# # Author: Joris Vankerschaver 2013 # import math import numpy as np from numpy import asarray_chkfinite, asarray from numpy.lib import NumpyVersion import scipy.linalg from scipy._lib import doccer from scipy.special import (gammaln, psi, multigammaln, xlogy, entr, betaln, ive, loggamma) from scipy._lib._util import check_random_state from scipy.linalg.blas import drot from scipy.linalg._misc import LinAlgError from scipy.linalg.lapack import get_lapack_funcs from ._discrete_distns import binom from . import _mvn, _covariance, _rcont from ._qmvnt import _qmvt from ._morestats import directional_stats from scipy.optimize import root_scalar __all__ = ['multivariate_normal', 'matrix_normal', 'dirichlet', 'dirichlet_multinomial', 'wishart', 'invwishart', 'multinomial', 'special_ortho_group', 'ortho_group', 'random_correlation', 'unitary_group', 'multivariate_t', 'multivariate_hypergeom', 'random_table', 'uniform_direction', 'vonmises_fisher'] _LOG_2PI = np.log(2 * np.pi) _LOG_2 = np.log(2) _LOG_PI = np.log(np.pi) _doc_random_state = """\ 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`. """ def _squeeze_output(out): """ Remove single-dimensional entries from array and convert to scalar, if necessary. """ out = out.squeeze() if out.ndim == 0: out = out[()] return out def _eigvalsh_to_eps(spectrum, cond=None, rcond=None): """Determine which eigenvalues are "small" given the spectrum. This is for compatibility across various linear algebra functions that should agree about whether or not a Hermitian matrix is numerically singular and what is its numerical matrix rank. This is designed to be compatible with scipy.linalg.pinvh. Parameters ---------- spectrum : 1d ndarray Array of eigenvalues of a Hermitian matrix. cond, rcond : float, optional Cutoff for small eigenvalues. Singular values smaller than rcond * largest_eigenvalue are considered zero. If None or -1, suitable machine precision is used. Returns ------- eps : float Magnitude cutoff for numerical negligibility. """ if rcond is not None: cond = rcond if cond in [None, -1]: t = spectrum.dtype.char.lower() factor = {'f': 1E3, 'd': 1E6} cond = factor[t] * np.finfo(t).eps eps = cond * np.max(abs(spectrum)) return eps def _pinv_1d(v, eps=1e-5): """A helper function for computing the pseudoinverse. Parameters ---------- v : iterable of numbers This may be thought of as a vector of eigenvalues or singular values. eps : float Values with magnitude no greater than eps are considered negligible. Returns ------- v_pinv : 1d float ndarray A vector of pseudo-inverted numbers. """ return np.array([0 if abs(x) <= eps else 1/x for x in v], dtype=float) class _PSD: """ Compute coordinated functions of a symmetric positive semidefinite matrix. This class addresses two issues. Firstly it allows the pseudoinverse, the logarithm of the pseudo-determinant, and the rank of the matrix to be computed using one call to eigh instead of three. Secondly it allows these functions to be computed in a way that gives mutually compatible results. All of the functions are computed with a common understanding as to which of the eigenvalues are to be considered negligibly small. The functions are designed to coordinate with scipy.linalg.pinvh() but not necessarily with np.linalg.det() or with np.linalg.matrix_rank(). Parameters ---------- M : array_like Symmetric positive semidefinite matrix (2-D). cond, rcond : float, optional Cutoff for small eigenvalues. Singular values smaller than rcond * largest_eigenvalue are considered zero. If None or -1, suitable machine precision is used. lower : bool, optional Whether the pertinent array data is taken from the lower or upper triangle of M. (Default: lower) check_finite : bool, optional Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. allow_singular : bool, optional Whether to allow a singular matrix. (Default: True) Notes ----- The arguments are similar to those of scipy.linalg.pinvh(). """ def __init__(self, M, cond=None, rcond=None, lower=True, check_finite=True, allow_singular=True): self._M = np.asarray(M) # Compute the symmetric eigendecomposition. # Note that eigh takes care of array conversion, chkfinite, # and assertion that the matrix is square. s, u = scipy.linalg.eigh(M, lower=lower, check_finite=check_finite) eps = _eigvalsh_to_eps(s, cond, rcond) if np.min(s) < -eps: msg = "The input matrix must be symmetric positive semidefinite." raise ValueError(msg) d = s[s > eps] if len(d) < len(s) and not allow_singular: msg = ("When `allow_singular is False`, the input matrix must be " "symmetric positive definite.") raise np.linalg.LinAlgError(msg) s_pinv = _pinv_1d(s, eps) U = np.multiply(u, np.sqrt(s_pinv)) # Save the eigenvector basis, and tolerance for testing support self.eps = 1e3*eps self.V = u[:, s <= eps] # Initialize the eagerly precomputed attributes. self.rank = len(d) self.U = U self.log_pdet = np.sum(np.log(d)) # Initialize attributes to be lazily computed. self._pinv = None def _support_mask(self, x): """ Check whether x lies in the support of the distribution. """ residual = np.linalg.norm(x @ self.V, axis=-1) in_support = residual < self.eps return in_support @property def pinv(self): if self._pinv is None: self._pinv = np.dot(self.U, self.U.T) return self._pinv class multi_rv_generic: """ Class which encapsulates common functionality between all multivariate distributions. """ def __init__(self, seed=None): super().__init__() self._random_state = check_random_state(seed) @property def random_state(self): """ Get or set the Generator object for generating random variates. 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. """ return self._random_state @random_state.setter def random_state(self, seed): self._random_state = check_random_state(seed) def _get_random_state(self, random_state): if random_state is not None: return check_random_state(random_state) else: return self._random_state class multi_rv_frozen: """ Class which encapsulates common functionality between all frozen multivariate distributions. """ @property def random_state(self): return self._dist._random_state @random_state.setter def random_state(self, seed): self._dist._random_state = check_random_state(seed) _mvn_doc_default_callparams = """\ mean : array_like, default: ``[0]`` Mean of the distribution. cov : array_like or `Covariance`, default: ``[1]`` Symmetric positive (semi)definite covariance matrix of the distribution. allow_singular : bool, default: ``False`` Whether to allow a singular covariance matrix. This is ignored if `cov` is a `Covariance` object. """ _mvn_doc_callparams_note = """\ Setting the parameter `mean` to `None` is equivalent to having `mean` be the zero-vector. The parameter `cov` can be a scalar, in which case the covariance matrix is the identity times that value, a vector of diagonal entries for the covariance matrix, a two-dimensional array_like, or a `Covariance` object. """ _mvn_doc_frozen_callparams = "" _mvn_doc_frozen_callparams_note = """\ See class definition for a detailed description of parameters.""" mvn_docdict_params = { '_mvn_doc_default_callparams': _mvn_doc_default_callparams, '_mvn_doc_callparams_note': _mvn_doc_callparams_note, '_doc_random_state': _doc_random_state } mvn_docdict_noparams = { '_mvn_doc_default_callparams': _mvn_doc_frozen_callparams, '_mvn_doc_callparams_note': _mvn_doc_frozen_callparams_note, '_doc_random_state': _doc_random_state } class multivariate_normal_gen(multi_rv_generic): r"""A multivariate normal random variable. The `mean` keyword specifies the mean. The `cov` keyword specifies the covariance matrix. Methods ------- pdf(x, mean=None, cov=1, allow_singular=False) Probability density function. logpdf(x, mean=None, cov=1, allow_singular=False) Log of the probability density function. cdf(x, mean=None, cov=1, allow_singular=False, maxpts=1000000*dim, abseps=1e-5, releps=1e-5, lower_limit=None) # noqa Cumulative distribution function. logcdf(x, mean=None, cov=1, allow_singular=False, maxpts=1000000*dim, abseps=1e-5, releps=1e-5) Log of the cumulative distribution function. rvs(mean=None, cov=1, size=1, random_state=None) Draw random samples from a multivariate normal distribution. entropy() Compute the differential entropy of the multivariate normal. Parameters ---------- %(_mvn_doc_default_callparams)s %(_doc_random_state)s Notes ----- %(_mvn_doc_callparams_note)s The covariance matrix `cov` may be an instance of a subclass of `Covariance`, e.g. `scipy.stats.CovViaPrecision`. If so, `allow_singular` is ignored. Otherwise, `cov` must be a symmetric positive semidefinite matrix when `allow_singular` is True; it must be (strictly) positive definite when `allow_singular` is False. Symmetry is not checked; only the lower triangular portion is used. The determinant and inverse of `cov` are computed as the pseudo-determinant and pseudo-inverse, respectively, so that `cov` does not need to have full rank. The probability density function for `multivariate_normal` is .. math:: f(x) = \frac{1}{\sqrt{(2 \pi)^k \det \Sigma}} \exp\left( -\frac{1}{2} (x - \mu)^T \Sigma^{-1} (x - \mu) \right), where :math:`\mu` is the mean, :math:`\Sigma` the covariance matrix, :math:`k` the rank of :math:`\Sigma`. In case of singular :math:`\Sigma`, SciPy extends this definition according to [1]_. .. versionadded:: 0.14.0 References ---------- .. [1] Multivariate Normal Distribution - Degenerate Case, Wikipedia, https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Degenerate_case Examples -------- >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.stats import multivariate_normal >>> x = np.linspace(0, 5, 10, endpoint=False) >>> y = multivariate_normal.pdf(x, mean=2.5, cov=0.5); y array([ 0.00108914, 0.01033349, 0.05946514, 0.20755375, 0.43939129, 0.56418958, 0.43939129, 0.20755375, 0.05946514, 0.01033349]) >>> fig1 = plt.figure() >>> ax = fig1.add_subplot(111) >>> ax.plot(x, y) >>> plt.show() Alternatively, the object may be called (as a function) to fix the mean and covariance parameters, returning a "frozen" multivariate normal random variable: >>> rv = multivariate_normal(mean=None, cov=1, allow_singular=False) >>> # Frozen object with the same methods but holding the given >>> # mean and covariance fixed. The input quantiles can be any shape of array, as long as the last axis labels the components. This allows us for instance to display the frozen pdf for a non-isotropic random variable in 2D as follows: >>> x, y = np.mgrid[-1:1:.01, -1:1:.01] >>> pos = np.dstack((x, y)) >>> rv = multivariate_normal([0.5, -0.2], [[2.0, 0.3], [0.3, 0.5]]) >>> fig2 = plt.figure() >>> ax2 = fig2.add_subplot(111) >>> ax2.contourf(x, y, rv.pdf(pos)) """ def __init__(self, seed=None): super().__init__(seed) self.__doc__ = doccer.docformat(self.__doc__, mvn_docdict_params) def __call__(self, mean=None, cov=1, allow_singular=False, seed=None): """Create a frozen multivariate normal distribution. See `multivariate_normal_frozen` for more information. """ return multivariate_normal_frozen(mean, cov, allow_singular=allow_singular, seed=seed) def _process_parameters(self, mean, cov, allow_singular=True): """ Infer dimensionality from mean or covariance matrix, ensure that mean and covariance are full vector resp. matrix. """ if isinstance(cov, _covariance.Covariance): return self._process_parameters_Covariance(mean, cov) else: # Before `Covariance` classes were introduced, # `multivariate_normal` accepted plain arrays as `cov` and used the # following input validation. To avoid disturbing the behavior of # `multivariate_normal` when plain arrays are used, we use the # original input validation here. dim, mean, cov = self._process_parameters_psd(None, mean, cov) # After input validation, some methods then processed the arrays # with a `_PSD` object and used that to perform computation. # To avoid branching statements in each method depending on whether # `cov` is an array or `Covariance` object, we always process the # array with `_PSD`, and then use wrapper that satisfies the # `Covariance` interface, `CovViaPSD`. psd = _PSD(cov, allow_singular=allow_singular) cov_object = _covariance.CovViaPSD(psd) return dim, mean, cov_object def _process_parameters_Covariance(self, mean, cov): dim = cov.shape[-1] mean = np.array([0.]) if mean is None else mean message = (f"`cov` represents a covariance matrix in {dim} dimensions," f"and so `mean` must be broadcastable to shape {(dim,)}") try: mean = np.broadcast_to(mean, dim) except ValueError as e: raise ValueError(message) from e return dim, mean, cov def _process_parameters_psd(self, dim, mean, cov): # Try to infer dimensionality if dim is None: if mean is None: if cov is None: dim = 1 else: cov = np.asarray(cov, dtype=float) if cov.ndim < 2: dim = 1 else: dim = cov.shape[0] else: mean = np.asarray(mean, dtype=float) dim = mean.size else: if not np.isscalar(dim): raise ValueError("Dimension of random variable must be " "a scalar.") # Check input sizes and return full arrays for mean and cov if # necessary if mean is None: mean = np.zeros(dim) mean = np.asarray(mean, dtype=float) if cov is None: cov = 1.0 cov = np.asarray(cov, dtype=float) if dim == 1: mean = mean.reshape(1) cov = cov.reshape(1, 1) if mean.ndim != 1 or mean.shape[0] != dim: raise ValueError("Array 'mean' must be a vector of length %d." % dim) if cov.ndim == 0: cov = cov * np.eye(dim) elif cov.ndim == 1: cov = np.diag(cov) elif cov.ndim == 2 and cov.shape != (dim, dim): rows, cols = cov.shape if rows != cols: msg = ("Array 'cov' must be square if it is two dimensional," " but cov.shape = %s." % str(cov.shape)) else: msg = ("Dimension mismatch: array 'cov' is of shape %s," " but 'mean' is a vector of length %d.") msg = msg % (str(cov.shape), len(mean)) raise ValueError(msg) elif cov.ndim > 2: raise ValueError("Array 'cov' must be at most two-dimensional," " but cov.ndim = %d" % cov.ndim) return dim, mean, cov 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 _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): """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`. 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) 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) # mvnun expects 1-d arguments, so process points sequentially def func1d(limits): return _mvn.mvnun(limits[:n], limits[n:], mean, cov, maxpts, abseps, releps)[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): """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`. 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 cdf = self._cdf(x, mean, cov, maxpts, abseps, releps, lower_limit) # 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): """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`. 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 out = self._cdf(x, mean, cov, maxpts, abseps, releps, lower_limit) 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) 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.]]) """ 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): cdf = self.cdf(x, lower_limit=lower_limit) # 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): x = self._dist._process_quantiles(x, self.dim) out = self._dist._cdf(x, self.mean, self.cov_object.covariance, self.maxpts, self.abseps, self.releps, lower_limit) 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, " "but a.shape = {}.".format(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, " "parameter vector 'a', but alpha.shape = {} " "and x.shape = {}.".format(alpha.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 " "simplex. but np.sum(x, 0) = %s." % 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 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 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 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', '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, but scale.scale = %s." % str(scale.shape)) elif scale.ndim > 2: raise ValueError("Array 'scale' must be at most two-dimensional," " but scale.ndim = %d" % 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, but x.shape = %s." % 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 if they are three dimensional" ", but x.shape = %s." % str(x.shape)) elif x.ndim > 3: raise ValueError("Quantiles must be at most two-dimensional with" " an additional dimension for multiple" "components, but x.ndim = %d" % 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' ' be {}, got {}.'.format((dim, dim), 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.' ' Got size.ndim = %s' % 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 triagular. 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) def _cho_inv_batch(a, check_finite=True): """ Invert the matrices a_i, using a Cholesky factorization of A, where a_i resides in the last two dimensions of a and the other indices describe the index i. Overwrites the data in a. Parameters ---------- a : array Array of matrices to invert, where the matrices themselves are stored in the last two dimensions. check_finite : bool, optional Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. Returns ------- x : array Array of inverses of the matrices ``a_i``. See Also -------- scipy.linalg.cholesky : Cholesky factorization of a matrix """ if check_finite: a1 = asarray_chkfinite(a) else: a1 = asarray(a) if len(a1.shape) < 2 or a1.shape[-2] != a1.shape[-1]: raise ValueError('expected square matrix in last two dimensions') potrf, potri = get_lapack_funcs(('potrf', 'potri'), (a1,)) triu_rows, triu_cols = np.triu_indices(a.shape[-2], k=1) for index in np.ndindex(a1.shape[:-2]): # Cholesky decomposition a1[index], info = potrf(a1[index], lower=True, overwrite_a=False, clean=False) if info > 0: raise LinAlgError("%d-th leading minor not positive definite" % info) if info < 0: raise ValueError('illegal value in %d-th argument of internal' ' potrf' % -info) # Inversion a1[index], info = potri(a1[index], lower=True, overwrite_c=False) if info > 0: raise LinAlgError("the inverse could not be computed") if info < 0: raise ValueError('illegal value in %d-th argument of internal' ' potrf' % -info) # Make symmetric (dpotri only fills in the lower triangle) a1[index][triu_rows, triu_cols] = a1[index][triu_cols, triu_rows] return a1 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}`. .. 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. 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, scale, log_det_scale): """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 scale : ndarray Scale matrix 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 'logpdf' instead. """ log_det_x = np.empty(x.shape[-1]) x_inv = np.copy(x).T if dim > 1: _cho_inv_batch(x_inv) # works in-place else: x_inv = 1./x_inv tr_scale_x_inv = np.empty(x.shape[-1]) for i in range(x.shape[-1]): C, lower = scipy.linalg.cho_factor(x[:, :, i], lower=True) log_det_x[i] = 2 * np.sum(np.log(C.diagonal())) tr_scale_x_inv[i] = np.dot(scale, x_inv[i]).trace() # 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) _, log_det_scale = self._cholesky_logdet(scale) out = self._logpdf(x, dim, df, scale, log_det_scale) 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 _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 triagular. %(_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 A ~ W(df, I) A = super()._standard_rvs(n, shape, dim, df, random_state) # Calculate SA = (CA)'^{-1} (CA)^{-1} ~ iW(df, scale) eye = np.eye(dim) trtrs = get_lapack_funcs(('trtrs'), (A,)) for index in np.ndindex(A.shape[:-2]): # Calculate CA CA = np.dot(C, A[index]) # Get (C A)^{-1} via triangular solver if dim > 1: CA, info = trtrs(CA, eye, lower=True) if info > 0: raise LinAlgError("Singular matrix.") if info < 0: raise ValueError('Illegal value in %d-th argument of' ' internal trtrs' % -info) else: CA = 1. / CA # Get SA A[index] = np.dot(CA.T, CA) 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) # Invert the scale eye = np.eye(dim) L, lower = scipy.linalg.cho_factor(scale, lower=True) inv_scale = scipy.linalg.cho_solve((L, lower), eye) # Cholesky decomposition of inverted scale C = scipy.linalg.cholesky(inv_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 C, lower = scipy.linalg.cho_factor(self.scale, lower=True) self.log_det_scale = 2 * np.sum(np.log(C.diagonal())) # Get the inverse using the Cholesky factorization eye = np.eye(self.dim) self.inv_scale = scipy.linalg.cho_solve((C, lower), eye) # Get the Cholesky factorization of the inverse scale self.C = scipy.linalg.cholesky(self.inv_scale, lower=True) 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) 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. """ # noqa: E501 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, eps=1e-15): """Returns: n_, p_, npcond. n_ and p_ are arrays of the correct shape; npcond is a boolean array flagging values out of the domain. """ p = np.array(p, dtype=np.float64, copy=True) p_adjusted = 1. - p[..., :-1].sum(axis=-1) i_adjusted = np.abs(p_adjusted) > eps p[i_adjusted, -1] = p_adjusted[i_adjusted] # true for bad p pcond = np.any(p < 0, axis=-1) pcond |= np.any(p > 1, axis=-1) n = np.array(n, dtype=np.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=np.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("Size of each quantile should be size of p: " "received %d, but expected %d." % (xx.shape[-1], 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 ----- This class is wrapping the random_rot code from the MDP Toolkit, https://github.com/mdp-toolkit/mdp-toolkit Return a random rotation matrix, drawn from the Haar distribution (the only uniform distribution on SO(N)). The algorithm is described in the paper Stewart, G.W., "The efficient generation of random orthogonal matrices with an application to condition estimators", SIAM Journal on Numerical Analysis, 17(3), pp. 403-409, 1980. For more information see https://en.wikipedia.org/wiki/Orthogonal_matrix#Randomization See also the similar `ortho_group`. 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 <= 1 or dim != int(dim): raise ValueError("""Dimension of rotation must be specified, and must be a scalar greater than 1.""") 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) size = int(size) size = (size,) if size > 1 else () dim = self._process_parameters(dim) # H represents a (dim, dim) matrix, while D represents the diagonal of # a (dim, dim) diagonal matrix. The algorithm that follows is # broadcasted on the leading shape in `size` to vectorize along # samples. H = np.empty(size + (dim, dim)) H[..., :, :] = np.eye(dim) D = np.empty(size + (dim,)) for n in range(dim-1): # x is a vector with length dim-n, xrow and xcol are views of it as # a row vector and column vector respectively. It's important they # are views and not copies because we are going to modify x # in-place. x = random_state.normal(size=size + (dim-n,)) xrow = x[..., None, :] xcol = x[..., :, None] # This is the squared norm of x, without vectorization it would be # dot(x, x), to have proper broadcasting we use matmul and squeeze # out (convert to scalar) the resulting 1x1 matrix norm2 = np.matmul(xrow, xcol).squeeze((-2, -1)) x0 = x[..., 0].copy() D[..., n] = np.where(x0 != 0, np.sign(x0), 1) x[..., 0] += D[..., n]*np.sqrt(norm2) # In renormalizing x we have to append an additional axis with # [..., None] to broadcast the scalar against the vector x x /= np.sqrt((norm2 - x0**2 + x[..., 0]**2) / 2.)[..., None] # Householder transformation, without vectorization the RHS can be # written as outer(H @ x, x) (apart from the slicing) H[..., :, n:] -= np.matmul(H[..., :, n:], xcol) * xrow D[..., -1] = (-1)**(dim-1)*D[..., :-1].prod(axis=-1) # Without vectorization this could be written as H = diag(D) @ H, # left-multiplication by a diagonal matrix amounts to multiplying each # row of H by an element of the diagonal, so we add a dummy axis for # the column index H *= D[..., :, None] return H 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() """ 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 <= 1 or dim != int(dim): raise ValueError("Dimension of rotation must be specified," "and must be a scalar greater than 1.") 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) if size > 1 and NumpyVersion(np.__version__) < '1.22.0': return np.array([self.rvs(dim, size=1, random_state=random_state) for i in range(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() """ 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 `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 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. """ 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]]) 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 <= 1 or dim != int(dim): raise ValueError("Dimension of rotation must be specified," "and must be a scalar greater than 1.") 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) if size > 1 and NumpyVersion(np.__version__) < '1.22.0': return np.array([self.rvs(dim, size=1, random_state=random_state) for i in range(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() """ 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) 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)) + 0.5 * shape_info.log_pdet) 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("Array 'loc' must be a vector of length %d." % 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," " but cov.shape = %s." % 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("Array 'cov' must be at most two-dimensional," " but cov.ndim = %d" % 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 >>> 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/_downloads/pdf/multi_hyper.pdf """ 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.float_) den = np.zeros_like(n, dtype=np.float_) 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 strictly positive 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, # noqa '_doc_random_state': _doc_random_state } dirichlet_mn_docdict_noparams = { '_dirichlet_mn_doc_default_callparams': _dirichlet_mn_doc_frozen_callparams, # noqa '_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 positive 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 dimenions 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)