a BCCf@sddlZddlZddlZddlmZddlmZm Z m Z m Z m Z m Z mZmZddlmZmZddlmZmZddlmZddlmZdd lmZmZmZdd lmZdd l m!Z!dd l"m#Z#gd Z$e%dej&Z'e%dZ(e%ej&Z)dZ*ddZ+dddZ,dddZ-GdddZ.GdddZ/GdddZ0dZ1dZ2dZ3d Z4e1e2e*d!Z5e3e4e*d!Z6Gd"d#d#e/Z7e7Z8Gd$d%d%e0Z9d&D]:Z:e7j;e:Zeeeeeed?d?e/ZZeZZ[Gd@dAdAe0Z\dBD]:Z:eZj;e:Zeeeeeeeeeeee|jj}ddd}||t|j}|tt|}|S)aDetermine 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. N)N@@g.A)fd)dtypecharlowernpZfinfoepsmaxabs)Zspectrumcondrcondtfactorr9r*r*r/_eigvalsh_to_eps@s  r@h㈵>cstjfdd|DtdS)aA 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. cs$g|]}t|krdnd|qS)rr)r;).0xr9r*r/ rz_pinv_1d..r5)r8arrayfloat)vr9r*rDr/_pinv_1dbsrKc@s.eZdZdZd ddZddZedd ZdS) _PSDaN Compute coordinated functions of a symmetric positive semidefinite matrix. This class addresses two issues. Firstly it allows the pseudoinverse, the logarithm of the pseudo-determinant, and the rank of the matrix to be computed using one call to eigh instead of three. Secondly it allows these functions to be computed in a way that gives mutually compatible results. All of the functions are computed with a common understanding as to which of the eigenvalues are to be considered negligibly small. The functions are designed to coordinate with scipy.linalg.pinvh() but not necessarily with np.linalg.det() or with np.linalg.matrix_rank(). Parameters ---------- M : array_like Symmetric positive semidefinite matrix (2-D). cond, rcond : float, optional Cutoff for small eigenvalues. Singular values smaller than rcond * largest_eigenvalue are considered zero. If None or -1, suitable machine precision is used. lower : bool, optional Whether the pertinent array data is taken from the lower or upper triangle of M. (Default: lower) check_finite : bool, optional Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. allow_singular : bool, optional Whether to allow a singular matrix. (Default: True) Notes ----- The arguments are similar to those of scipy.linalg.pinvh(). NTcCst||_tjj|||d\}}t|||} t|| krJd} t| ||| k} t | t |krz|szd} tj | t || } t |t | } d| |_|dd|| kf|_t | |_| |_tt| |_d|_dS)Nr7 check_finitez9The input matrix must be symmetric positive semidefinite.zUWhen `allow_singular is False`, the input matrix must be symmetric positive definite.r2)r8asarray_Mscipylinalgeighr@min ValueErrorlenZ LinAlgErrorrKmultiplysqrtr9VrankUsumloglog_pdet_pinv)selfMr<r=r7rNallow_singularsur9msgr4Zs_pinvr[r*r*r/__init__s$       z _PSD.__init__cCs$tjj||jdd}||jk}|S)zJ Check whether x lies in the support of the distribution. r1axis)r8rRrrYr9)r`rCZresidualZ in_supportr*r*r/ _support_masks z_PSD._support_maskcCs$|jdurt|j|jj|_|jSN)r_r8dotr[Tr`r*r*r/pinvs z _PSD.pinv)NNTTT)__name__ __module__ __qualname____doc__rfripropertyrnr*r*r*r/rLus' !rLcsDeZdZdZd fdd ZeddZejddZdd ZZ S) multi_rv_genericzc Class which encapsulates common functionality between all multivariate distributions. Ncstt||_dSrj)superrfr _random_stater`seed __class__r*r/rfs zmulti_rv_generic.__init__cCs|jS)a 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. )rvrmr*r*r/ random_states zmulti_rv_generic.random_statecCst||_dSrjr rvrwr*r*r/r{scCs|durt|S|jSdSrjr|r`r{r*r*r/_get_random_statesz"multi_rv_generic._get_random_state)N) rorprqrrrfrsr{setterr~ __classcell__r*r*ryr/rts  rtc@s*eZdZdZeddZejddZdS)multi_rv_frozenzj Class which encapsulates common functionality between all frozen multivariate distributions. cCs|jjSrj)_distrvrmr*r*r/r{szmulti_rv_frozen.random_statecCst||j_dSrj)r rrvrwr*r*r/r{sN)rorprqrrrsr{rr*r*r*r/rs  raVmean : 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. a5Setting 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. z>See class definition for a detailed description of parameters.)_mvn_doc_default_callparams_mvn_doc_callparams_note_doc_random_statecseZdZdZd&fdd Zd'ddZd(d d Zd d ZddZddZ ddZ d)ddZ d*ddZ ddZ d+ddddZd,ddddZd-d d!Zd.d"d#Zd/d$d%ZZS)0multivariate_normal_gena{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) 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(mean=None, cov=1) Compute the differential entropy of the multivariate normal. fit(x, fix_mean=None, fix_cov=None) Fit a multivariate normal distribution to data. 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)) Ncs t|t|jt|_dSrj)rurfr docformatrrmvn_docdict_paramsrwryr*r/rfs z multivariate_normal_gen.__init__rFcCst||||dS)zzCreate a frozen multivariate normal distribution. See `multivariate_normal_frozen` for more information. )rbrx)multivariate_normal_frozen)r`meancovrbrxr*r*r/__call__sz multivariate_normal_gen.__call__TcCsPt|tjr|||S|d||\}}}t||d}t|}|||fSdS)z Infer dimensionality from mean or covariance matrix, ensure that mean and covariance are full vector resp. matrix. Nrb) isinstancerZ Covariance_process_parameters_Covariance_process_parameters_psdrL CovViaPSD)r`rrrbdimZpsd cov_objectr*r*r/_process_parameterss     z+multivariate_normal_gen._process_parametersc Cs~|jd}|durtdgn|}d|d|f}zt||}Wn.tyr}zt||WYd}~n d}~00|||fS)Nr1z(`cov` represents a covariance matrix in z9 dimensions,and so `mean` must be broadcastable to shape )shaper8rHZ broadcast_torU)r`rrrmessageer*r*r/rs  z6multivariate_normal_gen._process_parameters_CovariancecCs|dur^|durH|durd}q\tj|td}|jdkrArray 'cov' must be at most two-dimensional, but cov.ndim = %d)r8rOrIr,rsizeisscalarrUzerosreshapeeyediagstrrV)r`rrrrowscolsrer*r*r/rsV             z/multivariate_normal_gen._process_parameters_psdcCs`tj|td}|jdkr$|tj}n8|jdkr\|dkrJ|ddtjf}n|tjddf}|Sl Adjust quantiles array so that last axis labels the components of each data point. rGrrNr8rOrIr,newaxisr`rCrr*r*r/_process_quantiless   z*multivariate_normal_gen._process_quantilescCsj|j|j}}||}|jdkr<|dtjf}|dtjf}tjt||dd}d|t||S)aLog 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. r.r1rg) r^rZr,r8rr\squareZwhiten_LOG_2PI)r`rCrrZ log_det_covrZdevmahar*r*r/_logpdfs zmultivariate_normal_gen._logpdfc Csf||||}|\}}}|||}||||}t|j|kr^|||} tj || <t|S)aLog 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 ) rrrr8anyrZriinfr0 r`rCrrrbparamsrrr. out_of_boundsr*r*r/logpdfs   zmultivariate_normal_gen.logpdfc Csh||||}|\}}}|||}t||||}t|j|kr`|||} d|| <t|S)aMultivariate 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 r) rrr8exprrrZrir0rr*r*r/pdf5s  zmultivariate_normal_gen.pdfcs|durtjtj n|}t||\} } | | k} d| jdd} | | } } | | | | | | <| | <|jdtj| | fdd} fdd}t|d| | }t |S)aMultivariate 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 Nr1rgc s*t|d|ddSNr)rZmvnun)limitsabsepsrmaxptsrnrelepsr*r/func1d~s z,multivariate_normal_gen._cdf..func1d) r8fullrrbroadcast_arraysr\copy concatenateapply_along_axisr0)r`rCrrrrr lower_limitr7bai_swapsignsrrr.r*rr/_cdfQs zmultivariate_normal_gen._cdfrArc Cst||||} | \} }} | j}||| }|s6d| }||||||||} t| dkrb| dn| } t| } | S)aLog 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 @Br)r covariancerrr8rr])r`rCrrrbrrrrrrrcdfr.r*r*r/logcdfs    zmultivariate_normal_gen.logcdfc CsP||||} | \} }} | j}||| }|s6d| }||||||||} | S)aMultivariate 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 r)rrrr) r`rCrrrbrrrrrrrr.r*r*r/rs   zmultivariate_normal_gen.cdfc Cs|||\}}}||}t|tjrF|j}||||}t|}nH|pNt}t |s`|f}t||j df}|j |d} || | }|S)aDraw 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 r1r)rr~rrrrrr0tupler8iterablernormalZcolorize) r`rrrr{rrr.rrCr*r*r/rvss      zmultivariate_normal_gen.rvscCs*|||\}}}d|jtd|jS)aGCompute 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 ?r)rrZrr^)r`rrrrr*r*r/entropyszmultivariate_normal_gen.entropyc Cst|}|jdkrtd|j\}}|durVt|}|j|fkrPd}t||}n |jdd}|durt|}|j||fkrd}t|tj j |ddd \}} t |} t || krd }t||} n||} | j | |} || fS) a#Fit a multivariate normal distribution to data. Parameters ---------- x : ndarray (m, n) Data the distribution is fitted to. Must have two axes. The first axis of length `m` represents the number of vectors the distribution is fitted to. The second axis of length `n` determines the dimensionality of the fitted distribution. fix_mean : ndarray(n, ) Fixed mean vector. Must have length `n`. fix_cov: ndarray (n, n) Fixed covariance matrix. Must have shape `(n, n)`. Returns ------- mean : ndarray (n, ) Maximum likelihood estimate of the mean vector cov : ndarray (n, n) Maximum likelihood estimate of the covariance matrix r)z`x` must be two-dimensional.Nzd`fix_mean` must be a one-dimensional array the same length as the dimensionality of the vectors `x`.rrgzn`fix_cov` must be a two-dimensional square array of same side length as the dimensionality of the vectors `x`.TrMz2`fix_cov` must be symmetric positive semidefinite.)r8rOr,rUr atleast_1drZ atleast_2drQrRrSr@rTrl) r`rCZfix_meanZfix_covZ n_vectorsrrerrcrdr9rZ centered_datar*r*r/fits2       zmultivariate_normal_gen.fit)N)NrFN)T)NrF)NrF)NrFNrArA)NrFNrArA)NrrN)Nr)NN)rorprqrrrfrrrrrrrrrrrrrrrr*r*ryr/r!s0b  9  4 - ) % rc@s`eZdZdddZeddZd d Zd d Zdd ddZdd ddZ dddZ ddZ dS)rNrFrAcCsXt||_|j|||\|_|_|_|p0|jj|_|sBd|j}||_||_ ||_ dS)afCreate 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.]]) rN) rrrrrrZ_allow_singularrbrrr)r`rrrbrxrrrr*r*r/rfZs+  z#multivariate_normal_frozen.__init__cCs|jjSrj)rrrmr*r*r/rszmultivariate_normal_frozen.covcCs`|j||j}|j||j|j}t|jj|jkrX|j ||j}tj ||<t |Srj) rrrrrrr8rrZrirr0)r`rCr.rr*r*r/rs  z!multivariate_normal_frozen.logpdfcCst||Srjr8rrr`rCr*r*r/rszmultivariate_normal_frozen.pdfrcCs6|j||d}t|dkr$|dn|}t|}|S)Nrrr)rr8rr])r`rCrrr.r*r*r/rs z!multivariate_normal_frozen.logcdfc Cs<|j||j}|j||j|jj|j|j|j |}t |Srj) rrrrrrrrrrr0)r`rCrr.r*r*r/rs  zmultivariate_normal_frozen.cdfcCs|j|j|j||Srj)rrrrr`rr{r*r*r/rszmultivariate_normal_frozen.rvscCs$|jj}|jj}d|td|S)zComputes the differential entropy of the multivariate normal. Returns ------- h : scalar Entropy of the multivariate normal distribution rr)rr^rZr)r`r^rZr*r*r/rs z"multivariate_normal_frozen.entropy)NrFNNrArA)rN) rorprqrfrsrrrrrrrr*r*r*r/rYs 5  r)rrrrramean : 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`) aIf `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_default_callparams_matnorm_doc_callparams_notercsteZdZdZdfdd ZdddZdd Zd d Zd d ZdddZ dddZ dddZ dddZ ddZ ZS)matrix_normal_genaT 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. Ncs t|t|jt|_dSrj)rurfrrrrmatnorm_docdict_paramsrwryr*r/rfSs zmatrix_normal_gen.__init__rcCst||||dS)zoCreate a frozen matrix normal distribution. See `matrix_normal_frozen` for more information. rx)matrix_normal_frozenr`rrowcovcolcovrxr*r*r/rWszmatrix_normal_gen.__call__c Cs|durFtj|td}|j}t|dkr0tdt|dkrFtdtj|td}|jdkr|durz|t|d}q|td}n|jdkrt |}|j}t|dkrtd|d|dkrtd |ddkrtd |d}tj|td}|jdkr4|dur$|t|d}n|td}n|jdkrJt |}|j}t|dkrftd |d|dkrtd |ddkrtd |d}|dur|d|krtd|d|krtdnt ||f}||f} | |||fS)z Infer dimensionality from mean or covariance matrices. Handle defaults. Ensure compatible dimensions. NrGr)z%Array `mean` must be two dimensional.rzArray `mean` has invalid shape.rz(`rowcov` must be a scalar or a 2D array.zArray `rowcov` must be square.z!Array `rowcov` has invalid shape.z(`colcov` must be a scalar or a 2D array.zArray `colcov` must be square.z!Array `colcov` has invalid shape.z=Arrays `mean` and `rowcov` must have the same number of rows.z@Arrays `mean` and `colcov` must have the same number of columns.) r8rOrIrrVrUrr,identityrr) r`rrrZ meanshapeZrowshapenumrowsZcolshapenumcolsdimsr*r*r/r_sZ            z%matrix_normal_gen._process_parameterscCsHtj|td}|jdkr*|tjddf}|jdd|krDtd|S)zp Adjust quantiles array so that last two axes labels the components of each data point. rGr)NzJThe shape of array `X` is not compatible with the distribution parameters.)r8rOrIr,rrrU)r`Xrr*r*r/rs  z$matrix_normal_gen._process_quantilesc Cst|\}} t||dd} t|jt| |d} tjtjt| dddd} d|| t| |||| S)aLog 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. r1rrrgr)r8ZmoveaxisZ tensordotrlrkr\rr) r`rrrZ row_prec_rtZlog_det_rowcovZ col_prec_rtZlog_det_colcovrrZroll_devZ scale_devrr*r*r/rs zmatrix_normal_gen._logpdfc Cs`||||\}}}}|||}t|dd}t|dd}|||||j|j|j|j}t|S)aLog 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 Fr)rrrLrr[r^r0) r`rrrrrrowpsdcolpsdr.r*r*r/rs    zmatrix_normal_gen.logpdfcCst|||||S)aMatrix 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 r)r`rrrrr*r*r/rszmatrix_normal_gen.pdfc Cst|}||||\}}}}tjj|dd}tjj|dd}||}|j|d||dfdddd} |tj d|| |dd} |dkr| |j } | S) aDraw 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 Tr7rrrr)zjp,ipq,kq->ijk)optimize) intrrQrRcholeskyr~standard_normal transposer8einsumrr) r`rrrrr{rZrowcholZcolcholZstd_normr.r*r*r/rs&   zmatrix_normal_gen.rvscCsZt|jd|jdf}||||\}}}}t|dd}t|dd}|||j|jS)aLog 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 rFr)r8rrrrL_entropyr^)r`rrZ dummy_meanr_rrr*r*r/r.s   zmatrix_normal_gen.entropycCs4|\}}d||dtd||d||SNrr)r)r`rZrow_cov_logdetZcol_cov_logdetrpr*r*r/rKs zmatrix_normal_gen._entropy)N)NrrN)Nrr)Nrr)NrrrN)rr)rorprqrrrfrrrrrrrrrrr*r*ryr/rsd B &   ) rc@s<eZdZdZdddZddZdd Zdd d Zd d ZdS)ra 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 NrcCsNt||_|j|||\|_|_|_|_t|jdd|_t|jdd|_ dS)NFr) rrrrrrrrLrrrr*r*r/rfss  zmatrix_normal_frozen.__init__c CsD|j||j}|j|j||j|jj|jj|jj|jj}t |Srj) rrrrrrr[r^rr0)r`rr.r*r*r/rzs  zmatrix_normal_frozen.logpdfcCst||Srjr)r`rr*r*r/rszmatrix_normal_frozen.pdfcCs|j|j|j|j||Srj)rrrrrrr*r*r/rszmatrix_normal_frozen.rvscCs|j|j|jj|jjSrj)rrrrr^rrmr*r*r/rszmatrix_normal_frozen.entropy)NrrN)rN) rorprqrrrfrrrrr*r*r*r/rTs   r)rrrrzalpha : array_like The concentration parameters. The number of entries determines the dimensionality of the distribution. )!_dirichlet_doc_default_callparamsrcCsBt|}t|dkr"tdn|jdkr>td|jd|S)Nrz%All parameters must be greater than 0rztd tt|dd dkrrtdt|d|S)Nrrz|Vector 'x' must have either the same number of entries as, or one entry fewer than, parameter vector 'a', but alpha.shape = z and x.shape = rr)zUThe input must be one dimensional or a two dimensional matrix containing the entries.z8Each entry in 'x' must be greater than or equal to zero.z/Each entry in 'x' must be smaller or equal one.r1rgzJEach entry in 'x' must be greater than zero if its alpha is less than one.rg& .>zOThe input vector 'x' must lie within the normal simplex. but np.sum(x, 0) = %s.)r8rOrrUrHr\r,appendZvstackrTr:repeatr logical_andr;r)rrCZxkZxeq0Zalphalt1Zchkr*r*r/_dirichlet_check_inputs< ,       rcCstt|tt|S)avInternal 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 )r8r\rrr*r*r/_lnBsrcsneZdZdZdfdd ZdddZddZd d Zd d Zd dZ ddZ ddZ ddZ dddZ ZS) dirichlet_gena A Dirichlet random variable. The ``alpha`` keyword specifies the concentration parameters of the distribution. .. versionadded:: 0.15.0 Methods ------- pdf(x, alpha) Probability density function. logpdf(x, alpha) Log of the probability density function. rvs(alpha, size=1, random_state=None) Draw random samples from a Dirichlet distribution. mean(alpha) The mean of the Dirichlet distribution var(alpha) The variance of the Dirichlet distribution cov(alpha) The covariance of the Dirichlet distribution entropy(alpha) Compute the differential entropy of the Dirichlet distribution. Parameters ---------- %(_dirichlet_doc_default_callparams)s %(_doc_random_state)s Notes ----- Each :math:`\alpha` entry must be positive. The distribution has only support on the simplex defined by .. math:: \sum_{i=1}^{K} x_i = 1 where :math:`0 < x_i < 1`. If the quantiles don't lie within the simplex, a ValueError is raised. The probability density function for `dirichlet` is .. math:: f(x) = \frac{1}{\mathrm{B}(\boldsymbol\alpha)} \prod_{i=1}^K x_i^{\alpha_i - 1} where .. math:: \mathrm{B}(\boldsymbol\alpha) = \frac{\prod_{i=1}^K \Gamma(\alpha_i)} {\Gamma\bigl(\sum_{i=1}^K \alpha_i\bigr)} and :math:`\boldsymbol\alpha=(\alpha_1,\ldots,\alpha_K)`, the concentration parameters and :math:`K` is the dimension of the space where :math:`x` takes values. Note that the `dirichlet` interface is somewhat inconsistent. The array returned by the rvs function is transposed with respect to the format expected by the pdf and logpdf. Examples -------- >>> import numpy as np >>> from scipy.stats import dirichlet Generate a dirichlet random variable >>> quantiles = np.array([0.2, 0.2, 0.6]) # specify quantiles >>> alpha = np.array([0.4, 5, 15]) # specify concentration parameters >>> dirichlet.pdf(quantiles, alpha) 0.2843831684937255 The same PDF but following a log scale >>> dirichlet.logpdf(quantiles, alpha) -1.2574327653159187 Once we specify the dirichlet distribution we can then calculate quantities of interest >>> dirichlet.mean(alpha) # get the mean of the distribution array([0.01960784, 0.24509804, 0.73529412]) >>> dirichlet.var(alpha) # get variance array([0.00089829, 0.00864603, 0.00909517]) >>> dirichlet.entropy(alpha) # calculate the differential entropy -4.3280162474082715 We can also return random samples from the distribution >>> dirichlet.rvs(alpha, size=1, random_state=1) array([[0.00766178, 0.24670518, 0.74563305]]) >>> dirichlet.rvs(alpha, size=2, random_state=2) array([[0.01639427, 0.1292273 , 0.85437844], [0.00156917, 0.19033695, 0.80809388]]) Alternatively, the object may be called (as a function) to fix concentration parameters, returning a "frozen" Dirichlet random variable: >>> rv = dirichlet(alpha) >>> # Frozen object with the same methods but holding the given >>> # concentration parameters fixed. Ncs t|t|jt|_dSrj)rurfrrrrdirichlet_docdict_paramsrwryr*r/rfbs zdirichlet_gen.__init__cCs t||dSNr)dirichlet_frozenr`rrxr*r*r/rfszdirichlet_gen.__call__cCs(t|}| tt|d|jjdS)aLog 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. rr)rr8r\rrl)r`rCrlnBr*r*r/riszdirichlet_gen._logpdfcCs&t|}t||}|||}t|S)ayLog 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`. )rrrr0r`rCrr.r*r*r/r|s  zdirichlet_gen.logpdfcCs,t|}t||}t|||}t|S)akThe 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`. )rrr8rrr0r r*r*r/rs zdirichlet_gen.pdfcCst|}|t|}t|S)zMean of the Dirichlet distribution. Parameters ---------- %(_dirichlet_doc_default_callparams)s Returns ------- mu : ndarray or scalar Mean of the Dirichlet distribution. rr8r\r0)r`rr.r*r*r/rs zdirichlet_gen.meancCs6t|}t|}||||||d}t|S)zVariance of the Dirichlet distribution. Parameters ---------- %(_dirichlet_doc_default_callparams)s Returns ------- v : ndarray or scalar Variance of the Dirichlet distribution. rr )r`ralpha0r.r*r*r/vars zdirichlet_gen.varcCs@t|}t|}||}t|t|||d}t|S)zCovariance matrix of the Dirichlet distribution. Parameters ---------- %(_dirichlet_doc_default_callparams)s Returns ------- cov : ndarray The covariance matrix of the distribution. r)rr8r\routerr0)r`rr rrr*r*r/rs  zdirichlet_gen.covcCs^t|}t|}t|}|jd}|||tj|t|dtj|}t|S)a Differential entropy of the Dirichlet distribution. Parameters ---------- %(_dirichlet_doc_default_callparams)s Returns ------- h : scalar Entropy of the Dirichlet distribution rr) rr8r\rrrQZspecialrr0)r`rr r Kr.r*r*r/rs  zdirichlet_gen.entropyrcCs t|}||}|j||dS)a 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. r)rr~r)r`rrr{r*r*r/rs zdirichlet_gen.rvs)N)N)rN)rorprqrrrfrrrrrrrrrrr*r*ryr/rsk rc@sPeZdZdddZddZddZdd Zd d Zd d ZddZ dddZ dS)rNcCst||_t||_dSrj)rrrrr r*r*r/rfs zdirichlet_frozen.__init__cCs|j||jSrj)rrrrr*r*r/rszdirichlet_frozen.logpdfcCs|j||jSrj)rrrrr*r*r/rszdirichlet_frozen.pdfcCs|j|jSrj)rrrrmr*r*r/rszdirichlet_frozen.meancCs|j|jSrj)rrrrmr*r*r/r!szdirichlet_frozen.varcCs|j|jSrj)rrrrmr*r*r/r$szdirichlet_frozen.covcCs|j|jSrj)rrrrmr*r*r/r'szdirichlet_frozen.entropyrcCs|j|j||Srj)rrrrr*r*r/r*szdirichlet_frozen.rvs)N)rN) rorprqrfrrrrrrrr*r*r*r/rs r)rrrrrrrzdf : int Degrees of freedom, must be greater than or equal to dimension of the scale matrix scale : array_like Symmetric positive definite scale matrix of the distribution )Z_doc_default_callparamsZ_doc_callparams_notercseZdZdZd,fdd Zd-ddZddZd d Zd d Zd dZ ddZ ddZ ddZ ddZ ddZddZddZddZdd Zd!d"Zd.d$d%Zd&d'Zd(d)Zd*d+ZZS)/ wishart_genaqA 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. Ncs t|t|jt|_dSrjrurfrrrrwishart_docdict_paramsrwryr*r/rfs zwishart_gen.__init__cCs t|||S)zbCreate a frozen Wishart distribution. See `wishart_frozen` for more information. )wishart_frozenr`dfscalerxr*r*r/rszwishart_gen.__call__cCs|dur d}tj|td}|jdkr6|tjtjf}n`|jdkrLt|}nJ|jdkr~|jd|jdks~tdt|jn|jdkrtd|j|jd}|dur|}n(t |stdn||dkrtd |||fS) NrrGrrr)zLArray 'scale' must be square if it is two dimensional, but scale.scale = %s.zBArray 'scale' must be at most two-dimensional, but scale.ndim = %dz$Degrees of freedom must be a scalar.zNDegrees of freedom must be greater than the dimension of scale matrix minus 1.) r8rOrIr,rrrrUrr)r`rrrr*r*r/rs.        zwishart_gen._process_parameterscCs^tj|td}|jdkr:|t|ddddtjf}|jdkr|dkrd|tjtjddf}nt|ddddtjf}n|jdkr|jd|jdkstdt |j|ddddtjf}nP|jdkr|jd|jdkstdt |jn|jdkrtd |j|jdd||fksZtd ||fd |jddd |S) rrGrNrr)zGQuantiles must be square if they are two dimensional, but x.shape = %s.zeQuantiles must be square in the first two dimensions if they are three dimensional, but x.shape = %s.znQuantiles must be at most two-dimensional with an additional dimension for multiplecomponents, but x.ndim = %dz2Quantiles have incompatible dimensions: should be z, got r) r8rOrIr,rrrrrUrrr*r*r/rs: "      zwishart_gen._process_quantilescCsVt|}|jdkr |tj}n|jdkr>tdtt||}t|}||fS)Nrrz_Size must be an integer or tuple of integers; thus must have dimension <= 1. Got size.ndim = %s)r8rOr,rrUrrprod)r`rrrr*r*r/ _process_sizes     zwishart_gen._process_sizec Cst|jd}t|j}t|jd} t|jdD]|} ||dddd| f\} || <tj|df|dddd| f|dddd| f<|dddd| f| | <q:d||d|d| d||t d||t d||} | S)aLog 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. r1NTrr) r8emptyrrange_cholesky_logdetrQrRZ cho_solvetrace_LOG_2r) r`rCrrr log_det_scaleC log_det_xZ scale_inv_xZtr_scale_inv_xirr.r*r*r/rs $6  zwishart_gen._logpdfcCsH|||\}}}|||}||\}}|||||||}t|S)aLog 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 rrrrr0r`rCrrrr!r r.r*r*r/rMs  zwishart_gen.logpdfcCst||||S)aWishart 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 rr`rCrrr*r*r/rjszwishart_gen.pdfcCs||S)aAMean 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. r*r`rrrr*r*r/_meanszwishart_gen._meancCs(|||\}}}||||}t|S)zMean of the Wishart distribution. Parameters ---------- %(_doc_default_callparams)s Returns ------- mean : float The mean of the distribution rr(r0r`rrrr.r*r*r/rs zwishart_gen.meancCs&||dkr||d|}nd}|S)aAMode 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. rNr*r`rrrr.r*r*r/_modes zwishart_gen._modecCs4|||\}}}||||}|dur0t|S|S)aEMode 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 Nrr,r0r*r*r*r/modeszwishart_gen.modecCs,|d}|}|t||7}||9}|S)aDVariance 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. r)diagonalr8rr`rrrrrr*r*r/_vars zwishart_gen._varcCs(|||\}}}||||}t|S)zVariance of the Wishart distribution. Parameters ---------- %(_doc_default_callparams)s Returns ------- var : float The variance of the distribution rr2r0r*r*r*r/rs zwishart_gen.varc s||dd}j|d||f}tjfddt|D|f|dddj}t|||f} ttdddgt |} tj |dd} || | | <t |} || | | <| S) ax 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. rr)rcs*g|]"}j|ddddqS)rrr) chisquarerBr#rrr{r*r/rE sz-wishart_gen._standard_rvs..Nr1k) rrr8r_rrlrrslicerV tril_indicesZ diag_indices) r`rrrrr{n_trilZ covariancesZ variancesAZsize_idxZtril_idxZdiag_idxr*r6r/ _standard_rvss(    zwishart_gen._standard_rvsc CsR||}||||||}t|D]&}t|||} t| | j||<q&|S)a]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. )r~r>r8ndindexrkrl) r`rrrrr!r{r=indexCAr*r*r/_rvs" s   zwishart_gen._rvsrc CsL||\}}|||\}}}tjj|dd}|||||||} t| S)aDraw 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 TrrrrQrRrrBr0 r`rrrr{rrrr!r.r*r*r/rO s zwishart_gen.rvscsjd|d|d||dttd|d|dtfddt|Dd|S)aCompute 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. rrcs$g|]}tdd|dqS)rr)rr5rr*r/rE rFz(wishart_gen._entropy..)rrr8r\r)r`rrr r*rEr/rn s  zwishart_gen._entropycCs.|||\}}}||\}}||||S)a'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 rrrr`rrrrr r*r*r/r szwishart_gen.entropycCs0tjj|dd}dtt|}||fS)aCompute 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. Trr))rQrRrr8r\r]r0)r`rZc_decompZlogdetr*r*r/r szwishart_gen._cholesky_logdet)N)NNN)rN)rorprqrrrfrrrrrrrr(rr,r.r2rr>rBrrrrrr*r*ryr/rTs*o %.4- rc@sTeZdZdZdddZddZddZd d Zd d Zd dZ dddZ ddZ dS)raXCreate 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. NcCs>t||_|j||\|_|_|_|j|j\|_|_dSrj) rrrrrrrr!r rr*r*r/rf s  zwishart_frozen.__init__cCs8|j||j}|j||j|j|j|j|j}t|Srj) rrrrrrr r!r0r`rCr.r*r*r/r s zwishart_frozen.logpdfcCst||Srjrrr*r*r/r szwishart_frozen.pdfcCs|j|j|j|j}t|Srjrr(rrrr0r`r.r*r*r/r szwishart_frozen.meancCs*|j|j|j|j}|dur&t|S|Srjrr,rrrr0rJr*r*r/r. szwishart_frozen.modecCs|j|j|j|j}t|Srjrr2rrrr0rJr*r*r/r szwishart_frozen.varrcCs4|j|\}}|j|||j|j|j|}t|SrjrrrBrrr!r0r`rr{rrr.r*r*r/r s zwishart_frozen.rvscCs|j|j|j|jSrjrrrrr rmr*r*r/r szwishart_frozen.entropy)N)rN) rorprqrrrfrrrr.rrrr*r*r*r/r s  r)rrrr.rrrcseZdZdZd$fdd Zd%ddZddZd d Zd d Zd dZ ddZ ddZ ddZ ddZ ddZddZddZd&ddZd d!Zd"d#ZZS)'invwishart_genacAn inverse Wishart random variable. The `df` keyword specifies the degrees of freedom. The `scale` keyword specifies the scale matrix, which must be symmetric and positive definite. In this context, the scale matrix is often interpreted in terms of a multivariate normal covariance matrix. Methods ------- pdf(x, df, scale) Probability density function. logpdf(x, df, scale) Log of the probability density function. rvs(df, scale, size=1, random_state=None) Draw random samples from an inverse Wishart distribution. entropy(df, scale) Differential entropy of the distribution. Parameters ---------- %(_doc_default_callparams)s %(_doc_random_state)s Raises ------ scipy.linalg.LinAlgError If the scale matrix `scale` is not positive definite. See Also -------- wishart Notes ----- %(_doc_callparams_note)s The scale matrix `scale` must be a symmetric positive definite matrix. Singular matrices, including the symmetric positive semi-definite case, are not supported. Symmetry is not checked; only the lower triangular portion is used. The inverse Wishart distribution is often denoted .. math:: W_p^{-1}(\nu, \Psi) where :math:`\nu` is the degrees of freedom and :math:`\Psi` is the :math:`p \times p` scale matrix. The probability density function for `invwishart` has support over positive definite matrices :math:`S`; if :math:`S \sim W^{-1}_p(\nu, \Sigma)`, then its PDF is given by: .. math:: f(S) = \frac{|\Sigma|^\frac{\nu}{2}}{2^{ \frac{\nu p}{2} } |S|^{\frac{\nu + p + 1}{2}} \Gamma_p \left(\frac{\nu}{2} \right)} \exp\left( -tr(\Sigma S^{-1}) / 2 \right) If :math:`S \sim W_p^{-1}(\nu, \Psi)` (inverse Wishart) then :math:`S^{-1} \sim W_p(\nu, \Psi^{-1})` (Wishart). If the scale matrix is 1-dimensional and equal to one, then the inverse Wishart distribution :math:`W_1(\nu, 1)` collapses to the inverse Gamma distribution with parameters shape = :math:`\frac{\nu}{2}` and scale = :math:`\frac{1}{2}`. Instead of inverting a randomly generated Wishart matrix as described in [2], here the algorithm in [4] is used to directly generate a random inverse-Wishart matrix without inversion. .. versionadded:: 0.16.0 References ---------- .. [1] M.L. Eaton, "Multivariate Statistics: A Vector Space Approach", Wiley, 1983. .. [2] M.C. Jones, "Generating Inverse Wishart Matrices", Communications in Statistics - Simulation and Computation, vol. 14.2, pp.511-514, 1985. .. [3] Gupta, M. and Srivastava, S. "Parametric Bayesian Estimation of Differential Entropy and Relative Entropy". Entropy 12, 818 - 843. 2010. .. [4] S.D. Axen, "Efficiently generating inverse-Wishart matrices and their Cholesky factors", :arXiv:`2310.15884v1`. 2023. Examples -------- >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.stats import invwishart, invgamma >>> x = np.linspace(0.01, 1, 100) >>> iw = invwishart.pdf(x, df=6, scale=1) >>> iw[:3] array([ 1.20546865e-15, 5.42497807e-06, 4.45813929e-03]) >>> ig = invgamma.pdf(x, 6/2., scale=1./2) >>> ig[:3] array([ 1.20546865e-15, 5.42497807e-06, 4.45813929e-03]) >>> plt.plot(x, iw) >>> plt.show() The input quantiles can be any shape of array, as long as the last axis labels the components. Alternatively, the object may be called (as a function) to fix the degrees of freedom and scale parameters, returning a "frozen" inverse Wishart random variable: >>> rv = invwishart(df=1, scale=1) >>> # Frozen object with the same methods but holding the given >>> # degrees of freedom and scale fixed. Ncs t|t|jt|_dSrjrrwryr*r/rfr s zinvwishart_gen.__init__cCs t|||S)znCreate a frozen inverse Wishart distribution. See `invwishart_frozen` for more information. )invwishart_frozenrr*r*r/rv szinvwishart_gen.__call__c Cst|jd}t|jd}td|f}|dkrt|jdD]N} ||dddd| f\} || <|d| |ddd} tj| d || <qBn2t|d |dd<|d d |d |dd<d ||d |d ||t d ||d|t d ||} | S) aLog of the inverse Wishart probability density function. Parameters ---------- x : ndarray Points at which to evaluate the log of the probability density function. dim : int Dimension of the scale matrix df : int Degrees of freedom log_det_scale : float Logarithm of the determinant of the scale matrix C : ndarray Cholesky factorization of the scale matrix, lower triagular. Notes ----- As this function does no argument checking, it should not be called directly; use 'logpdf' instead. r1trsmrNrrTsider7r)rrr) r8rrrrrrRrr]rr) r`rCrrr r!r"Ztr_scale_x_invrRr#ZCxr=r.r*r*r/r~ s  $" zinvwishart_gen._logpdfcCsF|||\}}}|||}||\}}||||||}t|S)aLog of the inverse Wishart probability density function. Parameters ---------- x : array_like Quantiles, with the last axis of `x` denoting the components. Each quantile must be a symmetric positive definite matrix. %(_doc_default_callparams)s Returns ------- pdf : ndarray Log of the probability density function evaluated at `x` Notes ----- %(_doc_callparams_note)s r$r%r*r*r/r s  zinvwishart_gen.logpdfcCst||||S)aInverse 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 rr&r*r*r/r szinvwishart_gen.pdfcCs&||dkr|||d}nd}|S)aIMean 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. rNr*r+r*r*r/r( s zinvwishart_gen._meancCs4|||\}}}||||}|dur0t|S|S)aXMean 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 Nr)r*r*r*r/r szinvwishart_gen.meancCs|||dS)aIMode 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. rr*r'r*r*r/r, szinvwishart_gen._modecCs(|||\}}}||||}t|S)zMode of the inverse Wishart distribution. Parameters ---------- %(_doc_default_callparams)s Returns ------- mode : float The Mode of the distribution r-r*r*r*r/r. s zinvwishart_gen.modecCsv||dkrn||d|d}|}|||dt||7}|||||dd||d}nd}|S)aLVariance 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. rrr)Nr/r1r*r*r/r2$ s *zinvwishart_gen._varcCs4|||\}}}||||}|dur0t|S|S)aXVariance 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 Nr3r*r*r*r/r< szinvwishart_gen.varc Cst|||f}tj|dd\}}||dd} |jg|| Rd|d||f<t|} ||d| } |j| g||Rdd|d| | f<|S) a Parameters ---------- n : integer Number of variates to generate shape : iterable Shape of the variates to generate dim : int Dimension of the scale matrix df : int Degrees of freedom random_state : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional If `seed` is None (or `np.random`), the `numpy.random.RandomState` singleton is used. If `seed` is an int, a new ``RandomState`` instance is used, seeded with `seed`. If `seed` is already a ``Generator`` or ``RandomState`` instance then that instance is used. Returns ------- A : ndarray Random variates of shape (`shape`) + (``dim``, ``dim``). Each slice `A[..., :, :]` is lower-triangular, and its inverse is the lower Cholesky factor of a draw from `invwishart(df, np.eye(dim))`. Notes ----- As this function does no argument checking, it should not be called directly; use 'rvs' instead. r1r7rr)r.)rrr)r8rr;raranger4) r`rrrrr{r=Ztri_rowsZtri_colsr<rZchi_dfsr*r*r/_inv_standard_rvsO s$  z invwishart_gen._inv_standard_rvsc Cs||}||||||}td|f}td|f} t|jddD]\} |dkr|d|| |ddd} | d| | dddd || <qH|d || d d || d <qH|S) aeDraw 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. rRtrmmNrrrTrS)rTr7Ztrans_arUr))r~rWrr8r?r) r`rrrrr!r{r=rRrXr@rAr*r*r/rB s   "zinvwishart_gen._rvsrc CsL||\}}|||\}}}tjj|dd}|||||||} t| S)aDraw 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 TrrCrDr*r*r/r s zinvwishart_gen.rvscsvfddtddD}t|}tdddd|tddt||dS)Ncsg|]}d|qS)rr*r5rrr*r/rE rFz+invwishart_gen._entropy..rrr-)rr8rOrrrr\)r`rrr Zpsi_eval_pointsr*rYr/r s zinvwishart_gen._entropycCs.|||\}}}||\}}||||SrjrFrGr*r*r/r szinvwishart_gen.entropy)N)NNN)rN)rorprqrrrfrrrrr(rr,r.r2rrWrBrrrrr*r*ryr/rP s"s +6+  rPc@sPeZdZdddZddZddZdd Zd d Zd d ZdddZ ddZ dS)rQNcCsXt||_|j||\|_|_|_tjj|jdd|_ dt t |j |_dS)aGCreate 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. Trr)N)rPrrrrrrQrRrr!r8r\r]r0r rr*r*r/rf s  zinvwishart_frozen.__init__cCs4|j||j}|j||j|j|j|j}t|Srj)rrrrrr r!r0rHr*r*r/r s zinvwishart_frozen.logpdfcCst||Srjrrr*r*r/r szinvwishart_frozen.pdfcCs*|j|j|j|j}|dur&t|S|SrjrIrJr*r*r/r szinvwishart_frozen.meancCs|j|j|j|j}t|SrjrKrJr*r*r/r. szinvwishart_frozen.modecCs*|j|j|j|j}|dur&t|S|SrjrLrJr*r*r/r szinvwishart_frozen.varrcCs4|j|\}}|j|||j|j|j|}t|SrjrMrNr*r*r/r s zinvwishart_frozen.rvscCs|j|j|j|jSrjrOrmr*r*r/r szinvwishart_frozen.entropy)N)rN) rorprqrfrrrr.rrrr*r*r*r/rQ s  rQ)rrrr.rrzsn : int Number of trials p : array_like Probability of a trial falling into each category; should sum to 1 a`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. cseZdZdZdfdd ZdddZddd Zd d Zd d ZddZ ddZ ddZ ddZ ddZ ddZdddZZS) multinomial_gena 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. Ncs t|t|jt|_dSrj)rurfrrrrmultinomial_docdict_paramsrwryr*r/rf s  zmultinomial_gen.__init__cCs t|||S)zjCreate a frozen multinomial distribution. See `multinomial_frozen` for more information. )multinomial_frozen)r`rrrxr*r*r/r szmultinomial_gen.__call__V瞯...jkr1)rr8rrrrreri)r`rrrjnnrdr#r*r*r/r: s $zmultinomial_gen.covcCs|||\}}}tjdt|d}|tjt|dd}|t|d8}|dtjf}t|j|j|jd}|j d|7_ tjt |||t|ddd|fd}| |||tj S)aCompute 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 rr1rg.)r)rr8r9r:r\rrrr,rrroreri)r`rrrjrCZterm1Znew_axes_neededZterm2r*r*r/rQ s zmultinomial_gen.entropycCs*|||\}}}||}||||S)aDraw 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 )rr~r)r`rrrr{rjr*r*r/rx s zmultinomial_gen.rvs)N)N)r])NN)rorprqrrrfrrrrergrmrorrrrrr*r*ryr/rZF sh    'rZc@sLeZdZdZdddZddZddZd d Zd d Zd dZ dddZ dS)r\aZCreate 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. Ncs<t|_j||\___fdd}|j_dS)NcsjjjfSrj)rrrj)rrrmr*r/r sz8multinomial_frozen.__init__.._process_parameters)rZrrrrrj)r`rrrxrr*rmr/rf s  zmultinomial_frozen.__init__cCs|j||j|jSrj)rrmrrrr*r*r/rm szmultinomial_frozen.logpmfcCs|j||j|jSrj)rrorrrr*r*r/ro szmultinomial_frozen.pmfcCs|j|j|jSrj)rrrrrmr*r*r/r szmultinomial_frozen.meancCs|j|j|jSrj)rrrrrmr*r*r/r szmultinomial_frozen.covcCs|j|j|jSrj)rrrrrmr*r*r/r szmultinomial_frozen.entropyrcCs|j|j|j||Srj)rrrrrr*r*r/r szmultinomial_frozen.rvs)N)rN) rorprqrrrfrmrorrrrr*r*r*r/r\ s r\)rmrorrrcs>eZdZdZd fdd Zd ddZddZdd d ZZS)special_ortho_group_gena 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 Ncst|t|j|_dSrjrurfrrrrrwryr*r/rfs z special_ortho_group_gen.__init__cCs t||dS)zlCreate a frozen SO(N) distribution. See `special_ortho_group_frozen` for more information. r)special_ortho_group_frozenr`rrxr*r*r/rsz special_ortho_group_gen.__call__cCs2|dus&t|r&|dks&|t|kr.td|S)5Dimension N must be specified; it cannot be inferred.NrzmDimension of rotation must be specified, and must be a scalar greater than 1.r8rrrUr`rr*r*r/r!s&z+special_ortho_group_gen._process_parametersrc Cs||}t|}|dkr |fnd}||}t|||f}t||dddddf<t||f}t|dD]}|j|||fd}|ddddf}|ddddf} t||  d} |d } t | dkt | d|d|f<|d|d|ft | 7<|t | | d |dd d d }|ddd|dft|ddd|df| |8<qxd |d|ddd fjd d |d<||ddddf9}|S)asDraw 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) rr*.Nr)rr1.rrr)@.Nr1rg).r1)r~rrr8rrrrmatmulr+rwheresignrXr) r`rrr{HDrrCZxrowZxcolZnorm2Zx0r*r*r/r)s(    "*>(zspecial_ortho_group_gen.rvs)N)NN)rN rorprqrrrfrrrrr*r*ryr/rq s I rqc@s eZdZdddZdddZdS) rsNcCst||_|j||_dS)aCreate 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() N)rqrrrrtr*r*r/rfqs z#special_ortho_group_frozen.__init__rcCs|j|j||Srjrrrrr*r*r/rszspecial_ortho_group_frozen.rvs)NN)rNrorprqrfrr*r*r*r/rsps rscs>eZdZdZd fdd Zd ddZddZdd d ZZS)ortho_group_genaAn 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 Ncst|t|j|_dSrjrrrwryr*r/rfs zortho_group_gen.__init__cCs t||dS)zcCreate a frozen O(N) distribution. See `ortho_group_frozen` for more information. r)ortho_group_frozenrtr*r*r/rszortho_group_gen.__call__cCs2|dus&t|r&|dks&|t|kr.td|SruNrzLDimension of rotation must be specified,and must be a scalar greater than 1.rvrwr*r*r/rs&z#ortho_group_gen._process_parametersrcCs||}t|}||}|dkr*|fnd}|j|||fd}tj|\}}|jdddd}||t|dtj d d f9}|S) arDraw 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) rr*rrrr1offsetZaxis1Zaxis2.N) r~rrrr8rRqrr0r;rr`rrr{zqrr4r*r*r/rs   zortho_group_gen.rvs)N)NN)rNrr*r*ryr/rs B rc@s eZdZdddZdddZdS) rNcCst||_|j||_dS)aCreate 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() N)rrrrrtr*r*r/rf s zortho_group_frozen.__init__rcCs|j|j||Srjrrr*r*r/r#szortho_group_frozen.rvs)NN)rNrr*r*r*r/r s rcsNeZdZdZdfdd ZdddZd d Zd d Zd dZdddZ Z S)random_correlation_gena 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]) Ncst|t|j|_dSrjrrrwryr*r/rfss zrandom_correlation_gen.__init__vIh%<=Hz>cCst||||dS)zrCreate a frozen random correlation matrix. See `random_correlation_frozen` for more information. )rxtoldiag_tol)random_correlation_frozen)r`eigsrxrrr*r*r/rwszrandom_correlation_gen.__call__cCstj|td}|j}|jdks4|jd|ks4|dkr= 1 and floating point issues do not occur. Otherwise, some other valid rotation is returned. When tr(M)==2, also bjj=1. rr)rrr))mathrXr:copysign) r`ZaiiZajjZaijZaiidZajjdddr>crcr*r*r/ _givens_to_1sz#random_correlation_gen._givens_to_1cCs>|jjr(|jtjkr(|jd|jdks.t|jd}t|dD]}|||fdkr\qDnb|||fdkrt|d|D]}|||fdkrzqqzn(t|d|D]}|||fdkrqq||||f|||f|||f\}}| }t |||| |||d||dddd t |||| |||||ddd qD|S)z 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 rrT)rZoffxZincxZoffyZincyZ overwrite_xZ overwrite_y) flags c_contiguousr5r8r_rrUrrZravelr )r`mr4r#jrrcmvr*r*r/_to_corrs4 *zrandom_correlation_gen._to_corrcCst|j||d\}}||}tj||d}tt|t||j}||}t | d |krpt d|S)aDraw 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. r)r{rz-Failed to generate a valid correlation matrix) rr~r!rr8rkrrlrr;r0r: RuntimeError)r`rr{rrrrr*r*r/rs  zrandom_correlation_gen.rvs)N)Nrr)Nrr) rorprqrrrfrrrrrrr*r*ryr/r'sK !)rc@s eZdZdddZd ddZdS) rNrrcCs2t||_||_||_|jj||jd\}|_dS)axCreate 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. rN)rrrrrr)r`rrxrrrr*r*r/rfs! z"random_correlation_frozen.__init__cCs|jj|j||j|jdS)N)r{rr)rrrrrr}r*r*r/r-s zrandom_correlation_frozen.rvs)Nrr)Nrr*r*r*r/rs &rcs>eZdZdZd fdd Zd ddZddZdd d ZZS)unitary_group_genaA 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, must be greater than 1. 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 Ncst|t|j|_dSrjrrrwryr*r/rfms zunitary_group_gen.__init__cCs t||dS)zCreate a frozen (U(N)) n-dimensional unitary matrix distribution. See `unitary_group_frozen` for more information. r)unitary_group_frozenrtr*r*r/rqszunitary_group_gen.__call__cCs2|dus&t|r&|dks&|t|kr.td|Srrvrwr*r*r/rxs&z%unitary_group_gen._process_parametersrcCs||}t|}||}|dkr*|fnd}dtd|j|||fdd|j|||fd}tj|\}}|j dddd }||t |d tj d d f9}|S) aiDraw 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) rr*r)ry?rrr1r.N) r~rrrrXrr8rRrr0r;rrr*r*r/rs   zunitary_group_gen.rvs)N)NN)rNrr*r*ryr/r2s : rc@s eZdZdddZdddZdS) rNcCst||_|j||_dS)aCreate 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() N)rrrrrtr*r*r/rfs zunitary_group_frozen.__init__rcCs|j|j||Srjrrr*r*r/rszunitary_group_frozen.rvs)NN)rNrr*r*r*r/rs raloc : 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``) aSetting 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_default_callparams_mvt_doc_callparams_notercseZdZdZdfdd ZdddZd d d Zd!d d Zd dZd"ddZ d#ddddddZ d$ddZ d%ddZ d&ddZ ddZddZZS)'multivariate_t_gena 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)) Ncs*t|t|jt|_t||_dS)zInitialize a multivariate t-distributed random variable. Parameters ---------- seed : Random state. N)rurfrrrrmvt_docdict_paramsr rvrwryr*r/rf:s zmultivariate_t_gen.__init__rFcCs,|tjkrt||||dSt|||||dS)zjCreate a frozen multivariate t-distribution. See `multivariate_t_frozen` for parameters. )rrrbrx)locrrrbrx)r8rrmultivariate_t_frozen)r`rrrrbrxr*r*r/rFs zmultivariate_t_gen.__call__c CsT||||\}}}}|||}t||d}||||j|j|||j}t|S)aZMultivariate 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 r) rrrLrr[r^rZr8r) r`rCrrrrbr shape_inforr*r*r/rSs  zmultivariate_t_gen.pdfc CsF||||\}}}}|||}t|}||||j|j|||jS)aLog 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. )rrrLrr[r^rZ)r`rCrrrrrr*r*r/rrs  zmultivariate_t_gen.logpdfcCs|tjkrt|||||S||}tt||jdd} d||} t| } td|} |dt|tj } d|}| tdd|| }t | | | ||S)aBUtility 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. r1rgrryrr) r8rrrrrkr\rr]pir0)r`rCrZprec_Ur^rrrZrrr>r=Br!rEr*r*r/rs   zmultivariate_t_gen._logpdfc s|durt|n|js$d||||}|durJt|jtj n|}||||}}t||\} } | | k} d| jdd} | | } } | | | | | | <| | <|jdtj | | fdd} fdd}t |d| | }t |S)Nir1rgcs0|d|d}}t||dSrr)rrrrrrrngrr*r/rsz'multivariate_t_gen._cdf..func1d) r rvrr8rrrrr\rrrr0)r`rCrrrrrrr{rrrrrrresr*rr/rs*   zmultivariate_t_gen._cdfrrr{c Cs<||||\} }}}t||dj}|||||| |||S)aMultivariate 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 r)rrLrPr) r`rCrrrrbrrr{rr*r*r/rs !zmultivariate_t_gen.cdfcsv|tjkrtd|dSt|}d|jfdd}fdd}|ddt|d }t||k||f||d S) N)rrcsVd||}d|}t| t|d|t|tj|t|t|S)Nr)rr8r]rr)rrZhalfsumZhalf_dfZ shape_termr*r/regulars z,multivariate_t_gen._entropy..regularcs|t||||d|dd|d|d|dd|d|dd|dd|d d |dd|dd |dd |d ddS)Nr)grg gr*)rr)rrrr*r/ asymptotics*.z/multivariate_t_gen._entropy..asymptoticdrr)r3f2)r8rrrrLr^r]r )r`rrrrrr thresholdr*rr/rs   zmultivariate_t_gen._entropycCs$|d||\}}}}||||S)zCalculate the differential entropy of a multivariate t-distribution. Parameters ---------- %(_mvt_doc_default_callparams)s Returns ------- h : float Differential entropy Nrr)r`rrrrr*r*r/r0szmultivariate_t_gen.entropyc Cs||||\}}}}|dur(t|}n|j}t|rDt|}n|j||d|}|jt|||d} || t |d} t | S)aDraw random samples from a multivariate t-distribution. Parameters ---------- %(_mvt_doc_default_callparams)s size : integer, optional Number of samples to draw (default 1). %(_doc_random_state)s Returns ------- rvs : ndarray or scalar Random variates of size (`size`, `P`), where `P` is the dimension of the random variable. Examples -------- >>> from scipy.stats import multivariate_t >>> x = [0.4, 5] >>> loc = [0, 1] >>> shape = [[1, 0.1], [0.1, 1]] >>> df = 7 >>> multivariate_t.rvs(loc, shape, df) array([[0.93477495, 3.00408716]]) Nrrz) rr rvr8isinfZonesr4rrrXr0) r`rrrrr{rrrCrsamplesr*r*r/rAs    zmultivariate_t_gen.rvscCs`tj|td}|jdkr$|tj}n8|jdkr\|dkrJ|ddtjf}n|tjddf}|Srrrr*r*r/rps   z%multivariate_t_gen._process_quantilescCs|dur2|dur2tjdtd}tjdtd}d}n|durntj|td}|jdkrXd}n |jd}t|}nJ|durtj|td}|j}t|}n"tj|td}tj|td}|j}|dkr|d}|dd}|jdks|jd|krt d||jdkr|t|}n|jdkr.t |}n~|jdkr|j||fkr|j\}}||krndt |j}nd}|t |jt |f}t |n|jdkrt d |j|durd}n(|dkrt d nt |rt d ||||fS) z Infer dimensionality from location array and shape matrix, handle defaults, and ensure compatible dimensions. NrrGrr)z*Array 'loc' must be a vector of length %d.rzSDimension mismatch: array 'cov' is of shape %s, but 'loc' is a vector of length %d.rz'df' must be greater than zero.z8'df' is 'nan' but must be greater than zero or 'np.inf'.)r8rOrIr,rrrrrrUrrrVisnan)r`rrrrrrrer*r*r/rs`                 z&multivariate_t_gen._process_parameters)N)NrrFN)NrrF)Nrr)NNN)NrrF)rr)Nrr)NrrrN)rorprqrrrfrrrrrrrrrrrrr*r*ryr/rs&R   #+ ' ' "  /rc@sJeZdZdddZddZddddd d Zd d Zdd dZddZdS)rNrFcCsPt||_|j|||\}}}}||||f\|_|_|_|_t||d|_dS)aCreate a frozen multivariate t distribution. Parameters ---------- %(_mvt_doc_default_callparams)s Examples -------- >>> import numpy as np >>> from scipy.stats import multivariate_t >>> loc = np.zeros(3) >>> shape = np.eye(3) >>> df = 10 >>> dist = multivariate_t(loc, shape, df) >>> dist.rvs() array([[ 0.81412036, -1.53612361, 0.42199647]]) >>> dist.pdf([1, 1, 1]) array([0.01237803]) rN) rrrrrrrrLr)r`rrrrbrxrr*r*r/rfs zmultivariate_t_frozen.__init__c CsB|j||j}|jj}|jj}|j||j|||j|j|jj Srj) rrrrr[r^rrrrZ)r`rCr[r^r*r*r/rs zmultivariate_t_frozen.logpdfrc Cs2|j||j}|j||j|j|j|j|||Srj)rrrrrrr)r`rCrrr{r*r*r/rszmultivariate_t_frozen.cdfcCst||Srjrrr*r*r/rszmultivariate_t_frozen.pdfcCs|jj|j|j|j||dS)N)rrrrr{)rrrrrrr*r*r/rs  zmultivariate_t_frozen.rvscCs|j|j|j|jSrj)rrrrrrmr*r*r/rszmultivariate_t_frozen.entropy)NrrFN)rN) rorprqrfrrrrrr*r*r*r/rs  r)rrrrrzm : 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. a`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. cs~eZdZdZdfdd ZdddZddZd d Zd d Zd dZ ddZ ddZ ddZ ddZ ddZdddZZS)multivariate_hypergeom_genaA multivariate hypergeometric random variable. Methods ------- pmf(x, m, n) Probability mass function. logpmf(x, m, n) Log of the probability mass function. rvs(m, n, size=1, random_state=None) Draw random samples from a multivariate hypergeometric distribution. mean(m, n) Mean of the multivariate hypergeometric distribution. var(m, n) Variance of the multivariate hypergeometric distribution. cov(m, n) Compute the covariance matrix of the multivariate hypergeometric distribution. Parameters ---------- %(_doc_default_callparams)s %(_doc_random_state)s Notes ----- %(_doc_callparams_note)s The probability mass function for `multivariate_hypergeom` is .. math:: P(X_1 = x_1, X_2 = x_2, \ldots, X_k = x_k) = \frac{\binom{m_1}{x_1} \binom{m_2}{x_2} \cdots \binom{m_k}{x_k}}{\binom{M}{n}}, \\ \quad (x_1, x_2, \ldots, x_k) \in \mathbb{N}^k \text{ with } \sum_{i=1}^k x_i = n where :math:`m_i` are the number of objects of type :math:`i`, :math:`M` is the total number of objects in the population (sum of all the :math:`m_i`), and :math:`n` is the size of the sample to be taken from the population. .. versionadded:: 1.6.0 Examples -------- To evaluate the probability mass function of the multivariate hypergeometric distribution, with a dichotomous population of size :math:`10` and :math:`20`, at a sample of size :math:`12` with :math:`8` objects of the first type and :math:`4` objects of the second type, use: >>> from scipy.stats import multivariate_hypergeom >>> multivariate_hypergeom.pmf(x=[8, 4], m=[10, 20], n=12) 0.0025207176631464523 The `multivariate_hypergeom` distribution is identical to the corresponding `hypergeom` distribution (tiny numerical differences notwithstanding) when only two types (good and bad) of objects are present in the population as in the example above. Consider another example for a comparison with the hypergeometric distribution: >>> from scipy.stats import hypergeom >>> multivariate_hypergeom.pmf(x=[3, 1], m=[10, 5], n=4) 0.4395604395604395 >>> hypergeom.pmf(k=3, M=15, n=4, N=10) 0.43956043956044005 The functions ``pmf``, ``logpmf``, ``mean``, ``var``, ``cov``, and ``rvs`` support broadcasting, under the convention that the vector parameters (``x``, ``m``, and ``n``) are interpreted as if each row along the last axis is a single object. For instance, we can combine the previous two calls to `multivariate_hypergeom` as >>> multivariate_hypergeom.pmf(x=[[8, 4], [3, 1]], m=[[10, 20], [10, 5]], ... n=[12, 4]) array([0.00252072, 0.43956044]) This broadcasting also works for ``cov``, where the output objects are square matrices of size ``m.shape[-1]``. For example: >>> multivariate_hypergeom.cov(m=[[7, 9], [10, 15]], n=[8, 12]) array([[[ 1.05, -1.05], [-1.05, 1.05]], [[ 1.56, -1.56], [-1.56, 1.56]]]) That is, ``result[0]`` is equal to ``multivariate_hypergeom.cov(m=[7, 9], n=8)`` and ``result[1]`` is equal to ``multivariate_hypergeom.cov(m=[10, 15], n=12)``. Alternatively, the object may be called (as a function) to fix the `m` and `n` parameters, returning a "frozen" multivariate hypergeometric random variable. >>> rv = multivariate_hypergeom(m=[10, 20], n=12) >>> rv.pmf(x=[8, 4]) 0.0025207176631464523 See Also -------- scipy.stats.hypergeom : The hypergeometric distribution. scipy.stats.multinomial : The multinomial distribution. References ---------- .. [1] The Multivariate Hypergeometric Distribution, http://www.randomservices.org/random/urn/MultiHypergeometric.html .. [2] Thomas J. Sargent and John Stachurski, 2020, Multivariate Hypergeometric Distribution https://python.quantecon.org/multi_hyper.html Ncs t|t|jt|_dSrj)rurfrrrrmhg_docdict_paramsrwryr*r/rfs z#multivariate_hypergeom_gen.__init__cCst|||dS)zCreate a frozen multivariate_hypergeom distribution. See `multivariate_hypergeom_frozen` for more information. r)multivariate_hypergeom_frozen)r`rrrxr*r*r/rsz#multivariate_hypergeom_gen.__call__c Cst|}t|}|jdkr(|t}|jdkr<|t}t|jtjsTtdt|jtjsltd|j dkr~t d|jdkr|dtj f}t ||\}}|jdkr|d}|dk}|j dd}|dk||kB}|||||tj|dd|BfS) Nrz'm' must an array of integers.z'n' must an array of integers.z1'm' must be an array with at least one dimension..rxr1rg)r8rOrastyper issubdtyper5integer TypeErrorr,rUrrr\r)r`rrmcondrar`r*r*r/rs*          z.multivariate_hypergeom_gen._process_parametersc Cst|}t|jtjs"td|jdkr4td|jd|jdksjtd|jdd|jdd|j dkr|dtj f}|dtj f}t ||||\}}}}|j dkr|d |d }}|dk||kB}|||||tj |dd |j dd |kBfS) Nz'x' must an array of integers.rz1'x' must be an array with at least one dimension.r1z4Size of each quantile must be size of 'm': received z, but expected r.rxrg)r8rOrr5rrr,rUrrrrrr\)r`rCrarrrkr*r*r/rs*      z-multivariate_hypergeom_gen._process_quantilescCs<t|}|jdkr|||<n|r&|S|jdkr8|dS|S)Nrr*rbrcr*r*r/res    z'multivariate_hypergeom_gen._checkresultc Cstj|tjd}tj|tjd}||||}}||||}}t|ddt|d||d||<t|ddt|d||d||<tj||<tj||<|jdd}||S)NrGrr1rg)r8Z zeros_liker_rrir\) r`rCrarrmxcondr`numZdenr*r*r/rgs**   z"multivariate_hypergeom_gen._logpmfcCs|||\}}}}}}|||||\}}}}}} ||B} |tj|jtjdB}|||||| |} | tj|jtjdB} || | tj } |tj| jtjdB} || | tj S)aLog 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 rG) rrr8rrrhrgrerri)r`rCrrrarr`mncondrkZ xcond_reducedrrdrlZmncond_r*r*r/rms z!multivariate_hypergeom_gen.logpmfcCst||||}|S)aMultivariate 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 rn)r`rCrrr.r*r*r/roszmultivariate_hypergeom_gen.pmfcCs|||\}}}}}}|jdkr@|dtjf|dtjf}}|dk}tjj||d}|||}|jdkr|dtjftj|jtjdB}| ||tj S)zMean of the multivariate hypergeometric distribution. Parameters ---------- %(_doc_default_callparams)s Returns ------- mean : array_like or scalar The mean of the distribution r.maskrG rrr8rma masked_arrayrrrhreri)r`rrrarrr<mur*r*r/r-s     zmultivariate_hypergeom_gen.meancCs|||\}}}}}}|jdkr@|dtjf|dtjf}}|dk|ddk@}tjj||d}|||||||||d}|jdkr|dtjftj|jtjdB}| ||tj S)aQVariance 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 r.rrrGr)r`rrrarrr<outputr*r*r/rEs  (  zmultivariate_hypergeom_gen.varc Cs|||\}}}}}}|jdkrF|dtjtjf}|dtjtjf}|dk|ddk@}tjj||d}| |||dtd|||d}|jdkr|d|d}}|d}|jd}t|D]v} ||||d| f||d| f|d| | f<|d| | f|d|d| | f<|d| | f|d|d| | f<q|jdkrt|dtjtjftj |jtj d B}| ||tj S) aCovariance matrix of the multivariate hypergeometric distribution. Parameters ---------- %(_doc_default_callparams)s Returns ------- cov : array_like The covariance matrix of the distribution r.rrz...i,...j->...ijr)).rrr1rG) rrr8rrrrrrrrhreri) r`rrrarrr<rrr#r*r*r/r^s0     2 " zmultivariate_hypergeom_gen.covc Cs|||\}}}}}}||}|dur:t|tr:|f}|durVtj|j|jd}ntj||jdf|jd}|}t|jddD]\} ||d| f}|dk} | |j |d| f|| || |d|d| f<||d| f}q||d|jddf<|S)aDraw 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. NrGr1r.rr) rr~rrr8rrr5rZhypergeometric) r`rrrr{rarrremrZn0maskr*r*r/rs*  zmultivariate_hypergeom_gen.rvs)N)N)NN)rorprqrrrfrrrrergrmrorrrrrr*r*ryr/r%sp   $&rc@sHeZdZdddZddZddZdd Zd d Zd d ZdddZ dS)rNcsHt|_j||\______fdd}|j_dS)NcsjjjjjjfSrj)rarrrr`r)rrrmr*r/rs zCmultivariate_hypergeom_frozen.__init__.._process_parameters) rrrrarrrr`r)r`rrrxrr*rmr/rfs   z&multivariate_hypergeom_frozen.__init__cCs|j||j|jSrj)rrmrrrr*r*r/rmsz$multivariate_hypergeom_frozen.logpmfcCs|j||j|jSrj)rrorrrr*r*r/rosz!multivariate_hypergeom_frozen.pmfcCs|j|j|jSrj)rrrrrmr*r*r/rsz"multivariate_hypergeom_frozen.meancCs|j|j|jSrj)rrrrrmr*r*r/rsz!multivariate_hypergeom_frozen.varcCs|j|j|jSrj)rrrrrmr*r*r/rsz!multivariate_hypergeom_frozen.covrcCs|jj|j|j||dS)N)rr{)rrrrrr*r*r/rsz!multivariate_hypergeom_frozen.rvs)N)rN) rorprqrfrmrorrrrr*r*r*r/rs r)rmrorrrrcseZdZdZdfdd ZddddZdd Zd d Zd d ZddddddZ e ddZ e ddZ e ddZe ddZe ddZe ddZZS)random_table_gena 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 Ncst|dSrjrurfrwryr*r/rf-szrandom_table_gen.__init__rcCst|||dS)z~Create a frozen distribution of tables with given marginals. See `random_table_frozen` for more information. r)random_table_frozen)r`rowcolrxr*r*r/r0szrandom_table_gen.__call__c Cs|||\}}}t|}|jdkr.tdt|jtj}tjdd0|slt | t |ksltdWdn1s0Yt |dkrtdtj |d d }tj |d d } |jd t|krtd | jd t|krtd t|jdd } tj ||kd d tj | |kd d @} dd} tj | |d d tj | |d d | |tj | || dd | | <tj | | <| dS)aLog-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)z$`x` must be at least two-dimensionalignore)invalidz%`x` must contain only integral valuesNrz)`x` must contain only non-negative valuesr1rgrz"shape of `x` must agree with `row`z"shape of `x` must agree with `col`cSs t|dS)Nr)r)rCr*r*r/lnfacysz&random_table_gen.logpmf..lnfac)r1rr*)rr8rOr,rUrr5rZerrstateallrrrr\rrVrr) r`rCrrrrrZ dtype_is_intr2c2rrrr*r*r/rm7s4&  &$"zrandom_table_gen.logpmfcCst||||S)aProbability 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 rn)r`rCrrr*r*r/ros&zrandom_table_gen.pmfcCs"|||\}}}t|||S)aAMean 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]]) )rr8r)r`rrrrrr*r*r/rszrandom_table_gen.meanrmethodr{c CsT|||\}}}||||\}} ||}|||||} | |||||| S)agDraw 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_process_size_shaper~_process_rvs_methodr) r`rrrrr{rrrrmethr*r*r/rs ( zrandom_table_gen.rvscCstj|tjdd}tj|tjdd}t|dkr:tdt|dkrPtdt|dkrftdt|dkr|tdt|}|t|krtd t|t|kstd t|t|kstd |||fS) z 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. Tr^rz`row` must be one-dimensionalz`col` must be one-dimensionalrz*each element of `row` must be non-negativez*each element of `col` must be non-negativez'sums over `row` and `col` must be equalz(each element of `row` must be an integerz(each element of `col` must be an integer) r8rHZint64r,rUrr\rrO)rrrrrr*r*r/rs$ z$random_table_gen._process_parameterscCsft|t|f}|dur d|fSt|}t|jtjrHt|dkrPtdt|t ||fS)zW Compute the number of samples to be drawn and the shape of the output Nrrz/`size` must be a non-negative integer or `None`) rVr8rrr5rrrUrr)rrrrr*r*r/rs z$random_table_gen._process_size_shapec CsT|||||j|jd}z ||WStyNtd|dt|Yn0dS)N)NZboyettZ patefield'z!' not recognized, must be one of ) _rvs_select _rvs_boyett_rvs_patefieldKeyErrorrUset)clsrrrrZ known_methodsr*r*r/r+s    z$random_table_gen._process_rvs_methodcCs:d}t|t|}||t|d|kr4|jS|jS)Nrr)rVr8r]rr)rrrrZfacr8r*r*r/r8s zrandom_table_gen._rvs_selectcCst|||||Srj)rZ rvs_rcont1rrZntotrr{r*r*r/rAszrandom_table_gen._rvs_boyettcCst|||||Srj)rZ rvs_rcont2rr*r*r/rEszrandom_table_gen._rvs_patefield)N)rorprqrrrfrrmrorr staticmethodrr classmethodrrrrrr*r*ryr/rs&BK("0     rc@s<eZdZddddZddZddZd d Zd d d ZdS)rNrcs2t|_j||_fdd}|j_dS)NcsjSrj)_params)rrrmr*r/rSsz9random_table_frozen.__init__.._process_parameters)rrrr)r`rrrxrr*rmr/rfNs  zrandom_table_frozen.__init__cCs|j|ddSrj)rrmrr*r*r/rmWszrandom_table_frozen.logpmfcCs|j|ddSrj)rrorr*r*r/roZszrandom_table_frozen.pmfcCs|jddSrj)rrrmr*r*r/r]szrandom_table_frozen.meancCs|jjdd|||dS)Nr)rr)r`rrr{r*r*r/r`szrandom_table_frozen.rvs)NNN)rorprqrfrmrorrr*r*r*r/rMs  rzprow : array_like Sum of table entries in each row. col : array_like Sum of table entries in each column.zx : array-like Two-dimensional table of non-negative integers, or a multi-dimensional array with the last two dimensions corresponding with the tables.zThe 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.z Parameters ---------- )r _doc_row_colZ_doc_x_doc_mean_params_doc_row_col_note)rrrcCst|p |j||_dSrj)rrrr)objZdocdicttemplater*r*r/_docfillsr)rmrorrcs>eZdZdZd fdd Zd ddZddZd d d ZZS)uniform_direction_genaA 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]]) Ncst|t|j|_dSrjrrrwryr*r/rfs zuniform_direction_gen.__init__cCs t||dS)z}Create a frozen n-dimensional uniform direction distribution. See `uniform_direction` for more information. r)uniform_direction_frozenrtr*r*r/rszuniform_direction_gen.__call__cCs6|dus&t|r&|dks&|t|kr.tdt|S)ruNrzMDimension of vector must be specified, and must be an integer greater than 0.rvrwr*r*r/rs&z)uniform_direction_gen._process_parameterscCsD||}|dur tjgtd}t|}||}t|||}|S)a{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 NrG)r~r8rHrrr_sample_uniform_direction)r`rrr{rr*r*r/rs    zuniform_direction_gen.rvs)N)NN)NNrr*r*ryr/rs A rc@s eZdZdddZdddZdS)rNcCst||_|j||_dS)aCreate 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() N)rrrrrtr*r*r/rf s z!uniform_direction_frozen.__init__cCs|j|j||Srjrrr*r*r/r;szuniform_direction_frozen.rvs)NN)NNrr*r*r*r/rs rcCs0t||}||}|tjj|ddd}|S)z 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. r1T)rhZkeepdims)r8rrrRr)rrr{Z samples_shaperr*r*r/r?s  ra"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_default_callparamsrc Cst|}t|}|durzt||\}}Wn2tyb}zd}t||WYd}~n d}~00t|}t|dkst||krtd|}t|dkrtdt|}t|dkst||krtd|}tj|dd}t||\}}|dur|||fS||||fS)Nz&`x` and `alpha` must be broadcastable.rz,`x` must contain only non-negative integers.z*`alpha` must contain only positive values.z`n` must be a positive integer.r1rg)r8rOrrUfloorrr\)rrrCrreZx_intZn_intZ sum_alphar*r*r/'_dirichlet_multinomial_check_parametersds*     rcsTeZdZdZdfdd ZdddZddZd d Zd d Zd dZ ddZ Z S)dirichlet_multinomial_gena| 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) Ncs t|t|jt|_dSrj)rurfrrrrdirichlet_mn_docdict_paramsrwryr*r/rfs z"dirichlet_multinomial_gen.__init__cCst|||dSr)dirichlet_multinomial_frozen)r`rrrxr*r*r/rsz"dirichlet_multinomial_gen.__call__cCst|||\}}}}tt|t|dt||}|t||t|t|djdd7}t|||jddktj |dS)aThe 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. rr1rgr*)rr8rOr r\Zplacer)r`rCrrrSar.r*r*r/rms &,z dirichlet_multinomial_gen.logpmfcCst||||S)aProbability 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. rn)r`rCrrr*r*r/roszdirichlet_multinomial_gen.pmfcCs:t||\}}}|dtjf|dtjf}}|||S)zMean of a Dirichlet multinomial distribution. Parameters ---------- %(_dirichlet_mn_doc_default_callparams)s Returns ------- out: ndarray Mean of a Dirichlet multinomial distribution. .rr8rr`rrrrr*r*r/r$s zdirichlet_multinomial_gen.meancCsVt||\}}}|dtjf|dtjf}}|||d||||d|S)abThe variance of the Dirichlet multinomial distribution. Parameters ---------- %(_dirichlet_mn_doc_default_callparams)s Returns ------- out: array_like The variances of the components of the distribution. This is the diagonal of the covariance matrix of the distribution. .rrrr*r*r/r5szdirichlet_multinomial_gen.varc Cst||\}}}t||}|dtjtjf|dtjtjf}}|dddtjf|dtjddf}| ||d||d|}t|jd}||d||f<|S)a Covariance matrix of a Dirichlet multinomial distribution. Parameters ---------- %(_dirichlet_mn_doc_default_callparams)s Returns ------- out : array_like The covariance matrix of the distribution. .Nr)rr1)rrrr8rrVr) r`rrrrrZaiajriir*r*r/rGs  &("zdirichlet_multinomial_gen.cov)N)N) rorprqrrrfrrmrorrrrr*r*ryr/rsj rc@s>eZdZdddZddZddZdd Zd d Zd d ZdS)rNcCs*t||\}}}||_||_t||_dSrj)rrrrr)r`rrrxrr*r*r/rfdsz%dirichlet_multinomial_frozen.__init__cCs|j||j|jSrj)rrmrrrr*r*r/rmjsz#dirichlet_multinomial_frozen.logpmfcCs|j||j|jSrj)rrorrrr*r*r/romsz dirichlet_multinomial_frozen.pmfcCs|j|j|jSrj)rrrrrmr*r*r/rpsz!dirichlet_multinomial_frozen.meancCs|j|j|jSrj)rrrrrmr*r*r/rssz dirichlet_multinomial_frozen.varcCs|j|j|jSrj)rrrrrmr*r*r/rvsz dirichlet_multinomial_frozen.cov)N) rorprqrfrmrorrrr*r*r*r/rcs  r)rmrorrrcseZdZdZd&fdd Zd'ddZdd Zd d Zd d ZddZ d(ddZ d)ddZ ddZ ddZ ddZddZddZd*ddZd d!Zd+d"d#Zd$d%ZZS),vonmises_fisher_gena$A von Mises-Fisher variable. The `mu` keyword specifies the mean direction vector. The `kappa` keyword specifies the concentration parameter. Methods ------- pdf(x, mu=None, kappa=1) Probability density function. logpdf(x, mu=None, kappa=1) Log of the probability density function. rvs(mu=None, kappa=1, size=1, random_state=None) Draw random samples from a von Mises-Fisher distribution. entropy(mu=None, kappa=1) Compute the differential entropy of the von Mises-Fisher distribution. fit(data) Fit a von Mises-Fisher distribution to data. Parameters ---------- mu : array_like Mean direction of the distribution. Must be a one-dimensional unit vector of norm 1. kappa : float Concentration parameter. Must be positive. seed : {None, int, np.random.RandomState, np.random.Generator}, optional Used for drawing random variates. If `seed` is `None`, the `~np.random.RandomState` singleton is used. If `seed` is an int, a new ``RandomState`` instance is used, seeded with seed. If `seed` is already a ``RandomState`` or ``Generator`` instance, then that object is used. Default is `None`. See Also -------- scipy.stats.vonmises : Von-Mises Fisher distribution in 2D on a circle uniform_direction : uniform distribution on the surface of a hypersphere Notes ----- The von Mises-Fisher distribution is a directional distribution on the surface of the unit hypersphere. The probability density function of a unit vector :math:`\mathbf{x}` is .. math:: f(\mathbf{x}) = \frac{\kappa^{d/2-1}}{(2\pi)^{d/2}I_{d/2-1}(\kappa)} \exp\left(\kappa \mathbf{\mu}^T\mathbf{x}\right), where :math:`\mathbf{\mu}` is the mean direction, :math:`\kappa` the concentration parameter, :math:`d` the dimension and :math:`I` the modified Bessel function of the first kind. As :math:`\mu` represents a direction, it must be a unit vector or in other words, a point on the hypersphere: :math:`\mathbf{\mu}\in S^{d-1}`. :math:`\kappa` is a concentration parameter, which means that it must be positive (:math:`\kappa>0`) and that the distribution becomes more narrow with increasing :math:`\kappa`. In that sense, the reciprocal value :math:`1/\kappa` resembles the variance parameter of the normal distribution. The von Mises-Fisher distribution often serves as an analogue of the normal distribution on the sphere. Intuitively, for unit vectors, a useful distance measure is given by the angle :math:`\alpha` between them. This is exactly what the scalar product :math:`\mathbf{\mu}^T\mathbf{x}=\cos(\alpha)` in the von Mises-Fisher probability density function describes: the angle between the mean direction :math:`\mathbf{\mu}` and the vector :math:`\mathbf{x}`. The larger the angle between them, the smaller the probability to observe :math:`\mathbf{x}` for this particular mean direction :math:`\mathbf{\mu}`. In dimensions 2 and 3, specialized algorithms are used for fast sampling [2]_, [3]_. For dimensions of 4 or higher the rejection sampling algorithm described in [4]_ is utilized. This implementation is partially based on the geomstats package [5]_, [6]_. .. versionadded:: 1.11 References ---------- .. [1] Von Mises-Fisher distribution, Wikipedia, https://en.wikipedia.org/wiki/Von_Mises%E2%80%93Fisher_distribution .. [2] Mardia, K., and Jupp, P. Directional statistics. Wiley, 2000. .. [3] J. Wenzel. Numerically stable sampling of the von Mises Fisher distribution on S2. https://www.mitsuba-renderer.org/~wenzel/files/vmf.pdf .. [4] Wood, A. Simulation of the von mises fisher distribution. Communications in statistics-simulation and computation 23, 1 (1994), 157-164. https://doi.org/10.1080/03610919408813161 .. [5] geomstats, Github. MIT License. Accessed: 06.01.2023. https://github.com/geomstats/geomstats .. [6] Miolane, N. et al. Geomstats: A Python Package for Riemannian Geometry in Machine Learning. Journal of Machine Learning Research 21 (2020). http://jmlr.org/papers/v21/19-027.html Examples -------- **Visualization of the probability density** Plot the probability density in three dimensions for increasing concentration parameter. The density is calculated by the ``pdf`` method. >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.stats import vonmises_fisher >>> from matplotlib.colors import Normalize >>> n_grid = 100 >>> u = np.linspace(0, np.pi, n_grid) >>> v = np.linspace(0, 2 * np.pi, n_grid) >>> u_grid, v_grid = np.meshgrid(u, v) >>> vertices = np.stack([np.cos(v_grid) * np.sin(u_grid), ... np.sin(v_grid) * np.sin(u_grid), ... np.cos(u_grid)], ... axis=2) >>> x = np.outer(np.cos(v), np.sin(u)) >>> y = np.outer(np.sin(v), np.sin(u)) >>> z = np.outer(np.ones_like(u), np.cos(u)) >>> def plot_vmf_density(ax, x, y, z, vertices, mu, kappa): ... vmf = vonmises_fisher(mu, kappa) ... pdf_values = vmf.pdf(vertices) ... pdfnorm = Normalize(vmin=pdf_values.min(), vmax=pdf_values.max()) ... ax.plot_surface(x, y, z, rstride=1, cstride=1, ... facecolors=plt.cm.viridis(pdfnorm(pdf_values)), ... linewidth=0) ... ax.set_aspect('equal') ... ax.view_init(azim=-130, elev=0) ... ax.axis('off') ... ax.set_title(rf"$\kappa={kappa}$") >>> fig, axes = plt.subplots(nrows=1, ncols=3, figsize=(9, 4), ... subplot_kw={"projection": "3d"}) >>> left, middle, right = axes >>> mu = np.array([-np.sqrt(0.5), -np.sqrt(0.5), 0]) >>> plot_vmf_density(left, x, y, z, vertices, mu, 5) >>> plot_vmf_density(middle, x, y, z, vertices, mu, 20) >>> plot_vmf_density(right, x, y, z, vertices, mu, 100) >>> plt.subplots_adjust(top=1, bottom=0.0, left=0.0, right=1.0, wspace=0.) >>> plt.show() As we increase the concentration parameter, the points are getting more clustered together around the mean direction. **Sampling** Draw 5 samples from the distribution using the ``rvs`` method resulting in a 5x3 array. >>> rng = np.random.default_rng() >>> mu = np.array([0, 0, 1]) >>> samples = vonmises_fisher(mu, 20).rvs(5, random_state=rng) >>> samples array([[ 0.3884594 , -0.32482588, 0.86231516], [ 0.00611366, -0.09878289, 0.99509023], [-0.04154772, -0.01637135, 0.99900239], [-0.14613735, 0.12553507, 0.98126695], [-0.04429884, -0.23474054, 0.97104814]]) These samples are unit vectors on the sphere :math:`S^2`. To verify, let us calculate their euclidean norms: >>> np.linalg.norm(samples, axis=1) array([1., 1., 1., 1., 1.]) Plot 20 observations drawn from the von Mises-Fisher distribution for increasing concentration parameter :math:`\kappa`. The red dot highlights the mean direction :math:`\mu`. >>> def plot_vmf_samples(ax, x, y, z, mu, kappa): ... vmf = vonmises_fisher(mu, kappa) ... samples = vmf.rvs(20) ... ax.plot_surface(x, y, z, rstride=1, cstride=1, linewidth=0, ... alpha=0.2) ... ax.scatter(samples[:, 0], samples[:, 1], samples[:, 2], c='k', s=5) ... ax.scatter(mu[0], mu[1], mu[2], c='r', s=30) ... ax.set_aspect('equal') ... ax.view_init(azim=-130, elev=0) ... ax.axis('off') ... ax.set_title(rf"$\kappa={kappa}$") >>> mu = np.array([-np.sqrt(0.5), -np.sqrt(0.5), 0]) >>> fig, axes = plt.subplots(nrows=1, ncols=3, ... subplot_kw={"projection": "3d"}, ... figsize=(9, 4)) >>> left, middle, right = axes >>> plot_vmf_samples(left, x, y, z, mu, 5) >>> plot_vmf_samples(middle, x, y, z, mu, 20) >>> plot_vmf_samples(right, x, y, z, mu, 100) >>> plt.subplots_adjust(top=1, bottom=0.0, left=0.0, ... right=1.0, wspace=0.) >>> plt.show() The plots show that with increasing concentration :math:`\kappa` the resulting samples are centered more closely around the mean direction. **Fitting the distribution parameters** The distribution can be fitted to data using the ``fit`` method returning the estimated parameters. As a toy example let's fit the distribution to samples drawn from a known von Mises-Fisher distribution. >>> mu, kappa = np.array([0, 0, 1]), 20 >>> samples = vonmises_fisher(mu, kappa).rvs(1000, random_state=rng) >>> mu_fit, kappa_fit = vonmises_fisher.fit(samples) >>> mu_fit, kappa_fit (array([0.01126519, 0.01044501, 0.99988199]), 19.306398751730995) We see that the estimated parameters `mu_fit` and `kappa_fit` are very close to the ground truth parameters. Ncst|dSrjrrwryr*r/rfXszvonmises_fisher_gen.__init__rcCst|||dS)zsCreate a frozen von Mises-Fisher distribution. See `vonmises_fisher_frozen` for more information. r)vonmises_fisher_frozenr`rkapparxr*r*r/r[szvonmises_fisher_gen.__call__cCst|}|jdkrtdttj|ds8td|jdksJtdd}t|r`|dkrht|t |dkr|td |j}|||fS) z~ Infer dimensionality from mu and ensure that mu is a one-dimensional unit vector and kappa positive. rz%'mu' must have one-dimensional shape.rz%'mu' must be a unit vector of norm 1.z$'mu' must have at least two entries.z"'kappa' must be a positive scalar.rrzFor 'kappa=0' the von Mises-Fisher distribution becomes the uniform distribution on the sphere surface. Consider using 'scipy.stats.uniform_direction' instead.) r8rOr,rUallcloserRrrrrI)r`rr Zkappa_error_msgrr*r*r/rbs    z'vonmises_fisher_gen._process_parameterscCs>|jd|krtdttjj|ddds:d}t|dS)Nr1znThe dimensionality of the last axis of 'x' must match the dimensionality of the von Mises Fisher distribution.rgr8'x' must be unit vectors of norm 1 along last dimension.)rrUr8r rRr)r`rCrrer*r*r/_check_data_vs_distzs z'vonmises_fisher_gen._check_data_vs_distcCs>d|}d|dt||ttt|d||S)Nrr)r)r8r]rr r`rr halfdimr*r*r/_log_norm_factors z$vonmises_fisher_gen._log_norm_factorcCs8t|}|||td||}|||||S)zLog of the von Mises-Fisher probability density function. As this function does no argument checking, it should not be called directly; use 'logpdf' instead. z i,...i->...)r8rOr rr)r`rCrrr Z dotproductsr*r*r/rs  zvonmises_fisher_gen._logpdfcCs"|||\}}}|||||S)aLog 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`. )rrr`rCrr rr*r*r/rszvonmises_fisher_gen.logpdfcCs(|||\}}}t|||||S)aVon 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`. )rr8rrrr*r*r/rszvonmises_fisher_gen.pdfcCsFt|d|d}|j|||d}tjt|t|gdd}|S)z 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. rrrr1rg)r8Zarctan2Zvonmisesstackcossin)r`rr rr{Z mean_angleZ angle_samplesrr*r*r/_rvs_2ds zvonmises_fisher_gen._rvs_2dc Cs|durd}n|}||}dt|d|td||}tdt|}td||}tj|||d||dgdd }|durt|}|S) ah 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 Nrrrr)rx).rr1rg) randomr8r]rrXrrrr+) r`r rr{Z sample_sizerCtempZ uniformcirclerr*r*r/_rvs_3ds  (   zvonmises_fisher_gen._rvs_3dcCs|d}|dur,t|s |f}t|}nd}td|d|d}d|||}|dkr|d|d|d d |d }d |d |} || |tdt|dt|} d} t|f} d |} | |kr|j| | || d}dd||dd||}| || }|||td||||d|| t|k}t |}||| | | |<| |7} qt || |}t dtd| d|}tj | d|gdd}|dur|||f}n t|}|S)a$ 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 rNrryr)rrgr@rrrrz...,...i->...irzrg)r8rrrrXr]log1prbetarr\rrrrr+)r`rr rr{Z dim_minus_oneZ n_samplesrXZ envelop_paramnodeZ correctionZ n_acceptedrCrZsym_betaZcoord_xZ accept_tol criterionZ accepted_iterZ coord_restrr*r*r/_rejection_samplingsl         z'vonmises_fisher_gen._rejection_samplingc Cst|f}d|d<t|dddft|d|fg}tjt|\}}tt||dddfdddf|rd}nd}td|||}|S)aA 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 rrNrr1z ij,...j->...i) r8rrrRrrr r{r) r`rrrZ base_pointZembeddedZ rotmatrixrZrotsignr*r*r/_rotate_samplesAs (,z#vonmises_fisher_gen._rotate_samplescCs\|dkr|||||}n(|dkr2||||}n|||||}|dkrX||||}|S)Nr)r)rrrr)r`rrr rr{rr*r*r/rB^s zvonmises_fisher_gen._rvscCs2|||\}}}||}||||||}|S)a(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. )rr~rB)r`rr rr{rrr*r*r/rks! zvonmises_fisher_gen.rvscCs2d|}||| |t||t|d|Sr)rr r r*r*r/rs  zvonmises_fisher_gen._entropycCs|||\}}}|||S)aCompute 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. r)r`rr rrr*r*r/rszvonmises_fisher_gen.entropyc st|}|jdkrtdttjj|ddds@d}t||jd}t|}|j }|j d|fdd }t |d d d }|j }||fS) aFit 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. r)z'x' must be two dimensional.r1rgrr rcs&tdg|}|d|dS)Nrr)r )r Z bessel_valsrrr*r/solve_for_kappasz0vonmises_fisher_gen.fit..solve_for_kappaZbrentq)g:0yE>geA)rZbracket) r8rOr,rUr rRrrrZmean_directionZmean_resultant_lengthrroot) r`rCrerZdirstatsrr Zroot_resr r*rr/rs"   zvonmises_fisher_gen.fit)N)NrN)Nr)Nr)NrrN)Nr)rorprqrrrfrrr rrrrrrrrrBrrrrrr*r*ryr/rs&S    H & rc@s8eZdZd ddZddZddZdd d Zd d ZdS)rNrcCs(t||_|j||\|_|_|_dS)aCreate 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. N)rrrrrr rr*r*r/rfs  zvonmises_fisher_frozen.__init__cCs|j||j|j|jS)a 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`. )rrrrr rr*r*r/rszvonmises_fisher_frozen.logpdfcCst||S)a 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`. rrr*r*r/r szvonmises_fisher_frozen.pdfcCs&|j|}|j|j|j|j||S)a?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. )rr~rBrrr rr*r*r/rs zvonmises_fisher_frozen.rvscCs|j|j|jS)z Calculate the differential entropy of the von Mises-Fisher distribution. Returns ------- h: float Entropy of the Von Mises-Fisher distribution. )rrrr rmr*r*r/r:s zvonmises_fisher_frozen.entropy)NrN)rN)rorprqrfrrrrr*r*r*r/rs   r)NN)rA)N)N)rnumpyr8Z scipy.linalgrQZ scipy._librZ scipy.specialrrrrrrr r Zscipy._lib._utilr r Zscipy.linalg.blasr rZ_continuous_distnsrZ_discrete_distnsrrrrrZ_qmvntrZ _morestatsrZscipy.optimizer__all__r]rrrZ_LOG_PIrr0r@rKrLrtrrrZ_mvn_doc_frozen_callparamsZ_mvn_doc_frozen_callparams_noterZmvn_docdict_noparamsrrrname__dict__rZ method_frozenrrrrrZ_matnorm_doc_frozen_callparamsZ#_matnorm_doc_frozen_callparams_noterZmatnorm_docdict_noparamsrrrrZ _dirichlet_doc_frozen_callparamsZ%_dirichlet_doc_frozen_callparams_noterZdirichlet_docdict_noparamsrrrrrrZ_wishart_doc_default_callparamsZ_wishart_doc_callparams_noteZ_wishart_doc_frozen_callparamsZ#_wishart_doc_frozen_callparams_noterZwishart_docdict_noparamsrrrrPrrQZ#_multinomial_doc_default_callparamsZ _multinomial_doc_callparams_noteZ"_multinomial_doc_frozen_callparamsZ'_multinomial_doc_frozen_callparams_noter[Zmultinomial_docdict_noparamsrZrr\rqr rsrr!rrr"rrr#rrrZ_mvt_doc_frozen_callparams_noterZmvt_docdict_noparamsrrr$Z_mhg_doc_default_callparamsZ_mhg_doc_callparams_noteZ_mhg_doc_frozen_callparamsZ_mhg_doc_frozen_callparams_noterZmhg_docdict_noparamsrr%rrr&rZ_ctab_doc_row_colZ _ctab_doc_xZ_ctab_doc_row_col_noteZ_ctab_doc_mean_paramsZ_ctab_doc_row_col_note_frozenZ _ctab_docdictrZ_ctab_docdict_frozenupdaterrr'rrrZ#_dirichlet_mn_doc_frozen_callparamsZ(_dirichlet_mn_doc_frozen_callparams_noterZdirichlet_mn_docdict_noparamsrrrrrr(rr*r*r*r/sH (         " X" 9g    e;   ,  i:  c>  L0   "x],q X7    %   b        !\   \