError("Array `rowcov` must be square.")
|
|
if rowshape[0] == 0:
|
|
raise ValueError("Array `rowcov` has invalid shape.")
|
|
numrows = rowshape[0]
|
|
|
|
# Process among-column covariance
|
|
colcov = np.asarray(colcov, dtype=float)
|
|
if colcov.ndim == 0:
|
|
if mean is not None:
|
|
colcov = colcov * np.identity(meanshape[1])
|
|
else:
|
|
colcov = colcov * np.identity(1)
|
|
elif colcov.ndim == 1:
|
|
colcov = np.diag(colcov)
|
|
colshape = colcov.shape
|
|
if len(colshape) != 2:
|
|
raise ValueError("`colcov` must be a scalar or a 2D array.")
|
|
if colshape[0] != colshape[1]:
|
|
raise ValueError("Array `colcov` must be square.")
|
|
if colshape[0] == 0:
|
|
raise ValueError("Array `colcov` has invalid shape.")
|
|
numcols = colshape[0]
|
|
|
|
# Ensure mean and covariances compatible
|
|
if mean is not None:
|
|
if meanshape[0] != numrows:
|
|
raise ValueError("Arrays `mean` and `rowcov` must have the "
|
|
"same number of rows.")
|
|
if meanshape[1] != numcols:
|
|
raise ValueError("Arrays `mean` and `colcov` must have the "
|
|
"same number of columns.")
|
|
else:
|
|
mean = np.zeros((numrows, numcols))
|
|
|
|
dims = (numrows, numcols)
|
|
|
|
return dims, mean, rowcov, colcov
|
|
|
|
def _process_quantiles(self, X, dims):
|
|
"""
|
|
Adjust quantiles array so that last two axes labels the components of
|
|
each data point.
|
|
"""
|
|
X = np.asarray(X, dtype=float)
|
|
if X.ndim == 2:
|
|
X = X[np.newaxis, :]
|
|
if X.shape[-2:] != dims:
|
|
raise ValueError("The shape of array `X` is not compatible "
|
|
"with the distribution parameters.")
|
|
return X
|
|
|
|
def _logpdf(self, dims, X, mean, row_prec_rt, log_det_rowcov,
|
|
col_prec_rt, log_det_colcov):
|
|
"""Log of the matrix normal probability density function.
|
|
|
|
Parameters
|
|
----------
|
|
dims : tuple
|
|
Dimensions of the matrix variates
|
|
X : ndarray
|
|
Points at which to evaluate the log of the probability
|
|
density function
|
|
mean : ndarray
|
|
Mean of the distribution
|
|
row_prec_rt : ndarray
|
|
A decomposition such that np.dot(row_prec_rt, row_prec_rt.T)
|
|
is the inverse of the among-row covariance matrix
|
|
log_det_rowcov : float
|
|
Logarithm of the determinant of the among-row covariance matrix
|
|
col_prec_rt : ndarray
|
|
A decomposition such that np.dot(col_prec_rt, col_prec_rt.T)
|
|
is the inverse of the among-column covariance matrix
|
|
log_det_colcov : float
|
|
Logarithm of the determinant of the among-column covariance matrix
|
|
|
|
Notes
|
|
-----
|
|
As this function does no argument checking, it should not be
|
|
called directly; use 'logpdf' instead.
|
|
|
|
"""
|
|
numrows, numcols = dims
|
|
roll_dev = np.moveaxis(X-mean, -1, 0)
|
|
scale_dev = np.tensordot(col_prec_rt.T,
|
|
np.dot(roll_dev, row_prec_rt), 1)
|
|
maha = np.sum(np.sum(np.square(scale_dev), axis=-1), axis=0)
|
|
return -0.5 * (numrows*numcols*_LOG_2PI + numcols*log_det_rowcov
|
|
+ numrows*log_det_colcov + maha)
|
|
|
|
def logpdf(self, X, mean=None, rowcov=1, colcov=1):
|
|
"""Log of the matrix normal probability density function.
|
|
|
|
Parameters
|
|
----------
|
|
X : array_like
|
|
Quantiles, with the last two axes of `X` denoting the components.
|
|
%(_matnorm_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
logpdf : ndarray
|
|
Log of the probability density function evaluated at `X`
|
|
|
|
Notes
|
|
-----
|
|
%(_matnorm_doc_callparams_note)s
|
|
|
|
"""
|
|
dims, mean, rowcov, colcov = self._process_parameters(mean, rowcov,
|
|
colcov)
|
|
X = self._process_quantiles(X, dims)
|
|
rowpsd = _PSD(rowcov, allow_singular=False)
|
|
colpsd = _PSD(colcov, allow_singular=False)
|
|
out = self._logpdf(dims, X, mean, rowpsd.U, rowpsd.log_pdet, colpsd.U,
|
|
colpsd.log_pdet)
|
|
return _squeeze_output(out)
|
|
|
|
def pdf(self, X, mean=None, rowcov=1, colcov=1):
|
|
"""Matrix normal probability density function.
|
|
|
|
Parameters
|
|
----------
|
|
X : array_like
|
|
Quantiles, with the last two axes of `X` denoting the components.
|
|
%(_matnorm_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
pdf : ndarray
|
|
Probability density function evaluated at `X`
|
|
|
|
Notes
|
|
-----
|
|
%(_matnorm_doc_callparams_note)s
|
|
|
|
"""
|
|
return np.exp(self.logpdf(X, mean, rowcov, colcov))
|
|
|
|
def rvs(self, mean=None, rowcov=1, colcov=1, size=1, random_state=None):
|
|
"""Draw random samples from a matrix normal distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_matnorm_doc_default_callparams)s
|
|
size : integer, optional
|
|
Number of samples to draw (default 1).
|
|
%(_doc_random_state)s
|
|
|
|
Returns
|
|
-------
|
|
rvs : ndarray or scalar
|
|
Random variates of size (`size`, `dims`), where `dims` is the
|
|
dimension of the random matrices.
|
|
|
|
Notes
|
|
-----
|
|
%(_matnorm_doc_callparams_note)s
|
|
|
|
"""
|
|
size = int(size)
|
|
dims, mean, rowcov, colcov = self._process_parameters(mean, rowcov,
|
|
colcov)
|
|
rowchol = scipy.linalg.cholesky(rowcov, lower=True)
|
|
colchol = scipy.linalg.cholesky(colcov, lower=True)
|
|
random_state = self._get_random_state(random_state)
|
|
# We aren't generating standard normal variates with size=(size,
|
|
# dims[0], dims[1]) directly to ensure random variates remain backwards
|
|
# compatible. See https://github.com/scipy/scipy/pull/12312 for more
|
|
# details.
|
|
std_norm = random_state.standard_normal(
|
|
size=(dims[1], size, dims[0])
|
|
).transpose(1, 2, 0)
|
|
out = mean + np.einsum('jp,ipq,kq->ijk',
|
|
rowchol, std_norm, colchol,
|
|
optimize=True)
|
|
if size == 1:
|
|
out = out.reshape(mean.shape)
|
|
return out
|
|
|
|
def entropy(self, rowcov=1, colcov=1):
|
|
"""Log of the matrix normal probability density function.
|
|
|
|
Parameters
|
|
----------
|
|
rowcov : array_like, optional
|
|
Among-row covariance matrix of the distribution (default: ``1``)
|
|
colcov : array_like, optional
|
|
Among-column covariance matrix of the distribution (default: ``1``)
|
|
|
|
Returns
|
|
-------
|
|
entropy : float
|
|
Entropy of the distribution
|
|
|
|
Notes
|
|
-----
|
|
%(_matnorm_doc_callparams_note)s
|
|
|
|
"""
|
|
dummy_mean = np.zeros((rowcov.shape[0], colcov.shape[0]))
|
|
dims, _, rowcov, colcov = self._process_parameters(dummy_mean,
|
|
rowcov,
|
|
colcov)
|
|
rowpsd = _PSD(rowcov, allow_singular=False)
|
|
colpsd = _PSD(colcov, allow_singular=False)
|
|
|
|
return self._entropy(dims, rowpsd.log_pdet, colpsd.log_pdet)
|
|
|
|
def _entropy(self, dims, row_cov_logdet, col_cov_logdet):
|
|
n, p = dims
|
|
return (0.5 * n * p * (1 + _LOG_2PI) + 0.5 * p * row_cov_logdet +
|
|
0.5 * n * col_cov_logdet)
|
|
|
|
|
|
matrix_normal = matrix_normal_gen()
|
|
|
|
|
|
class matrix_normal_frozen(multi_rv_frozen):
|
|
"""
|
|
Create a frozen matrix normal distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_matnorm_doc_default_callparams)s
|
|
seed : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional
|
|
If `seed` is `None` the `~np.random.RandomState` singleton is used.
|
|
If `seed` is an int, a new ``RandomState`` instance is used, seeded
|
|
with seed.
|
|
If `seed` is already a ``RandomState`` or ``Generator`` instance,
|
|
then that object is used.
|
|
Default is `None`.
|
|
|
|
Examples
|
|
--------
|
|
>>> import numpy as np
|
|
>>> from scipy.stats import matrix_normal
|
|
|
|
>>> distn = matrix_normal(mean=np.zeros((3,3)))
|
|
>>> X = distn.rvs(); X
|
|
array([[-0.02976962, 0.93339138, -0.09663178],
|
|
[ 0.67405524, 0.28250467, -0.93308929],
|
|
[-0.31144782, 0.74535536, 1.30412916]])
|
|
>>> distn.pdf(X)
|
|
2.5160642368346784e-05
|
|
>>> distn.logpdf(X)
|
|
-10.590229595124615
|
|
"""
|
|
|
|
def __init__(self, mean=None, rowcov=1, colcov=1, seed=None):
|
|
self._dist = matrix_normal_gen(seed)
|
|
self.dims, self.mean, self.rowcov, self.colcov = \
|
|
self._dist._process_parameters(mean, rowcov, colcov)
|
|
self.rowpsd = _PSD(self.rowcov, allow_singular=False)
|
|
self.colpsd = _PSD(self.colcov, allow_singular=False)
|
|
|
|
def logpdf(self, X):
|
|
X = self._dist._process_quantiles(X, self.dims)
|
|
out = self._dist._logpdf(self.dims, X, self.mean, self.rowpsd.U,
|
|
self.rowpsd.log_pdet, self.colpsd.U,
|
|
self.colpsd.log_pdet)
|
|
return _squeeze_output(out)
|
|
|
|
def pdf(self, X):
|
|
return np.exp(self.logpdf(X))
|
|
|
|
def rvs(self, size=1, random_state=None):
|
|
return self._dist.rvs(self.mean, self.rowcov, self.colcov, size,
|
|
random_state)
|
|
|
|
def entropy(self):
|
|
return self._dist._entropy(self.dims, self.rowpsd.log_pdet,
|
|
self.colpsd.log_pdet)
|
|
|
|
|
|
# Set frozen generator docstrings from corresponding docstrings in
|
|
# matrix_normal_gen and fill in default strings in class docstrings
|
|
for name in ['logpdf', 'pdf', 'rvs', 'entropy']:
|
|
method = matrix_normal_gen.__dict__[name]
|
|
method_frozen = matrix_normal_frozen.__dict__[name]
|
|
method_frozen.__doc__ = doccer.docformat(method.__doc__,
|
|
matnorm_docdict_noparams)
|
|
method.__doc__ = doccer.docformat(method.__doc__, matnorm_docdict_params)
|
|
|
|
_dirichlet_doc_default_callparams = """\
|
|
alpha : array_like
|
|
The concentration parameters. The number of entries determines the
|
|
dimensionality of the distribution.
|
|
"""
|
|
_dirichlet_doc_frozen_callparams = ""
|
|
|
|
_dirichlet_doc_frozen_callparams_note = """\
|
|
See class definition for a detailed description of parameters."""
|
|
|
|
dirichlet_docdict_params = {
|
|
'_dirichlet_doc_default_callparams': _dirichlet_doc_default_callparams,
|
|
'_doc_random_state': _doc_random_state
|
|
}
|
|
|
|
dirichlet_docdict_noparams = {
|
|
'_dirichlet_doc_default_callparams': _dirichlet_doc_frozen_callparams,
|
|
'_doc_random_state': _doc_random_state
|
|
}
|
|
|
|
|
|
def _dirichlet_check_parameters(alpha):
|
|
alpha = np.asarray(alpha)
|
|
if np.min(alpha) <= 0:
|
|
raise ValueError("All parameters must be greater than 0")
|
|
elif alpha.ndim != 1:
|
|
raise ValueError("Parameter vector 'a' must be one dimensional, "
|
|
f"but a.shape = {alpha.shape}.")
|
|
return alpha
|
|
|
|
|
|
def _dirichlet_check_input(alpha, x):
|
|
x = np.asarray(x)
|
|
|
|
if x.shape[0] + 1 != alpha.shape[0] and x.shape[0] != alpha.shape[0]:
|
|
raise ValueError("Vector 'x' must have either the same number "
|
|
"of entries as, or one entry fewer than, "
|
|
f"parameter vector 'a', but alpha.shape = {alpha.shape} "
|
|
f"and x.shape = {x.shape}.")
|
|
|
|
if x.shape[0] != alpha.shape[0]:
|
|
xk = np.array([1 - np.sum(x, 0)])
|
|
if xk.ndim == 1:
|
|
x = np.append(x, xk)
|
|
elif xk.ndim == 2:
|
|
x = np.vstack((x, xk))
|
|
else:
|
|
raise ValueError("The input must be one dimensional or a two "
|
|
"dimensional matrix containing the entries.")
|
|
|
|
if np.min(x) < 0:
|
|
raise ValueError("Each entry in 'x' must be greater than or equal "
|
|
"to zero.")
|
|
|
|
if np.max(x) > 1:
|
|
raise ValueError("Each entry in 'x' must be smaller or equal one.")
|
|
|
|
# Check x_i > 0 or alpha_i > 1
|
|
xeq0 = (x == 0)
|
|
alphalt1 = (alpha < 1)
|
|
if x.shape != alpha.shape:
|
|
alphalt1 = np.repeat(alphalt1, x.shape[-1], axis=-1).reshape(x.shape)
|
|
chk = np.logical_and(xeq0, alphalt1)
|
|
|
|
if np.sum(chk):
|
|
raise ValueError("Each entry in 'x' must be greater than zero if its "
|
|
"alpha is less than one.")
|
|
|
|
if (np.abs(np.sum(x, 0) - 1.0) > 10e-10).any():
|
|
raise ValueError("The input vector 'x' must lie within the normal "
|
|
f"simplex. but np.sum(x, 0) = {np.sum(x, 0)}.")
|
|
|
|
return x
|
|
|
|
|
|
def _lnB(alpha):
|
|
r"""Internal helper function to compute the log of the useful quotient.
|
|
|
|
.. math::
|
|
|
|
B(\alpha) = \frac{\prod_{i=1}{K}\Gamma(\alpha_i)}
|
|
{\Gamma\left(\sum_{i=1}^{K} \alpha_i \right)}
|
|
|
|
Parameters
|
|
----------
|
|
%(_dirichlet_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
B : scalar
|
|
Helper quotient, internal use only
|
|
|
|
"""
|
|
return np.sum(gammaln(alpha)) - gammaln(np.sum(alpha))
|
|
|
|
|
|
class dirichlet_gen(multi_rv_generic):
|
|
r"""A Dirichlet random variable.
|
|
|
|
The ``alpha`` keyword specifies the concentration parameters of the
|
|
distribution.
|
|
|
|
.. versionadded:: 0.15.0
|
|
|
|
Methods
|
|
-------
|
|
pdf(x, alpha)
|
|
Probability density function.
|
|
logpdf(x, alpha)
|
|
Log of the probability density function.
|
|
rvs(alpha, size=1, random_state=None)
|
|
Draw random samples from a Dirichlet distribution.
|
|
mean(alpha)
|
|
The mean of the Dirichlet distribution
|
|
var(alpha)
|
|
The variance of the Dirichlet distribution
|
|
cov(alpha)
|
|
The covariance of the Dirichlet distribution
|
|
entropy(alpha)
|
|
Compute the differential entropy of the Dirichlet distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_dirichlet_doc_default_callparams)s
|
|
%(_doc_random_state)s
|
|
|
|
Notes
|
|
-----
|
|
Each :math:`\alpha` entry must be positive. The distribution has only
|
|
support on the simplex defined by
|
|
|
|
.. math::
|
|
\sum_{i=1}^{K} x_i = 1
|
|
|
|
where :math:`0 < x_i < 1`.
|
|
|
|
If the quantiles don't lie within the simplex, a ValueError is raised.
|
|
|
|
The probability density function for `dirichlet` is
|
|
|
|
.. math::
|
|
|
|
f(x) = \frac{1}{\mathrm{B}(\boldsymbol\alpha)} \prod_{i=1}^K x_i^{\alpha_i - 1}
|
|
|
|
where
|
|
|
|
.. math::
|
|
|
|
\mathrm{B}(\boldsymbol\alpha) = \frac{\prod_{i=1}^K \Gamma(\alpha_i)}
|
|
{\Gamma\bigl(\sum_{i=1}^K \alpha_i\bigr)}
|
|
|
|
and :math:`\boldsymbol\alpha=(\alpha_1,\ldots,\alpha_K)`, the
|
|
concentration parameters and :math:`K` is the dimension of the space
|
|
where :math:`x` takes values.
|
|
|
|
Note that the `dirichlet` interface is somewhat inconsistent.
|
|
The array returned by the rvs function is transposed
|
|
with respect to the format expected by the pdf and logpdf.
|
|
|
|
Examples
|
|
--------
|
|
>>> import numpy as np
|
|
>>> from scipy.stats import dirichlet
|
|
|
|
Generate a dirichlet random variable
|
|
|
|
>>> quantiles = np.array([0.2, 0.2, 0.6]) # specify quantiles
|
|
>>> alpha = np.array([0.4, 5, 15]) # specify concentration parameters
|
|
>>> dirichlet.pdf(quantiles, alpha)
|
|
0.2843831684937255
|
|
|
|
The same PDF but following a log scale
|
|
|
|
>>> dirichlet.logpdf(quantiles, alpha)
|
|
-1.2574327653159187
|
|
|
|
Once we specify the dirichlet distribution
|
|
we can then calculate quantities of interest
|
|
|
|
>>> dirichlet.mean(alpha) # get the mean of the distribution
|
|
array([0.01960784, 0.24509804, 0.73529412])
|
|
>>> dirichlet.var(alpha) # get variance
|
|
array([0.00089829, 0.00864603, 0.00909517])
|
|
>>> dirichlet.entropy(alpha) # calculate the differential entropy
|
|
-4.3280162474082715
|
|
|
|
We can also return random samples from the distribution
|
|
|
|
>>> dirichlet.rvs(alpha, size=1, random_state=1)
|
|
array([[0.00766178, 0.24670518, 0.74563305]])
|
|
>>> dirichlet.rvs(alpha, size=2, random_state=2)
|
|
array([[0.01639427, 0.1292273 , 0.85437844],
|
|
[0.00156917, 0.19033695, 0.80809388]])
|
|
|
|
Alternatively, the object may be called (as a function) to fix
|
|
concentration parameters, returning a "frozen" Dirichlet
|
|
random variable:
|
|
|
|
>>> rv = dirichlet(alpha)
|
|
>>> # Frozen object with the same methods but holding the given
|
|
>>> # concentration parameters fixed.
|
|
|
|
"""
|
|
|
|
def __init__(self, seed=None):
|
|
super().__init__(seed)
|
|
self.__doc__ = doccer.docformat(self.__doc__, dirichlet_docdict_params)
|
|
|
|
def __call__(self, alpha, seed=None):
|
|
return dirichlet_frozen(alpha, seed=seed)
|
|
|
|
def _logpdf(self, x, alpha):
|
|
"""Log of the Dirichlet probability density function.
|
|
|
|
Parameters
|
|
----------
|
|
x : ndarray
|
|
Points at which to evaluate the log of the probability
|
|
density function
|
|
%(_dirichlet_doc_default_callparams)s
|
|
|
|
Notes
|
|
-----
|
|
As this function does no argument checking, it should not be
|
|
called directly; use 'logpdf' instead.
|
|
|
|
"""
|
|
lnB = _lnB(alpha)
|
|
return - lnB + np.sum((xlogy(alpha - 1, x.T)).T, 0)
|
|
|
|
def logpdf(self, x, alpha):
|
|
"""Log of the Dirichlet probability density function.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
Quantiles, with the last axis of `x` denoting the components.
|
|
%(_dirichlet_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
pdf : ndarray or scalar
|
|
Log of the probability density function evaluated at `x`.
|
|
|
|
"""
|
|
alpha = _dirichlet_check_parameters(alpha)
|
|
x = _dirichlet_check_input(alpha, x)
|
|
|
|
out = self._logpdf(x, alpha)
|
|
return _squeeze_output(out)
|
|
|
|
def pdf(self, x, alpha):
|
|
"""The Dirichlet probability density function.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
Quantiles, with the last axis of `x` denoting the components.
|
|
%(_dirichlet_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
pdf : ndarray or scalar
|
|
The probability density function evaluated at `x`.
|
|
|
|
"""
|
|
alpha = _dirichlet_check_parameters(alpha)
|
|
x = _dirichlet_check_input(alpha, x)
|
|
|
|
out = np.exp(self._logpdf(x, alpha))
|
|
return _squeeze_output(out)
|
|
|
|
def mean(self, alpha):
|
|
"""Mean of the Dirichlet distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_dirichlet_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
mu : ndarray or scalar
|
|
Mean of the Dirichlet distribution.
|
|
|
|
"""
|
|
alpha = _dirichlet_check_parameters(alpha)
|
|
|
|
out = alpha / (np.sum(alpha))
|
|
return _squeeze_output(out)
|
|
|
|
def var(self, alpha):
|
|
"""Variance of the Dirichlet distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_dirichlet_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
v : ndarray or scalar
|
|
Variance of the Dirichlet distribution.
|
|
|
|
"""
|
|
|
|
alpha = _dirichlet_check_parameters(alpha)
|
|
|
|
alpha0 = np.sum(alpha)
|
|
out = (alpha * (alpha0 - alpha)) / ((alpha0 * alpha0) * (alpha0 + 1))
|
|
return _squeeze_output(out)
|
|
|
|
def cov(self, alpha):
|
|
"""Covariance matrix of the Dirichlet distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_dirichlet_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
cov : ndarray
|
|
The covariance matrix of the distribution.
|
|
"""
|
|
|
|
alpha = _dirichlet_check_parameters(alpha)
|
|
alpha0 = np.sum(alpha)
|
|
a = alpha / alpha0
|
|
|
|
cov = (np.diag(a) - np.outer(a, a)) / (alpha0 + 1)
|
|
return _squeeze_output(cov)
|
|
|
|
def entropy(self, alpha):
|
|
"""
|
|
Differential entropy of the Dirichlet distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_dirichlet_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
h : scalar
|
|
Entropy of the Dirichlet distribution
|
|
|
|
"""
|
|
|
|
alpha = _dirichlet_check_parameters(alpha)
|
|
|
|
alpha0 = np.sum(alpha)
|
|
lnB = _lnB(alpha)
|
|
K = alpha.shape[0]
|
|
|
|
out = lnB + (alpha0 - K) * scipy.special.psi(alpha0) - np.sum(
|
|
(alpha - 1) * scipy.special.psi(alpha))
|
|
return _squeeze_output(out)
|
|
|
|
def rvs(self, alpha, size=1, random_state=None):
|
|
"""
|
|
Draw random samples from a Dirichlet distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_dirichlet_doc_default_callparams)s
|
|
size : int, optional
|
|
Number of samples to draw (default 1).
|
|
%(_doc_random_state)s
|
|
|
|
Returns
|
|
-------
|
|
rvs : ndarray or scalar
|
|
Random variates of size (`size`, `N`), where `N` is the
|
|
dimension of the random variable.
|
|
|
|
"""
|
|
alpha = _dirichlet_check_parameters(alpha)
|
|
random_state = self._get_random_state(random_state)
|
|
return random_state.dirichlet(alpha, size=size)
|
|
|
|
|
|
dirichlet = dirichlet_gen()
|
|
|
|
|
|
class dirichlet_frozen(multi_rv_frozen):
|
|
def __init__(self, alpha, seed=None):
|
|
self.alpha = _dirichlet_check_parameters(alpha)
|
|
self._dist = dirichlet_gen(seed)
|
|
|
|
def logpdf(self, x):
|
|
return self._dist.logpdf(x, self.alpha)
|
|
|
|
def pdf(self, x):
|
|
return self._dist.pdf(x, self.alpha)
|
|
|
|
def mean(self):
|
|
return self._dist.mean(self.alpha)
|
|
|
|
def var(self):
|
|
return self._dist.var(self.alpha)
|
|
|
|
def cov(self):
|
|
return self._dist.cov(self.alpha)
|
|
|
|
def entropy(self):
|
|
return self._dist.entropy(self.alpha)
|
|
|
|
def rvs(self, size=1, random_state=None):
|
|
return self._dist.rvs(self.alpha, size, random_state)
|
|
|
|
|
|
# Set frozen generator docstrings from corresponding docstrings in
|
|
# multivariate_normal_gen and fill in default strings in class docstrings
|
|
for name in ['logpdf', 'pdf', 'rvs', 'mean', 'var', 'cov', 'entropy']:
|
|
method = dirichlet_gen.__dict__[name]
|
|
method_frozen = dirichlet_frozen.__dict__[name]
|
|
method_frozen.__doc__ = doccer.docformat(
|
|
method.__doc__, dirichlet_docdict_noparams)
|
|
method.__doc__ = doccer.docformat(method.__doc__, dirichlet_docdict_params)
|
|
|
|
|
|
_wishart_doc_default_callparams = """\
|
|
df : int
|
|
Degrees of freedom, must be greater than or equal to dimension of the
|
|
scale matrix
|
|
scale : array_like
|
|
Symmetric positive definite scale matrix of the distribution
|
|
"""
|
|
|
|
_wishart_doc_callparams_note = ""
|
|
|
|
_wishart_doc_frozen_callparams = ""
|
|
|
|
_wishart_doc_frozen_callparams_note = """\
|
|
See class definition for a detailed description of parameters."""
|
|
|
|
wishart_docdict_params = {
|
|
'_doc_default_callparams': _wishart_doc_default_callparams,
|
|
'_doc_callparams_note': _wishart_doc_callparams_note,
|
|
'_doc_random_state': _doc_random_state
|
|
}
|
|
|
|
wishart_docdict_noparams = {
|
|
'_doc_default_callparams': _wishart_doc_frozen_callparams,
|
|
'_doc_callparams_note': _wishart_doc_frozen_callparams_note,
|
|
'_doc_random_state': _doc_random_state
|
|
}
|
|
|
|
|
|
class wishart_gen(multi_rv_generic):
|
|
r"""A Wishart random variable.
|
|
|
|
The `df` keyword specifies the degrees of freedom. The `scale` keyword
|
|
specifies the scale matrix, which must be symmetric and positive definite.
|
|
In this context, the scale matrix is often interpreted in terms of a
|
|
multivariate normal precision matrix (the inverse of the covariance
|
|
matrix). These arguments must satisfy the relationship
|
|
``df > scale.ndim - 1``, but see notes on using the `rvs` method with
|
|
``df < scale.ndim``.
|
|
|
|
Methods
|
|
-------
|
|
pdf(x, df, scale)
|
|
Probability density function.
|
|
logpdf(x, df, scale)
|
|
Log of the probability density function.
|
|
rvs(df, scale, size=1, random_state=None)
|
|
Draw random samples from a Wishart distribution.
|
|
entropy()
|
|
Compute the differential entropy of the Wishart distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_doc_default_callparams)s
|
|
%(_doc_random_state)s
|
|
|
|
Raises
|
|
------
|
|
scipy.linalg.LinAlgError
|
|
If the scale matrix `scale` is not positive definite.
|
|
|
|
See Also
|
|
--------
|
|
invwishart, chi2
|
|
|
|
Notes
|
|
-----
|
|
%(_doc_callparams_note)s
|
|
|
|
The scale matrix `scale` must be a symmetric positive definite
|
|
matrix. Singular matrices, including the symmetric positive semi-definite
|
|
case, are not supported. Symmetry is not checked; only the lower triangular
|
|
portion is used.
|
|
|
|
The Wishart distribution is often denoted
|
|
|
|
.. math::
|
|
|
|
W_p(\nu, \Sigma)
|
|
|
|
where :math:`\nu` is the degrees of freedom and :math:`\Sigma` is the
|
|
:math:`p \times p` scale matrix.
|
|
|
|
The probability density function for `wishart` has support over positive
|
|
definite matrices :math:`S`; if :math:`S \sim W_p(\nu, \Sigma)`, then
|
|
its PDF is given by:
|
|
|
|
.. math::
|
|
|
|
f(S) = \frac{|S|^{\frac{\nu - p - 1}{2}}}{2^{ \frac{\nu p}{2} }
|
|
|\Sigma|^\frac{\nu}{2} \Gamma_p \left ( \frac{\nu}{2} \right )}
|
|
\exp\left( -tr(\Sigma^{-1} S) / 2 \right)
|
|
|
|
If :math:`S \sim W_p(\nu, \Sigma)` (Wishart) then
|
|
:math:`S^{-1} \sim W_p^{-1}(\nu, \Sigma^{-1})` (inverse Wishart).
|
|
|
|
If the scale matrix is 1-dimensional and equal to one, then the Wishart
|
|
distribution :math:`W_1(\nu, 1)` collapses to the :math:`\chi^2(\nu)`
|
|
distribution.
|
|
|
|
The algorithm [2]_ implemented by the `rvs` method may
|
|
produce numerically singular matrices with :math:`p - 1 < \nu < p`; the
|
|
user may wish to check for this condition and generate replacement samples
|
|
as necessary.
|
|
|
|
|
|
.. versionadded:: 0.16.0
|
|
|
|
References
|
|
----------
|
|
.. [1] M.L. Eaton, "Multivariate Statistics: A Vector Space Approach",
|
|
Wiley, 1983.
|
|
.. [2] W.B. Smith and R.R. Hocking, "Algorithm AS 53: Wishart Variate
|
|
Generator", Applied Statistics, vol. 21, pp. 341-345, 1972.
|
|
|
|
Examples
|
|
--------
|
|
>>> import numpy as np
|
|
>>> import matplotlib.pyplot as plt
|
|
>>> from scipy.stats import wishart, chi2
|
|
>>> x = np.linspace(1e-5, 8, 100)
|
|
>>> w = wishart.pdf(x, df=3, scale=1); w[:5]
|
|
array([ 0.00126156, 0.10892176, 0.14793434, 0.17400548, 0.1929669 ])
|
|
>>> c = chi2.pdf(x, 3); c[:5]
|
|
array([ 0.00126156, 0.10892176, 0.14793434, 0.17400548, 0.1929669 ])
|
|
>>> plt.plot(x, w)
|
|
>>> plt.show()
|
|
|
|
The input quantiles can be any shape of array, as long as the last
|
|
axis labels the components.
|
|
|
|
Alternatively, the object may be called (as a function) to fix the degrees
|
|
of freedom and scale parameters, returning a "frozen" Wishart random
|
|
variable:
|
|
|
|
>>> rv = wishart(df=1, scale=1)
|
|
>>> # Frozen object with the same methods but holding the given
|
|
>>> # degrees of freedom and scale fixed.
|
|
|
|
"""
|
|
|
|
def __init__(self, seed=None):
|
|
super().__init__(seed)
|
|
self.__doc__ = doccer.docformat(self.__doc__, wishart_docdict_params)
|
|
|
|
def __call__(self, df=None, scale=None, seed=None):
|
|
"""Create a frozen Wishart distribution.
|
|
|
|
See `wishart_frozen` for more information.
|
|
"""
|
|
return wishart_frozen(df, scale, seed)
|
|
|
|
def _process_parameters(self, df, scale):
|
|
if scale is None:
|
|
scale = 1.0
|
|
scale = np.asarray(scale, dtype=float)
|
|
|
|
if scale.ndim == 0:
|
|
scale = scale[np.newaxis, np.newaxis]
|
|
elif scale.ndim == 1:
|
|
scale = np.diag(scale)
|
|
elif scale.ndim == 2 and not scale.shape[0] == scale.shape[1]:
|
|
raise ValueError("Array 'scale' must be square if it is two dimensional,"
|
|
f" but scale.scale = {str(scale.shape)}.")
|
|
elif scale.ndim > 2:
|
|
raise ValueError(f"Array 'scale' must be at most two-dimensional, "
|
|
f"but scale.ndim = {scale.ndim}")
|
|
|
|
dim = scale.shape[0]
|
|
|
|
if df is None:
|
|
df = dim
|
|
elif not np.isscalar(df):
|
|
raise ValueError("Degrees of freedom must be a scalar.")
|
|
elif df <= dim - 1:
|
|
raise ValueError("Degrees of freedom must be greater than the "
|
|
"dimension of scale matrix minus 1.")
|
|
|
|
return dim, df, scale
|
|
|
|
def _process_quantiles(self, x, dim):
|
|
"""
|
|
Adjust quantiles array so that last axis labels the components of
|
|
each data point.
|
|
"""
|
|
x = np.asarray(x, dtype=float)
|
|
|
|
if x.ndim == 0:
|
|
x = x * np.eye(dim)[:, :, np.newaxis]
|
|
if x.ndim == 1:
|
|
if dim == 1:
|
|
x = x[np.newaxis, np.newaxis, :]
|
|
else:
|
|
x = np.diag(x)[:, :, np.newaxis]
|
|
elif x.ndim == 2:
|
|
if not x.shape[0] == x.shape[1]:
|
|
raise ValueError(
|
|
"Quantiles must be square if they are two dimensional,"
|
|
f" but x.shape = {str(x.shape)}.")
|
|
x = x[:, :, np.newaxis]
|
|
elif x.ndim == 3:
|
|
if not x.shape[0] == x.shape[1]:
|
|
raise ValueError(
|
|
"Quantiles must be square in the first two dimensions "
|
|
f"if they are three dimensional, but x.shape = {str(x.shape)}.")
|
|
elif x.ndim > 3:
|
|
raise ValueError(f"Quantiles must be at most two-dimensional with an "
|
|
f"additional dimension for multiple components, "
|
|
f"but x.ndim = {x.ndim}")
|
|
|
|
# Now we have 3-dim array; should have shape [dim, dim, *]
|
|
if not x.shape[0:2] == (dim, dim):
|
|
raise ValueError('Quantiles have incompatible dimensions: should'
|
|
f' be {(dim, dim)}, got {x.shape[0:2]}.')
|
|
|
|
return x
|
|
|
|
def _process_size(self, size):
|
|
size = np.asarray(size)
|
|
|
|
if size.ndim == 0:
|
|
size = size[np.newaxis]
|
|
elif size.ndim > 1:
|
|
raise ValueError('Size must be an integer or tuple of integers;'
|
|
' thus must have dimension <= 1.'
|
|
f' Got size.ndim = {str(tuple(size))}')
|
|
n = size.prod()
|
|
shape = tuple(size)
|
|
|
|
return n, shape
|
|
|
|
def _logpdf(self, x, dim, df, scale, log_det_scale, C):
|
|
"""Log of the Wishart probability density function.
|
|
|
|
Parameters
|
|
----------
|
|
x : ndarray
|
|
Points at which to evaluate the log of the probability
|
|
density function
|
|
dim : int
|
|
Dimension of the scale matrix
|
|
df : int
|
|
Degrees of freedom
|
|
scale : ndarray
|
|
Scale matrix
|
|
log_det_scale : float
|
|
Logarithm of the determinant of the scale matrix
|
|
C : ndarray
|
|
Cholesky factorization of the scale matrix, lower triangular.
|
|
|
|
Notes
|
|
-----
|
|
As this function does no argument checking, it should not be
|
|
called directly; use 'logpdf' instead.
|
|
|
|
"""
|
|
# log determinant of x
|
|
# Note: x has components along the last axis, so that x.T has
|
|
# components alone the 0-th axis. Then since det(A) = det(A'), this
|
|
# gives us a 1-dim vector of determinants
|
|
|
|
# Retrieve tr(scale^{-1} x)
|
|
log_det_x = np.empty(x.shape[-1])
|
|
scale_inv_x = np.empty(x.shape)
|
|
tr_scale_inv_x = np.empty(x.shape[-1])
|
|
for i in range(x.shape[-1]):
|
|
_, log_det_x[i] = self._cholesky_logdet(x[:, :, i])
|
|
scale_inv_x[:, :, i] = scipy.linalg.cho_solve((C, True), x[:, :, i])
|
|
tr_scale_inv_x[i] = scale_inv_x[:, :, i].trace()
|
|
|
|
# Log PDF
|
|
out = ((0.5 * (df - dim - 1) * log_det_x - 0.5 * tr_scale_inv_x) -
|
|
(0.5 * df * dim * _LOG_2 + 0.5 * df * log_det_scale +
|
|
multigammaln(0.5*df, dim)))
|
|
|
|
return out
|
|
|
|
def logpdf(self, x, df, scale):
|
|
"""Log of the Wishart probability density function.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
Quantiles, with the last axis of `x` denoting the components.
|
|
Each quantile must be a symmetric positive definite matrix.
|
|
%(_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
pdf : ndarray
|
|
Log of the probability density function evaluated at `x`
|
|
|
|
Notes
|
|
-----
|
|
%(_doc_callparams_note)s
|
|
|
|
"""
|
|
dim, df, scale = self._process_parameters(df, scale)
|
|
x = self._process_quantiles(x, dim)
|
|
|
|
# Cholesky decomposition of scale, get log(det(scale))
|
|
C, log_det_scale = self._cholesky_logdet(scale)
|
|
|
|
out = self._logpdf(x, dim, df, scale, log_det_scale, C)
|
|
return _squeeze_output(out)
|
|
|
|
def pdf(self, x, df, scale):
|
|
"""Wishart probability density function.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
Quantiles, with the last axis of `x` denoting the components.
|
|
Each quantile must be a symmetric positive definite matrix.
|
|
%(_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
pdf : ndarray
|
|
Probability density function evaluated at `x`
|
|
|
|
Notes
|
|
-----
|
|
%(_doc_callparams_note)s
|
|
|
|
"""
|
|
return np.exp(self.logpdf(x, df, scale))
|
|
|
|
def _mean(self, dim, df, scale):
|
|
"""Mean of the Wishart distribution.
|
|
|
|
Parameters
|
|
----------
|
|
dim : int
|
|
Dimension of the scale matrix
|
|
%(_doc_default_callparams)s
|
|
|
|
Notes
|
|
-----
|
|
As this function does no argument checking, it should not be
|
|
called directly; use 'mean' instead.
|
|
|
|
"""
|
|
return df * scale
|
|
|
|
def mean(self, df, scale):
|
|
"""Mean of the Wishart distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
mean : float
|
|
The mean of the distribution
|
|
"""
|
|
dim, df, scale = self._process_parameters(df, scale)
|
|
out = self._mean(dim, df, scale)
|
|
return _squeeze_output(out)
|
|
|
|
def _mode(self, dim, df, scale):
|
|
"""Mode of the Wishart distribution.
|
|
|
|
Parameters
|
|
----------
|
|
dim : int
|
|
Dimension of the scale matrix
|
|
%(_doc_default_callparams)s
|
|
|
|
Notes
|
|
-----
|
|
As this function does no argument checking, it should not be
|
|
called directly; use 'mode' instead.
|
|
|
|
"""
|
|
if df >= dim + 1:
|
|
out = (df-dim-1) * scale
|
|
else:
|
|
out = None
|
|
return out
|
|
|
|
def mode(self, df, scale):
|
|
"""Mode of the Wishart distribution
|
|
|
|
Only valid if the degrees of freedom are greater than the dimension of
|
|
the scale matrix.
|
|
|
|
Parameters
|
|
----------
|
|
%(_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
mode : float or None
|
|
The Mode of the distribution
|
|
"""
|
|
dim, df, scale = self._process_parameters(df, scale)
|
|
out = self._mode(dim, df, scale)
|
|
return _squeeze_output(out) if out is not None else out
|
|
|
|
def _var(self, dim, df, scale):
|
|
"""Variance of the Wishart distribution.
|
|
|
|
Parameters
|
|
----------
|
|
dim : int
|
|
Dimension of the scale matrix
|
|
%(_doc_default_callparams)s
|
|
|
|
Notes
|
|
-----
|
|
As this function does no argument checking, it should not be
|
|
called directly; use 'var' instead.
|
|
|
|
"""
|
|
var = scale**2
|
|
diag = scale.diagonal() # 1 x dim array
|
|
var += np.outer(diag, diag)
|
|
var *= df
|
|
return var
|
|
|
|
def var(self, df, scale):
|
|
"""Variance of the Wishart distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
var : float
|
|
The variance of the distribution
|
|
"""
|
|
dim, df, scale = self._process_parameters(df, scale)
|
|
out = self._var(dim, df, scale)
|
|
return _squeeze_output(out)
|
|
|
|
def _standard_rvs(self, n, shape, dim, df, random_state):
|
|
"""
|
|
Parameters
|
|
----------
|
|
n : integer
|
|
Number of variates to generate
|
|
shape : iterable
|
|
Shape of the variates to generate
|
|
dim : int
|
|
Dimension of the scale matrix
|
|
df : int
|
|
Degrees of freedom
|
|
random_state : {None, int, `numpy.random.Generator`,
|
|
`numpy.random.RandomState`}, optional
|
|
|
|
If `seed` is None (or `np.random`), the `numpy.random.RandomState`
|
|
singleton is used.
|
|
If `seed` is an int, a new ``RandomState`` instance is used,
|
|
seeded with `seed`.
|
|
If `seed` is already a ``Generator`` or ``RandomState`` instance
|
|
then that instance is used.
|
|
|
|
Notes
|
|
-----
|
|
As this function does no argument checking, it should not be
|
|
called directly; use 'rvs' instead.
|
|
|
|
"""
|
|
# Random normal variates for off-diagonal elements
|
|
n_tril = dim * (dim-1) // 2
|
|
covariances = random_state.normal(
|
|
size=n*n_tril).reshape(shape+(n_tril,))
|
|
|
|
# Random chi-square variates for diagonal elements
|
|
variances = (np.r_[[random_state.chisquare(df-(i+1)+1, size=n)**0.5
|
|
for i in range(dim)]].reshape((dim,) +
|
|
shape[::-1]).T)
|
|
|
|
# Create the A matri(ces) - lower triangular
|
|
A = np.zeros(shape + (dim, dim))
|
|
|
|
# Input the covariances
|
|
size_idx = tuple([slice(None, None, None)]*len(shape))
|
|
tril_idx = np.tril_indices(dim, k=-1)
|
|
A[size_idx + tril_idx] = covariances
|
|
|
|
# Input the variances
|
|
diag_idx = np.diag_indices(dim)
|
|
A[size_idx + diag_idx] = variances
|
|
|
|
return A
|
|
|
|
def _rvs(self, n, shape, dim, df, C, random_state):
|
|
"""Draw random samples from a Wishart distribution.
|
|
|
|
Parameters
|
|
----------
|
|
n : integer
|
|
Number of variates to generate
|
|
shape : iterable
|
|
Shape of the variates to generate
|
|
dim : int
|
|
Dimension of the scale matrix
|
|
df : int
|
|
Degrees of freedom
|
|
C : ndarray
|
|
Cholesky factorization of the scale matrix, lower triangular.
|
|
%(_doc_random_state)s
|
|
|
|
Notes
|
|
-----
|
|
As this function does no argument checking, it should not be
|
|
called directly; use 'rvs' instead.
|
|
|
|
"""
|
|
random_state = self._get_random_state(random_state)
|
|
# Calculate the matrices A, which are actually lower triangular
|
|
# Cholesky factorizations of a matrix B such that B ~ W(df, I)
|
|
A = self._standard_rvs(n, shape, dim, df, random_state)
|
|
|
|
# Calculate SA = C A A' C', where SA ~ W(df, scale)
|
|
# Note: this is the product of a (lower) (lower) (lower)' (lower)'
|
|
# or, denoting B = AA', it is C B C' where C is the lower
|
|
# triangular Cholesky factorization of the scale matrix.
|
|
# this appears to conflict with the instructions in [1]_, which
|
|
# suggest that it should be D' B D where D is the lower
|
|
# triangular factorization of the scale matrix. However, it is
|
|
# meant to refer to the Bartlett (1933) representation of a
|
|
# Wishart random variate as L A A' L' where L is lower triangular
|
|
# so it appears that understanding D' to be upper triangular
|
|
# is either a typo in or misreading of [1]_.
|
|
for index in np.ndindex(shape):
|
|
CA = np.dot(C, A[index])
|
|
A[index] = np.dot(CA, CA.T)
|
|
|
|
return A
|
|
|
|
def rvs(self, df, scale, size=1, random_state=None):
|
|
"""Draw random samples from a Wishart distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_doc_default_callparams)s
|
|
size : integer or iterable of integers, optional
|
|
Number of samples to draw (default 1).
|
|
%(_doc_random_state)s
|
|
|
|
Returns
|
|
-------
|
|
rvs : ndarray
|
|
Random variates of shape (`size`) + (``dim``, ``dim``), where
|
|
``dim`` is the dimension of the scale matrix.
|
|
|
|
Notes
|
|
-----
|
|
%(_doc_callparams_note)s
|
|
|
|
"""
|
|
n, shape = self._process_size(size)
|
|
dim, df, scale = self._process_parameters(df, scale)
|
|
|
|
# Cholesky decomposition of scale
|
|
C = scipy.linalg.cholesky(scale, lower=True)
|
|
|
|
out = self._rvs(n, shape, dim, df, C, random_state)
|
|
|
|
return _squeeze_output(out)
|
|
|
|
def _entropy(self, dim, df, log_det_scale):
|
|
"""Compute the differential entropy of the Wishart.
|
|
|
|
Parameters
|
|
----------
|
|
dim : int
|
|
Dimension of the scale matrix
|
|
df : int
|
|
Degrees of freedom
|
|
log_det_scale : float
|
|
Logarithm of the determinant of the scale matrix
|
|
|
|
Notes
|
|
-----
|
|
As this function does no argument checking, it should not be
|
|
called directly; use 'entropy' instead.
|
|
|
|
"""
|
|
return (
|
|
0.5 * (dim+1) * log_det_scale +
|
|
0.5 * dim * (dim+1) * _LOG_2 +
|
|
multigammaln(0.5*df, dim) -
|
|
0.5 * (df - dim - 1) * np.sum(
|
|
[psi(0.5*(df + 1 - (i+1))) for i in range(dim)]
|
|
) +
|
|
0.5 * df * dim
|
|
)
|
|
|
|
def entropy(self, df, scale):
|
|
"""Compute the differential entropy of the Wishart.
|
|
|
|
Parameters
|
|
----------
|
|
%(_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
h : scalar
|
|
Entropy of the Wishart distribution
|
|
|
|
Notes
|
|
-----
|
|
%(_doc_callparams_note)s
|
|
|
|
"""
|
|
dim, df, scale = self._process_parameters(df, scale)
|
|
_, log_det_scale = self._cholesky_logdet(scale)
|
|
return self._entropy(dim, df, log_det_scale)
|
|
|
|
def _cholesky_logdet(self, scale):
|
|
"""Compute Cholesky decomposition and determine (log(det(scale)).
|
|
|
|
Parameters
|
|
----------
|
|
scale : ndarray
|
|
Scale matrix.
|
|
|
|
Returns
|
|
-------
|
|
c_decomp : ndarray
|
|
The Cholesky decomposition of `scale`.
|
|
logdet : scalar
|
|
The log of the determinant of `scale`.
|
|
|
|
Notes
|
|
-----
|
|
This computation of ``logdet`` is equivalent to
|
|
``np.linalg.slogdet(scale)``. It is ~2x faster though.
|
|
|
|
"""
|
|
c_decomp = scipy.linalg.cholesky(scale, lower=True)
|
|
logdet = 2 * np.sum(np.log(c_decomp.diagonal()))
|
|
return c_decomp, logdet
|
|
|
|
|
|
wishart = wishart_gen()
|
|
|
|
|
|
class wishart_frozen(multi_rv_frozen):
|
|
"""Create a frozen Wishart distribution.
|
|
|
|
Parameters
|
|
----------
|
|
df : array_like
|
|
Degrees of freedom of the distribution
|
|
scale : array_like
|
|
Scale matrix of the distribution
|
|
seed : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional
|
|
If `seed` is None (or `np.random`), the `numpy.random.RandomState`
|
|
singleton is used.
|
|
If `seed` is an int, a new ``RandomState`` instance is used,
|
|
seeded with `seed`.
|
|
If `seed` is already a ``Generator`` or ``RandomState`` instance then
|
|
that instance is used.
|
|
|
|
"""
|
|
def __init__(self, df, scale, seed=None):
|
|
self._dist = wishart_gen(seed)
|
|
self.dim, self.df, self.scale = self._dist._process_parameters(
|
|
df, scale)
|
|
self.C, self.log_det_scale = self._dist._cholesky_logdet(self.scale)
|
|
|
|
def logpdf(self, x):
|
|
x = self._dist._process_quantiles(x, self.dim)
|
|
|
|
out = self._dist._logpdf(x, self.dim, self.df, self.scale,
|
|
self.log_det_scale, self.C)
|
|
return _squeeze_output(out)
|
|
|
|
def pdf(self, x):
|
|
return np.exp(self.logpdf(x))
|
|
|
|
def mean(self):
|
|
out = self._dist._mean(self.dim, self.df, self.scale)
|
|
return _squeeze_output(out)
|
|
|
|
def mode(self):
|
|
out = self._dist._mode(self.dim, self.df, self.scale)
|
|
return _squeeze_output(out) if out is not None else out
|
|
|
|
def var(self):
|
|
out = self._dist._var(self.dim, self.df, self.scale)
|
|
return _squeeze_output(out)
|
|
|
|
def rvs(self, size=1, random_state=None):
|
|
n, shape = self._dist._process_size(size)
|
|
out = self._dist._rvs(n, shape, self.dim, self.df,
|
|
self.C, random_state)
|
|
return _squeeze_output(out)
|
|
|
|
def entropy(self):
|
|
return self._dist._entropy(self.dim, self.df, self.log_det_scale)
|
|
|
|
|
|
# Set frozen generator docstrings from corresponding docstrings in
|
|
# Wishart and fill in default strings in class docstrings
|
|
for name in ['logpdf', 'pdf', 'mean', 'mode', 'var', 'rvs', 'entropy']:
|
|
method = wishart_gen.__dict__[name]
|
|
method_frozen = wishart_frozen.__dict__[name]
|
|
method_frozen.__doc__ = doccer.docformat(
|
|
method.__doc__, wishart_docdict_noparams)
|
|
method.__doc__ = doccer.docformat(method.__doc__, wishart_docdict_params)
|
|
|
|
|
|
class invwishart_gen(wishart_gen):
|
|
r"""An inverse Wishart random variable.
|
|
|
|
The `df` keyword specifies the degrees of freedom. The `scale` keyword
|
|
specifies the scale matrix, which must be symmetric and positive definite.
|
|
In this context, the scale matrix is often interpreted in terms of a
|
|
multivariate normal covariance matrix.
|
|
|
|
Methods
|
|
-------
|
|
pdf(x, df, scale)
|
|
Probability density function.
|
|
logpdf(x, df, scale)
|
|
Log of the probability density function.
|
|
rvs(df, scale, size=1, random_state=None)
|
|
Draw random samples from an inverse Wishart distribution.
|
|
entropy(df, scale)
|
|
Differential entropy of the distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_doc_default_callparams)s
|
|
%(_doc_random_state)s
|
|
|
|
Raises
|
|
------
|
|
scipy.linalg.LinAlgError
|
|
If the scale matrix `scale` is not positive definite.
|
|
|
|
See Also
|
|
--------
|
|
wishart
|
|
|
|
Notes
|
|
-----
|
|
%(_doc_callparams_note)s
|
|
|
|
The scale matrix `scale` must be a symmetric positive definite
|
|
matrix. Singular matrices, including the symmetric positive semi-definite
|
|
case, are not supported. Symmetry is not checked; only the lower triangular
|
|
portion is used.
|
|
|
|
The inverse Wishart distribution is often denoted
|
|
|
|
.. math::
|
|
|
|
W_p^{-1}(\nu, \Psi)
|
|
|
|
where :math:`\nu` is the degrees of freedom and :math:`\Psi` is the
|
|
:math:`p \times p` scale matrix.
|
|
|
|
The probability density function for `invwishart` has support over positive
|
|
definite matrices :math:`S`; if :math:`S \sim W^{-1}_p(\nu, \Sigma)`,
|
|
then its PDF is given by:
|
|
|
|
.. math::
|
|
|
|
f(S) = \frac{|\Sigma|^\frac{\nu}{2}}{2^{ \frac{\nu p}{2} }
|
|
|S|^{\frac{\nu + p + 1}{2}} \Gamma_p \left(\frac{\nu}{2} \right)}
|
|
\exp\left( -tr(\Sigma S^{-1}) / 2 \right)
|
|
|
|
If :math:`S \sim W_p^{-1}(\nu, \Psi)` (inverse Wishart) then
|
|
:math:`S^{-1} \sim W_p(\nu, \Psi^{-1})` (Wishart).
|
|
|
|
If the scale matrix is 1-dimensional and equal to one, then the inverse
|
|
Wishart distribution :math:`W_1(\nu, 1)` collapses to the
|
|
inverse Gamma distribution with parameters shape = :math:`\frac{\nu}{2}`
|
|
and scale = :math:`\frac{1}{2}`.
|
|
|
|
Instead of inverting a randomly generated Wishart matrix as described in [2],
|
|
here the algorithm in [4] is used to directly generate a random inverse-Wishart
|
|
matrix without inversion.
|
|
|
|
.. versionadded:: 0.16.0
|
|
|
|
References
|
|
----------
|
|
.. [1] M.L. Eaton, "Multivariate Statistics: A Vector Space Approach",
|
|
Wiley, 1983.
|
|
.. [2] M.C. Jones, "Generating Inverse Wishart Matrices", Communications
|
|
in Statistics - Simulation and Computation, vol. 14.2, pp.511-514,
|
|
1985.
|
|
.. [3] Gupta, M. and Srivastava, S. "Parametric Bayesian Estimation of
|
|
Differential Entropy and Relative Entropy". Entropy 12, 818 - 843.
|
|
2010.
|
|
.. [4] S.D. Axen, "Efficiently generating inverse-Wishart matrices and
|
|
their Cholesky factors", :arXiv:`2310.15884v1`. 2023.
|
|
|
|
Examples
|
|
--------
|
|
>>> import numpy as np
|
|
>>> import matplotlib.pyplot as plt
|
|
>>> from scipy.stats import invwishart, invgamma
|
|
>>> x = np.linspace(0.01, 1, 100)
|
|
>>> iw = invwishart.pdf(x, df=6, scale=1)
|
|
>>> iw[:3]
|
|
array([ 1.20546865e-15, 5.42497807e-06, 4.45813929e-03])
|
|
>>> ig = invgamma.pdf(x, 6/2., scale=1./2)
|
|
>>> ig[:3]
|
|
array([ 1.20546865e-15, 5.42497807e-06, 4.45813929e-03])
|
|
>>> plt.plot(x, iw)
|
|
>>> plt.show()
|
|
|
|
The input quantiles can be any shape of array, as long as the last
|
|
axis labels the components.
|
|
|
|
Alternatively, the object may be called (as a function) to fix the degrees
|
|
of freedom and scale parameters, returning a "frozen" inverse Wishart
|
|
random variable:
|
|
|
|
>>> rv = invwishart(df=1, scale=1)
|
|
>>> # Frozen object with the same methods but holding the given
|
|
>>> # degrees of freedom and scale fixed.
|
|
|
|
"""
|
|
|
|
def __init__(self, seed=None):
|
|
super().__init__(seed)
|
|
self.__doc__ = doccer.docformat(self.__doc__, wishart_docdict_params)
|
|
|
|
def __call__(self, df=None, scale=None, seed=None):
|
|
"""Create a frozen inverse Wishart distribution.
|
|
|
|
See `invwishart_frozen` for more information.
|
|
|
|
"""
|
|
return invwishart_frozen(df, scale, seed)
|
|
|
|
def _logpdf(self, x, dim, df, log_det_scale, C):
|
|
"""Log of the inverse Wishart probability density function.
|
|
|
|
Parameters
|
|
----------
|
|
x : ndarray
|
|
Points at which to evaluate the log of the probability
|
|
density function.
|
|
dim : int
|
|
Dimension of the scale matrix
|
|
df : int
|
|
Degrees of freedom
|
|
log_det_scale : float
|
|
Logarithm of the determinant of the scale matrix
|
|
C : ndarray
|
|
Cholesky factorization of the scale matrix, lower triangular.
|
|
|
|
Notes
|
|
-----
|
|
As this function does no argument checking, it should not be
|
|
called directly; use 'logpdf' instead.
|
|
|
|
"""
|
|
# Retrieve tr(scale x^{-1})
|
|
log_det_x = np.empty(x.shape[-1])
|
|
tr_scale_x_inv = np.empty(x.shape[-1])
|
|
trsm = get_blas_funcs(('trsm'), (x,))
|
|
if dim > 1:
|
|
for i in range(x.shape[-1]):
|
|
Cx, log_det_x[i] = self._cholesky_logdet(x[:, :, i])
|
|
A = trsm(1., Cx, C, side=0, lower=True)
|
|
tr_scale_x_inv[i] = np.linalg.norm(A)**2
|
|
else:
|
|
log_det_x[:] = np.log(x[0, 0])
|
|
tr_scale_x_inv[:] = C[0, 0]**2 / x[0, 0]
|
|
|
|
# Log PDF
|
|
out = ((0.5 * df * log_det_scale - 0.5 * tr_scale_x_inv) -
|
|
(0.5 * df * dim * _LOG_2 + 0.5 * (df + dim + 1) * log_det_x) -
|
|
multigammaln(0.5*df, dim))
|
|
|
|
return out
|
|
|
|
def logpdf(self, x, df, scale):
|
|
"""Log of the inverse Wishart probability density function.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
Quantiles, with the last axis of `x` denoting the components.
|
|
Each quantile must be a symmetric positive definite matrix.
|
|
%(_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
pdf : ndarray
|
|
Log of the probability density function evaluated at `x`
|
|
|
|
Notes
|
|
-----
|
|
%(_doc_callparams_note)s
|
|
|
|
"""
|
|
dim, df, scale = self._process_parameters(df, scale)
|
|
x = self._process_quantiles(x, dim)
|
|
C, log_det_scale = self._cholesky_logdet(scale)
|
|
out = self._logpdf(x, dim, df, log_det_scale, C)
|
|
return _squeeze_output(out)
|
|
|
|
def pdf(self, x, df, scale):
|
|
"""Inverse Wishart probability density function.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
Quantiles, with the last axis of `x` denoting the components.
|
|
Each quantile must be a symmetric positive definite matrix.
|
|
%(_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
pdf : ndarray
|
|
Probability density function evaluated at `x`
|
|
|
|
Notes
|
|
-----
|
|
%(_doc_callparams_note)s
|
|
|
|
"""
|
|
return np.exp(self.logpdf(x, df, scale))
|
|
|
|
def _mean(self, dim, df, scale):
|
|
"""Mean of the inverse Wishart distribution.
|
|
|
|
Parameters
|
|
----------
|
|
dim : int
|
|
Dimension of the scale matrix
|
|
%(_doc_default_callparams)s
|
|
|
|
Notes
|
|
-----
|
|
As this function does no argument checking, it should not be
|
|
called directly; use 'mean' instead.
|
|
|
|
"""
|
|
if df > dim + 1:
|
|
out = scale / (df - dim - 1)
|
|
else:
|
|
out = None
|
|
return out
|
|
|
|
def mean(self, df, scale):
|
|
"""Mean of the inverse Wishart distribution.
|
|
|
|
Only valid if the degrees of freedom are greater than the dimension of
|
|
the scale matrix plus one.
|
|
|
|
Parameters
|
|
----------
|
|
%(_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
mean : float or None
|
|
The mean of the distribution
|
|
|
|
"""
|
|
dim, df, scale = self._process_parameters(df, scale)
|
|
out = self._mean(dim, df, scale)
|
|
return _squeeze_output(out) if out is not None else out
|
|
|
|
def _mode(self, dim, df, scale):
|
|
"""Mode of the inverse Wishart distribution.
|
|
|
|
Parameters
|
|
----------
|
|
dim : int
|
|
Dimension of the scale matrix
|
|
%(_doc_default_callparams)s
|
|
|
|
Notes
|
|
-----
|
|
As this function does no argument checking, it should not be
|
|
called directly; use 'mode' instead.
|
|
|
|
"""
|
|
return scale / (df + dim + 1)
|
|
|
|
def mode(self, df, scale):
|
|
"""Mode of the inverse Wishart distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
mode : float
|
|
The Mode of the distribution
|
|
|
|
"""
|
|
dim, df, scale = self._process_parameters(df, scale)
|
|
out = self._mode(dim, df, scale)
|
|
return _squeeze_output(out)
|
|
|
|
def _var(self, dim, df, scale):
|
|
"""Variance of the inverse Wishart distribution.
|
|
|
|
Parameters
|
|
----------
|
|
dim : int
|
|
Dimension of the scale matrix
|
|
%(_doc_default_callparams)s
|
|
|
|
Notes
|
|
-----
|
|
As this function does no argument checking, it should not be
|
|
called directly; use 'var' instead.
|
|
|
|
"""
|
|
if df > dim + 3:
|
|
var = (df - dim + 1) * scale**2
|
|
diag = scale.diagonal() # 1 x dim array
|
|
var += (df - dim - 1) * np.outer(diag, diag)
|
|
var /= (df - dim) * (df - dim - 1)**2 * (df - dim - 3)
|
|
else:
|
|
var = None
|
|
return var
|
|
|
|
def var(self, df, scale):
|
|
"""Variance of the inverse Wishart distribution.
|
|
|
|
Only valid if the degrees of freedom are greater than the dimension of
|
|
the scale matrix plus three.
|
|
|
|
Parameters
|
|
----------
|
|
%(_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
var : float
|
|
The variance of the distribution
|
|
"""
|
|
dim, df, scale = self._process_parameters(df, scale)
|
|
out = self._var(dim, df, scale)
|
|
return _squeeze_output(out) if out is not None else out
|
|
|
|
def _inv_standard_rvs(self, n, shape, dim, df, random_state):
|
|
"""
|
|
Parameters
|
|
----------
|
|
n : integer
|
|
Number of variates to generate
|
|
shape : iterable
|
|
Shape of the variates to generate
|
|
dim : int
|
|
Dimension of the scale matrix
|
|
df : int
|
|
Degrees of freedom
|
|
random_state : {None, int, `numpy.random.Generator`,
|
|
`numpy.random.RandomState`}, optional
|
|
|
|
If `seed` is None (or `np.random`), the `numpy.random.RandomState`
|
|
singleton is used.
|
|
If `seed` is an int, a new ``RandomState`` instance is used,
|
|
seeded with `seed`.
|
|
If `seed` is already a ``Generator`` or ``RandomState`` instance
|
|
then that instance is used.
|
|
|
|
Returns
|
|
-------
|
|
A : ndarray
|
|
Random variates of shape (`shape`) + (``dim``, ``dim``).
|
|
Each slice `A[..., :, :]` is lower-triangular, and its
|
|
inverse is the lower Cholesky factor of a draw from
|
|
`invwishart(df, np.eye(dim))`.
|
|
|
|
Notes
|
|
-----
|
|
As this function does no argument checking, it should not be
|
|
called directly; use 'rvs' instead.
|
|
|
|
"""
|
|
A = np.zeros(shape + (dim, dim))
|
|
|
|
# Random normal variates for off-diagonal elements
|
|
tri_rows, tri_cols = np.tril_indices(dim, k=-1)
|
|
n_tril = dim * (dim-1) // 2
|
|
A[..., tri_rows, tri_cols] = random_state.normal(
|
|
size=(*shape, n_tril),
|
|
)
|
|
|
|
# Random chi variates for diagonal elements
|
|
rows = np.arange(dim)
|
|
chi_dfs = (df - dim + 1) + rows
|
|
A[..., rows, rows] = random_state.chisquare(
|
|
df=chi_dfs, size=(*shape, dim),
|
|
)**0.5
|
|
|
|
return A
|
|
|
|
def _rvs(self, n, shape, dim, df, C, random_state):
|
|
"""Draw random samples from an inverse Wishart distribution.
|
|
|
|
Parameters
|
|
----------
|
|
n : integer
|
|
Number of variates to generate
|
|
shape : iterable
|
|
Shape of the variates to generate
|
|
dim : int
|
|
Dimension of the scale matrix
|
|
df : int
|
|
Degrees of freedom
|
|
C : ndarray
|
|
Cholesky factorization of the scale matrix, lower triangular.
|
|
%(_doc_random_state)s
|
|
|
|
Notes
|
|
-----
|
|
As this function does no argument checking, it should not be
|
|
called directly; use 'rvs' instead.
|
|
|
|
"""
|
|
random_state = self._get_random_state(random_state)
|
|
# Get random draws A such that inv(A) ~ iW(df, I)
|
|
A = self._inv_standard_rvs(n, shape, dim, df, random_state)
|
|
|
|
# Calculate SA = (CA)'^{-1} (CA)^{-1} ~ iW(df, scale)
|
|
trsm = get_blas_funcs(('trsm'), (A,))
|
|
trmm = get_blas_funcs(('trmm'), (A,))
|
|
|
|
for index in np.ndindex(A.shape[:-2]):
|
|
if dim > 1:
|
|
# Calculate CA
|
|
# Get CA = C A^{-1} via triangular solver
|
|
CA = trsm(1., A[index], C, side=1, lower=True)
|
|
# get SA
|
|
A[index] = trmm(1., CA, CA, side=1, lower=True, trans_a=True)
|
|
else:
|
|
A[index][0, 0] = (C[0, 0] / A[index][0, 0])**2
|
|
|
|
return A
|
|
|
|
def rvs(self, df, scale, size=1, random_state=None):
|
|
"""Draw random samples from an inverse Wishart distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_doc_default_callparams)s
|
|
size : integer or iterable of integers, optional
|
|
Number of samples to draw (default 1).
|
|
%(_doc_random_state)s
|
|
|
|
Returns
|
|
-------
|
|
rvs : ndarray
|
|
Random variates of shape (`size`) + (``dim``, ``dim``), where
|
|
``dim`` is the dimension of the scale matrix.
|
|
|
|
Notes
|
|
-----
|
|
%(_doc_callparams_note)s
|
|
|
|
"""
|
|
n, shape = self._process_size(size)
|
|
dim, df, scale = self._process_parameters(df, scale)
|
|
|
|
# Cholesky decomposition of scale
|
|
C = scipy.linalg.cholesky(scale, lower=True)
|
|
|
|
out = self._rvs(n, shape, dim, df, C, random_state)
|
|
|
|
return _squeeze_output(out)
|
|
|
|
def _entropy(self, dim, df, log_det_scale):
|
|
# reference: eq. (17) from ref. 3
|
|
psi_eval_points = [0.5 * (df - dim + i) for i in range(1, dim + 1)]
|
|
psi_eval_points = np.asarray(psi_eval_points)
|
|
return multigammaln(0.5 * df, dim) + 0.5 * dim * df + \
|
|
0.5 * (dim + 1) * (log_det_scale - _LOG_2) - \
|
|
0.5 * (df + dim + 1) * \
|
|
psi(psi_eval_points, out=psi_eval_points).sum()
|
|
|
|
def entropy(self, df, scale):
|
|
dim, df, scale = self._process_parameters(df, scale)
|
|
_, log_det_scale = self._cholesky_logdet(scale)
|
|
return self._entropy(dim, df, log_det_scale)
|
|
|
|
|
|
invwishart = invwishart_gen()
|
|
|
|
|
|
class invwishart_frozen(multi_rv_frozen):
|
|
def __init__(self, df, scale, seed=None):
|
|
"""Create a frozen inverse Wishart distribution.
|
|
|
|
Parameters
|
|
----------
|
|
df : array_like
|
|
Degrees of freedom of the distribution
|
|
scale : array_like
|
|
Scale matrix of the distribution
|
|
seed : {None, int, `numpy.random.Generator`}, optional
|
|
If `seed` is None the `numpy.random.Generator` singleton is used.
|
|
If `seed` is an int, a new ``Generator`` instance is used,
|
|
seeded with `seed`.
|
|
If `seed` is already a ``Generator`` instance then that instance is
|
|
used.
|
|
|
|
"""
|
|
self._dist = invwishart_gen(seed)
|
|
self.dim, self.df, self.scale = self._dist._process_parameters(
|
|
df, scale
|
|
)
|
|
|
|
# Get the determinant via Cholesky factorization
|
|
self.C = scipy.linalg.cholesky(self.scale, lower=True)
|
|
self.log_det_scale = 2 * np.sum(np.log(self.C.diagonal()))
|
|
|
|
def logpdf(self, x):
|
|
x = self._dist._process_quantiles(x, self.dim)
|
|
out = self._dist._logpdf(x, self.dim, self.df,
|
|
self.log_det_scale, self.C)
|
|
return _squeeze_output(out)
|
|
|
|
def pdf(self, x):
|
|
return np.exp(self.logpdf(x))
|
|
|
|
def mean(self):
|
|
out = self._dist._mean(self.dim, self.df, self.scale)
|
|
return _squeeze_output(out) if out is not None else out
|
|
|
|
def mode(self):
|
|
out = self._dist._mode(self.dim, self.df, self.scale)
|
|
return _squeeze_output(out)
|
|
|
|
def var(self):
|
|
out = self._dist._var(self.dim, self.df, self.scale)
|
|
return _squeeze_output(out) if out is not None else out
|
|
|
|
def rvs(self, size=1, random_state=None):
|
|
n, shape = self._dist._process_size(size)
|
|
|
|
out = self._dist._rvs(n, shape, self.dim, self.df,
|
|
self.C, random_state)
|
|
|
|
return _squeeze_output(out)
|
|
|
|
def entropy(self):
|
|
return self._dist._entropy(self.dim, self.df, self.log_det_scale)
|
|
|
|
|
|
# Set frozen generator docstrings from corresponding docstrings in
|
|
# inverse Wishart and fill in default strings in class docstrings
|
|
for name in ['logpdf', 'pdf', 'mean', 'mode', 'var', 'rvs']:
|
|
method = invwishart_gen.__dict__[name]
|
|
method_frozen = wishart_frozen.__dict__[name]
|
|
method_frozen.__doc__ = doccer.docformat(
|
|
method.__doc__, wishart_docdict_noparams)
|
|
method.__doc__ = doccer.docformat(method.__doc__, wishart_docdict_params)
|
|
|
|
_multinomial_doc_default_callparams = """\
|
|
n : int
|
|
Number of trials
|
|
p : array_like
|
|
Probability of a trial falling into each category; should sum to 1
|
|
"""
|
|
|
|
_multinomial_doc_callparams_note = """\
|
|
`n` should be a nonnegative integer. Each element of `p` should be in the
|
|
interval :math:`[0,1]` and the elements should sum to 1. If they do not sum to
|
|
1, the last element of the `p` array is not used and is replaced with the
|
|
remaining probability left over from the earlier elements.
|
|
"""
|
|
|
|
_multinomial_doc_frozen_callparams = ""
|
|
|
|
_multinomial_doc_frozen_callparams_note = """\
|
|
See class definition for a detailed description of parameters."""
|
|
|
|
multinomial_docdict_params = {
|
|
'_doc_default_callparams': _multinomial_doc_default_callparams,
|
|
'_doc_callparams_note': _multinomial_doc_callparams_note,
|
|
'_doc_random_state': _doc_random_state
|
|
}
|
|
|
|
multinomial_docdict_noparams = {
|
|
'_doc_default_callparams': _multinomial_doc_frozen_callparams,
|
|
'_doc_callparams_note': _multinomial_doc_frozen_callparams_note,
|
|
'_doc_random_state': _doc_random_state
|
|
}
|
|
|
|
|
|
class multinomial_gen(multi_rv_generic):
|
|
r"""A multinomial random variable.
|
|
|
|
Methods
|
|
-------
|
|
pmf(x, n, p)
|
|
Probability mass function.
|
|
logpmf(x, n, p)
|
|
Log of the probability mass function.
|
|
rvs(n, p, size=1, random_state=None)
|
|
Draw random samples from a multinomial distribution.
|
|
entropy(n, p)
|
|
Compute the entropy of the multinomial distribution.
|
|
cov(n, p)
|
|
Compute the covariance matrix of the multinomial distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_doc_default_callparams)s
|
|
%(_doc_random_state)s
|
|
|
|
Notes
|
|
-----
|
|
%(_doc_callparams_note)s
|
|
|
|
The probability mass function for `multinomial` is
|
|
|
|
.. math::
|
|
|
|
f(x) = \frac{n!}{x_1! \cdots x_k!} p_1^{x_1} \cdots p_k^{x_k},
|
|
|
|
supported on :math:`x=(x_1, \ldots, x_k)` where each :math:`x_i` is a
|
|
nonnegative integer and their sum is :math:`n`.
|
|
|
|
.. versionadded:: 0.19.0
|
|
|
|
Examples
|
|
--------
|
|
|
|
>>> from scipy.stats import multinomial
|
|
>>> rv = multinomial(8, [0.3, 0.2, 0.5])
|
|
>>> rv.pmf([1, 3, 4])
|
|
0.042000000000000072
|
|
|
|
The multinomial distribution for :math:`k=2` is identical to the
|
|
corresponding binomial distribution (tiny numerical differences
|
|
notwithstanding):
|
|
|
|
>>> from scipy.stats import binom
|
|
>>> multinomial.pmf([3, 4], n=7, p=[0.4, 0.6])
|
|
0.29030399999999973
|
|
>>> binom.pmf(3, 7, 0.4)
|
|
0.29030400000000012
|
|
|
|
The functions ``pmf``, ``logpmf``, ``entropy``, and ``cov`` support
|
|
broadcasting, under the convention that the vector parameters (``x`` and
|
|
``p``) are interpreted as if each row along the last axis is a single
|
|
object. For instance:
|
|
|
|
>>> multinomial.pmf([[3, 4], [3, 5]], n=[7, 8], p=[.3, .7])
|
|
array([0.2268945, 0.25412184])
|
|
|
|
Here, ``x.shape == (2, 2)``, ``n.shape == (2,)``, and ``p.shape == (2,)``,
|
|
but following the rules mentioned above they behave as if the rows
|
|
``[3, 4]`` and ``[3, 5]`` in ``x`` and ``[.3, .7]`` in ``p`` were a single
|
|
object, and as if we had ``x.shape = (2,)``, ``n.shape = (2,)``, and
|
|
``p.shape = ()``. To obtain the individual elements without broadcasting,
|
|
we would do this:
|
|
|
|
>>> multinomial.pmf([3, 4], n=7, p=[.3, .7])
|
|
0.2268945
|
|
>>> multinomial.pmf([3, 5], 8, p=[.3, .7])
|
|
0.25412184
|
|
|
|
This broadcasting also works for ``cov``, where the output objects are
|
|
square matrices of size ``p.shape[-1]``. For example:
|
|
|
|
>>> multinomial.cov([4, 5], [[.3, .7], [.4, .6]])
|
|
array([[[ 0.84, -0.84],
|
|
[-0.84, 0.84]],
|
|
[[ 1.2 , -1.2 ],
|
|
[-1.2 , 1.2 ]]])
|
|
|
|
In this example, ``n.shape == (2,)`` and ``p.shape == (2, 2)``, and
|
|
following the rules above, these broadcast as if ``p.shape == (2,)``.
|
|
Thus the result should also be of shape ``(2,)``, but since each output is
|
|
a :math:`2 \times 2` matrix, the result in fact has shape ``(2, 2, 2)``,
|
|
where ``result[0]`` is equal to ``multinomial.cov(n=4, p=[.3, .7])`` and
|
|
``result[1]`` is equal to ``multinomial.cov(n=5, p=[.4, .6])``.
|
|
|
|
Alternatively, the object may be called (as a function) to fix the `n` and
|
|
`p` parameters, returning a "frozen" multinomial random variable:
|
|
|
|
>>> rv = multinomial(n=7, p=[.3, .7])
|
|
>>> # Frozen object with the same methods but holding the given
|
|
>>> # degrees of freedom and scale fixed.
|
|
|
|
See also
|
|
--------
|
|
scipy.stats.binom : The binomial distribution.
|
|
numpy.random.Generator.multinomial : Sampling from the multinomial distribution.
|
|
scipy.stats.multivariate_hypergeom :
|
|
The multivariate hypergeometric distribution.
|
|
"""
|
|
|
|
def __init__(self, seed=None):
|
|
super().__init__(seed)
|
|
self.__doc__ = \
|
|
doccer.docformat(self.__doc__, multinomial_docdict_params)
|
|
|
|
def __call__(self, n, p, seed=None):
|
|
"""Create a frozen multinomial distribution.
|
|
|
|
See `multinomial_frozen` for more information.
|
|
"""
|
|
return multinomial_frozen(n, p, seed)
|
|
|
|
def _process_parameters(self, n, p):
|
|
"""Returns: n_, p_, npcond.
|
|
|
|
n_ and p_ are arrays of the correct shape; npcond is a boolean array
|
|
flagging values out of the domain.
|
|
"""
|
|
eps = np.finfo(np.result_type(np.asarray(p), np.float32)).eps * 10
|
|
p = np.array(p, dtype=np.float64, copy=True)
|
|
p_adjusted = 1. - p[..., :-1].sum(axis=-1)
|
|
# only make adjustment when it's significant
|
|
i_adjusted = np.abs(1 - p.sum(axis=-1)) > eps
|
|
p[i_adjusted, -1] = p_adjusted[i_adjusted]
|
|
|
|
if np.any(i_adjusted):
|
|
message = ("Some rows of `p` do not sum to 1.0 within tolerance of "
|
|
f"{eps=}. Currently, the last element of these rows is adjusted "
|
|
"to compensate, but this condition will produce NaNs "
|
|
"beginning in SciPy 1.18.0. Please ensure that rows of `p` sum "
|
|
"to 1.0 to avoid futher disruption.")
|
|
warnings.warn(message, FutureWarning, stacklevel=3)
|
|
|
|
# true for bad p
|
|
pcond = np.any(p < 0, axis=-1)
|
|
pcond |= np.any(p > 1, axis=-1)
|
|
|
|
n = np.array(n, dtype=int, copy=True)
|
|
|
|
# true for bad n
|
|
ncond = n < 0
|
|
|
|
return n, p, ncond | pcond
|
|
|
|
def _process_quantiles(self, x, n, p):
|
|
"""Returns: x_, xcond.
|
|
|
|
x_ is an int array; xcond is a boolean array flagging values out of the
|
|
domain.
|
|
"""
|
|
xx = np.asarray(x, dtype=int)
|
|
|
|
if xx.ndim == 0:
|
|
raise ValueError("x must be an array.")
|
|
|
|
if xx.size != 0 and not xx.shape[-1] == p.shape[-1]:
|
|
raise ValueError(f"Size of each quantile should be size of p: "
|
|
f"received {xx.shape[-1]}, but expected "
|
|
f"{p.shape[-1]}.")
|
|
|
|
# true for x out of the domain
|
|
cond = np.any(xx != x, axis=-1)
|
|
cond |= np.any(xx < 0, axis=-1)
|
|
cond = cond | (np.sum(xx, axis=-1) != n)
|
|
|
|
return xx, cond
|
|
|
|
def _checkresult(self, result, cond, bad_value):
|
|
result = np.asarray(result)
|
|
|
|
if cond.ndim != 0:
|
|
result[cond] = bad_value
|
|
elif cond:
|
|
if result.ndim == 0:
|
|
return bad_value
|
|
result[...] = bad_value
|
|
return result
|
|
|
|
def _logpmf(self, x, n, p):
|
|
return gammaln(n+1) + np.sum(xlogy(x, p) - gammaln(x+1), axis=-1)
|
|
|
|
def logpmf(self, x, n, p):
|
|
"""Log of the Multinomial probability mass function.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
Quantiles, with the last axis of `x` denoting the components.
|
|
%(_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
logpmf : ndarray or scalar
|
|
Log of the probability mass function evaluated at `x`
|
|
|
|
Notes
|
|
-----
|
|
%(_doc_callparams_note)s
|
|
"""
|
|
n, p, npcond = self._process_parameters(n, p)
|
|
x, xcond = self._process_quantiles(x, n, p)
|
|
|
|
result = self._logpmf(x, n, p)
|
|
|
|
# replace values for which x was out of the domain; broadcast
|
|
# xcond to the right shape
|
|
xcond_ = xcond | np.zeros(npcond.shape, dtype=np.bool_)
|
|
result = self._checkresult(result, xcond_, -np.inf)
|
|
|
|
# replace values bad for n or p; broadcast npcond to the right shape
|
|
npcond_ = npcond | np.zeros(xcond.shape, dtype=np.bool_)
|
|
return self._checkresult(result, npcond_, np.nan)
|
|
|
|
def pmf(self, x, n, p):
|
|
"""Multinomial probability mass function.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
Quantiles, with the last axis of `x` denoting the components.
|
|
%(_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
pmf : ndarray or scalar
|
|
Probability density function evaluated at `x`
|
|
|
|
Notes
|
|
-----
|
|
%(_doc_callparams_note)s
|
|
"""
|
|
return np.exp(self.logpmf(x, n, p))
|
|
|
|
def mean(self, n, p):
|
|
"""Mean of the Multinomial distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
mean : float
|
|
The mean of the distribution
|
|
"""
|
|
n, p, npcond = self._process_parameters(n, p)
|
|
result = n[..., np.newaxis]*p
|
|
return self._checkresult(result, npcond, np.nan)
|
|
|
|
def cov(self, n, p):
|
|
"""Covariance matrix of the multinomial distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
cov : ndarray
|
|
The covariance matrix of the distribution
|
|
"""
|
|
n, p, npcond = self._process_parameters(n, p)
|
|
|
|
nn = n[..., np.newaxis, np.newaxis]
|
|
result = nn * np.einsum('...j,...k->...jk', -p, p)
|
|
|
|
# change the diagonal
|
|
for i in range(p.shape[-1]):
|
|
result[..., i, i] += n*p[..., i]
|
|
|
|
return self._checkresult(result, npcond, np.nan)
|
|
|
|
def entropy(self, n, p):
|
|
r"""Compute the entropy of the multinomial distribution.
|
|
|
|
The entropy is computed using this expression:
|
|
|
|
.. math::
|
|
|
|
f(x) = - \log n! - n\sum_{i=1}^k p_i \log p_i +
|
|
\sum_{i=1}^k \sum_{x=0}^n \binom n x p_i^x(1-p_i)^{n-x} \log x!
|
|
|
|
Parameters
|
|
----------
|
|
%(_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
h : scalar
|
|
Entropy of the Multinomial distribution
|
|
|
|
Notes
|
|
-----
|
|
%(_doc_callparams_note)s
|
|
"""
|
|
n, p, npcond = self._process_parameters(n, p)
|
|
|
|
x = np.r_[1:np.max(n)+1]
|
|
|
|
term1 = n*np.sum(entr(p), axis=-1)
|
|
term1 -= gammaln(n+1)
|
|
|
|
n = n[..., np.newaxis]
|
|
new_axes_needed = max(p.ndim, n.ndim) - x.ndim + 1
|
|
x.shape += (1,)*new_axes_needed
|
|
|
|
term2 = np.sum(binom.pmf(x, n, p)*gammaln(x+1),
|
|
axis=(-1, -1-new_axes_needed))
|
|
|
|
return self._checkresult(term1 + term2, npcond, np.nan)
|
|
|
|
def rvs(self, n, p, size=None, random_state=None):
|
|
"""Draw random samples from a Multinomial distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_doc_default_callparams)s
|
|
size : integer or iterable of integers, optional
|
|
Number of samples to draw (default 1).
|
|
%(_doc_random_state)s
|
|
|
|
Returns
|
|
-------
|
|
rvs : ndarray or scalar
|
|
Random variates of shape (`size`, `len(p)`)
|
|
|
|
Notes
|
|
-----
|
|
%(_doc_callparams_note)s
|
|
"""
|
|
n, p, npcond = self._process_parameters(n, p)
|
|
random_state = self._get_random_state(random_state)
|
|
return random_state.multinomial(n, p, size)
|
|
|
|
|
|
multinomial = multinomial_gen()
|
|
|
|
|
|
class multinomial_frozen(multi_rv_frozen):
|
|
r"""Create a frozen Multinomial distribution.
|
|
|
|
Parameters
|
|
----------
|
|
n : int
|
|
number of trials
|
|
p: array_like
|
|
probability of a trial falling into each category; should sum to 1
|
|
seed : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional
|
|
If `seed` is None (or `np.random`), the `numpy.random.RandomState`
|
|
singleton is used.
|
|
If `seed` is an int, a new ``RandomState`` instance is used,
|
|
seeded with `seed`.
|
|
If `seed` is already a ``Generator`` or ``RandomState`` instance then
|
|
that instance is used.
|
|
"""
|
|
def __init__(self, n, p, seed=None):
|
|
self._dist = multinomial_gen(seed)
|
|
self.n, self.p, self.npcond = self._dist._process_parameters(n, p)
|
|
|
|
# monkey patch self._dist
|
|
def _process_parameters(n, p):
|
|
return self.n, self.p, self.npcond
|
|
|
|
self._dist._process_parameters = _process_parameters
|
|
|
|
def logpmf(self, x):
|
|
return self._dist.logpmf(x, self.n, self.p)
|
|
|
|
def pmf(self, x):
|
|
return self._dist.pmf(x, self.n, self.p)
|
|
|
|
def mean(self):
|
|
return self._dist.mean(self.n, self.p)
|
|
|
|
def cov(self):
|
|
return self._dist.cov(self.n, self.p)
|
|
|
|
def entropy(self):
|
|
return self._dist.entropy(self.n, self.p)
|
|
|
|
def rvs(self, size=1, random_state=None):
|
|
return self._dist.rvs(self.n, self.p, size, random_state)
|
|
|
|
|
|
# Set frozen generator docstrings from corresponding docstrings in
|
|
# multinomial and fill in default strings in class docstrings
|
|
for name in ['logpmf', 'pmf', 'mean', 'cov', 'rvs']:
|
|
method = multinomial_gen.__dict__[name]
|
|
method_frozen = multinomial_frozen.__dict__[name]
|
|
method_frozen.__doc__ = doccer.docformat(
|
|
method.__doc__, multinomial_docdict_noparams)
|
|
method.__doc__ = doccer.docformat(method.__doc__,
|
|
multinomial_docdict_params)
|
|
|
|
|
|
class special_ortho_group_gen(multi_rv_generic):
|
|
r"""A Special Orthogonal matrix (SO(N)) random variable.
|
|
|
|
Return a random rotation matrix, drawn from the Haar distribution
|
|
(the only uniform distribution on SO(N)) with a determinant of +1.
|
|
|
|
The `dim` keyword specifies the dimension N.
|
|
|
|
Methods
|
|
-------
|
|
rvs(dim=None, size=1, random_state=None)
|
|
Draw random samples from SO(N).
|
|
|
|
Parameters
|
|
----------
|
|
dim : scalar
|
|
Dimension of matrices
|
|
seed : {None, int, np.random.RandomState, np.random.Generator}, optional
|
|
Used for drawing random variates.
|
|
If `seed` is `None`, the `~np.random.RandomState` singleton is used.
|
|
If `seed` is an int, a new ``RandomState`` instance is used, seeded
|
|
with seed.
|
|
If `seed` is already a ``RandomState`` or ``Generator`` instance,
|
|
then that object is used.
|
|
Default is `None`.
|
|
|
|
Notes
|
|
-----
|
|
The ``rvs`` method returns a random rotation matrix drawn from the Haar
|
|
distribution, the only uniform distribution on SO(N). The algorithm generates
|
|
a Haar-distributed orthogonal matrix in O(N) using the ``rvs`` method of
|
|
`ortho_group`, then adjusts the matrix to ensure that the determinant is +1.
|
|
|
|
For a random rotation in three dimensions, see
|
|
`scipy.spatial.transform.Rotation.random`.
|
|
|
|
Examples
|
|
--------
|
|
>>> import numpy as np
|
|
>>> from scipy.stats import special_ortho_group
|
|
>>> x = special_ortho_group.rvs(3)
|
|
|
|
>>> np.dot(x, x.T)
|
|
array([[ 1.00000000e+00, 1.13231364e-17, -2.86852790e-16],
|
|
[ 1.13231364e-17, 1.00000000e+00, -1.46845020e-16],
|
|
[ -2.86852790e-16, -1.46845020e-16, 1.00000000e+00]])
|
|
|
|
>>> import scipy.linalg
|
|
>>> scipy.linalg.det(x)
|
|
1.0
|
|
|
|
This generates one random matrix from SO(3). It is orthogonal and
|
|
has a determinant of 1.
|
|
|
|
Alternatively, the object may be called (as a function) to fix the `dim`
|
|
parameter, returning a "frozen" special_ortho_group random variable:
|
|
|
|
>>> rv = special_ortho_group(5)
|
|
>>> # Frozen object with the same methods but holding the
|
|
>>> # dimension parameter fixed.
|
|
|
|
See Also
|
|
--------
|
|
ortho_group, scipy.spatial.transform.Rotation.random
|
|
|
|
"""
|
|
|
|
def __init__(self, seed=None):
|
|
super().__init__(seed)
|
|
self.__doc__ = doccer.docformat(self.__doc__)
|
|
|
|
def __call__(self, dim=None, seed=None):
|
|
"""Create a frozen SO(N) distribution.
|
|
|
|
See `special_ortho_group_frozen` for more information.
|
|
"""
|
|
return special_ortho_group_frozen(dim, seed=seed)
|
|
|
|
def _process_parameters(self, dim):
|
|
"""Dimension N must be specified; it cannot be inferred."""
|
|
if dim is None or not np.isscalar(dim) or dim < 0 or dim != int(dim):
|
|
raise ValueError("""Dimension of rotation must be specified,
|
|
and must be a scalar nonnegative integer.""")
|
|
|
|
return dim
|
|
|
|
def rvs(self, dim, size=1, random_state=None):
|
|
"""Draw random samples from SO(N).
|
|
|
|
Parameters
|
|
----------
|
|
dim : integer
|
|
Dimension of rotation space (N).
|
|
size : integer, optional
|
|
Number of samples to draw (default 1).
|
|
|
|
Returns
|
|
-------
|
|
rvs : ndarray or scalar
|
|
Random size N-dimensional matrices, dimension (size, dim, dim)
|
|
|
|
"""
|
|
random_state = self._get_random_state(random_state)
|
|
|
|
q = ortho_group.rvs(dim, size, random_state)
|
|
dets = np.linalg.det(q)
|
|
if dim:
|
|
q[..., 0, :] /= dets[..., np.newaxis]
|
|
return q
|
|
|
|
special_ortho_group = special_ortho_group_gen()
|
|
|
|
|
|
class special_ortho_group_frozen(multi_rv_frozen):
|
|
def __init__(self, dim=None, seed=None):
|
|
"""Create a frozen SO(N) distribution.
|
|
|
|
Parameters
|
|
----------
|
|
dim : scalar
|
|
Dimension of matrices
|
|
seed : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional
|
|
If `seed` is None (or `np.random`), the `numpy.random.RandomState`
|
|
singleton is used.
|
|
If `seed` is an int, a new ``RandomState`` instance is used,
|
|
seeded with `seed`.
|
|
If `seed` is already a ``Generator`` or ``RandomState`` instance
|
|
then that instance is used.
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy.stats import special_ortho_group
|
|
>>> g = special_ortho_group(5)
|
|
>>> x = g.rvs()
|
|
|
|
""" # numpy/numpydoc#87 # noqa: E501
|
|
self._dist = special_ortho_group_gen(seed)
|
|
self.dim = self._dist._process_parameters(dim)
|
|
|
|
def rvs(self, size=1, random_state=None):
|
|
return self._dist.rvs(self.dim, size, random_state)
|
|
|
|
|
|
class ortho_group_gen(multi_rv_generic):
|
|
r"""An Orthogonal matrix (O(N)) random variable.
|
|
|
|
Return a random orthogonal matrix, drawn from the O(N) Haar
|
|
distribution (the only uniform distribution on O(N)).
|
|
|
|
The `dim` keyword specifies the dimension N.
|
|
|
|
Methods
|
|
-------
|
|
rvs(dim=None, size=1, random_state=None)
|
|
Draw random samples from O(N).
|
|
|
|
Parameters
|
|
----------
|
|
dim : scalar
|
|
Dimension of matrices
|
|
seed : {None, int, np.random.RandomState, np.random.Generator}, optional
|
|
Used for drawing random variates.
|
|
If `seed` is `None`, the `~np.random.RandomState` singleton is used.
|
|
If `seed` is an int, a new ``RandomState`` instance is used, seeded
|
|
with seed.
|
|
If `seed` is already a ``RandomState`` or ``Generator`` instance,
|
|
then that object is used.
|
|
Default is `None`.
|
|
|
|
Notes
|
|
-----
|
|
This class is closely related to `special_ortho_group`.
|
|
|
|
Some care is taken to avoid numerical error, as per the paper by Mezzadri.
|
|
|
|
References
|
|
----------
|
|
.. [1] F. Mezzadri, "How to generate random matrices from the classical
|
|
compact groups", :arXiv:`math-ph/0609050v2`.
|
|
|
|
Examples
|
|
--------
|
|
>>> import numpy as np
|
|
>>> from scipy.stats import ortho_group
|
|
>>> x = ortho_group.rvs(3)
|
|
|
|
>>> np.dot(x, x.T)
|
|
array([[ 1.00000000e+00, 1.13231364e-17, -2.86852790e-16],
|
|
[ 1.13231364e-17, 1.00000000e+00, -1.46845020e-16],
|
|
[ -2.86852790e-16, -1.46845020e-16, 1.00000000e+00]])
|
|
|
|
>>> import scipy.linalg
|
|
>>> np.fabs(scipy.linalg.det(x))
|
|
1.0
|
|
|
|
This generates one random matrix from O(3). It is orthogonal and
|
|
has a determinant of +1 or -1.
|
|
|
|
Alternatively, the object may be called (as a function) to fix the `dim`
|
|
parameter, returning a "frozen" ortho_group random variable:
|
|
|
|
>>> rv = ortho_group(5)
|
|
>>> # Frozen object with the same methods but holding the
|
|
>>> # dimension parameter fixed.
|
|
|
|
See Also
|
|
--------
|
|
special_ortho_group
|
|
"""
|
|
|
|
def __init__(self, seed=None):
|
|
super().__init__(seed)
|
|
self.__doc__ = doccer.docformat(self.__doc__)
|
|
|
|
def __call__(self, dim=None, seed=None):
|
|
"""Create a frozen O(N) distribution.
|
|
|
|
See `ortho_group_frozen` for more information.
|
|
"""
|
|
return ortho_group_frozen(dim, seed=seed)
|
|
|
|
def _process_parameters(self, dim):
|
|
"""Dimension N must be specified; it cannot be inferred."""
|
|
if dim is None or not np.isscalar(dim) or dim < 0 or dim != int(dim):
|
|
raise ValueError("Dimension of rotation must be specified,"
|
|
"and must be a scalar nonnegative integer.")
|
|
|
|
return dim
|
|
|
|
def rvs(self, dim, size=1, random_state=None):
|
|
"""Draw random samples from O(N).
|
|
|
|
Parameters
|
|
----------
|
|
dim : integer
|
|
Dimension of rotation space (N).
|
|
size : integer, optional
|
|
Number of samples to draw (default 1).
|
|
|
|
Returns
|
|
-------
|
|
rvs : ndarray or scalar
|
|
Random size N-dimensional matrices, dimension (size, dim, dim)
|
|
|
|
"""
|
|
random_state = self._get_random_state(random_state)
|
|
|
|
size = int(size)
|
|
|
|
dim = self._process_parameters(dim)
|
|
|
|
size = (size,) if size > 1 else ()
|
|
z = random_state.normal(size=size + (dim, dim))
|
|
q, r = np.linalg.qr(z)
|
|
# The last two dimensions are the rows and columns of R matrices.
|
|
# Extract the diagonals. Note that this eliminates a dimension.
|
|
d = r.diagonal(offset=0, axis1=-2, axis2=-1)
|
|
# Add back a dimension for proper broadcasting: we're dividing
|
|
# each row of each R matrix by the diagonal of the R matrix.
|
|
q *= (d/abs(d))[..., np.newaxis, :] # to broadcast properly
|
|
return q
|
|
|
|
|
|
ortho_group = ortho_group_gen()
|
|
|
|
|
|
class ortho_group_frozen(multi_rv_frozen):
|
|
def __init__(self, dim=None, seed=None):
|
|
"""Create a frozen O(N) distribution.
|
|
|
|
Parameters
|
|
----------
|
|
dim : scalar
|
|
Dimension of matrices
|
|
seed : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional
|
|
If `seed` is None (or `np.random`), the `numpy.random.RandomState`
|
|
singleton is used.
|
|
If `seed` is an int, a new ``RandomState`` instance is used,
|
|
seeded with `seed`.
|
|
If `seed` is already a ``Generator`` or ``RandomState`` instance
|
|
then that instance is used.
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy.stats import ortho_group
|
|
>>> g = ortho_group(5)
|
|
>>> x = g.rvs()
|
|
|
|
""" # numpy/numpydoc#87 # noqa: E501
|
|
self._dist = ortho_group_gen(seed)
|
|
self.dim = self._dist._process_parameters(dim)
|
|
|
|
def rvs(self, size=1, random_state=None):
|
|
return self._dist.rvs(self.dim, size, random_state)
|
|
|
|
|
|
class random_correlation_gen(multi_rv_generic):
|
|
r"""A random correlation matrix.
|
|
|
|
Return a random correlation matrix, given a vector of eigenvalues.
|
|
The returned matrix is symmetric positive semidefinite with unit diagonal.
|
|
|
|
The `eigs` keyword specifies the eigenvalues of the correlation matrix,
|
|
and implies the dimension.
|
|
|
|
Methods
|
|
-------
|
|
rvs(eigs=None, random_state=None)
|
|
Draw random correlation matrices, all with eigenvalues eigs.
|
|
|
|
Parameters
|
|
----------
|
|
eigs : 1d ndarray
|
|
Eigenvalues of correlation matrix. All eigenvalues need to be non-negative and
|
|
need to sum to the number of eigenvalues.
|
|
seed : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional
|
|
If `seed` is None (or `np.random`), the `numpy.random.RandomState`
|
|
singleton is used.
|
|
If `seed` is an int, a new ``RandomState`` instance is used,
|
|
seeded with `seed`.
|
|
If `seed` is already a ``Generator`` or ``RandomState`` instance
|
|
then that instance is used.
|
|
tol : float, optional
|
|
Tolerance for input parameter checks
|
|
diag_tol : float, optional
|
|
Tolerance for deviation of the diagonal of the resulting
|
|
matrix. Default: 1e-7
|
|
|
|
Raises
|
|
------
|
|
RuntimeError
|
|
Floating point error prevented generating a valid correlation
|
|
matrix.
|
|
|
|
Returns
|
|
-------
|
|
rvs : ndarray or scalar
|
|
Random size N-dimensional matrices, dimension (size, dim, dim),
|
|
each having eigenvalues eigs.
|
|
|
|
Notes
|
|
-----
|
|
|
|
Generates a random correlation matrix following a numerically stable
|
|
algorithm spelled out by Davies & Higham. This algorithm uses a single O(N)
|
|
similarity transformation to construct a symmetric positive semi-definite
|
|
matrix, and applies a series of Givens rotations to scale it to have ones
|
|
on the diagonal.
|
|
|
|
References
|
|
----------
|
|
|
|
.. [1] Davies, Philip I; Higham, Nicholas J; "Numerically stable generation
|
|
of correlation matrices and their factors", BIT 2000, Vol. 40,
|
|
No. 4, pp. 640 651
|
|
|
|
Examples
|
|
--------
|
|
>>> import numpy as np
|
|
>>> from scipy.stats import random_correlation
|
|
>>> rng = np.random.default_rng()
|
|
>>> x = random_correlation.rvs((.5, .8, 1.2, 1.5), random_state=rng)
|
|
>>> x
|
|
array([[ 1. , -0.02423399, 0.03130519, 0.4946965 ],
|
|
[-0.02423399, 1. , 0.20334736, 0.04039817],
|
|
[ 0.03130519, 0.20334736, 1. , 0.02694275],
|
|
[ 0.4946965 , 0.04039817, 0.02694275, 1. ]])
|
|
>>> import scipy.linalg
|
|
>>> e, v = scipy.linalg.eigh(x)
|
|
>>> e
|
|
array([ 0.5, 0.8, 1.2, 1.5])
|
|
|
|
"""
|
|
|
|
def __init__(self, seed=None):
|
|
super().__init__(seed)
|
|
self.__doc__ = doccer.docformat(self.__doc__)
|
|
|
|
def __call__(self, eigs, seed=None, tol=1e-13, diag_tol=1e-7):
|
|
"""Create a frozen random correlation matrix.
|
|
|
|
See `random_correlation_frozen` for more information.
|
|
"""
|
|
return random_correlation_frozen(eigs, seed=seed, tol=tol,
|
|
diag_tol=diag_tol)
|
|
|
|
def _process_parameters(self, eigs, tol):
|
|
eigs = np.asarray(eigs, dtype=float)
|
|
dim = eigs.size
|
|
|
|
if eigs.ndim != 1 or eigs.shape[0] != dim or dim <= 1:
|
|
raise ValueError("Array 'eigs' must be a vector of length "
|
|
"greater than 1.")
|
|
|
|
if np.fabs(np.sum(eigs) - dim) > tol:
|
|
raise ValueError("Sum of eigenvalues must equal dimensionality.")
|
|
|
|
for x in eigs:
|
|
if x < -tol:
|
|
raise ValueError("All eigenvalues must be non-negative.")
|
|
|
|
return dim, eigs
|
|
|
|
def _givens_to_1(self, aii, ajj, aij):
|
|
"""Computes a 2x2 Givens matrix to put 1's on the diagonal.
|
|
|
|
The input matrix is a 2x2 symmetric matrix M = [ aii aij ; aij ajj ].
|
|
|
|
The output matrix g is a 2x2 anti-symmetric matrix of the form
|
|
[ c s ; -s c ]; the elements c and s are returned.
|
|
|
|
Applying the output matrix to the input matrix (as b=g.T M g)
|
|
results in a matrix with bii=1, provided tr(M) - det(M) >= 1
|
|
and floating point issues do not occur. Otherwise, some other
|
|
valid rotation is returned. When tr(M)==2, also bjj=1.
|
|
|
|
"""
|
|
aiid = aii - 1.
|
|
ajjd = ajj - 1.
|
|
|
|
if ajjd == 0:
|
|
# ajj==1, so swap aii and ajj to avoid division by zero
|
|
return 0., 1.
|
|
|
|
dd = math.sqrt(max(aij**2 - aiid*ajjd, 0))
|
|
|
|
# The choice of t should be chosen to avoid cancellation [1]
|
|
t = (aij + math.copysign(dd, aij)) / ajjd
|
|
c = 1. / math.sqrt(1. + t*t)
|
|
if c == 0:
|
|
# Underflow
|
|
s = 1.0
|
|
else:
|
|
s = c*t
|
|
return c, s
|
|
|
|
def _to_corr(self, m):
|
|
"""
|
|
Given a psd matrix m, rotate to put one's on the diagonal, turning it
|
|
into a correlation matrix. This also requires the trace equal the
|
|
dimensionality. Note: modifies input matrix
|
|
"""
|
|
# Check requirements for in-place Givens
|
|
if not (m.flags.c_contiguous and m.dtype == np.float64 and
|
|
m.shape[0] == m.shape[1]):
|
|
raise ValueError()
|
|
|
|
d = m.shape[0]
|
|
for i in range(d-1):
|
|
if m[i, i] == 1:
|
|
continue
|
|
elif m[i, i] > 1:
|
|
for j in range(i+1, d):
|
|
if m[j, j] < 1:
|
|
break
|
|
else:
|
|
for j in range(i+1, d):
|
|
if m[j, j] > 1:
|
|
break
|
|
|
|
c, s = self._givens_to_1(m[i, i], m[j, j], m[i, j])
|
|
|
|
# Use BLAS to apply Givens rotations in-place. Equivalent to:
|
|
# g = np.eye(d)
|
|
# g[i, i] = g[j,j] = c
|
|
# g[j, i] = -s; g[i, j] = s
|
|
# m = np.dot(g.T, np.dot(m, g))
|
|
mv = m.ravel()
|
|
drot(mv, mv, c, -s, n=d,
|
|
offx=i*d, incx=1, offy=j*d, incy=1,
|
|
overwrite_x=True, overwrite_y=True)
|
|
drot(mv, mv, c, -s, n=d,
|
|
offx=i, incx=d, offy=j, incy=d,
|
|
overwrite_x=True, overwrite_y=True)
|
|
|
|
return m
|
|
|
|
def rvs(self, eigs, random_state=None, tol=1e-13, diag_tol=1e-7):
|
|
"""Draw random correlation matrices.
|
|
|
|
Parameters
|
|
----------
|
|
eigs : 1d ndarray
|
|
Eigenvalues of correlation matrix
|
|
tol : float, optional
|
|
Tolerance for input parameter checks
|
|
diag_tol : float, optional
|
|
Tolerance for deviation of the diagonal of the resulting
|
|
matrix. Default: 1e-7
|
|
|
|
Raises
|
|
------
|
|
RuntimeError
|
|
Floating point error prevented generating a valid correlation
|
|
matrix.
|
|
|
|
Returns
|
|
-------
|
|
rvs : ndarray or scalar
|
|
Random size N-dimensional matrices, dimension (size, dim, dim),
|
|
each having eigenvalues eigs.
|
|
|
|
"""
|
|
dim, eigs = self._process_parameters(eigs, tol=tol)
|
|
|
|
random_state = self._get_random_state(random_state)
|
|
|
|
m = ortho_group.rvs(dim, random_state=random_state)
|
|
m = np.dot(np.dot(m, np.diag(eigs)), m.T) # Set the trace of m
|
|
m = self._to_corr(m) # Carefully rotate to unit diagonal
|
|
|
|
# Check diagonal
|
|
if abs(m.diagonal() - 1).max() > diag_tol:
|
|
raise RuntimeError("Failed to generate a valid correlation matrix")
|
|
|
|
return m
|
|
|
|
|
|
random_correlation = random_correlation_gen()
|
|
|
|
|
|
class random_correlation_frozen(multi_rv_frozen):
|
|
def __init__(self, eigs, seed=None, tol=1e-13, diag_tol=1e-7):
|
|
"""Create a frozen random correlation matrix distribution.
|
|
|
|
Parameters
|
|
----------
|
|
eigs : 1d ndarray
|
|
Eigenvalues of correlation matrix
|
|
seed : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional
|
|
If `seed` is None (or `np.random`), the `numpy.random.RandomState`
|
|
singleton is used.
|
|
If `seed` is an int, a new ``RandomState`` instance is used,
|
|
seeded with `seed`.
|
|
If `seed` is already a ``Generator`` or ``RandomState`` instance
|
|
then that instance is used.
|
|
tol : float, optional
|
|
Tolerance for input parameter checks
|
|
diag_tol : float, optional
|
|
Tolerance for deviation of the diagonal of the resulting
|
|
matrix. Default: 1e-7
|
|
|
|
Raises
|
|
------
|
|
RuntimeError
|
|
Floating point error prevented generating a valid correlation
|
|
matrix.
|
|
|
|
Returns
|
|
-------
|
|
rvs : ndarray or scalar
|
|
Random size N-dimensional matrices, dimension (size, dim, dim),
|
|
each having eigenvalues eigs.
|
|
""" # numpy/numpydoc#87 # noqa: E501
|
|
|
|
self._dist = random_correlation_gen(seed)
|
|
self.tol = tol
|
|
self.diag_tol = diag_tol
|
|
_, self.eigs = self._dist._process_parameters(eigs, tol=self.tol)
|
|
|
|
def rvs(self, random_state=None):
|
|
return self._dist.rvs(self.eigs, random_state=random_state,
|
|
tol=self.tol, diag_tol=self.diag_tol)
|
|
|
|
|
|
class unitary_group_gen(multi_rv_generic):
|
|
r"""A matrix-valued U(N) random variable.
|
|
|
|
Return a random unitary matrix.
|
|
|
|
The `dim` keyword specifies the dimension N.
|
|
|
|
Methods
|
|
-------
|
|
rvs(dim=None, size=1, random_state=None)
|
|
Draw random samples from U(N).
|
|
|
|
Parameters
|
|
----------
|
|
dim : scalar
|
|
Dimension of matrices.
|
|
seed : {None, int, np.random.RandomState, np.random.Generator}, optional
|
|
Used for drawing random variates.
|
|
If `seed` is `None`, the `~np.random.RandomState` singleton is used.
|
|
If `seed` is an int, a new ``RandomState`` instance is used, seeded
|
|
with seed.
|
|
If `seed` is already a ``RandomState`` or ``Generator`` instance,
|
|
then that object is used.
|
|
Default is `None`.
|
|
|
|
Notes
|
|
-----
|
|
This class is similar to `ortho_group`.
|
|
|
|
References
|
|
----------
|
|
.. [1] F. Mezzadri, "How to generate random matrices from the classical
|
|
compact groups", :arXiv:`math-ph/0609050v2`.
|
|
|
|
Examples
|
|
--------
|
|
>>> import numpy as np
|
|
>>> from scipy.stats import unitary_group
|
|
>>> x = unitary_group.rvs(3)
|
|
|
|
>>> np.dot(x, x.conj().T)
|
|
array([[ 1.00000000e+00, 1.13231364e-17, -2.86852790e-16],
|
|
[ 1.13231364e-17, 1.00000000e+00, -1.46845020e-16],
|
|
[ -2.86852790e-16, -1.46845020e-16, 1.00000000e+00]]) # may vary
|
|
|
|
This generates one random matrix from U(3). The dot product confirms that
|
|
it is unitary up to machine precision.
|
|
|
|
Alternatively, the object may be called (as a function) to fix the `dim`
|
|
parameter, return a "frozen" unitary_group random variable:
|
|
|
|
>>> rv = unitary_group(5)
|
|
|
|
See Also
|
|
--------
|
|
ortho_group
|
|
|
|
"""
|
|
|
|
def __init__(self, seed=None):
|
|
super().__init__(seed)
|
|
self.__doc__ = doccer.docformat(self.__doc__)
|
|
|
|
def __call__(self, dim=None, seed=None):
|
|
"""Create a frozen (U(N)) n-dimensional unitary matrix distribution.
|
|
|
|
See `unitary_group_frozen` for more information.
|
|
"""
|
|
return unitary_group_frozen(dim, seed=seed)
|
|
|
|
def _process_parameters(self, dim):
|
|
"""Dimension N must be specified; it cannot be inferred."""
|
|
if dim is None or not np.isscalar(dim) or dim < 0 or dim != int(dim):
|
|
raise ValueError("Dimension of rotation must be specified,"
|
|
"and must be a scalar nonnegative integer.")
|
|
|
|
return dim
|
|
|
|
def rvs(self, dim, size=1, random_state=None):
|
|
"""Draw random samples from U(N).
|
|
|
|
Parameters
|
|
----------
|
|
dim : integer
|
|
Dimension of space (N).
|
|
size : integer, optional
|
|
Number of samples to draw (default 1).
|
|
|
|
Returns
|
|
-------
|
|
rvs : ndarray or scalar
|
|
Random size N-dimensional matrices, dimension (size, dim, dim)
|
|
|
|
"""
|
|
random_state = self._get_random_state(random_state)
|
|
|
|
size = int(size)
|
|
|
|
dim = self._process_parameters(dim)
|
|
|
|
size = (size,) if size > 1 else ()
|
|
z = 1/math.sqrt(2)*(random_state.normal(size=size + (dim, dim)) +
|
|
1j*random_state.normal(size=size + (dim, dim)))
|
|
q, r = np.linalg.qr(z)
|
|
# The last two dimensions are the rows and columns of R matrices.
|
|
# Extract the diagonals. Note that this eliminates a dimension.
|
|
d = r.diagonal(offset=0, axis1=-2, axis2=-1)
|
|
# Add back a dimension for proper broadcasting: we're dividing
|
|
# each row of each R matrix by the diagonal of the R matrix.
|
|
q *= (d/abs(d))[..., np.newaxis, :] # to broadcast properly
|
|
return q
|
|
|
|
|
|
unitary_group = unitary_group_gen()
|
|
|
|
|
|
class unitary_group_frozen(multi_rv_frozen):
|
|
def __init__(self, dim=None, seed=None):
|
|
"""Create a frozen (U(N)) n-dimensional unitary matrix distribution.
|
|
|
|
Parameters
|
|
----------
|
|
dim : scalar
|
|
Dimension of matrices
|
|
seed : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional
|
|
If `seed` is None (or `np.random`), the `numpy.random.RandomState`
|
|
singleton is used.
|
|
If `seed` is an int, a new ``RandomState`` instance is used,
|
|
seeded with `seed`.
|
|
If `seed` is already a ``Generator`` or ``RandomState`` instance
|
|
then that instance is used.
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy.stats import unitary_group
|
|
>>> x = unitary_group(3)
|
|
>>> x.rvs()
|
|
|
|
""" # numpy/numpydoc#87 # noqa: E501
|
|
self._dist = unitary_group_gen(seed)
|
|
self.dim = self._dist._process_parameters(dim)
|
|
|
|
def rvs(self, size=1, random_state=None):
|
|
return self._dist.rvs(self.dim, size, random_state)
|
|
|
|
|
|
_mvt_doc_default_callparams = """\
|
|
loc : array_like, optional
|
|
Location of the distribution. (default ``0``)
|
|
shape : array_like, optional
|
|
Positive semidefinite matrix of the distribution. (default ``1``)
|
|
df : float, optional
|
|
Degrees of freedom of the distribution; must be greater than zero.
|
|
If ``np.inf`` then results are multivariate normal. The default is ``1``.
|
|
allow_singular : bool, optional
|
|
Whether to allow a singular matrix. (default ``False``)
|
|
"""
|
|
|
|
_mvt_doc_callparams_note = """\
|
|
Setting the parameter `loc` to ``None`` is equivalent to having `loc`
|
|
be the zero-vector. The parameter `shape` can be a scalar, in which case
|
|
the shape matrix is the identity times that value, a vector of
|
|
diagonal entries for the shape matrix, or a two-dimensional array_like.
|
|
"""
|
|
|
|
_mvt_doc_frozen_callparams_note = """\
|
|
See class definition for a detailed description of parameters."""
|
|
|
|
mvt_docdict_params = {
|
|
'_mvt_doc_default_callparams': _mvt_doc_default_callparams,
|
|
'_mvt_doc_callparams_note': _mvt_doc_callparams_note,
|
|
'_doc_random_state': _doc_random_state
|
|
}
|
|
|
|
mvt_docdict_noparams = {
|
|
'_mvt_doc_default_callparams': "",
|
|
'_mvt_doc_callparams_note': _mvt_doc_frozen_callparams_note,
|
|
'_doc_random_state': _doc_random_state
|
|
}
|
|
|
|
|
|
class multivariate_t_gen(multi_rv_generic):
|
|
r"""A multivariate t-distributed random variable.
|
|
|
|
The `loc` parameter specifies the location. The `shape` parameter specifies
|
|
the positive semidefinite shape matrix. The `df` parameter specifies the
|
|
degrees of freedom.
|
|
|
|
In addition to calling the methods below, the object itself may be called
|
|
as a function to fix the location, shape matrix, and degrees of freedom
|
|
parameters, returning a "frozen" multivariate t-distribution random.
|
|
|
|
Methods
|
|
-------
|
|
pdf(x, loc=None, shape=1, df=1, allow_singular=False)
|
|
Probability density function.
|
|
logpdf(x, loc=None, shape=1, df=1, allow_singular=False)
|
|
Log of the probability density function.
|
|
cdf(x, loc=None, shape=1, df=1, allow_singular=False, *,
|
|
maxpts=None, lower_limit=None, random_state=None)
|
|
Cumulative distribution function.
|
|
rvs(loc=None, shape=1, df=1, size=1, random_state=None)
|
|
Draw random samples from a multivariate t-distribution.
|
|
entropy(loc=None, shape=1, df=1)
|
|
Differential entropy of a multivariate t-distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_mvt_doc_default_callparams)s
|
|
%(_doc_random_state)s
|
|
|
|
Notes
|
|
-----
|
|
%(_mvt_doc_callparams_note)s
|
|
The matrix `shape` must be a (symmetric) positive semidefinite matrix. The
|
|
determinant and inverse of `shape` are computed as the pseudo-determinant
|
|
and pseudo-inverse, respectively, so that `shape` does not need to have
|
|
full rank.
|
|
|
|
The probability density function for `multivariate_t` is
|
|
|
|
.. math::
|
|
|
|
f(x) = \frac{\Gamma((\nu + p)/2)}{\Gamma(\nu/2)\nu^{p/2}\pi^{p/2}|\Sigma|^{1/2}}
|
|
\left[1 + \frac{1}{\nu} (\mathbf{x} - \boldsymbol{\mu})^{\top}
|
|
\boldsymbol{\Sigma}^{-1}
|
|
(\mathbf{x} - \boldsymbol{\mu}) \right]^{-(\nu + p)/2},
|
|
|
|
where :math:`p` is the dimension of :math:`\mathbf{x}`,
|
|
:math:`\boldsymbol{\mu}` is the :math:`p`-dimensional location,
|
|
:math:`\boldsymbol{\Sigma}` the :math:`p \times p`-dimensional shape
|
|
matrix, and :math:`\nu` is the degrees of freedom.
|
|
|
|
.. versionadded:: 1.6.0
|
|
|
|
References
|
|
----------
|
|
.. [1] Arellano-Valle et al. "Shannon Entropy and Mutual Information for
|
|
Multivariate Skew-Elliptical Distributions". Scandinavian Journal
|
|
of Statistics. Vol. 40, issue 1.
|
|
|
|
Examples
|
|
--------
|
|
The object may be called (as a function) to fix the `loc`, `shape`,
|
|
`df`, and `allow_singular` parameters, returning a "frozen"
|
|
multivariate_t random variable:
|
|
|
|
>>> import numpy as np
|
|
>>> from scipy.stats import multivariate_t
|
|
>>> rv = multivariate_t([1.0, -0.5], [[2.1, 0.3], [0.3, 1.5]], df=2)
|
|
>>> # Frozen object with the same methods but holding the given location,
|
|
>>> # scale, and degrees of freedom fixed.
|
|
|
|
Create a contour plot of the PDF.
|
|
|
|
>>> import matplotlib.pyplot as plt
|
|
>>> x, y = np.mgrid[-1:3:.01, -2:1.5:.01]
|
|
>>> pos = np.dstack((x, y))
|
|
>>> fig, ax = plt.subplots(1, 1)
|
|
>>> ax.set_aspect('equal')
|
|
>>> plt.contourf(x, y, rv.pdf(pos))
|
|
|
|
"""
|
|
|
|
def __init__(self, seed=None):
|
|
"""Initialize a multivariate t-distributed random variable.
|
|
|
|
Parameters
|
|
----------
|
|
seed : Random state.
|
|
|
|
"""
|
|
super().__init__(seed)
|
|
self.__doc__ = doccer.docformat(self.__doc__, mvt_docdict_params)
|
|
self._random_state = check_random_state(seed)
|
|
|
|
def __call__(self, loc=None, shape=1, df=1, allow_singular=False,
|
|
seed=None):
|
|
"""Create a frozen multivariate t-distribution.
|
|
|
|
See `multivariate_t_frozen` for parameters.
|
|
"""
|
|
if df == np.inf:
|
|
return multivariate_normal_frozen(mean=loc, cov=shape,
|
|
allow_singular=allow_singular,
|
|
seed=seed)
|
|
return multivariate_t_frozen(loc=loc, shape=shape, df=df,
|
|
allow_singular=allow_singular, seed=seed)
|
|
|
|
def pdf(self, x, loc=None, shape=1, df=1, allow_singular=False):
|
|
"""Multivariate t-distribution probability density function.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
Points at which to evaluate the probability density function.
|
|
%(_mvt_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
pdf : Probability density function evaluated at `x`.
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy.stats import multivariate_t
|
|
>>> x = [0.4, 5]
|
|
>>> loc = [0, 1]
|
|
>>> shape = [[1, 0.1], [0.1, 1]]
|
|
>>> df = 7
|
|
>>> multivariate_t.pdf(x, loc, shape, df)
|
|
0.00075713
|
|
|
|
"""
|
|
dim, loc, shape, df = self._process_parameters(loc, shape, df)
|
|
x = self._process_quantiles(x, dim)
|
|
shape_info = _PSD(shape, allow_singular=allow_singular)
|
|
logpdf = self._logpdf(x, loc, shape_info.U, shape_info.log_pdet, df,
|
|
dim, shape_info.rank)
|
|
return np.exp(logpdf)
|
|
|
|
def logpdf(self, x, loc=None, shape=1, df=1):
|
|
"""Log of the multivariate t-distribution probability density function.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
Points at which to evaluate the log of the probability density
|
|
function.
|
|
%(_mvt_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
logpdf : Log of the probability density function evaluated at `x`.
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy.stats import multivariate_t
|
|
>>> x = [0.4, 5]
|
|
>>> loc = [0, 1]
|
|
>>> shape = [[1, 0.1], [0.1, 1]]
|
|
>>> df = 7
|
|
>>> multivariate_t.logpdf(x, loc, shape, df)
|
|
-7.1859802
|
|
|
|
See Also
|
|
--------
|
|
pdf : Probability density function.
|
|
|
|
"""
|
|
dim, loc, shape, df = self._process_parameters(loc, shape, df)
|
|
x = self._process_quantiles(x, dim)
|
|
shape_info = _PSD(shape)
|
|
return self._logpdf(x, loc, shape_info.U, shape_info.log_pdet, df, dim,
|
|
shape_info.rank)
|
|
|
|
def _logpdf(self, x, loc, prec_U, log_pdet, df, dim, rank):
|
|
"""Utility method `pdf`, `logpdf` for parameters.
|
|
|
|
Parameters
|
|
----------
|
|
x : ndarray
|
|
Points at which to evaluate the log of the probability density
|
|
function.
|
|
loc : ndarray
|
|
Location of the distribution.
|
|
prec_U : ndarray
|
|
A decomposition such that `np.dot(prec_U, prec_U.T)` is the inverse
|
|
of the shape matrix.
|
|
log_pdet : float
|
|
Logarithm of the determinant of the shape matrix.
|
|
df : float
|
|
Degrees of freedom of the distribution.
|
|
dim : int
|
|
Dimension of the quantiles x.
|
|
rank : int
|
|
Rank of the shape matrix.
|
|
|
|
Notes
|
|
-----
|
|
As this function does no argument checking, it should not be called
|
|
directly; use 'logpdf' instead.
|
|
|
|
"""
|
|
if df == np.inf:
|
|
return multivariate_normal._logpdf(x, loc, prec_U, log_pdet, rank)
|
|
|
|
dev = x - loc
|
|
maha = np.square(np.dot(dev, prec_U)).sum(axis=-1)
|
|
|
|
t = 0.5 * (df + dim)
|
|
A = gammaln(t)
|
|
B = gammaln(0.5 * df)
|
|
C = dim/2. * np.log(df * np.pi)
|
|
D = 0.5 * log_pdet
|
|
E = -t * np.log(1 + (1./df) * maha)
|
|
|
|
return _squeeze_output(A - B - C - D + E)
|
|
|
|
def _cdf(self, x, loc, shape, df, dim, maxpts=None, lower_limit=None,
|
|
random_state=None):
|
|
|
|
# All of this - random state validation, maxpts, apply_along_axis,
|
|
# etc. needs to go in this private method unless we want
|
|
# frozen distribution's `cdf` method to duplicate it or call `cdf`,
|
|
# which would require re-processing parameters
|
|
if random_state is not None:
|
|
rng = check_random_state(random_state)
|
|
else:
|
|
rng = self._random_state
|
|
|
|
if not maxpts:
|
|
maxpts = 1000 * dim
|
|
|
|
x = self._process_quantiles(x, dim)
|
|
lower_limit = (np.full(loc.shape, -np.inf)
|
|
if lower_limit is None else lower_limit)
|
|
|
|
# remove the mean
|
|
x, lower_limit = x - loc, lower_limit - loc
|
|
|
|
b, a = np.broadcast_arrays(x, lower_limit)
|
|
i_swap = b < a
|
|
signs = (-1)**(i_swap.sum(axis=-1)) # odd # of swaps -> negative
|
|
a, b = a.copy(), b.copy()
|
|
a[i_swap], b[i_swap] = b[i_swap], a[i_swap]
|
|
n = x.shape[-1]
|
|
limits = np.concatenate((a, b), axis=-1)
|
|
|
|
def func1d(limits):
|
|
a, b = limits[:n], limits[n:]
|
|
return _qmvt(maxpts, df, shape, a, b, rng)[0]
|
|
|
|
res = np.apply_along_axis(func1d, -1, limits) * signs
|
|
# Fixing the output shape for existing distributions is a separate
|
|
# issue. For now, let's keep this consistent with pdf.
|
|
return _squeeze_output(res)
|
|
|
|
def cdf(self, x, loc=None, shape=1, df=1, allow_singular=False, *,
|
|
maxpts=None, lower_limit=None, random_state=None):
|
|
"""Multivariate t-distribution cumulative distribution function.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
Points at which to evaluate the cumulative distribution function.
|
|
%(_mvt_doc_default_callparams)s
|
|
maxpts : int, optional
|
|
Maximum number of points to use for integration. The default is
|
|
1000 times the number of dimensions.
|
|
lower_limit : array_like, optional
|
|
Lower limit of integration of the cumulative distribution function.
|
|
Default is negative infinity. Must be broadcastable with `x`.
|
|
%(_doc_random_state)s
|
|
|
|
Returns
|
|
-------
|
|
cdf : ndarray or scalar
|
|
Cumulative distribution function evaluated at `x`.
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy.stats import multivariate_t
|
|
>>> x = [0.4, 5]
|
|
>>> loc = [0, 1]
|
|
>>> shape = [[1, 0.1], [0.1, 1]]
|
|
>>> df = 7
|
|
>>> multivariate_t.cdf(x, loc, shape, df)
|
|
0.64798491
|
|
|
|
"""
|
|
dim, loc, shape, df = self._process_parameters(loc, shape, df)
|
|
shape = _PSD(shape, allow_singular=allow_singular)._M
|
|
|
|
return self._cdf(x, loc, shape, df, dim, maxpts,
|
|
lower_limit, random_state)
|
|
|
|
def _entropy(self, dim, df=1, shape=1):
|
|
if df == np.inf:
|
|
return multivariate_normal(None, cov=shape).entropy()
|
|
|
|
shape_info = _PSD(shape)
|
|
shape_term = 0.5 * shape_info.log_pdet
|
|
|
|
def regular(dim, df):
|
|
halfsum = 0.5 * (dim + df)
|
|
half_df = 0.5 * df
|
|
return (
|
|
-gammaln(halfsum) + gammaln(half_df)
|
|
+ 0.5 * dim * np.log(df * np.pi) + halfsum
|
|
* (psi(halfsum) - psi(half_df))
|
|
+ shape_term
|
|
)
|
|
|
|
def asymptotic(dim, df):
|
|
# Formula from Wolfram Alpha:
|
|
# "asymptotic expansion -gammaln((m+d)/2) + gammaln(d/2) + (m*log(d*pi))/2
|
|
# + ((m+d)/2) * (digamma((m+d)/2) - digamma(d/2))"
|
|
return (
|
|
dim * norm._entropy() + dim / df
|
|
- dim * (dim - 2) * df**-2.0 / 4
|
|
+ dim**2 * (dim - 2) * df**-3.0 / 6
|
|
+ dim * (-3 * dim**3 + 8 * dim**2 - 8) * df**-4.0 / 24
|
|
+ dim**2 * (3 * dim**3 - 10 * dim**2 + 16) * df**-5.0 / 30
|
|
+ shape_term
|
|
)[()]
|
|
|
|
# preserves ~12 digits accuracy up to at least `dim=1e5`. See gh-18465.
|
|
threshold = dim * 100 * 4 / (np.log(dim) + 1)
|
|
return xpx.apply_where(df >= threshold, (dim, df), asymptotic, regular)
|
|
|
|
def entropy(self, loc=None, shape=1, df=1):
|
|
"""Calculate the differential entropy of a multivariate
|
|
t-distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_mvt_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
h : float
|
|
Differential entropy
|
|
|
|
"""
|
|
dim, loc, shape, df = self._process_parameters(None, shape, df)
|
|
return self._entropy(dim, df, shape)
|
|
|
|
def rvs(self, loc=None, shape=1, df=1, size=1, random_state=None):
|
|
"""Draw random samples from a multivariate t-distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_mvt_doc_default_callparams)s
|
|
size : integer, optional
|
|
Number of samples to draw (default 1).
|
|
%(_doc_random_state)s
|
|
|
|
Returns
|
|
-------
|
|
rvs : ndarray or scalar
|
|
Random variates of size (`size`, `P`), where `P` is the
|
|
dimension of the random variable.
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy.stats import multivariate_t
|
|
>>> x = [0.4, 5]
|
|
>>> loc = [0, 1]
|
|
>>> shape = [[1, 0.1], [0.1, 1]]
|
|
>>> df = 7
|
|
>>> multivariate_t.rvs(loc, shape, df)
|
|
array([[0.93477495, 3.00408716]])
|
|
|
|
"""
|
|
# For implementation details, see equation (3):
|
|
#
|
|
# Hofert, "On Sampling from the Multivariatet Distribution", 2013
|
|
# http://rjournal.github.io/archive/2013-2/hofert.pdf
|
|
#
|
|
dim, loc, shape, df = self._process_parameters(loc, shape, df)
|
|
if random_state is not None:
|
|
rng = check_random_state(random_state)
|
|
else:
|
|
rng = self._random_state
|
|
|
|
if np.isinf(df):
|
|
x = np.ones(size)
|
|
else:
|
|
x = rng.chisquare(df, size=size) / df
|
|
|
|
z = rng.multivariate_normal(np.zeros(dim), shape, size=size)
|
|
samples = loc + z / np.sqrt(x)[..., None]
|
|
return _squeeze_output(samples)
|
|
|
|
def _process_quantiles(self, x, dim):
|
|
"""
|
|
Adjust quantiles array so that last axis labels the components of
|
|
each data point.
|
|
"""
|
|
x = np.asarray(x, dtype=float)
|
|
if x.ndim == 0:
|
|
x = x[np.newaxis]
|
|
elif x.ndim == 1:
|
|
if dim == 1:
|
|
x = x[:, np.newaxis]
|
|
else:
|
|
x = x[np.newaxis, :]
|
|
return x
|
|
|
|
def _process_parameters(self, loc, shape, df):
|
|
"""
|
|
Infer dimensionality from location array and shape matrix, handle
|
|
defaults, and ensure compatible dimensions.
|
|
"""
|
|
if loc is None and shape is None:
|
|
loc = np.asarray(0, dtype=float)
|
|
shape = np.asarray(1, dtype=float)
|
|
dim = 1
|
|
elif loc is None:
|
|
shape = np.asarray(shape, dtype=float)
|
|
if shape.ndim < 2:
|
|
dim = 1
|
|
else:
|
|
dim = shape.shape[0]
|
|
loc = np.zeros(dim)
|
|
elif shape is None:
|
|
loc = np.asarray(loc, dtype=float)
|
|
dim = loc.size
|
|
shape = np.eye(dim)
|
|
else:
|
|
shape = np.asarray(shape, dtype=float)
|
|
loc = np.asarray(loc, dtype=float)
|
|
dim = loc.size
|
|
|
|
if dim == 1:
|
|
loc = loc.reshape(1)
|
|
shape = shape.reshape(1, 1)
|
|
|
|
if loc.ndim != 1 or loc.shape[0] != dim:
|
|
raise ValueError(f"Array 'loc' must be a vector of length {dim}.")
|
|
if shape.ndim == 0:
|
|
shape = shape * np.eye(dim)
|
|
elif shape.ndim == 1:
|
|
shape = np.diag(shape)
|
|
elif shape.ndim == 2 and shape.shape != (dim, dim):
|
|
rows, cols = shape.shape
|
|
if rows != cols:
|
|
msg = ("Array 'cov' must be square if it is two dimensional,"
|
|
f" but cov.shape = {str(shape.shape)}.")
|
|
else:
|
|
msg = ("Dimension mismatch: array 'cov' is of shape %s,"
|
|
" but 'loc' is a vector of length %d.")
|
|
msg = msg % (str(shape.shape), len(loc))
|
|
raise ValueError(msg)
|
|
elif shape.ndim > 2:
|
|
raise ValueError(f"Array 'cov' must be at most two-dimensional, "
|
|
f"but cov.ndim = {shape.ndim}")
|
|
|
|
# Process degrees of freedom.
|
|
if df is None:
|
|
df = 1
|
|
elif df <= 0:
|
|
raise ValueError("'df' must be greater than zero.")
|
|
elif np.isnan(df):
|
|
raise ValueError("'df' is 'nan' but must be greater than zero or 'np.inf'.")
|
|
|
|
return dim, loc, shape, df
|
|
|
|
|
|
class multivariate_t_frozen(multi_rv_frozen):
|
|
|
|
def __init__(self, loc=None, shape=1, df=1, allow_singular=False,
|
|
seed=None):
|
|
"""Create a frozen multivariate t distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_mvt_doc_default_callparams)s
|
|
|
|
Examples
|
|
--------
|
|
>>> import numpy as np
|
|
>>> from scipy.stats import multivariate_t
|
|
>>> loc = np.zeros(3)
|
|
>>> shape = np.eye(3)
|
|
>>> df = 10
|
|
>>> dist = multivariate_t(loc, shape, df)
|
|
>>> dist.rvs()
|
|
array([[ 0.81412036, -1.53612361, 0.42199647]])
|
|
>>> dist.pdf([1, 1, 1])
|
|
array([0.01237803])
|
|
|
|
"""
|
|
self._dist = multivariate_t_gen(seed)
|
|
dim, loc, shape, df = self._dist._process_parameters(loc, shape, df)
|
|
self.dim, self.loc, self.shape, self.df = dim, loc, shape, df
|
|
self.shape_info = _PSD(shape, allow_singular=allow_singular)
|
|
|
|
def logpdf(self, x):
|
|
x = self._dist._process_quantiles(x, self.dim)
|
|
U = self.shape_info.U
|
|
log_pdet = self.shape_info.log_pdet
|
|
return self._dist._logpdf(x, self.loc, U, log_pdet, self.df, self.dim,
|
|
self.shape_info.rank)
|
|
|
|
def cdf(self, x, *, maxpts=None, lower_limit=None, random_state=None):
|
|
x = self._dist._process_quantiles(x, self.dim)
|
|
return self._dist._cdf(x, self.loc, self.shape, self.df, self.dim,
|
|
maxpts, lower_limit, random_state)
|
|
|
|
def pdf(self, x):
|
|
return np.exp(self.logpdf(x))
|
|
|
|
def rvs(self, size=1, random_state=None):
|
|
return self._dist.rvs(loc=self.loc,
|
|
shape=self.shape,
|
|
df=self.df,
|
|
size=size,
|
|
random_state=random_state)
|
|
|
|
def entropy(self):
|
|
return self._dist._entropy(self.dim, self.df, self.shape)
|
|
|
|
|
|
multivariate_t = multivariate_t_gen()
|
|
|
|
|
|
# Set frozen generator docstrings from corresponding docstrings in
|
|
# multivariate_t_gen and fill in default strings in class docstrings
|
|
for name in ['logpdf', 'pdf', 'rvs', 'cdf', 'entropy']:
|
|
method = multivariate_t_gen.__dict__[name]
|
|
method_frozen = multivariate_t_frozen.__dict__[name]
|
|
method_frozen.__doc__ = doccer.docformat(method.__doc__,
|
|
mvt_docdict_noparams)
|
|
method.__doc__ = doccer.docformat(method.__doc__, mvt_docdict_params)
|
|
|
|
|
|
_mhg_doc_default_callparams = """\
|
|
m : array_like
|
|
The number of each type of object in the population.
|
|
That is, :math:`m[i]` is the number of objects of
|
|
type :math:`i`.
|
|
n : array_like
|
|
The number of samples taken from the population.
|
|
"""
|
|
|
|
_mhg_doc_callparams_note = """\
|
|
`m` must be an array of positive integers. If the quantile
|
|
:math:`i` contains values out of the range :math:`[0, m_i]`
|
|
where :math:`m_i` is the number of objects of type :math:`i`
|
|
in the population or if the parameters are inconsistent with one
|
|
another (e.g. ``x.sum() != n``), methods return the appropriate
|
|
value (e.g. ``0`` for ``pmf``). If `m` or `n` contain negative
|
|
values, the result will contain ``nan`` there.
|
|
"""
|
|
|
|
_mhg_doc_frozen_callparams = ""
|
|
|
|
_mhg_doc_frozen_callparams_note = """\
|
|
See class definition for a detailed description of parameters."""
|
|
|
|
mhg_docdict_params = {
|
|
'_doc_default_callparams': _mhg_doc_default_callparams,
|
|
'_doc_callparams_note': _mhg_doc_callparams_note,
|
|
'_doc_random_state': _doc_random_state
|
|
}
|
|
|
|
mhg_docdict_noparams = {
|
|
'_doc_default_callparams': _mhg_doc_frozen_callparams,
|
|
'_doc_callparams_note': _mhg_doc_frozen_callparams_note,
|
|
'_doc_random_state': _doc_random_state
|
|
}
|
|
|
|
|
|
class multivariate_hypergeom_gen(multi_rv_generic):
|
|
r"""A multivariate hypergeometric random variable.
|
|
|
|
Methods
|
|
-------
|
|
pmf(x, m, n)
|
|
Probability mass function.
|
|
logpmf(x, m, n)
|
|
Log of the probability mass function.
|
|
rvs(m, n, size=1, random_state=None)
|
|
Draw random samples from a multivariate hypergeometric
|
|
distribution.
|
|
mean(m, n)
|
|
Mean of the multivariate hypergeometric distribution.
|
|
var(m, n)
|
|
Variance of the multivariate hypergeometric distribution.
|
|
cov(m, n)
|
|
Compute the covariance matrix of the multivariate
|
|
hypergeometric distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_doc_default_callparams)s
|
|
%(_doc_random_state)s
|
|
|
|
Notes
|
|
-----
|
|
%(_doc_callparams_note)s
|
|
|
|
The probability mass function for `multivariate_hypergeom` is
|
|
|
|
.. math::
|
|
|
|
P(X_1 = x_1, X_2 = x_2, \ldots, X_k = x_k) = \frac{\binom{m_1}{x_1}
|
|
\binom{m_2}{x_2} \cdots \binom{m_k}{x_k}}{\binom{M}{n}}, \\ \quad
|
|
(x_1, x_2, \ldots, x_k) \in \mathbb{N}^k \text{ with }
|
|
\sum_{i=1}^k x_i = n
|
|
|
|
where :math:`m_i` are the number of objects of type :math:`i`, :math:`M`
|
|
is the total number of objects in the population (sum of all the
|
|
:math:`m_i`), and :math:`n` is the size of the sample to be taken
|
|
from the population.
|
|
|
|
.. versionadded:: 1.6.0
|
|
|
|
Examples
|
|
--------
|
|
To evaluate the probability mass function of the multivariate
|
|
hypergeometric distribution, with a dichotomous population of size
|
|
:math:`10` and :math:`20`, at a sample of size :math:`12` with
|
|
:math:`8` objects of the first type and :math:`4` objects of the
|
|
second type, use:
|
|
|
|
>>> from scipy.stats import multivariate_hypergeom
|
|
>>> multivariate_hypergeom.pmf(x=[8, 4], m=[10, 20], n=12)
|
|
0.0025207176631464523
|
|
|
|
The `multivariate_hypergeom` distribution is identical to the
|
|
corresponding `hypergeom` distribution (tiny numerical differences
|
|
notwithstanding) when only two types (good and bad) of objects
|
|
are present in the population as in the example above. Consider
|
|
another example for a comparison with the hypergeometric distribution:
|
|
|
|
>>> from scipy.stats import hypergeom
|
|
>>> multivariate_hypergeom.pmf(x=[3, 1], m=[10, 5], n=4)
|
|
0.4395604395604395
|
|
>>> hypergeom.pmf(k=3, M=15, n=4, N=10)
|
|
0.43956043956044005
|
|
|
|
The functions ``pmf``, ``logpmf``, ``mean``, ``var``, ``cov``, and ``rvs``
|
|
support broadcasting, under the convention that the vector parameters
|
|
(``x``, ``m``, and ``n``) are interpreted as if each row along the last
|
|
axis is a single object. For instance, we can combine the previous two
|
|
calls to `multivariate_hypergeom` as
|
|
|
|
>>> multivariate_hypergeom.pmf(x=[[8, 4], [3, 1]], m=[[10, 20], [10, 5]],
|
|
... n=[12, 4])
|
|
array([0.00252072, 0.43956044])
|
|
|
|
This broadcasting also works for ``cov``, where the output objects are
|
|
square matrices of size ``m.shape[-1]``. For example:
|
|
|
|
>>> multivariate_hypergeom.cov(m=[[7, 9], [10, 15]], n=[8, 12])
|
|
array([[[ 1.05, -1.05],
|
|
[-1.05, 1.05]],
|
|
[[ 1.56, -1.56],
|
|
[-1.56, 1.56]]])
|
|
|
|
That is, ``result[0]`` is equal to
|
|
``multivariate_hypergeom.cov(m=[7, 9], n=8)`` and ``result[1]`` is equal
|
|
to ``multivariate_hypergeom.cov(m=[10, 15], n=12)``.
|
|
|
|
Alternatively, the object may be called (as a function) to fix the `m`
|
|
and `n` parameters, returning a "frozen" multivariate hypergeometric
|
|
random variable.
|
|
|
|
>>> rv = multivariate_hypergeom(m=[10, 20], n=12)
|
|
>>> rv.pmf(x=[8, 4])
|
|
0.0025207176631464523
|
|
|
|
See Also
|
|
--------
|
|
scipy.stats.hypergeom : The hypergeometric distribution.
|
|
scipy.stats.multinomial : The multinomial distribution.
|
|
|
|
References
|
|
----------
|
|
.. [1] The Multivariate Hypergeometric Distribution,
|
|
http://www.randomservices.org/random/urn/MultiHypergeometric.html
|
|
.. [2] Thomas J. Sargent and John Stachurski, 2020,
|
|
Multivariate Hypergeometric Distribution
|
|
https://python.quantecon.org/multi_hyper.html
|
|
"""
|
|
def __init__(self, seed=None):
|
|
super().__init__(seed)
|
|
self.__doc__ = doccer.docformat(self.__doc__, mhg_docdict_params)
|
|
|
|
def __call__(self, m, n, seed=None):
|
|
"""Create a frozen multivariate_hypergeom distribution.
|
|
|
|
See `multivariate_hypergeom_frozen` for more information.
|
|
"""
|
|
return multivariate_hypergeom_frozen(m, n, seed=seed)
|
|
|
|
def _process_parameters(self, m, n):
|
|
m = np.asarray(m)
|
|
n = np.asarray(n)
|
|
if m.size == 0:
|
|
m = m.astype(int)
|
|
if n.size == 0:
|
|
n = n.astype(int)
|
|
if not np.issubdtype(m.dtype, np.integer):
|
|
raise TypeError("'m' must an array of integers.")
|
|
if not np.issubdtype(n.dtype, np.integer):
|
|
raise TypeError("'n' must an array of integers.")
|
|
if m.ndim == 0:
|
|
raise ValueError("'m' must be an array with"
|
|
" at least one dimension.")
|
|
|
|
# check for empty arrays
|
|
if m.size != 0:
|
|
n = n[..., np.newaxis]
|
|
|
|
m, n = np.broadcast_arrays(m, n)
|
|
|
|
# check for empty arrays
|
|
if m.size != 0:
|
|
n = n[..., 0]
|
|
|
|
mcond = m < 0
|
|
|
|
M = m.sum(axis=-1)
|
|
|
|
ncond = (n < 0) | (n > M)
|
|
return M, m, n, mcond, ncond, np.any(mcond, axis=-1) | ncond
|
|
|
|
def _process_quantiles(self, x, M, m, n):
|
|
x = np.asarray(x)
|
|
if not np.issubdtype(x.dtype, np.integer):
|
|
raise TypeError("'x' must an array of integers.")
|
|
if x.ndim == 0:
|
|
raise ValueError("'x' must be an array with"
|
|
" at least one dimension.")
|
|
if not x.shape[-1] == m.shape[-1]:
|
|
raise ValueError(f"Size of each quantile must be size of 'm': "
|
|
f"received {x.shape[-1]}, "
|
|
f"but expected {m.shape[-1]}.")
|
|
|
|
# check for empty arrays
|
|
if m.size != 0:
|
|
n = n[..., np.newaxis]
|
|
M = M[..., np.newaxis]
|
|
|
|
x, m, n, M = np.broadcast_arrays(x, m, n, M)
|
|
|
|
# check for empty arrays
|
|
if m.size != 0:
|
|
n, M = n[..., 0], M[..., 0]
|
|
|
|
xcond = (x < 0) | (x > m)
|
|
return (x, M, m, n, xcond,
|
|
np.any(xcond, axis=-1) | (x.sum(axis=-1) != n))
|
|
|
|
def _checkresult(self, result, cond, bad_value):
|
|
result = np.asarray(result)
|
|
if cond.ndim != 0:
|
|
result[cond] = bad_value
|
|
elif cond:
|
|
return bad_value
|
|
if result.ndim == 0:
|
|
return result[()]
|
|
return result
|
|
|
|
def _logpmf(self, x, M, m, n, mxcond, ncond):
|
|
# This equation of the pmf comes from the relation,
|
|
# n combine r = beta(n+1, 1) / beta(r+1, n-r+1)
|
|
num = np.zeros_like(m, dtype=np.float64)
|
|
den = np.zeros_like(n, dtype=np.float64)
|
|
m, x = m[~mxcond], x[~mxcond]
|
|
M, n = M[~ncond], n[~ncond]
|
|
num[~mxcond] = (betaln(m+1, 1) - betaln(x+1, m-x+1))
|
|
den[~ncond] = (betaln(M+1, 1) - betaln(n+1, M-n+1))
|
|
num[mxcond] = np.nan
|
|
den[ncond] = np.nan
|
|
num = num.sum(axis=-1)
|
|
return num - den
|
|
|
|
def logpmf(self, x, m, n):
|
|
"""Log of the multivariate hypergeometric probability mass function.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
Quantiles, with the last axis of `x` denoting the components.
|
|
%(_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
logpmf : ndarray or scalar
|
|
Log of the probability mass function evaluated at `x`
|
|
|
|
Notes
|
|
-----
|
|
%(_doc_callparams_note)s
|
|
"""
|
|
M, m, n, mcond, ncond, mncond = self._process_parameters(m, n)
|
|
(x, M, m, n, xcond,
|
|
xcond_reduced) = self._process_quantiles(x, M, m, n)
|
|
mxcond = mcond | xcond
|
|
ncond = ncond | np.zeros(n.shape, dtype=np.bool_)
|
|
|
|
result = self._logpmf(x, M, m, n, mxcond, ncond)
|
|
|
|
# replace values for which x was out of the domain; broadcast
|
|
# xcond to the right shape
|
|
xcond_ = xcond_reduced | np.zeros(mncond.shape, dtype=np.bool_)
|
|
result = self._checkresult(result, xcond_, -np.inf)
|
|
|
|
# replace values bad for n or m; broadcast
|
|
# mncond to the right shape
|
|
mncond_ = mncond | np.zeros(xcond_reduced.shape, dtype=np.bool_)
|
|
return self._checkresult(result, mncond_, np.nan)
|
|
|
|
def pmf(self, x, m, n):
|
|
"""Multivariate hypergeometric probability mass function.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
Quantiles, with the last axis of `x` denoting the components.
|
|
%(_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
pmf : ndarray or scalar
|
|
Probability density function evaluated at `x`
|
|
|
|
Notes
|
|
-----
|
|
%(_doc_callparams_note)s
|
|
"""
|
|
out = np.exp(self.logpmf(x, m, n))
|
|
return out
|
|
|
|
def mean(self, m, n):
|
|
"""Mean of the multivariate hypergeometric distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
mean : array_like or scalar
|
|
The mean of the distribution
|
|
"""
|
|
M, m, n, _, _, mncond = self._process_parameters(m, n)
|
|
# check for empty arrays
|
|
if m.size != 0:
|
|
M, n = M[..., np.newaxis], n[..., np.newaxis]
|
|
cond = (M == 0)
|
|
M = np.ma.masked_array(M, mask=cond)
|
|
mu = n*(m/M)
|
|
if m.size != 0:
|
|
mncond = (mncond[..., np.newaxis] |
|
|
np.zeros(mu.shape, dtype=np.bool_))
|
|
return self._checkresult(mu, mncond, np.nan)
|
|
|
|
def var(self, m, n):
|
|
"""Variance of the multivariate hypergeometric distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
array_like
|
|
The variances of the components of the distribution. This is
|
|
the diagonal of the covariance matrix of the distribution
|
|
"""
|
|
M, m, n, _, _, mncond = self._process_parameters(m, n)
|
|
# check for empty arrays
|
|
if m.size != 0:
|
|
M, n = M[..., np.newaxis], n[..., np.newaxis]
|
|
cond = (M == 0) & (M-1 == 0)
|
|
M = np.ma.masked_array(M, mask=cond)
|
|
output = n * m/M * (M-m)/M * (M-n)/(M-1)
|
|
if m.size != 0:
|
|
mncond = (mncond[..., np.newaxis] |
|
|
np.zeros(output.shape, dtype=np.bool_))
|
|
return self._checkresult(output, mncond, np.nan)
|
|
|
|
def cov(self, m, n):
|
|
"""Covariance matrix of the multivariate hypergeometric distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
cov : array_like
|
|
The covariance matrix of the distribution
|
|
"""
|
|
# see [1]_ for the formula and [2]_ for implementation
|
|
# cov( x_i,x_j ) = -n * (M-n)/(M-1) * (K_i*K_j) / (M**2)
|
|
M, m, n, _, _, mncond = self._process_parameters(m, n)
|
|
# check for empty arrays
|
|
if m.size != 0:
|
|
M = M[..., np.newaxis, np.newaxis]
|
|
n = n[..., np.newaxis, np.newaxis]
|
|
cond = (M == 0) & (M-1 == 0)
|
|
M = np.ma.masked_array(M, mask=cond)
|
|
output = (-n * (M-n)/(M-1) *
|
|
np.einsum("...i,...j->...ij", m, m) / (M**2))
|
|
# check for empty arrays
|
|
if m.size != 0:
|
|
M, n = M[..., 0, 0], n[..., 0, 0]
|
|
cond = cond[..., 0, 0]
|
|
dim = m.shape[-1]
|
|
# diagonal entries need to be computed differently
|
|
for i in range(dim):
|
|
output[..., i, i] = (n * (M-n) * m[..., i]*(M-m[..., i]))
|
|
output[..., i, i] = output[..., i, i] / (M-1)
|
|
output[..., i, i] = output[..., i, i] / (M**2)
|
|
if m.size != 0:
|
|
mncond = (mncond[..., np.newaxis, np.newaxis] |
|
|
np.zeros(output.shape, dtype=np.bool_))
|
|
return self._checkresult(output, mncond, np.nan)
|
|
|
|
def rvs(self, m, n, size=None, random_state=None):
|
|
"""Draw random samples from a multivariate hypergeometric distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_doc_default_callparams)s
|
|
size : integer or iterable of integers, optional
|
|
Number of samples to draw. Default is ``None``, in which case a
|
|
single variate is returned as an array with shape ``m.shape``.
|
|
%(_doc_random_state)s
|
|
|
|
Returns
|
|
-------
|
|
rvs : array_like
|
|
Random variates of shape ``size`` or ``m.shape``
|
|
(if ``size=None``).
|
|
|
|
Notes
|
|
-----
|
|
%(_doc_callparams_note)s
|
|
|
|
Also note that NumPy's `multivariate_hypergeometric` sampler is not
|
|
used as it doesn't support broadcasting.
|
|
"""
|
|
M, m, n, _, _, _ = self._process_parameters(m, n)
|
|
|
|
random_state = self._get_random_state(random_state)
|
|
|
|
if size is not None and isinstance(size, int):
|
|
size = (size, )
|
|
|
|
if size is None:
|
|
rvs = np.empty(m.shape, dtype=m.dtype)
|
|
else:
|
|
rvs = np.empty(size + (m.shape[-1], ), dtype=m.dtype)
|
|
rem = M
|
|
|
|
# This sampler has been taken from numpy gh-13794
|
|
# https://github.com/numpy/numpy/pull/13794
|
|
for c in range(m.shape[-1] - 1):
|
|
rem = rem - m[..., c]
|
|
n0mask = n == 0
|
|
rvs[..., c] = (~n0mask *
|
|
random_state.hypergeometric(m[..., c],
|
|
rem + n0mask,
|
|
n + n0mask,
|
|
size=size))
|
|
n = n - rvs[..., c]
|
|
rvs[..., m.shape[-1] - 1] = n
|
|
|
|
return rvs
|
|
|
|
|
|
multivariate_hypergeom = multivariate_hypergeom_gen()
|
|
|
|
|
|
class multivariate_hypergeom_frozen(multi_rv_frozen):
|
|
def __init__(self, m, n, seed=None):
|
|
self._dist = multivariate_hypergeom_gen(seed)
|
|
(self.M, self.m, self.n,
|
|
self.mcond, self.ncond,
|
|
self.mncond) = self._dist._process_parameters(m, n)
|
|
|
|
# monkey patch self._dist
|
|
def _process_parameters(m, n):
|
|
return (self.M, self.m, self.n,
|
|
self.mcond, self.ncond,
|
|
self.mncond)
|
|
self._dist._process_parameters = _process_parameters
|
|
|
|
def logpmf(self, x):
|
|
return self._dist.logpmf(x, self.m, self.n)
|
|
|
|
def pmf(self, x):
|
|
return self._dist.pmf(x, self.m, self.n)
|
|
|
|
def mean(self):
|
|
return self._dist.mean(self.m, self.n)
|
|
|
|
def var(self):
|
|
return self._dist.var(self.m, self.n)
|
|
|
|
def cov(self):
|
|
return self._dist.cov(self.m, self.n)
|
|
|
|
def rvs(self, size=1, random_state=None):
|
|
return self._dist.rvs(self.m, self.n,
|
|
size=size,
|
|
random_state=random_state)
|
|
|
|
|
|
# Set frozen generator docstrings from corresponding docstrings in
|
|
# multivariate_hypergeom and fill in default strings in class docstrings
|
|
for name in ['logpmf', 'pmf', 'mean', 'var', 'cov', 'rvs']:
|
|
method = multivariate_hypergeom_gen.__dict__[name]
|
|
method_frozen = multivariate_hypergeom_frozen.__dict__[name]
|
|
method_frozen.__doc__ = doccer.docformat(
|
|
method.__doc__, mhg_docdict_noparams)
|
|
method.__doc__ = doccer.docformat(method.__doc__,
|
|
mhg_docdict_params)
|
|
|
|
|
|
class random_table_gen(multi_rv_generic):
|
|
r"""Contingency tables from independent samples with fixed marginal sums.
|
|
|
|
This is the distribution of random tables with given row and column vector
|
|
sums. This distribution represents the set of random tables under the null
|
|
hypothesis that rows and columns are independent. It is used in hypothesis
|
|
tests of independence.
|
|
|
|
Because of assumed independence, the expected frequency of each table
|
|
element can be computed from the row and column sums, so that the
|
|
distribution is completely determined by these two vectors.
|
|
|
|
Methods
|
|
-------
|
|
logpmf(x)
|
|
Log-probability of table `x` to occur in the distribution.
|
|
pmf(x)
|
|
Probability of table `x` to occur in the distribution.
|
|
mean(row, col)
|
|
Mean table.
|
|
rvs(row, col, size=None, method=None, random_state=None)
|
|
Draw random tables with given row and column vector sums.
|
|
|
|
Parameters
|
|
----------
|
|
%(_doc_row_col)s
|
|
%(_doc_random_state)s
|
|
|
|
Notes
|
|
-----
|
|
%(_doc_row_col_note)s
|
|
|
|
Random elements from the distribution are generated either with Boyett's
|
|
[1]_ or Patefield's algorithm [2]_. Boyett's algorithm has
|
|
O(N) time and space complexity, where N is the total sum of entries in the
|
|
table. Patefield's algorithm has O(K x log(N)) time complexity, where K is
|
|
the number of cells in the table and requires only a small constant work
|
|
space. By default, the `rvs` method selects the fastest algorithm based on
|
|
the input, but you can specify the algorithm with the keyword `method`.
|
|
Allowed values are "boyett" and "patefield".
|
|
|
|
.. versionadded:: 1.10.0
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy.stats import random_table
|
|
|
|
>>> row = [1, 5]
|
|
>>> col = [2, 3, 1]
|
|
>>> random_table.mean(row, col)
|
|
array([[0.33333333, 0.5 , 0.16666667],
|
|
[1.66666667, 2.5 , 0.83333333]])
|
|
|
|
Alternatively, the object may be called (as a function) to fix the row
|
|
and column vector sums, returning a "frozen" distribution.
|
|
|
|
>>> dist = random_table(row, col)
|
|
>>> dist.rvs(random_state=123)
|
|
array([[1, 0, 0],
|
|
[1, 3, 1]])
|
|
|
|
References
|
|
----------
|
|
.. [1] J. Boyett, AS 144 Appl. Statist. 28 (1979) 329-332
|
|
.. [2] W.M. Patefield, AS 159 Appl. Statist. 30 (1981) 91-97
|
|
"""
|
|
|
|
def __init__(self, seed=None):
|
|
super().__init__(seed)
|
|
|
|
def __call__(self, row, col, *, seed=None):
|
|
"""Create a frozen distribution of tables with given marginals.
|
|
|
|
See `random_table_frozen` for more information.
|
|
"""
|
|
return random_table_frozen(row, col, seed=seed)
|
|
|
|
def logpmf(self, x, row, col):
|
|
"""Log-probability of table to occur in the distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_doc_x)s
|
|
%(_doc_row_col)s
|
|
|
|
Returns
|
|
-------
|
|
logpmf : ndarray or scalar
|
|
Log of the probability mass function evaluated at `x`.
|
|
|
|
Notes
|
|
-----
|
|
%(_doc_row_col_note)s
|
|
|
|
If row and column marginals of `x` do not match `row` and `col`,
|
|
negative infinity is returned.
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy.stats import random_table
|
|
>>> import numpy as np
|
|
|
|
>>> x = [[1, 5, 1], [2, 3, 1]]
|
|
>>> row = np.sum(x, axis=1)
|
|
>>> col = np.sum(x, axis=0)
|
|
>>> random_table.logpmf(x, row, col)
|
|
-1.6306401200847027
|
|
|
|
Alternatively, the object may be called (as a function) to fix the row
|
|
and column vector sums, returning a "frozen" distribution.
|
|
|
|
>>> d = random_table(row, col)
|
|
>>> d.logpmf(x)
|
|
-1.6306401200847027
|
|
"""
|
|
r, c, n = self._process_parameters(row, col)
|
|
x = np.asarray(x)
|
|
|
|
if x.ndim < 2:
|
|
raise ValueError("`x` must be at least two-dimensional")
|
|
|
|
dtype_is_int = np.issubdtype(x.dtype, np.integer)
|
|
with np.errstate(invalid='ignore'):
|
|
if not dtype_is_int and not np.all(x.astype(int) == x):
|
|
raise ValueError("`x` must contain only integral values")
|
|
|
|
# x does not contain NaN if we arrive here
|
|
if np.any(x < 0):
|
|
raise ValueError("`x` must contain only non-negative values")
|
|
|
|
r2 = np.sum(x, axis=-1)
|
|
c2 = np.sum(x, axis=-2)
|
|
|
|
if r2.shape[-1] != len(r):
|
|
raise ValueError("shape of `x` must agree with `row`")
|
|
|
|
if c2.shape[-1] != len(c):
|
|
raise ValueError("shape of `x` must agree with `col`")
|
|
|
|
res = np.empty(x.shape[:-2])
|
|
|
|
mask = np.all(r2 == r, axis=-1) & np.all(c2 == c, axis=-1)
|
|
|
|
def lnfac(x):
|
|
return gammaln(x + 1)
|
|
|
|
res[mask] = (np.sum(lnfac(r), axis=-1) + np.sum(lnfac(c), axis=-1)
|
|
- lnfac(n) - np.sum(lnfac(x[mask]), axis=(-1, -2)))
|
|
res[~mask] = -np.inf
|
|
|
|
return res[()]
|
|
|
|
def pmf(self, x, row, col):
|
|
"""Probability of table to occur in the distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_doc_x)s
|
|
%(_doc_row_col)s
|
|
|
|
Returns
|
|
-------
|
|
pmf : ndarray or scalar
|
|
Probability mass function evaluated at `x`.
|
|
|
|
Notes
|
|
-----
|
|
%(_doc_row_col_note)s
|
|
|
|
If row and column marginals of `x` do not match `row` and `col`,
|
|
zero is returned.
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy.stats import random_table
|
|
>>> import numpy as np
|
|
|
|
>>> x = [[1, 5, 1], [2, 3, 1]]
|
|
>>> row = np.sum(x, axis=1)
|
|
>>> col = np.sum(x, axis=0)
|
|
>>> random_table.pmf(x, row, col)
|
|
0.19580419580419592
|
|
|
|
Alternatively, the object may be called (as a function) to fix the row
|
|
and column vector sums, returning a "frozen" distribution.
|
|
|
|
>>> d = random_table(row, col)
|
|
>>> d.pmf(x)
|
|
0.19580419580419592
|
|
"""
|
|
return np.exp(self.logpmf(x, row, col))
|
|
|
|
def mean(self, row, col):
|
|
"""Mean of distribution of conditional tables.
|
|
%(_doc_mean_params)s
|
|
|
|
Returns
|
|
-------
|
|
mean: ndarray
|
|
Mean of the distribution.
|
|
|
|
Notes
|
|
-----
|
|
%(_doc_row_col_note)s
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy.stats import random_table
|
|
|
|
>>> row = [1, 5]
|
|
>>> col = [2, 3, 1]
|
|
>>> random_table.mean(row, col)
|
|
array([[0.33333333, 0.5 , 0.16666667],
|
|
[1.66666667, 2.5 , 0.83333333]])
|
|
|
|
Alternatively, the object may be called (as a function) to fix the row
|
|
and column vector sums, returning a "frozen" distribution.
|
|
|
|
>>> d = random_table(row, col)
|
|
>>> d.mean()
|
|
array([[0.33333333, 0.5 , 0.16666667],
|
|
[1.66666667, 2.5 , 0.83333333]])
|
|
"""
|
|
r, c, n = self._process_parameters(row, col)
|
|
return np.outer(r, c) / n
|
|
|
|
def rvs(self, row, col, *, size=None, method=None, random_state=None):
|
|
"""Draw random tables with fixed column and row marginals.
|
|
|
|
Parameters
|
|
----------
|
|
%(_doc_row_col)s
|
|
size : integer, optional
|
|
Number of samples to draw (default 1).
|
|
method : str, optional
|
|
Which method to use, "boyett" or "patefield". If None (default),
|
|
selects the fastest method for this input.
|
|
%(_doc_random_state)s
|
|
|
|
Returns
|
|
-------
|
|
rvs : ndarray
|
|
Random 2D tables of shape (`size`, `len(row)`, `len(col)`).
|
|
|
|
Notes
|
|
-----
|
|
%(_doc_row_col_note)s
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy.stats import random_table
|
|
|
|
>>> row = [1, 5]
|
|
>>> col = [2, 3, 1]
|
|
>>> random_table.rvs(row, col, random_state=123)
|
|
array([[1., 0., 0.],
|
|
[1., 3., 1.]])
|
|
|
|
Alternatively, the object may be called (as a function) to fix the row
|
|
and column vector sums, returning a "frozen" distribution.
|
|
|
|
>>> d = random_table(row, col)
|
|
>>> d.rvs(random_state=123)
|
|
array([[1., 0., 0.],
|
|
[1., 3., 1.]])
|
|
"""
|
|
r, c, n = self._process_parameters(row, col)
|
|
size, shape = self._process_size_shape(size, r, c)
|
|
|
|
random_state = self._get_random_state(random_state)
|
|
meth = self._process_rvs_method(method, r, c, n)
|
|
|
|
return meth(r, c, n, size, random_state).reshape(shape)
|
|
|
|
@staticmethod
|
|
def _process_parameters(row, col):
|
|
"""
|
|
Check that row and column vectors are one-dimensional, that they do
|
|
not contain negative or non-integer entries, and that the sums over
|
|
both vectors are equal.
|
|
"""
|
|
r = np.array(row, dtype=np.int64, copy=True)
|
|
c = np.array(col, dtype=np.int64, copy=True)
|
|
|
|
if np.ndim(r) != 1:
|
|
raise ValueError("`row` must be one-dimensional")
|
|
if np.ndim(c) != 1:
|
|
raise ValueError("`col` must be one-dimensional")
|
|
|
|
if np.any(r < 0):
|
|
raise ValueError("each element of `row` must be non-negative")
|
|
if np.any(c < 0):
|
|
raise ValueError("each element of `col` must be non-negative")
|
|
|
|
n = np.sum(r)
|
|
if n != np.sum(c):
|
|
raise ValueError("sums over `row` and `col` must be equal")
|
|
|
|
if not np.all(r == np.asarray(row)):
|
|
raise ValueError("each element of `row` must be an integer")
|
|
if not np.all(c == np.asarray(col)):
|
|
raise ValueError("each element of `col` must be an integer")
|
|
|
|
return r, c, n
|
|
|
|
@staticmethod
|
|
def _process_size_shape(size, r, c):
|
|
"""
|
|
Compute the number of samples to be drawn and the shape of the output
|
|
"""
|
|
shape = (len(r), len(c))
|
|
|
|
if size is None:
|
|
return 1, shape
|
|
|
|
size = np.atleast_1d(size)
|
|
if not np.issubdtype(size.dtype, np.integer) or np.any(size < 0):
|
|
raise ValueError("`size` must be a non-negative integer or `None`")
|
|
|
|
return np.prod(size), tuple(size) + shape
|
|
|
|
@classmethod
|
|
def _process_rvs_method(cls, method, r, c, n):
|
|
known_methods = {
|
|
None: cls._rvs_select(r, c, n),
|
|
"boyett": cls._rvs_boyett,
|
|
"patefield": cls._rvs_patefield,
|
|
}
|
|
try:
|
|
return known_methods[method]
|
|
except KeyError:
|
|
raise ValueError(f"'{method}' not recognized, "
|
|
f"must be one of {set(known_methods)}")
|
|
|
|
@classmethod
|
|
def _rvs_select(cls, r, c, n):
|
|
fac = 1.0 # benchmarks show that this value is about 1
|
|
k = len(r) * len(c) # number of cells
|
|
# n + 1 guards against failure if n == 0
|
|
if n > fac * np.log(n + 1) * k:
|
|
return cls._rvs_patefield
|
|
return cls._rvs_boyett
|
|
|
|
@staticmethod
|
|
def _rvs_boyett(row, col, ntot, size, random_state):
|
|
return _rcont.rvs_rcont1(row, col, ntot, size, random_state)
|
|
|
|
@staticmethod
|
|
def _rvs_patefield(row, col, ntot, size, random_state):
|
|
return _rcont.rvs_rcont2(row, col, ntot, size, random_state)
|
|
|
|
|
|
random_table = random_table_gen()
|
|
|
|
|
|
class random_table_frozen(multi_rv_frozen):
|
|
def __init__(self, row, col, *, seed=None):
|
|
self._dist = random_table_gen(seed)
|
|
self._params = self._dist._process_parameters(row, col)
|
|
|
|
# monkey patch self._dist
|
|
def _process_parameters(r, c):
|
|
return self._params
|
|
self._dist._process_parameters = _process_parameters
|
|
|
|
def logpmf(self, x):
|
|
return self._dist.logpmf(x, None, None)
|
|
|
|
def pmf(self, x):
|
|
return self._dist.pmf(x, None, None)
|
|
|
|
def mean(self):
|
|
return self._dist.mean(None, None)
|
|
|
|
def rvs(self, size=None, method=None, random_state=None):
|
|
# optimisations are possible here
|
|
return self._dist.rvs(None, None, size=size, method=method,
|
|
random_state=random_state)
|
|
|
|
|
|
_ctab_doc_row_col = """\
|
|
row : array_like
|
|
Sum of table entries in each row.
|
|
col : array_like
|
|
Sum of table entries in each column."""
|
|
|
|
_ctab_doc_x = """\
|
|
x : array-like
|
|
Two-dimensional table of non-negative integers, or a
|
|
multi-dimensional array with the last two dimensions
|
|
corresponding with the tables."""
|
|
|
|
_ctab_doc_row_col_note = """\
|
|
The row and column vectors must be one-dimensional, not empty,
|
|
and each sum up to the same value. They cannot contain negative
|
|
or noninteger entries."""
|
|
|
|
_ctab_doc_mean_params = f"""
|
|
Parameters
|
|
----------
|
|
{_ctab_doc_row_col}"""
|
|
|
|
_ctab_doc_row_col_note_frozen = """\
|
|
See class definition for a detailed description of parameters."""
|
|
|
|
_ctab_docdict = {
|
|
"_doc_random_state": _doc_random_state,
|
|
"_doc_row_col": _ctab_doc_row_col,
|
|
"_doc_x": _ctab_doc_x,
|
|
"_doc_mean_params": _ctab_doc_mean_params,
|
|
"_doc_row_col_note": _ctab_doc_row_col_note,
|
|
}
|
|
|
|
_ctab_docdict_frozen = _ctab_docdict.copy()
|
|
_ctab_docdict_frozen.update({
|
|
"_doc_row_col": "",
|
|
"_doc_mean_params": "",
|
|
"_doc_row_col_note": _ctab_doc_row_col_note_frozen,
|
|
})
|
|
|
|
|
|
def _docfill(obj, docdict, template=None):
|
|
obj.__doc__ = doccer.docformat(template or obj.__doc__, docdict)
|
|
|
|
|
|
# Set frozen generator docstrings from corresponding docstrings in
|
|
# random_table and fill in default strings in class docstrings
|
|
_docfill(random_table_gen, _ctab_docdict)
|
|
for name in ['logpmf', 'pmf', 'mean', 'rvs']:
|
|
method = random_table_gen.__dict__[name]
|
|
method_frozen = random_table_frozen.__dict__[name]
|
|
_docfill(method_frozen, _ctab_docdict_frozen, method.__doc__)
|
|
_docfill(method, _ctab_docdict)
|
|
|
|
|
|
class uniform_direction_gen(multi_rv_generic):
|
|
r"""A vector-valued uniform direction.
|
|
|
|
Return a random direction (unit vector). The `dim` keyword specifies
|
|
the dimensionality of the space.
|
|
|
|
Methods
|
|
-------
|
|
rvs(dim=None, size=1, random_state=None)
|
|
Draw random directions.
|
|
|
|
Parameters
|
|
----------
|
|
dim : scalar
|
|
Dimension of directions.
|
|
seed : {None, int, `numpy.random.Generator`,
|
|
`numpy.random.RandomState`}, optional
|
|
|
|
Used for drawing random variates.
|
|
If `seed` is `None`, the `~np.random.RandomState` singleton is used.
|
|
If `seed` is an int, a new ``RandomState`` instance is used, seeded
|
|
with seed.
|
|
If `seed` is already a ``RandomState`` or ``Generator`` instance,
|
|
then that object is used.
|
|
Default is `None`.
|
|
|
|
Notes
|
|
-----
|
|
This distribution generates unit vectors uniformly distributed on
|
|
the surface of a hypersphere. These can be interpreted as random
|
|
directions.
|
|
For example, if `dim` is 3, 3D vectors from the surface of :math:`S^2`
|
|
will be sampled.
|
|
|
|
References
|
|
----------
|
|
.. [1] Marsaglia, G. (1972). "Choosing a Point from the Surface of a
|
|
Sphere". Annals of Mathematical Statistics. 43 (2): 645-646.
|
|
|
|
Examples
|
|
--------
|
|
>>> import numpy as np
|
|
>>> from scipy.stats import uniform_direction
|
|
>>> x = uniform_direction.rvs(3)
|
|
>>> np.linalg.norm(x)
|
|
1.
|
|
|
|
This generates one random direction, a vector on the surface of
|
|
:math:`S^2`.
|
|
|
|
Alternatively, the object may be called (as a function) to return a frozen
|
|
distribution with fixed `dim` parameter. Here,
|
|
we create a `uniform_direction` with ``dim=3`` and draw 5 observations.
|
|
The samples are then arranged in an array of shape 5x3.
|
|
|
|
>>> rng = np.random.default_rng()
|
|
>>> uniform_sphere_dist = uniform_direction(3)
|
|
>>> unit_vectors = uniform_sphere_dist.rvs(5, random_state=rng)
|
|
>>> unit_vectors
|
|
array([[ 0.56688642, -0.1332634 , -0.81294566],
|
|
[-0.427126 , -0.74779278, 0.50830044],
|
|
[ 0.3793989 , 0.92346629, 0.05715323],
|
|
[ 0.36428383, -0.92449076, -0.11231259],
|
|
[-0.27733285, 0.94410968, -0.17816678]])
|
|
"""
|
|
|
|
def __init__(self, seed=None):
|
|
super().__init__(seed)
|
|
self.__doc__ = doccer.docformat(self.__doc__)
|
|
|
|
def __call__(self, dim=None, seed=None):
|
|
"""Create a frozen n-dimensional uniform direction distribution.
|
|
|
|
See `uniform_direction` for more information.
|
|
"""
|
|
return uniform_direction_frozen(dim, seed=seed)
|
|
|
|
def _process_parameters(self, dim):
|
|
"""Dimension N must be specified; it cannot be inferred."""
|
|
if dim is None or not np.isscalar(dim) or dim < 1 or dim != int(dim):
|
|
raise ValueError("Dimension of vector must be specified, "
|
|
"and must be an integer greater than 0.")
|
|
|
|
return int(dim)
|
|
|
|
def rvs(self, dim, size=None, random_state=None):
|
|
"""Draw random samples from S(N-1).
|
|
|
|
Parameters
|
|
----------
|
|
dim : integer
|
|
Dimension of space (N).
|
|
size : int or tuple of ints, optional
|
|
Given a shape of, for example, (m,n,k), m*n*k samples are
|
|
generated, and packed in an m-by-n-by-k arrangement.
|
|
Because each sample is N-dimensional, the output shape
|
|
is (m,n,k,N). If no shape is specified, a single (N-D)
|
|
sample is returned.
|
|
random_state : {None, int, `numpy.random.Generator`,
|
|
`numpy.random.RandomState`}, optional
|
|
|
|
Pseudorandom number generator state used to generate resamples.
|
|
|
|
If `random_state` is ``None`` (or `np.random`), the
|
|
`numpy.random.RandomState` singleton is used.
|
|
If `random_state` is an int, a new ``RandomState`` instance is
|
|
used, seeded with `random_state`.
|
|
If `random_state` is already a ``Generator`` or ``RandomState``
|
|
instance then that instance is used.
|
|
|
|
Returns
|
|
-------
|
|
rvs : ndarray
|
|
Random direction vectors
|
|
|
|
"""
|
|
random_state = self._get_random_state(random_state)
|
|
if size is None:
|
|
size = np.array([], dtype=int)
|
|
size = np.atleast_1d(size)
|
|
|
|
dim = self._process_parameters(dim)
|
|
|
|
samples = _sample_uniform_direction(dim, size, random_state)
|
|
return samples
|
|
|
|
|
|
uniform_direction = uniform_direction_gen()
|
|
|
|
|
|
class uniform_direction_frozen(multi_rv_frozen):
|
|
def __init__(self, dim=None, seed=None):
|
|
"""Create a frozen n-dimensional uniform direction distribution.
|
|
|
|
Parameters
|
|
----------
|
|
dim : int
|
|
Dimension of matrices
|
|
seed : {None, int, `numpy.random.Generator`,
|
|
`numpy.random.RandomState`}, optional
|
|
|
|
If `seed` is None (or `np.random`), the `numpy.random.RandomState`
|
|
singleton is used.
|
|
If `seed` is an int, a new ``RandomState`` instance is used,
|
|
seeded with `seed`.
|
|
If `seed` is already a ``Generator`` or ``RandomState`` instance
|
|
then that instance is used.
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy.stats import uniform_direction
|
|
>>> x = uniform_direction(3)
|
|
>>> x.rvs()
|
|
|
|
"""
|
|
self._dist = uniform_direction_gen(seed)
|
|
self.dim = self._dist._process_parameters(dim)
|
|
|
|
def rvs(self, size=None, random_state=None):
|
|
return self._dist.rvs(self.dim, size, random_state)
|
|
|
|
|
|
def _sample_uniform_direction(dim, size, random_state):
|
|
"""
|
|
Private method to generate uniform directions
|
|
Reference: Marsaglia, G. (1972). "Choosing a Point from the Surface of a
|
|
Sphere". Annals of Mathematical Statistics. 43 (2): 645-646.
|
|
"""
|
|
samples_shape = np.append(size, dim)
|
|
samples = random_state.standard_normal(samples_shape)
|
|
samples /= np.linalg.norm(samples, axis=-1, keepdims=True)
|
|
return samples
|
|
|
|
|
|
_dirichlet_mn_doc_default_callparams = """\
|
|
alpha : array_like
|
|
The concentration parameters. The number of entries along the last axis
|
|
determines the dimensionality of the distribution. Each entry must be
|
|
strictly positive.
|
|
n : int or array_like
|
|
The number of trials. Each element must be a non-negative integer.
|
|
"""
|
|
|
|
_dirichlet_mn_doc_frozen_callparams = ""
|
|
|
|
_dirichlet_mn_doc_frozen_callparams_note = """\
|
|
See class definition for a detailed description of parameters."""
|
|
|
|
dirichlet_mn_docdict_params = {
|
|
'_dirichlet_mn_doc_default_callparams': _dirichlet_mn_doc_default_callparams,
|
|
'_doc_random_state': _doc_random_state
|
|
}
|
|
|
|
dirichlet_mn_docdict_noparams = {
|
|
'_dirichlet_mn_doc_default_callparams': _dirichlet_mn_doc_frozen_callparams,
|
|
'_doc_random_state': _doc_random_state
|
|
}
|
|
|
|
|
|
def _dirichlet_multinomial_check_parameters(alpha, n, x=None):
|
|
|
|
alpha = np.asarray(alpha)
|
|
n = np.asarray(n)
|
|
|
|
if x is not None:
|
|
# Ensure that `x` and `alpha` are arrays. If the shapes are
|
|
# incompatible, NumPy will raise an appropriate error.
|
|
try:
|
|
x, alpha = np.broadcast_arrays(x, alpha)
|
|
except ValueError as e:
|
|
msg = "`x` and `alpha` must be broadcastable."
|
|
raise ValueError(msg) from e
|
|
|
|
x_int = np.floor(x)
|
|
if np.any(x < 0) or np.any(x != x_int):
|
|
raise ValueError("`x` must contain only non-negative integers.")
|
|
x = x_int
|
|
|
|
if np.any(alpha <= 0):
|
|
raise ValueError("`alpha` must contain only positive values.")
|
|
|
|
n_int = np.floor(n)
|
|
if np.any(n < 0) or np.any(n != n_int):
|
|
raise ValueError("`n` must be a non-negative integer.")
|
|
n = n_int
|
|
|
|
sum_alpha = np.sum(alpha, axis=-1)
|
|
sum_alpha, n = np.broadcast_arrays(sum_alpha, n)
|
|
|
|
return (alpha, sum_alpha, n) if x is None else (alpha, sum_alpha, n, x)
|
|
|
|
|
|
class dirichlet_multinomial_gen(multi_rv_generic):
|
|
r"""A Dirichlet multinomial random variable.
|
|
|
|
The Dirichlet multinomial distribution is a compound probability
|
|
distribution: it is the multinomial distribution with number of trials
|
|
`n` and class probabilities ``p`` randomly sampled from a Dirichlet
|
|
distribution with concentration parameters ``alpha``.
|
|
|
|
Methods
|
|
-------
|
|
logpmf(x, alpha, n):
|
|
Log of the probability mass function.
|
|
pmf(x, alpha, n):
|
|
Probability mass function.
|
|
mean(alpha, n):
|
|
Mean of the Dirichlet multinomial distribution.
|
|
var(alpha, n):
|
|
Variance of the Dirichlet multinomial distribution.
|
|
cov(alpha, n):
|
|
The covariance of the Dirichlet multinomial distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_dirichlet_mn_doc_default_callparams)s
|
|
%(_doc_random_state)s
|
|
|
|
See Also
|
|
--------
|
|
scipy.stats.dirichlet : The dirichlet distribution.
|
|
scipy.stats.multinomial : The multinomial distribution.
|
|
|
|
References
|
|
----------
|
|
.. [1] Dirichlet-multinomial distribution, Wikipedia,
|
|
https://www.wikipedia.org/wiki/Dirichlet-multinomial_distribution
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy.stats import dirichlet_multinomial
|
|
|
|
Get the PMF
|
|
|
|
>>> n = 6 # number of trials
|
|
>>> alpha = [3, 4, 5] # concentration parameters
|
|
>>> x = [1, 2, 3] # counts
|
|
>>> dirichlet_multinomial.pmf(x, alpha, n)
|
|
0.08484162895927604
|
|
|
|
If the sum of category counts does not equal the number of trials,
|
|
the probability mass is zero.
|
|
|
|
>>> dirichlet_multinomial.pmf(x, alpha, n=7)
|
|
0.0
|
|
|
|
Get the log of the PMF
|
|
|
|
>>> dirichlet_multinomial.logpmf(x, alpha, n)
|
|
-2.4669689491013327
|
|
|
|
Get the mean
|
|
|
|
>>> dirichlet_multinomial.mean(alpha, n)
|
|
array([1.5, 2. , 2.5])
|
|
|
|
Get the variance
|
|
|
|
>>> dirichlet_multinomial.var(alpha, n)
|
|
array([1.55769231, 1.84615385, 2.01923077])
|
|
|
|
Get the covariance
|
|
|
|
>>> dirichlet_multinomial.cov(alpha, n)
|
|
array([[ 1.55769231, -0.69230769, -0.86538462],
|
|
[-0.69230769, 1.84615385, -1.15384615],
|
|
[-0.86538462, -1.15384615, 2.01923077]])
|
|
|
|
Alternatively, the object may be called (as a function) to fix the
|
|
`alpha` and `n` parameters, returning a "frozen" Dirichlet multinomial
|
|
random variable.
|
|
|
|
>>> dm = dirichlet_multinomial(alpha, n)
|
|
>>> dm.pmf(x)
|
|
0.08484162895927579
|
|
|
|
All methods are fully vectorized. Each element of `x` and `alpha` is
|
|
a vector (along the last axis), each element of `n` is an
|
|
integer (scalar), and the result is computed element-wise.
|
|
|
|
>>> x = [[1, 2, 3], [4, 5, 6]]
|
|
>>> alpha = [[1, 2, 3], [4, 5, 6]]
|
|
>>> n = [6, 15]
|
|
>>> dirichlet_multinomial.pmf(x, alpha, n)
|
|
array([0.06493506, 0.02626937])
|
|
|
|
>>> dirichlet_multinomial.cov(alpha, n).shape # both covariance matrices
|
|
(2, 3, 3)
|
|
|
|
Broadcasting according to standard NumPy conventions is supported. Here,
|
|
we have four sets of concentration parameters (each a two element vector)
|
|
for each of three numbers of trials (each a scalar).
|
|
|
|
>>> alpha = [[3, 4], [4, 5], [5, 6], [6, 7]]
|
|
>>> n = [[6], [7], [8]]
|
|
>>> dirichlet_multinomial.mean(alpha, n).shape
|
|
(3, 4, 2)
|
|
|
|
"""
|
|
def __init__(self, seed=None):
|
|
super().__init__(seed)
|
|
self.__doc__ = doccer.docformat(self.__doc__,
|
|
dirichlet_mn_docdict_params)
|
|
|
|
def __call__(self, alpha, n, seed=None):
|
|
return dirichlet_multinomial_frozen(alpha, n, seed=seed)
|
|
|
|
def logpmf(self, x, alpha, n):
|
|
"""The log of the probability mass function.
|
|
|
|
Parameters
|
|
----------
|
|
x: ndarray
|
|
Category counts (non-negative integers). Must be broadcastable
|
|
with shape parameter ``alpha``. If multidimensional, the last axis
|
|
must correspond with the categories.
|
|
%(_dirichlet_mn_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
out: ndarray or scalar
|
|
Log of the probability mass function.
|
|
|
|
"""
|
|
|
|
a, Sa, n, x = _dirichlet_multinomial_check_parameters(alpha, n, x)
|
|
|
|
out = np.asarray(loggamma(Sa) + loggamma(n + 1) - loggamma(n + Sa))
|
|
out += (loggamma(x + a) - (loggamma(a) + loggamma(x + 1))).sum(axis=-1)
|
|
np.place(out, n != x.sum(axis=-1), -np.inf)
|
|
return out[()]
|
|
|
|
def pmf(self, x, alpha, n):
|
|
"""Probability mass function for a Dirichlet multinomial distribution.
|
|
|
|
Parameters
|
|
----------
|
|
x: ndarray
|
|
Category counts (non-negative integers). Must be broadcastable
|
|
with shape parameter ``alpha``. If multidimensional, the last axis
|
|
must correspond with the categories.
|
|
%(_dirichlet_mn_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
out: ndarray or scalar
|
|
Probability mass function.
|
|
|
|
"""
|
|
return np.exp(self.logpmf(x, alpha, n))
|
|
|
|
def mean(self, alpha, n):
|
|
"""Mean of a Dirichlet multinomial distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_dirichlet_mn_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
out: ndarray
|
|
Mean of a Dirichlet multinomial distribution.
|
|
|
|
"""
|
|
a, Sa, n = _dirichlet_multinomial_check_parameters(alpha, n)
|
|
n, Sa = n[..., np.newaxis], Sa[..., np.newaxis]
|
|
return n * a / Sa
|
|
|
|
def var(self, alpha, n):
|
|
"""The variance of the Dirichlet multinomial distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_dirichlet_mn_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
out: array_like
|
|
The variances of the components of the distribution. This is
|
|
the diagonal of the covariance matrix of the distribution.
|
|
|
|
"""
|
|
a, Sa, n = _dirichlet_multinomial_check_parameters(alpha, n)
|
|
n, Sa = n[..., np.newaxis], Sa[..., np.newaxis]
|
|
return n * a / Sa * (1 - a/Sa) * (n + Sa) / (1 + Sa)
|
|
|
|
def cov(self, alpha, n):
|
|
"""Covariance matrix of a Dirichlet multinomial distribution.
|
|
|
|
Parameters
|
|
----------
|
|
%(_dirichlet_mn_doc_default_callparams)s
|
|
|
|
Returns
|
|
-------
|
|
out : array_like
|
|
The covariance matrix of the distribution.
|
|
|
|
"""
|
|
a, Sa, n = _dirichlet_multinomial_check_parameters(alpha, n)
|
|
var = dirichlet_multinomial.var(a, n)
|
|
|
|
n, Sa = n[..., np.newaxis, np.newaxis], Sa[..., np.newaxis, np.newaxis]
|
|
aiaj = a[..., :, np.newaxis] * a[..., np.newaxis, :]
|
|
cov = -n * aiaj / Sa ** 2 * (n + Sa) / (1 + Sa)
|
|
|
|
ii = np.arange(cov.shape[-1])
|
|
cov[..., ii, ii] = var
|
|
return cov
|
|
|
|
|
|
dirichlet_multinomial = dirichlet_multinomial_gen()
|
|
|
|
|
|
class dirichlet_multinomial_frozen(multi_rv_frozen):
|
|
def __init__(self, alpha, n, seed=None):
|
|
alpha, Sa, n = _dirichlet_multinomial_check_parameters(alpha, n)
|
|
self.alpha = alpha
|
|
self.n = n
|
|
self._dist = dirichlet_multinomial_gen(seed)
|
|
|
|
def logpmf(self, x):
|
|
return self._dist.logpmf(x, self.alpha, self.n)
|
|
|
|
def pmf(self, x):
|
|
return self._dist.pmf(x, self.alpha, self.n)
|
|
|
|
def mean(self):
|
|
return self._dist.mean(self.alpha, self.n)
|
|
|
|
def var(self):
|
|
return self._dist.var(self.alpha, self.n)
|
|
|
|
def cov(self):
|
|
return self._dist.cov(self.alpha, self.n)
|
|
|
|
|
|
# Set frozen generator docstrings from corresponding docstrings in
|
|
# dirichlet_multinomial and fill in default strings in class docstrings.
|
|
for name in ['logpmf', 'pmf', 'mean', 'var', 'cov']:
|
|
method = dirichlet_multinomial_gen.__dict__[name]
|
|
method_frozen = dirichlet_multinomial_frozen.__dict__[name]
|
|
method_frozen.__doc__ = doccer.docformat(
|
|
method.__doc__, dirichlet_mn_docdict_noparams)
|
|
method.__doc__ = doccer.docformat(method.__doc__,
|
|
dirichlet_mn_docdict_params)
|
|
|
|
|
|
class vonmises_fisher_gen(multi_rv_generic):
|
|
r"""A von Mises-Fisher variable.
|
|
|
|
The `mu` keyword specifies the mean direction vector. The `kappa` keyword
|
|
specifies the concentration parameter.
|
|
|
|
Methods
|
|
-------
|
|
pdf(x, mu=None, kappa=1)
|
|
Probability density function.
|
|
logpdf(x, mu=None, kappa=1)
|
|
Log of the probability density function.
|
|
rvs(mu=None, kappa=1, size=1, random_state=None)
|
|
Draw random samples from a von Mises-Fisher distribution.
|
|
entropy(mu=None, kappa=1)
|
|
Compute the differential entropy of the von Mises-Fisher distribution.
|
|
fit(data)
|
|
Fit a von Mises-Fisher distribution to data.
|
|
|
|
Parameters
|
|
----------
|
|
mu : array_like
|
|
Mean direction of the distribution. Must be a one-dimensional unit
|
|
vector of norm 1.
|
|
kappa : float
|
|
Concentration parameter. Must be positive.
|
|
seed : {None, int, np.random.RandomState, np.random.Generator}, optional
|
|
Used for drawing random variates.
|
|
If `seed` is `None`, the `~np.random.RandomState` singleton is used.
|
|
If `seed` is an int, a new ``RandomState`` instance is used, seeded
|
|
with seed.
|
|
If `seed` is already a ``RandomState`` or ``Generator`` instance,
|
|
then that object is used.
|
|
Default is `None`.
|
|
|
|
See Also
|
|
--------
|
|
scipy.stats.vonmises : Von-Mises Fisher distribution in 2D on a circle
|
|
uniform_direction : uniform distribution on the surface of a hypersphere
|
|
|
|
Notes
|
|
-----
|
|
The von Mises-Fisher distribution is a directional distribution on the
|
|
surface of the unit hypersphere. The probability density
|
|
function of a unit vector :math:`\mathbf{x}` is
|
|
|
|
.. math::
|
|
|
|
f(\mathbf{x}) = \frac{\kappa^{d/2-1}}{(2\pi)^{d/2}I_{d/2-1}(\kappa)}
|
|
\exp\left(\kappa \mathbf{\mu}^T\mathbf{x}\right),
|
|
|
|
where :math:`\mathbf{\mu}` is the mean direction, :math:`\kappa` the
|
|
concentration parameter, :math:`d` the dimension and :math:`I` the
|
|
modified Bessel function of the first kind. As :math:`\mu` represents
|
|
a direction, it must be a unit vector or in other words, a point
|
|
on the hypersphere: :math:`\mathbf{\mu}\in S^{d-1}`. :math:`\kappa` is a
|
|
concentration parameter, which means that it must be positive
|
|
(:math:`\kappa>0`) and that the distribution becomes more narrow with
|
|
increasing :math:`\kappa`. In that sense, the reciprocal value
|
|
:math:`1/\kappa` resembles the variance parameter of the normal
|
|
distribution.
|
|
|
|
The von Mises-Fisher distribution often serves as an analogue of the
|
|
normal distribution on the sphere. Intuitively, for unit vectors, a
|
|
useful distance measure is given by the angle :math:`\alpha` between
|
|
them. This is exactly what the scalar product
|
|
:math:`\mathbf{\mu}^T\mathbf{x}=\cos(\alpha)` in the
|
|
von Mises-Fisher probability density function describes: the angle
|
|
between the mean direction :math:`\mathbf{\mu}` and the vector
|
|
:math:`\mathbf{x}`. The larger the angle between them, the smaller the
|
|
probability to observe :math:`\mathbf{x}` for this particular mean
|
|
direction :math:`\mathbf{\mu}`.
|
|
|
|
In dimensions 2 and 3, specialized algorithms are used for fast sampling
|
|
[2]_, [3]_. For dimensions of 4 or higher the rejection sampling algorithm
|
|
described in [4]_ is utilized. This implementation is partially based on
|
|
the geomstats package [5]_, [6]_.
|
|
|
|
.. versionadded:: 1.11
|
|
|
|
References
|
|
----------
|
|
.. [1] Von Mises-Fisher distribution, Wikipedia,
|
|
https://en.wikipedia.org/wiki/Von_Mises%E2%80%93Fisher_distribution
|
|
.. [2] Mardia, K., and Jupp, P. Directional statistics. Wiley, 2000.
|
|
.. [3] J. Wenzel. Numerically stable sampling of the von Mises Fisher
|
|
distribution on S2.
|
|
https://www.mitsuba-renderer.org/~wenzel/files/vmf.pdf
|
|
.. [4] Wood, A. Simulation of the von mises fisher distribution.
|
|
Communications in statistics-simulation and computation 23,
|
|
1 (1994), 157-164. https://doi.org/10.1080/03610919408813161
|
|
.. [5] geomstats, Github. MIT License. Accessed: 06.01.2023.
|
|
https://github.com/geomstats/geomstats
|
|
.. [6] Miolane, N. et al. Geomstats: A Python Package for Riemannian
|
|
Geometry in Machine Learning. Journal of Machine Learning Research
|
|
21 (2020). http://jmlr.org/papers/v21/19-027.html
|
|
|
|
Examples
|
|
--------
|
|
**Visualization of the probability density**
|
|
|
|
Plot the probability density in three dimensions for increasing
|
|
concentration parameter. The density is calculated by the ``pdf``
|
|
method.
|
|
|
|
>>> import numpy as np
|
|
>>> import matplotlib.pyplot as plt
|
|
>>> from scipy.stats import vonmises_fisher
|
|
>>> from matplotlib.colors import Normalize
|
|
>>> n_grid = 100
|
|
>>> u = np.linspace(0, np.pi, n_grid)
|
|
>>> v = np.linspace(0, 2 * np.pi, n_grid)
|
|
>>> u_grid, v_grid = np.meshgrid(u, v)
|
|
>>> vertices = np.stack([np.cos(v_grid) * np.sin(u_grid),
|
|
... np.sin(v_grid) * np.sin(u_grid),
|
|
... np.cos(u_grid)],
|
|
... axis=2)
|
|
>>> x = np.outer(np.cos(v), np.sin(u))
|
|
>>> y = np.outer(np.sin(v), np.sin(u))
|
|
>>> z = np.outer(np.ones_like(u), np.cos(u))
|
|
>>> def plot_vmf_density(ax, x, y, z, vertices, mu, kappa):
|
|
... vmf = vonmises_fisher(mu, kappa)
|
|
... pdf_values = vmf.pdf(vertices)
|
|
... pdfnorm = Normalize(vmin=pdf_values.min(), vmax=pdf_values.max())
|
|
... ax.plot_surface(x, y, z, rstride=1, cstride=1,
|
|
... facecolors=plt.cm.viridis(pdfnorm(pdf_values)),
|
|
... linewidth=0)
|
|
... ax.set_aspect('equal')
|
|
... ax.view_init(azim=-130, elev=0)
|
|
... ax.axis('off')
|
|
... ax.set_title(rf"$\kappa={kappa}$")
|
|
>>> fig, axes = plt.subplots(nrows=1, ncols=3, figsize=(9, 4),
|
|
... subplot_kw={"projection": "3d"})
|
|
>>> left, middle, right = axes
|
|
>>> mu = np.array([-np.sqrt(0.5), -np.sqrt(0.5), 0])
|
|
>>> plot_vmf_density(left, x, y, z, vertices, mu, 5)
|
|
>>> plot_vmf_density(middle, x, y, z, vertices, mu, 20)
|
|
>>> plot_vmf_density(right, x, y, z, vertices, mu, 100)
|
|
>>> plt.subplots_adjust(top=1, bottom=0.0, left=0.0, right=1.0, wspace=0.)
|
|
>>> plt.show()
|
|
|
|
As we increase the concentration parameter, the points are getting more
|
|
clustered together around the mean direction.
|
|
|
|
**Sampling**
|
|
|
|
Draw 5 samples from the distribution using the ``rvs`` method resulting
|
|
in a 5x3 array.
|
|
|
|
>>> rng = np.random.default_rng()
|
|
>>> mu = np.array([0, 0, 1])
|
|
>>> samples = vonmises_fisher(mu, 20).rvs(5, random_state=rng)
|
|
>>> samples
|
|
array([[ 0.3884594 , -0.32482588, 0.86231516],
|
|
[ 0.00611366, -0.09878289, 0.99509023],
|
|
[-0.04154772, -0.01637135, 0.99900239],
|
|
[-0.14613735, 0.12553507, 0.98126695],
|
|
[-0.04429884, -0.23474054, 0.97104814]])
|
|
|
|
These samples are unit vectors on the sphere :math:`S^2`. To verify,
|
|
let us calculate their euclidean norms:
|
|
|
|
>>> np.linalg.norm(samples, axis=1)
|
|
array([1., 1., 1., 1., 1.])
|
|
|
|
Plot 20 observations drawn from the von Mises-Fisher distribution for
|
|
increasing concentration parameter :math:`\kappa`. The red dot highlights
|
|
the mean direction :math:`\mu`.
|
|
|
|
>>> def plot_vmf_samples(ax, x, y, z, mu, kappa):
|
|
... vmf = vonmises_fisher(mu, kappa)
|
|
... samples = vmf.rvs(20)
|
|
... ax.plot_surface(x, y, z, rstride=1, cstride=1, linewidth=0,
|
|
... alpha=0.2)
|
|
... ax.scatter(samples[:, 0], samples[:, 1], samples[:, 2], c='k', s=5)
|
|
... ax.scatter(mu[0], mu[1], mu[2], c='r', s=30)
|
|
... ax.set_aspect('equal')
|
|
... ax.view_init(azim=-130, elev=0)
|
|
... ax.axis('off')
|
|
... ax.set_title(rf"$\kappa={kappa}$")
|
|
>>> mu = np.array([-np.sqrt(0.5), -np.sqrt(0.5), 0])
|
|
>>> fig, axes = plt.subplots(nrows=1, ncols=3,
|
|
... subplot_kw={"projection": "3d"},
|
|
... figsize=(9, 4))
|
|
>>> left, middle, right = axes
|
|
>>> plot_vmf_samples(left, x, y, z, mu, 5)
|
|
>>> plot_vmf_samples(middle, x, y, z, mu, 20)
|
|
>>> plot_vmf_samples(right, x, y, z, mu, 100)
|
|
>>> plt.subplots_adjust(top=1, bottom=0.0, left=0.0,
|
|
... right=1.0, wspace=0.)
|
|
>>> plt.show()
|
|
|
|
The plots show that with increasing concentration :math:`\kappa` the
|
|
resulting samples are centered more closely around the mean direction.
|
|
|
|
**Fitting the distribution parameters**
|
|
|
|
The distribution can be fitted to data using the ``fit`` method returning
|
|
the estimated parameters. As a toy example let's fit the distribution to
|
|
samples drawn from a known von Mises-Fisher distribution.
|
|
|
|
>>> mu, kappa = np.array([0, 0, 1]), 20
|
|
>>> samples = vonmises_fisher(mu, kappa).rvs(1000, random_state=rng)
|
|
>>> mu_fit, kappa_fit = vonmises_fisher.fit(samples)
|
|
>>> mu_fit, kappa_fit
|
|
(array([0.01126519, 0.01044501, 0.99988199]), 19.306398751730995)
|
|
|
|
We see that the estimated parameters `mu_fit` and `kappa_fit` are
|
|
very close to the ground truth parameters.
|
|
|
|
"""
|
|
def __init__(self, seed=None):
|
|
super().__init__(seed)
|
|
|
|
def __call__(self, mu=None, kappa=1, seed=None):
|
|
"""Create a frozen von Mises-Fisher distribution.
|
|
|
|
See `vonmises_fisher_frozen` for more information.
|
|
"""
|
|
return vonmises_fisher_frozen(mu, kappa, seed=seed)
|
|
|
|
def _process_parameters(self, mu, kappa):
|
|
"""
|
|
Infer dimensionality from mu and ensure that mu is a one-dimensional
|
|
unit vector and kappa positive.
|
|
"""
|
|
mu = np.asarray(mu)
|
|
if mu.ndim > 1:
|
|
raise ValueError("'mu' must have one-dimensional shape.")
|
|
if not np.allclose(np.linalg.norm(mu), 1.):
|
|
raise ValueError("'mu' must be a unit vector of norm 1.")
|
|
if not mu.size > 1:
|
|
raise ValueError("'mu' must have at least two entries.")
|
|
kappa_error_msg = "'kappa' must be a positive scalar."
|
|
if not np.isscalar(kappa) or kappa < 0:
|
|
raise ValueError(kappa_error_msg)
|
|
if float(kappa) == 0.:
|
|
raise ValueError("For 'kappa=0' the von Mises-Fisher distribution "
|
|
"becomes the uniform distribution on the sphere "
|
|
"surface. Consider using "
|
|
"'scipy.stats.uniform_direction' instead.")
|
|
dim = mu.size
|
|
|
|
return dim, mu, kappa
|
|
|
|
def _check_data_vs_dist(self, x, dim):
|
|
if x.shape[-1] != dim:
|
|
raise ValueError("The dimensionality of the last axis of 'x' must "
|
|
"match the dimensionality of the "
|
|
"von Mises Fisher distribution.")
|
|
if not np.allclose(np.linalg.norm(x, axis=-1), 1.):
|
|
msg = "'x' must be unit vectors of norm 1 along last dimension."
|
|
raise ValueError(msg)
|
|
|
|
def _log_norm_factor(self, dim, kappa):
|
|
# normalization factor is given by
|
|
# c = kappa**(dim/2-1)/((2*pi)**(dim/2)*I[dim/2-1](kappa))
|
|
# = kappa**(dim/2-1)*exp(-kappa) /
|
|
# ((2*pi)**(dim/2)*I[dim/2-1](kappa)*exp(-kappa)
|
|
# = kappa**(dim/2-1)*exp(-kappa) /
|
|
# ((2*pi)**(dim/2)*ive[dim/2-1](kappa)
|
|
# Then the log is given by
|
|
# log c = 1/2*(dim -1)*log(kappa) - kappa - -1/2*dim*ln(2*pi) -
|
|
# ive[dim/2-1](kappa)
|
|
halfdim = 0.5 * dim
|
|
return (0.5 * (dim - 2)*np.log(kappa) - halfdim * _LOG_2PI -
|
|
np.log(ive(halfdim - 1, kappa)) - kappa)
|
|
|
|
def _logpdf(self, x, dim, mu, kappa):
|
|
"""Log of the von Mises-Fisher probability density function.
|
|
|
|
As this function does no argument checking, it should not be
|
|
called directly; use 'logpdf' instead.
|
|
|
|
"""
|
|
x = np.asarray(x)
|
|
self._check_data_vs_dist(x, dim)
|
|
dotproducts = np.einsum('i,...i->...', mu, x)
|
|
return self._log_norm_factor(dim, kappa) + kappa * dotproducts
|
|
|
|
def logpdf(self, x, mu=None, kappa=1):
|
|
"""Log of the von Mises-Fisher probability density function.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
Points at which to evaluate the log of the probability
|
|
density function. The last axis of `x` must correspond
|
|
to unit vectors of the same dimensionality as the distribution.
|
|
mu : array_like, default: None
|
|
Mean direction of the distribution. Must be a one-dimensional unit
|
|
vector of norm 1.
|
|
kappa : float, default: 1
|
|
Concentration parameter. Must be positive.
|
|
|
|
Returns
|
|
-------
|
|
logpdf : ndarray or scalar
|
|
Log of the probability density function evaluated at `x`.
|
|
|
|
"""
|
|
dim, mu, kappa = self._process_parameters(mu, kappa)
|
|
return self._logpdf(x, dim, mu, kappa)
|
|
|
|
def pdf(self, x, mu=None, kappa=1):
|
|
"""Von Mises-Fisher probability density function.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
Points at which to evaluate the probability
|
|
density function. The last axis of `x` must correspond
|
|
to unit vectors of the same dimensionality as the distribution.
|
|
mu : array_like
|
|
Mean direction of the distribution. Must be a one-dimensional unit
|
|
vector of norm 1.
|
|
kappa : float
|
|
Concentration parameter. Must be positive.
|
|
|
|
Returns
|
|
-------
|
|
pdf : ndarray or scalar
|
|
Probability density function evaluated at `x`.
|
|
|
|
"""
|
|
dim, mu, kappa = self._process_parameters(mu, kappa)
|
|
return np.exp(self._logpdf(x, dim, mu, kappa))
|
|
|
|
def _rvs_2d(self, mu, kappa, size, random_state):
|
|
"""
|
|
In 2D, the von Mises-Fisher distribution reduces to the
|
|
von Mises distribution which can be efficiently sampled by numpy.
|
|
This method is much faster than the general rejection
|
|
sampling based algorithm.
|
|
|
|
"""
|
|
mean_angle = np.arctan2(mu[1], mu[0])
|
|
angle_samples = random_state.vonmises(mean_angle, kappa, size=size)
|
|
samples = np.stack([np.cos(angle_samples), np.sin(angle_samples)],
|
|
axis=-1)
|
|
return samples
|
|
|
|
def _rvs_3d(self, kappa, size, random_state):
|
|
"""
|
|
Generate samples from a von Mises-Fisher distribution
|
|
with mu = [1, 0, 0] and kappa. Samples then have to be
|
|
rotated towards the desired mean direction mu.
|
|
This method is much faster than the general rejection
|
|
sampling based algorithm.
|
|
Reference: https://www.mitsuba-renderer.org/~wenzel/files/vmf.pdf
|
|
|
|
"""
|
|
if size is None:
|
|
sample_size = 1
|
|
else:
|
|
sample_size = size
|
|
|
|
# compute x coordinate acc. to equation from section 3.1
|
|
x = random_state.random(sample_size)
|
|
x = 1. + np.log(x + (1. - x) * np.exp(-2 * kappa))/kappa
|
|
|
|
# (y, z) are random 2D vectors that only have to be
|
|
# normalized accordingly. Then (x, y z) follow a VMF distribution
|
|
temp = np.sqrt(1. - np.square(x))
|
|
uniformcircle = _sample_uniform_direction(2, sample_size, random_state)
|
|
samples = np.stack([x, temp * uniformcircle[..., 0],
|
|
temp * uniformcircle[..., 1]],
|
|
axis=-1)
|
|
if size is None:
|
|
samples = np.squeeze(samples)
|
|
return samples
|
|
|
|
def _rejection_sampling(self, dim, kappa, size, random_state):
|
|
"""
|
|
Generate samples from a n-dimensional von Mises-Fisher distribution
|
|
with mu = [1, 0, ..., 0] and kappa via rejection sampling.
|
|
Samples then have to be rotated towards the desired mean direction mu.
|
|
Reference: https://doi.org/10.1080/03610919408813161
|
|
"""
|
|
dim_minus_one = dim - 1
|
|
# calculate number of requested samples
|
|
if size is not None:
|
|
if not np.iterable(size):
|
|
size = (size, )
|
|
n_samples = math.prod(size)
|
|
else:
|
|
n_samples = 1
|
|
# calculate envelope for rejection sampler (eq. 4)
|
|
sqrt = np.sqrt(4 * kappa ** 2. + dim_minus_one ** 2)
|
|
envelop_param = (-2 * kappa + sqrt) / dim_minus_one
|
|
if envelop_param == 0:
|
|
# the regular formula suffers from loss of precision for high
|
|
# kappa. This can only be detected by checking for 0 here.
|
|
# Workaround: expansion for sqrt variable
|
|
# https://www.wolframalpha.com/input?i=sqrt%284*x%5E2%2Bd%5E2%29
|
|
# e = (-2 * k + sqrt(k**2 + d**2)) / d
|
|
# ~ (-2 * k + 2 * k + d**2/(4 * k) - d**4/(64 * k**3)) / d
|
|
# = d/(4 * k) - d**3/(64 * k**3)
|
|
envelop_param = (dim_minus_one/4 * kappa**-1.
|
|
- dim_minus_one**3/64 * kappa**-3.)
|
|
# reference step 0
|
|
node = (1. - envelop_param) / (1. + envelop_param)
|
|
# t = ln(1 - ((1-x)/(1+x))**2)
|
|
# = ln(4 * x / (1+x)**2)
|
|
# = ln(4) + ln(x) - 2*log1p(x)
|
|
correction = (kappa * node + dim_minus_one
|
|
* (np.log(4) + np.log(envelop_param)
|
|
- 2 * np.log1p(envelop_param)))
|
|
n_accepted = 0
|
|
x = np.zeros((n_samples, ))
|
|
halfdim = 0.5 * dim_minus_one
|
|
# main loop
|
|
while n_accepted < n_samples:
|
|
# generate candidates acc. to reference step 1
|
|
sym_beta = random_state.beta(halfdim, halfdim,
|
|
size=n_samples - n_accepted)
|
|
coord_x = (1 - (1 + envelop_param) * sym_beta) / (
|
|
1 - (1 - envelop_param) * sym_beta)
|
|
# accept or reject: reference step 2
|
|
# reformulation for numerical stability:
|
|
# t = ln(1 - (1-x)/(1+x) * y)
|
|
# = ln((1 + x - y +x*y)/(1 +x))
|
|
accept_tol = random_state.random(n_samples - n_accepted)
|
|
criterion = (
|
|
kappa * coord_x
|
|
+ dim_minus_one * (np.log((1 + envelop_param - coord_x
|
|
+ coord_x * envelop_param) / (1 + envelop_param)))
|
|
- correction) > np.log(accept_tol)
|
|
accepted_iter = np.sum(criterion)
|
|
x[n_accepted:n_accepted + accepted_iter] = coord_x[criterion]
|
|
n_accepted += accepted_iter
|
|
# concatenate x and remaining coordinates: step 3
|
|
coord_rest = _sample_uniform_direction(dim_minus_one, n_accepted,
|
|
random_state)
|
|
coord_rest = np.einsum(
|
|
'...,...i->...i', np.sqrt(1 - x ** 2), coord_rest)
|
|
samples = np.concatenate([x[..., None], coord_rest], axis=1)
|
|
# reshape output to (size, dim)
|
|
if size is not None:
|
|
samples = samples.reshape(size + (dim, ))
|
|
else:
|
|
samples = np.squeeze(samples)
|
|
return samples
|
|
|
|
def _rotate_samples(self, samples, mu, dim):
|
|
"""A QR decomposition is used to find the rotation that maps the
|
|
north pole (1, 0,...,0) to the vector mu. This rotation is then
|
|
applied to all samples.
|
|
|
|
Parameters
|
|
----------
|
|
samples: array_like, shape = [..., n]
|
|
mu : array-like, shape=[n, ]
|
|
Point to parametrise the rotation.
|
|
|
|
Returns
|
|
-------
|
|
samples : rotated samples
|
|
|
|
"""
|
|
base_point = np.zeros((dim, ))
|
|
base_point[0] = 1.
|
|
embedded = np.concatenate([mu[None, :], np.zeros((dim - 1, dim))])
|
|
rotmatrix, _ = np.linalg.qr(np.transpose(embedded))
|
|
if np.allclose(np.matmul(rotmatrix, base_point[:, None])[:, 0], mu):
|
|
rotsign = 1
|
|
else:
|
|
rotsign = -1
|
|
|
|
# apply rotation
|
|
samples = np.einsum('ij,...j->...i', rotmatrix, samples) * rotsign
|
|
return samples
|
|
|
|
def _rvs(self, dim, mu, kappa, size, random_state):
|
|
if dim == 2:
|
|
samples = self._rvs_2d(mu, kappa, size, random_state)
|
|
elif dim == 3:
|
|
samples = self._rvs_3d(kappa, size, random_state)
|
|
else:
|
|
samples = self._rejection_sampling(dim, kappa, size,
|
|
random_state)
|
|
|
|
if dim != 2:
|
|
samples = self._rotate_samples(samples, mu, dim)
|
|
return samples
|
|
|
|
def rvs(self, mu=None, kappa=1, size=1, random_state=None):
|
|
"""Draw random samples from a von Mises-Fisher distribution.
|
|
|
|
Parameters
|
|
----------
|
|
mu : array_like
|
|
Mean direction of the distribution. Must be a one-dimensional unit
|
|
vector of norm 1.
|
|
kappa : float
|
|
Concentration parameter. Must be positive.
|
|
size : int or tuple of ints, optional
|
|
Given a shape of, for example, (m,n,k), m*n*k samples are
|
|
generated, and packed in an m-by-n-by-k arrangement.
|
|
Because each sample is N-dimensional, the output shape
|
|
is (m,n,k,N). If no shape is specified, a single (N-D)
|
|
sample is returned.
|
|
random_state : {None, int, np.random.RandomState, np.random.Generator},
|
|
optional
|
|
Used for drawing random variates.
|
|
If `seed` is `None`, the `~np.random.RandomState` singleton is used.
|
|
If `seed` is an int, a new ``RandomState`` instance is used, seeded
|
|
with seed.
|
|
If `seed` is already a ``RandomState`` or ``Generator`` instance,
|
|
then that object is used.
|
|
Default is `None`.
|
|
|
|
Returns
|
|
-------
|
|
rvs : ndarray
|
|
Random variates of shape (`size`, `N`), where `N` is the
|
|
dimension of the distribution.
|
|
|
|
"""
|
|
dim, mu, kappa = self._process_parameters(mu, kappa)
|
|
random_state = self._get_random_state(random_state)
|
|
samples = self._rvs(dim, mu, kappa, size, random_state)
|
|
return samples
|
|
|
|
def _entropy(self, dim, kappa):
|
|
halfdim = 0.5 * dim
|
|
return (-self._log_norm_factor(dim, kappa) - kappa *
|
|
ive(halfdim, kappa) / ive(halfdim - 1, kappa))
|
|
|
|
def entropy(self, mu=None, kappa=1):
|
|
"""Compute the differential entropy of the von Mises-Fisher
|
|
distribution.
|
|
|
|
Parameters
|
|
----------
|
|
mu : array_like, default: None
|
|
Mean direction of the distribution. Must be a one-dimensional unit
|
|
vector of norm 1.
|
|
kappa : float, default: 1
|
|
Concentration parameter. Must be positive.
|
|
|
|
Returns
|
|
-------
|
|
h : scalar
|
|
Entropy of the von Mises-Fisher distribution.
|
|
|
|
"""
|
|
dim, _, kappa = self._process_parameters(mu, kappa)
|
|
return self._entropy(dim, kappa)
|
|
|
|
def fit(self, x):
|
|
"""Fit the von Mises-Fisher distribution to data.
|
|
|
|
Parameters
|
|
----------
|
|
x : array-like
|
|
Data the distribution is fitted to. Must be two dimensional.
|
|
The second axis of `x` must be unit vectors of norm 1 and
|
|
determine the dimensionality of the fitted
|
|
von Mises-Fisher distribution.
|
|
|
|
Returns
|
|
-------
|
|
mu : ndarray
|
|
Estimated mean direction.
|
|
kappa : float
|
|
Estimated concentration parameter.
|
|
|
|
"""
|
|
# validate input data
|
|
x = np.asarray(x)
|
|
if x.ndim != 2:
|
|
raise ValueError("'x' must be two dimensional.")
|
|
if not np.allclose(np.linalg.norm(x, axis=-1), 1.):
|
|
msg = "'x' must be unit vectors of norm 1 along last dimension."
|
|
raise ValueError(msg)
|
|
dim = x.shape[-1]
|
|
|
|
# mu is simply the directional mean
|
|
dirstats = directional_stats(x)
|
|
mu = dirstats.mean_direction
|
|
r = dirstats.mean_resultant_length
|
|
|
|
# kappa is the solution to the equation:
|
|
# r = I[dim/2](kappa) / I[dim/2 -1](kappa)
|
|
# = I[dim/2](kappa) * exp(-kappa) / I[dim/2 -1](kappa) * exp(-kappa)
|
|
# = ive(dim/2, kappa) / ive(dim/2 -1, kappa)
|
|
|
|
halfdim = 0.5 * dim
|
|
|
|
def solve_for_kappa(kappa):
|
|
bessel_vals = ive([halfdim, halfdim - 1], kappa)
|
|
return bessel_vals[0]/bessel_vals[1] - r
|
|
|
|
root_res = root_scalar(solve_for_kappa, method="brentq",
|
|
bracket=(1e-8, 1e9))
|
|
kappa = root_res.root
|
|
return mu, kappa
|
|
|
|
|
|
vonmises_fisher = vonmises_fisher_gen()
|
|
|
|
|
|
class vonmises_fisher_frozen(multi_rv_frozen):
|
|
def __init__(self, mu=None, kappa=1, seed=None):
|
|
"""Create a frozen von Mises-Fisher distribution.
|
|
|
|
Parameters
|
|
----------
|
|
mu : array_like, default: None
|
|
Mean direction of the distribution.
|
|
kappa : float, default: 1
|
|
Concentration parameter. Must be positive.
|
|
seed : {None, int, `numpy.random.Generator`,
|
|
`numpy.random.RandomState`}, optional
|
|
If `seed` is None (or `np.random`), the `numpy.random.RandomState`
|
|
singleton is used.
|
|
If `seed` is an int, a new ``RandomState`` instance is used,
|
|
seeded with `seed`.
|
|
If `seed` is already a ``Generator`` or ``RandomState`` instance
|
|
then that instance is used.
|
|
|
|
"""
|
|
self._dist = vonmises_fisher_gen(seed)
|
|
self.dim, self.mu, self.kappa = (
|
|
self._dist._process_parameters(mu, kappa)
|
|
)
|
|
|
|
def logpdf(self, x):
|
|
"""
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
Points at which to evaluate the log of the probability
|
|
density function. The last axis of `x` must correspond
|
|
to unit vectors of the same dimensionality as the distribution.
|
|
|
|
Returns
|
|
-------
|
|
logpdf : ndarray or scalar
|
|
Log of probability density function evaluated at `x`.
|
|
|
|
"""
|
|
return self._dist._logpdf(x, self.dim, self.mu, self.kappa)
|
|
|
|
def pdf(self, x):
|
|
"""
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
Points at which to evaluate the log of the probability
|
|
density function. The last axis of `x` must correspond
|
|
to unit vectors of the same dimensionality as the distribution.
|
|
|
|
Returns
|
|
-------
|
|
pdf : ndarray or scalar
|
|
Probability density function evaluated at `x`.
|
|
|
|
"""
|
|
return np.exp(self.logpdf(x))
|
|
|
|
def rvs(self, size=1, random_state=None):
|
|
"""Draw random variates from the Von Mises-Fisher distribution.
|
|
|
|
Parameters
|
|
----------
|
|
size : int or tuple of ints, optional
|
|
Given a shape of, for example, (m,n,k), m*n*k samples are
|
|
generated, and packed in an m-by-n-by-k arrangement.
|
|
Because each sample is N-dimensional, the output shape
|
|
is (m,n,k,N). If no shape is specified, a single (N-D)
|
|
sample is returned.
|
|
random_state : {None, int, `numpy.random.Generator`,
|
|
`numpy.random.RandomState`}, optional
|
|
If `seed` is None (or `np.random`), the `numpy.random.RandomState`
|
|
singleton is used.
|
|
If `seed` is an int, a new ``RandomState`` instance is used,
|
|
seeded with `seed`.
|
|
If `seed` is already a ``Generator`` or ``RandomState`` instance
|
|
then that instance is used.
|
|
|
|
Returns
|
|
-------
|
|
rvs : ndarray or scalar
|
|
Random variates of size (`size`, `N`), where `N` is the
|
|
dimension of the distribution.
|
|
|
|
"""
|
|
random_state = self._dist._get_random_state(random_state)
|
|
return self._dist._rvs(self.dim, self.mu, self.kappa, size,
|
|
random_state)
|
|
|
|
def entropy(self):
|
|
"""
|
|
Calculate the differential entropy of the von Mises-Fisher
|
|
distribution.
|
|
|
|
Returns
|
|
-------
|
|
h: float
|
|
Entropy of the Von Mises-Fisher distribution.
|
|
|
|
"""
|
|
return self._dist._entropy(self.dim, self.kappa)
|
|
|
|
|
|
class normal_inverse_gamma_gen(multi_rv_generic):
|
|
r"""Normal-inverse-gamma distribution.
|
|
|
|
The normal-inverse-gamma distribution is the conjugate prior of a normal
|
|
distribution with unknown mean and variance.
|
|
|
|
Methods
|
|
-------
|
|
pdf(x, s2, mu=0, lmbda=1, a=1, b=1)
|
|
Probability density function.
|
|
logpdf(x, s2, mu=0, lmbda=1, a=1, b=1)
|
|
Log of the probability density function.
|
|
mean(mu=0, lmbda=1, a=1, b=1)
|
|
Distribution mean.
|
|
var(mu=0, lmbda=1, a=1, b=1)
|
|
Distribution variance.
|
|
rvs(mu=0, lmbda=1, a=1, b=1, size=None, random_state=None)
|
|
Draw random samples.
|
|
|
|
Parameters
|
|
----------
|
|
mu, lmbda, a, b : array_like
|
|
Shape parameters of the distribution. See notes.
|
|
seed : {None, int, np.random.RandomState, np.random.Generator}, optional
|
|
Used for drawing random variates.
|
|
If `seed` is `None`, the `~np.random.RandomState` singleton is used.
|
|
If `seed` is an int, a new ``RandomState`` instance is used, seeded
|
|
with seed.
|
|
If `seed` is already a ``RandomState`` or ``Generator`` instance,
|
|
then that object is used.
|
|
Default is `None`.
|
|
|
|
See Also
|
|
--------
|
|
norm
|
|
invgamma
|
|
|
|
Notes
|
|
-----
|
|
|
|
The probability density function of `normal_inverse_gamma` is:
|
|
|
|
.. math::
|
|
|
|
f(x, \sigma^2; \mu, \lambda, \alpha, \beta) =
|
|
\frac{\sqrt{\lambda}}{\sqrt{2 \pi \sigma^2}}
|
|
\frac{\beta^\alpha}{\Gamma(\alpha)}
|
|
\left( \frac{1}{\sigma^2} \right)^{\alpha + 1}
|
|
\exp \left(- \frac{2 \beta + \lambda (x - \mu)^2} {2 \sigma^2} \right)
|
|
|
|
where all parameters are real and finite, and :math:`\sigma^2 > 0`,
|
|
:math:`\lambda > 0`, :math:`\alpha > 0`, and :math:`\beta > 0`.
|
|
|
|
Methods ``normal_inverse_gamma.pdf`` and ``normal_inverse_gamma.logpdf``
|
|
accept `x` and `s2` for arguments :math:`x` and :math:`\sigma^2`.
|
|
All methods accept `mu`, `lmbda`, `a`, and `b` for shape parameters
|
|
:math:`\mu`, :math:`\lambda`, :math:`\alpha`, and :math:`\beta`,
|
|
respectively.
|
|
|
|
.. versionadded:: 1.15
|
|
|
|
References
|
|
----------
|
|
.. [1] Normal-inverse-gamma distribution, Wikipedia,
|
|
https://en.wikipedia.org/wiki/Normal-inverse-gamma_distribution
|
|
|
|
Examples
|
|
--------
|
|
Suppose we wish to investigate the relationship between the
|
|
normal-inverse-gamma distribution and the inverse gamma distribution.
|
|
|
|
>>> import numpy as np
|
|
>>> from scipy import stats
|
|
>>> import matplotlib.pyplot as plt
|
|
>>> rng = np.random.default_rng(527484872345)
|
|
>>> mu, lmbda, a, b = 0, 1, 20, 20
|
|
>>> norm_inv_gamma = stats.normal_inverse_gamma(mu, lmbda, a, b)
|
|
>>> inv_gamma = stats.invgamma(a, scale=b)
|
|
|
|
One approach is to compare the distribution of the `s2` elements of
|
|
random variates against the PDF of an inverse gamma distribution.
|
|
|
|
>>> _, s2 = norm_inv_gamma.rvs(size=10000, random_state=rng)
|
|
>>> bins = np.linspace(s2.min(), s2.max(), 50)
|
|
>>> plt.hist(s2, bins=bins, density=True, label='Frequency density')
|
|
>>> s2 = np.linspace(s2.min(), s2.max(), 300)
|
|
>>> plt.plot(s2, inv_gamma.pdf(s2), label='PDF')
|
|
>>> plt.xlabel(r'$\sigma^2$')
|
|
>>> plt.ylabel('Frequency density / PMF')
|
|
>>> plt.show()
|
|
|
|
Similarly, we can compare the marginal distribution of `s2` against
|
|
an inverse gamma distribution.
|
|
|
|
>>> from scipy.integrate import quad_vec
|
|
>>> from scipy import integrate
|
|
>>> s2 = np.linspace(0.5, 3, 6)
|
|
>>> res = quad_vec(lambda x: norm_inv_gamma.pdf(x, s2), -np.inf, np.inf)[0]
|
|
>>> np.allclose(res, inv_gamma.pdf(s2))
|
|
True
|
|
|
|
The sample mean is comparable to the mean of the distribution.
|
|
|
|
>>> x, s2 = norm_inv_gamma.rvs(size=10000, random_state=rng)
|
|
>>> x.mean(), s2.mean()
|
|
(np.float64(-0.005254750127304425), np.float64(1.050438111436508))
|
|
>>> norm_inv_gamma.mean()
|
|
(np.float64(0.0), np.float64(1.0526315789473684))
|
|
|
|
Similarly, for the variance:
|
|
|
|
>>> x.var(ddof=1), s2.var(ddof=1)
|
|
(np.float64(1.0546150578185023), np.float64(0.061829865266330754))
|
|
>>> norm_inv_gamma.var()
|
|
(np.float64(1.0526315789473684), np.float64(0.061557402277623886))
|
|
|
|
"""
|
|
def rvs(self, mu=0, lmbda=1, a=1, b=1, size=None, random_state=None):
|
|
"""Draw random samples from the distribution.
|
|
|
|
Parameters
|
|
----------
|
|
mu, lmbda, a, b : array_like, optional
|
|
Shape parameters. `lmbda`, `a`, and `b` must be greater
|
|
than zero.
|
|
size : int or tuple of ints, optional
|
|
Shape of samples to draw.
|
|
random_state : {None, int, np.random.RandomState, np.random.Generator}, optional
|
|
Used for drawing random variates.
|
|
If `random_state` is `None`, the `~np.random.RandomState` singleton is used.
|
|
If `random_state` is an int, a new ``RandomState`` instance is used, seeded
|
|
with `random_state`.
|
|
If `random_state` is already a ``RandomState`` or ``Generator`` instance,
|
|
then that object is used.
|
|
Default is `None`.
|
|
|
|
Returns
|
|
-------
|
|
x, s2 : ndarray
|
|
Random variates.
|
|
|
|
"""
|
|
random_state = self._get_random_state(random_state)
|
|
s2 = invgamma(a, scale=b).rvs(size=size, random_state=random_state)
|
|
scale = (s2 / lmbda)**0.5
|
|
x = norm(loc=mu, scale=scale).rvs(size=size, random_state=random_state)
|
|
dtype = np.result_type(1.0, mu, lmbda, a, b)
|
|
return x.astype(dtype), s2.astype(dtype)
|
|
|
|
def _logpdf(self, x, s2, mu, lmbda, a, b):
|
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t1 = 0.5 * (np.log(lmbda) - np.log(2 * np.pi * s2))
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t2 = a*np.log(b) - special.gammaln(a).astype(a.dtype)
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t3 = -(a + 1) * np.log(s2)
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t4 = -(2*b + lmbda*(x - mu)**2) / (2*s2)
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return t1 + t2 + t3 + t4
|
|
|
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def logpdf(self, x, s2, mu=0, lmbda=1, a=1, b=1):
|
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"""Log of the probability density function.
|
|
|
|
Parameters
|
|
----------
|
|
x, s2 : array_like
|
|
Arguments. `s2` must be greater than zero.
|
|
mu, lmbda, a, b : array_like, optional
|
|
Shape parameters. `lmbda`, `a`, and `b` must be greater
|
|
than zero.
|
|
|
|
Returns
|
|
-------
|
|
logpdf : ndarray or scalar
|
|
Log of the probability density function.
|
|
|
|
"""
|
|
invalid, args = self._process_parameters_pdf(x, s2, mu, lmbda, a, b)
|
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s2 = args[1]
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|
# Keep it simple for now; lazyselect later, perhaps.
|
|
with np.errstate(all='ignore'):
|
|
logpdf = np.asarray(self._logpdf(*args))
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logpdf[s2 <= 0] = -np.inf
|
|
logpdf[invalid] = np.nan
|
|
return logpdf[()]
|
|
|
|
def _pdf(self, x, s2, mu, lmbda, a, b):
|
|
t1 = np.sqrt(lmbda / (2 * np.pi * s2))
|
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t2 = b**a / special.gamma(a).astype(a.dtype)
|
|
t3 = (1 / s2)**(a + 1)
|
|
t4 = np.exp(-(2*b + lmbda*(x - mu)**2) / (2*s2))
|
|
return t1 * t2 * t3 * t4
|
|
|
|
def pdf(self, x, s2, mu=0, lmbda=1, a=1, b=1):
|
|
"""The probability density function.
|
|
|
|
Parameters
|
|
----------
|
|
x, s2 : array_like
|
|
Arguments. `s2` must be greater than zero.
|
|
mu, lmbda, a, b : array_like, optional
|
|
Shape parameters. `lmbda`, `a`, and `b` must be greater
|
|
than zero.
|
|
|
|
Returns
|
|
-------
|
|
logpdf : ndarray or scalar
|
|
The probability density function.
|
|
|
|
"""
|
|
invalid, args = self._process_parameters_pdf(x, s2, mu, lmbda, a, b)
|
|
s2 = args[1]
|
|
# Keep it simple for now; lazyselect later, perhaps.
|
|
with np.errstate(all='ignore'):
|
|
pdf = np.asarray(self._pdf(*args))
|
|
pdf[s2 <= 0] = 0
|
|
pdf[invalid] = np.nan
|
|
return pdf[()]
|
|
|
|
def mean(self, mu=0, lmbda=1, a=1, b=1):
|
|
"""The mean of the distribution.
|
|
|
|
Parameters
|
|
----------
|
|
mu, lmbda, a, b : array_like, optional
|
|
Shape parameters. `lmbda` and `b` must be greater
|
|
than zero, and `a` must be greater than one.
|
|
|
|
Returns
|
|
-------
|
|
x, s2 : ndarray
|
|
The mean of the distribution.
|
|
|
|
"""
|
|
invalid, args = self._process_shapes(mu, lmbda, a, b)
|
|
mu, lmbda, a, b = args
|
|
invalid |= ~(a > 1)
|
|
mean_x = np.asarray(mu).copy()
|
|
mean_s2 = np.asarray(b / (a - 1))
|
|
mean_x[invalid] = np.nan
|
|
mean_s2[invalid] = np.nan
|
|
return mean_x[()], mean_s2[()]
|
|
|
|
def var(self, mu=0, lmbda=1, a=1, b=1):
|
|
"""The variance of the distribution.
|
|
|
|
Parameters
|
|
----------
|
|
mu, lmbda, a, b : array_like, optional
|
|
Shape parameters. `lmbda` and `b` must be greater
|
|
than zero, and `a` must be greater than two.
|
|
|
|
Returns
|
|
-------
|
|
x, s2 : ndarray
|
|
The variance of the distribution.
|
|
|
|
"""
|
|
invalid, args = self._process_shapes(mu, lmbda, a, b)
|
|
mu, lmbda, a, b = args
|
|
invalid_x = invalid | ~(a > 1)
|
|
invalid_s2 = invalid | ~(a > 2)
|
|
var_x = b / ((a - 1) * lmbda)
|
|
var_s2 = b**2 / ((a - 1)**2 * (a - 2))
|
|
var_x, var_s2 = np.asarray(var_x), np.asarray(var_s2)
|
|
var_x[invalid_x] = np.nan
|
|
var_s2[invalid_s2] = np.nan
|
|
return var_x[()], var_s2[()]
|
|
|
|
def _process_parameters_pdf(self, x, s2, mu, lmbda, a, b):
|
|
args = np.broadcast_arrays(x, s2, mu, lmbda, a, b)
|
|
dtype = np.result_type(1.0, *(arg.dtype for arg in args))
|
|
args = [arg.astype(dtype, copy=False) for arg in args]
|
|
x, s2, mu, lmbda, a, b = args
|
|
invalid = ~((lmbda > 0) & (a > 0) & (b > 0))
|
|
return invalid, args
|
|
|
|
def _process_shapes(self, mu, lmbda, a, b):
|
|
args = np.broadcast_arrays(mu, lmbda, a, b)
|
|
dtype = np.result_type(1.0, *(arg.dtype for arg in args))
|
|
args = [arg.astype(dtype, copy=False) for arg in args]
|
|
mu, lmbda, a, b = args
|
|
invalid = ~((lmbda > 0) & (a > 0) & (b > 0))
|
|
return invalid, args
|
|
|
|
def __call__(self, mu=0, lmbda=1, a=1, b=1, seed=None):
|
|
return normal_inverse_gamma_frozen(mu, lmbda, a, b, seed=seed)
|
|
|
|
|
|
normal_inverse_gamma = normal_inverse_gamma_gen()
|
|
|
|
|
|
class normal_inverse_gamma_frozen(multi_rv_frozen):
|
|
|
|
def __init__(self, mu=0, lmbda=1, a=1, b=1, seed=None):
|
|
self._dist = normal_inverse_gamma_gen(seed)
|
|
self._shapes = mu, lmbda, a, b
|
|
|
|
def logpdf(self, x, s2):
|
|
return self._dist.logpdf(x, s2, *self._shapes)
|
|
|
|
def pdf(self, x, s2):
|
|
return self._dist.pdf(x, s2, *self._shapes)
|
|
|
|
def mean(self):
|
|
return self._dist.mean(*self._shapes)
|
|
|
|
def var(self):
|
|
return self._dist.var(*self._shapes)
|
|
|
|
def rvs(self, size=None, random_state=None):
|
|
return self._dist.rvs(*self._shapes, size=size, random_state=random_state)
|
|
|
|
|
|
# Set frozen generator docstrings from corresponding docstrings in
|
|
# normal_inverse_gamma_gen and fill in default strings in class docstrings
|
|
for name in ['logpdf', 'pdf', 'mean', 'var', 'rvs']:
|
|
method = normal_inverse_gamma_gen.__dict__[name]
|
|
method_frozen = normal_inverse_gamma_frozen.__dict__[name]
|
|
method_frozen.__doc__ = doccer.docformat(method.__doc__,
|
|
mvn_docdict_noparams)
|
|
method.__doc__ = doccer.docformat(method.__doc__, mvn_docdict_params)
|