M/Phٖr dZddlZddlmZddlmZddlmZddfd Z d dddd d fd Z dZ GddZ GddZ GddZGdde ZejZejZejZGdde ZdddfdZedkrddlmZmZdgZd Zejd!d"gd"d!ggZgd#Z ejgd$gd%gd&gZ!devre eeZ"e"#d'(Z#e$e#%de$ej&e#d)e$e"'e$e"(ddge eej)d*Z*e ed+ej)d*zZ+d,Z,e$e*'e,e$e*-e+e*.e+e$e+-e*e+.e*e$e*-e"e*.e"eeeZ/e/0ddge/0ej1d-ejgd$gd%gd&gZ!gd#Z ee e!Z2e20d.e23d.gd/Z4ee23e!e4d01gd2Z4ee20e!e4d31eejgd4e!Z5gd5Z4ee50e!e4d61ed4dZ6d7Z,e$e2 e,e$e2 e,d8(ed dd9Z7ee73ejd:d:gd;d<1ee70ejd:d:gd=d<1e73ejd!d!gd>z ed ddZ8e83ejd!d!gd?z e7#d@Z#eej&e#d)e7j&d1ee e!dZ9ee90e!gdAdB1ee e!dCZ7ee70e!gdDd31dSdS)Ea Multivariate Normal and t distributions Created on Sat May 28 15:38:23 2011 @author: Josef Perktold TODO: * renaming, - after adding t distribution, cov does not make sense for Sigma DONE - should mean also be renamed to mu, if there will be distributions with mean != mu * not sure about corner cases - behavior with (almost) singular sigma or transforms - df <= 2, is everything correct if variance is not finite or defined ? * check to return possibly univariate distribution for marginals or conditional distributions, does univariate special case work? seems ok for conditional * are all the extra transformation methods useful outside of testing ? - looks like I have some mixup in definitions of standardize, normalize * new methods marginal, conditional, ... just added, typos ? - largely tested for MVNormal, not yet for MVT DONE * conditional: reusing, vectorizing, should we reuse a projection matrix or allow for a vectorized, conditional_mean similar to OLS.predict * add additional things similar to LikelihoodModelResults? quadratic forms, F distribution, and others ??? * add Delta method for nonlinear functions here, current function is hidden somewhere in miscmodels * raise ValueErrors for wrong input shapes, currently only partially checked * quantile method (ppf for equal bounds for multiple testing) is missing http://svitsrv25.epfl.ch/R-doc/library/mvtnorm/html/qmvt.html seems to use just a root finder for inversion of cdf * normalize has ambiguous definition, and mixing it up in different versions std from sigma or std from cov ? I would like to get what I need for mvt-cdf, or not univariate standard t distribution has scale=1 but std>1 FIXED: add std_sigma, and normalize uses std_sigma * more work: bivariate distributions, inherit from multivariate but overwrite some methods for better efficiency, e.g. cdf and expect I kept the original MVNormal0 class as reference, can be deleted See Also -------- sandbox/examples/ex_mvelliptical.py Examples -------- Note, several parts of these examples are random and the numbers will not be (exactly) the same. >>> import numpy as np >>> import statsmodels.sandbox.distributions.mv_normal as mvd >>> >>> from numpy.testing import assert_array_almost_equal >>> >>> cov3 = np.array([[ 1. , 0.5 , 0.75], ... [ 0.5 , 1.5 , 0.6 ], ... [ 0.75, 0.6 , 2. ]]) >>> mu = np.array([-1, 0.0, 2.0]) multivariate normal distribution -------------------------------- >>> mvn3 = mvd.MVNormal(mu, cov3) >>> mvn3.rvs(size=3) array([[-0.08559948, -1.0319881 , 1.76073533], [ 0.30079522, 0.55859618, 4.16538667], [-1.36540091, -1.50152847, 3.87571161]]) >>> mvn3.std array([ 1. , 1.22474487, 1.41421356]) >>> a = [0.0, 1.0, 1.5] >>> mvn3.pdf(a) 0.013867410439318712 >>> mvn3.cdf(a) 0.31163181123730122 Monte Carlo integration >>> mvn3.expect_mc(lambda x: (x>> mvn3.expect_mc(lambda x: (x>> mvt3 = mvd.MVT(mu, cov3, 4) >>> mvt3.rvs(size=4) array([[-0.94185437, 0.3933273 , 2.40005487], [ 0.07563648, 0.06655433, 7.90752238], [ 1.06596474, 0.32701158, 2.03482886], [ 3.80529746, 7.0192967 , 8.41899229]]) >>> mvt3.pdf(a) 0.010402959362646937 >>> mvt3.cdf(a) 0.30269483623249821 >>> mvt3.expect_mc(lambda x: (x>> mvt3.cov array([[ 2. , 1. , 1.5], [ 1. , 3. , 1.2], [ 1.5, 1.2, 4. ]]) >>> mvt3.corr array([[ 1. , 0.40824829, 0.53033009], [ 0.40824829, 1. , 0.34641016], [ 0.53033009, 0.34641016, 1. ]]) get normalized distribution >>> mvt3n = mvt3.normalized() >>> mvt3n.sigma array([[ 1. , 0.40824829, 0.53033009], [ 0.40824829, 1. , 0.34641016], [ 0.53033009, 0.34641016, 1. ]]) >>> mvt3n.cov array([[ 2. , 0.81649658, 1.06066017], [ 0.81649658, 2. , 0.69282032], [ 1.06066017, 0.69282032, 2. ]]) What's currently there? >>> [i for i in dir(mvn3) if not i[0]=='_'] ['affine_transformed', 'cdf', 'cholsigmainv', 'conditional', 'corr', 'cov', 'expect_mc', 'extra_args', 'logdetsigma', 'logpdf', 'marginal', 'mean', 'normalize', 'normalized', 'normalized2', 'nvars', 'pdf', 'rvs', 'sigma', 'sigmainv', 'standardize', 'standardized', 'std', 'std_sigma', 'whiten'] >>> [i for i in dir(mvt3) if not i[0]=='_'] ['affine_transformed', 'cdf', 'cholsigmainv', 'corr', 'cov', 'df', 'expect_mc', 'extra_args', 'logdetsigma', 'logpdf', 'marginal', 'mean', 'normalize', 'normalized', 'normalized2', 'nvars', 'pdf', 'rvs', 'sigma', 'sigmainv', 'standardize', 'standardized', 'std', 'std_sigma', 'whiten'] N)special) mvstdtprob) mvnormcdfcdSNrxs k/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/statsmodels/sandbox/distributions/mv_normal.pyr s1iPcvfd}||}||dS)acalculate expected value of function by Monte Carlo integration Parameters ---------- dist : distribution instance needs to have rvs defined as a method for drawing random numbers func : callable function for which expectation is calculated, this function needs to be vectorized, integration is over axis=0 size : int number of random samples to use in the Monte Carlo integration, Notes ----- this does not batch Returns ------- expected value : ndarray return of function func integrated over axis=0 by MonteCarlo, this will have the same shape as the return of func without axis=0 Examples -------- integrate probability that both observations are negative >>> mvn = mve.MVNormal([0,0],2.) >>> mve.expect_mc(mvn, lambda x: (x>> c = stats.norm.isf(0.05, scale=np.sqrt(2.)) >>> expect_mc(mvn, lambda x: (np.abs(x)>np.array([c, c])), size=100000) array([ 0.09969, 0.0986 ]) or calling the method >>> mvn.expect_mc(lambda x: (np.abs(x)>np.array([c, c])), size=100000) array([ 0.09937, 0.10075]) c|SNr r funcs r funzexpect_mc..funtAwwrsizerrvsmean)distrrrrs ` r expect_mcrsJ\ (((  C 3s88==  rcdSrr r s r r r s!rFg333333?c||djd}|#tj tj|z}ntj|}|"tjtj|z}ntj|}fd}g} d} d} || z } |t | |z} | t ||zz } | | |k| |kzd}tj|}| |jdz } | |t| | |krntj | } t| j| | jdksJ||  d}|r|S|| dz| z zS) acalculate expected value of function by Monte Carlo integration Parameters ---------- dist : distribution instance needs to have rvs defined as a method for drawing random numbers func : callable function for which expectation is calculated, this function needs to be vectorized, integration is over axis=0 size : int minimum number of random samples to use in the Monte Carlo integration, the actual number used can be larger because of oversampling. lower : None or array_like lower integration bounds, if None, then it is set to -inf upper : None or array_like upper integration bounds, if None, then it is set to +inf conditional : bool If true, then the expectation is conditional on being in within [lower, upper] bounds, otherwise it is unconditional overfact : float oversampling factor, the actual number of random variables drawn in each attempt are overfact * remaining draws. Extra draws are also used in the integration. Notes ----- this does not batch Returns ------- expected value : ndarray return of function func integrated over axis=0 by MonteCarlo, this will have the same shape as the return of func without axis=0 Examples -------- >>> mvn = mve.MVNormal([0,0],2.) >>> mve.expect_mc_bounds(mvn, lambda x: np.ones(x.shape[0]), lower=[-10,-10],upper=[0,0]) 0.24990416666666668 get 3 marginal moments with one integration >>> mvn = mve.MVNormal([0,0],1.) >>> mve.expect_mc_bounds(mvn, lambda x: np.dstack([x, x**2, x**3, x**4]), lower=[-np.inf,-np.inf], upper=[np.inf,np.inf]) array([[ 2.88629497e-03, 9.96706297e-01, -2.51005344e-03, 2.95240921e+00], [ -5.48020088e-03, 9.96004409e-01, -2.23803072e-02, 2.96289203e+00]]) >>> from scipy import stats >>> [stats.norm.moment(i) for i in [1,2,3,4]] [0.0, 1.0, 0.0, 3.0] rrNc|Srr rs r rzexpect_mc_bounds..funrrrT?) rshapenpinfonesasarrayintall atleast_2dappendprintvstackr)rrrlowerupper conditionaloverfactrvsdimrrvsliusedtotalremainrrvsokmean_conditionals ` r expect_mc_boundsr8sxXX1X   #B 'F }"'&//) 5!! }( 5!! E D E hhC 122h33 TH_%%%cUlse|499"==> e$$  A U d 4<<   )E  C #) 39Q<    s3xx}}Q''64"9u#455rctj|\}}|\}}tj|\}}} } tj|tj| } }||z } ||z } ||| zz } | dz|dzz | dz| dzz zd| z| z| z|| zz z }dtjz|z| ztjd| dzz z}tj| dd| dzz zz |z S)z Bivariate Gaussian distribution for equal shape *X*, *Y*. See `bivariate normal `_ at mathworld. r)r# transposeravelsqrtpiexp)r mucovXYmuxmuysigmaxsigmaxytmpsigmayXmuYmurhozdenoms r bivariate_normalrO0s <??DAqHC#%8C== FGS&WV__bgfooFF C%C C%C 6&= !C Qvqy36&!)++aeCimVF].KKA beGFN6 !"'!CF("3"3 3E 6A2q!CF(|$ % % --rcLeZdZdZddZdZdZdZddd fd Zd Z dd Z dS)BivariateNormalct|_||_tj|\|_|_}|_d|_dS)Nr:) r@rrAr#r<rFrGrInvars)selfrrArHs r __init__zBivariateNormal.__init__Ls7 68hsmm3 T\3  rrcZtj|j|j|S)Nrr#randommultivariate_normalrrArTrs r rzBivariateNormal.rvsRs#y,,TYt,LLLrc8t||j|jSr)rOrrArTr s r pdfzBivariateNormal.pdfUs49dh777rcPtj||Sr)r#logr]r\s r logpdfzBivariateNormal.logpdfXsvdhhqkk"""rc.||S)N)r.expectr\s r cdfzBivariateNormal.cdf\s{{{###rcdSrr r s r r zBivariateNormal._sArrg ricdfd}ddlm}||ddfdfdS)Ncrtj||f}||zSr)r# column_stackr])r yrrTs r rz#BivariateNormal.expect..fun`s31&&A477TXXa[[( (rrdblquadcdSrr rmr-s r r z(BivariateNormal.expect..ds %(rcdSrr rmr.s r r z(BivariateNormal.expect..es qrscipy.integratero)rTrr-r.rros```` r rczBivariateNormal.expect_sr ) ) ) ) ) ) ,+++++wsE!HeAh0B0B0B0B))))++ +rc<fd}|S)aKullback-Leibler divergence between this and another distribution int f(x) (log f(x) - log g(x)) dx where f is the pdf of self, and g is the pdf of other uses double integration with scipy.integrate.dblquad limits currently hardcoded cZ||z Srr`r otherrTs r r z$BivariateNormal.kl..s Q%,,q//9rrb)rTrzrs`` r klzBivariateNormal.klgs):9999{{3r cxfd}|}||S)NcZ||z Srrxrys r r z'BivariateNormal.kl_mc..wr{rrr)rTrzrrrs`` r kl_mczBivariateNormal.kl_mcvs>99999hhDh!!s3xx}}rNr)r}) __name__ __module__ __qualname__rUrr]r`rdrcr|rr rr rQrQFs  MMMM888###$$$&+Yg++++    rrQceZdZdZdZddZdZdZdZdZ d Z d Z d Z d Z ddZddZedZedZedZeZdZdS) MVEllipticalzBase Class for multivariate elliptical distributions, normal and t contains common initialization, and some common methods subclass needs to implement at least rvs and logpdf methods c g|_tj||_tj|x|_}tj|}t |x|_}|jdkrgtj ||z|_tj ||z |_ tj |tj |z |_ n|j dkrwt ||krdtj||_tjd|z |_ tjdtj |z |_ no|j||fkrStj||_ tj|j j|_ nt'dtjtj|j|_dS)initialize instance Parameters ---------- mean : array_like parameter mu (might be renamed), for symmetric distributions this is the mean sigma : array_like, 2d dispersion matrix, covariance matrix in normal distribution, but only proportional to covariance matrix in t distribution args : list distribution specific arguments, e.g. df for t distribution kwds : dict currently not used r rr!zsigma has invalid shapeN) extra_argsr#r&rsigmasqueezelenrSr"eyesigmainvr= cholsigmainvndimdiaglinalgpinvcholeskyT ValueErrorr_det logdetsigma)rTrrargskwdsrSs r rUzMVElliptical.__init__s$Jt$$ Z... U 5!! YY& U ;"  .DJF5MME1DM "u  >D  jAooCJJ%$7$7DJGBJ//DM "bgenn)< = =D   [UEN * *INN511DM " 2 24= A A CD  677 76")-- ";";<<rrct)a~random variable Parameters ---------- size : int or tuple the number and shape of random variables to draw. Returns ------- rvs : ndarray the returned random variables with shape given by size and the dimension of the multivariate random vector as additional last dimension NotImplementedErrorrZs r rzMVElliptical.rvs ""!rct) logarithm of probability density function Parameters ---------- x : array_like can be 1d or 2d, if 2d, then each row is taken as independent multivariate random vector Returns ------- logpdf : float or array probability density value of each random vector this should be made to work with 2d x, with multivariate normal vector in each row and iid across rows does not work now because of dot in whiten rr\s r r`zMVElliptical.logpdfs ,"!rc tacumulative distribution function Parameters ---------- x : array_like can be 1d or 2d, if 2d, then each row is taken as independent multivariate random vector kwds : dict contains options for the numerical calculation of the cdf Returns ------- cdf : float or array probability density value of each random vector rrTr rs r rdzMVElliptical.cdfrrct)z^affine transformation define in subclass because of distribution specific restrictionsr)rTshift scale_matrixs r affine_transformedzMVElliptical.affine_transformeds "!rchtj|}tj||jjS)aG whiten the data by linear transformation Parameters ---------- x : array_like, 1d or 2d Data to be whitened, if 2d then each row contains an independent sample of the multivariate random vector Returns ------- np.dot(x, self.cholsigmainv.T) Notes ----- This only does rescaling, it does not subtract the mean, use standardize for this instead See Also -------- standardize : subtract mean and rescale to standardized random variable. )r#r&dotrrr\s r whitenzMVElliptical.whitens). JqMMva*,---rcPtj||SaOprobability density function Parameters ---------- x : array_like can be 1d or 2d, if 2d, then each row is taken as independent multivariate random vector Returns ------- pdf : float or array probability density value of each random vector r#r?r`r\s r r]zMVElliptical.pdfsvdkk!nn%%%rc<|||jz S)astandardize the random variable, i.e. subtract mean and whiten Parameters ---------- x : array_like, 1d or 2d Data to be whitened, if 2d then each row contains an independent sample of the multivariate random vector Returns ------- np.dot(x - self.mean, self.cholsigmainv.T) Notes ----- See Also -------- whiten : rescale random variable, standardize without subtracting mean. )rrr\s r standardizezMVElliptical.standardize$s.{{1ty=)))rcD||j |jS)z2return new standardized MVNormal instance )rrrrTs r standardizedzMVElliptical.standardized=s!&& z43DEEErcNtj|j}||jz |z S)a3normalize the random variable, i.e. subtract mean and rescale The distribution will have zero mean and sigma equal to correlation Parameters ---------- x : array_like, 1d or 2d Data to be whitened, if 2d then each row contains an independent sample of the multivariate random vector Returns ------- (x - self.mean)/std_sigma Notes ----- See Also -------- whiten : rescale random variable, standardize without subtracting mean. )r#r) std_sigmar)rTr std_s r normalizezMVElliptical.normalizeCs&2}T^,,DI t##rTc|rtjj}njjz }j}fdjD}j||g|RS)zwreturn a normalized distribution where sigma=corr if demeaned is True, then mean will be set to zero c0g|]}t|Sr getattr.0earTs r z+MVElliptical.normalized..j#<<1HI <<<6I.J.JKKKrcXtjtj|jS)zDstandard deviation, square root of diagonal elements of cov )r#r=rrArs r stdzMVElliptical.std~s wrwtx(()))rcXtjtj|jS)zFstandard deviation, square root of diagonal elements of sigma )r#r=rrrs r rzMVElliptical.std_sigmas wrwtz**+++rcP|jtj|j|jz S)zcorrelation matrix)rAr#outerrrs r rzMVElliptical.corrs!x"(48TX6666rctj|}j|}j|dddf|f}fdjD}j||g|RS)areturn marginal distribution for variables given by indices this should be correct for normal and t distribution Parameters ---------- indices : array_like, int list of indices of variables in the marginal distribution Returns ------- mvdist : instance new instance of the same multivariate distribution class that contains the marginal distribution of the variables given in indices Nc0g|]}t|Sr rrs r rz)MVElliptical.marginal..rr)r#r&rrrr)rTindicesrrrs` r marginalzMVElliptical.marginalst$*W%%9W%Jwqqqv78 <<<rrTr x_whitenedSSRllfs r r`zMVNormal0.logpdfs( JqMM[[TY// fZ]B''d tzBF2:.... t~ s  rNr) rrrrrUrrr]r`rr rr rrss9998<MMMM,&&&$:IIIrrcTeZdZdZedkd dZdZdZedZdZ d Z d S) MVNormalzClass for Multivariate Normal Distribution uses Cholesky decomposition of covariance matrix for the transformation of the data z Multivariate Normal DistributionrcZtj|j|j|Sr)r#rXrYrrrZs r rz MVNormal.rvsDs%(y,,TY ,NNNrctj|}|||jz }tj|dzd}| }||jtjdtjzzz}||jz}|dz}|Sr) r#r&rrrrSr_r>rrs r r`zMVNormal.logpdfZs( JqMM[[TY// fZ]B''d tzBF2:.... t s  rc 4t||j|jfi|Sr)rrrArs r rdz MVNormal.cdfws"&DItx884888rc|jS)zcovariance matrix)rrs r rAz MVNormal.covs zrc|}tj||j|z}tjtj||j|j}t ||S)a,return distribution of an affine transform for full rank scale_matrix only Parameters ---------- shift : array_like shift of mean scale_matrix : array_like linear transformation matrix Returns ------- mvt : instance of MVNormal instance of multivariate normal distribution given by affine transformation Notes ----- the affine transformation is defined by y = a + B x where a is shift, B is a scale matrix for the linear transformation Notes ----- This should also work to select marginal distributions, but not tested for this case yet. currently only tested because it's called by standardized )r#rrrrrrTrrBrrs r rzMVNormal.affine_transformedsSD 6!TY''%/F26!TZ00!#66 ),,,rc  tj| tj fdt|jD}|j dddf f}|j|dddf|f}|j dddf|f}|j|dddf f}|tj|tj||z }|j tj|tj|||j|z z} t| |S)aireturn conditional distribution indices are the variables to keep, the complement is the conditioning set values are the values of the conditioning variables \bar{\mu} = \mu_1 + \Sigma_{12} \Sigma_{22}^{-1} \left( a - \mu_2 \right) and covariance matrix \overline{\Sigma} = \Sigma_{11} - \Sigma_{12} \Sigma_{22}^{-1} \Sigma_{21}.T Parameters ---------- indices : array_like, int list of indices of variables in the marginal distribution given : array_like values of the conditioning variables Returns ------- mvn : instance of MVNormal new instance of the MVNormal class that contains the conditional distribution of the variables given in indices for given values of the excluded variables. cg|]}|v| Sr r )rikeeps r rz(MVNormal.conditional..sJJJ!ATMMAMMMrN) r#r&rangerSrrrsolverr) rTrvaluesgivensigmakksigmaggsigmakgsigmagkrrrs @r r/zMVNormal.conditionals,<z'"" JJJJuTZ'8'8JJJKK*T!!!T']D01*U111d7^U23*T!!!T']E12*U111d7^T12bfWbioogw.O.OPPP 9T? F7BIOOGVDIe1. Notes ----- generated as a chi-square mixture of multivariate normal random variables. does this require df>2 ? r)multivariate_t_rvs)rn) multivariaterrrr)rTrrs r rzMVT.rvss60 544444!!$)TZDGtLLLLrc tj|}|j}|j}|||jz }| t |tzz}||jz}|||zt dtj |dzd|z zzz}|dz}|t||zdz t|dz z z }|S)a_logarithm of probability density function Parameters ---------- x : array_like can be 1d or 2d, if 2d, then each row is taken as independent multivariate random vector Returns ------- logpdf : float or array probability density value of each random vector rr:rrr) r#r&rrSrrnp_lognp_pirr sps_gamln)rTr rrSrrs r r`z MVT.logpdf#s JqMM W [[TY// grEz*** t U fQ A b)A)AB)F%FGGGG s  y"u**++iR.@.@@@ rc tj tj|z}||jz |jz }t |||j|jfi|Sr)r#r$ ones_likerrrrr)rTr rr-r.s r rdzMVT.cdfBsL"",q//)TY.% 47CCdCCCrc|jdkr&tjtj|jzS|j|jdz z |jzS)zcovariance matrix The covariance matrix for the t distribution does not exist for df<=2, and is equal to sigma * df/(df-2) for df>2 r:r)rr#nanr rrs r rAzMVT.covZsB 7a<<6BL444 47dgl+dj8 8rc|}|j|j|jfksDtj|dkrt dtj||j|z}tjtj||j |j }t|||j S)a^return distribution of a full rank affine transform for full rank scale_matrix only Parameters ---------- shift : array_like shift of mean scale_matrix : array_like linear transformation matrix Returns ------- mvt : instance of MVT instance of multivariate t distribution given by affine transformation Notes ----- This checks for eigvals<=0, so there are possible problems for cases with positive eigenvalues close to zero. see: http://www.statlect.com/mcdstu1.htm I'm not sure about general case, non-full rank transformation are not multivariate t distributed. y = a + B x where a is shift, B is full rank scale matrix with same dimension as sigma rz$affine transform has to be full rank) r"rSr#reigvalsrrrrrrrrrs r rzMVT.affine_transformedgsL w4:tz222 !!!$$)..00 I !GHHH6!TY''%/F26!TZ00!#66 8Y000rr) rrrrUrr`rdrrAr __classcell__)rs@r rrs 555*MMMM8>DDD0 9 9X 9-1-1-1-1-1-1-1rrcdSrr r s r r r s!rrfrhc`fd}ddlm}||ddfdfdS)NcFtj||f}|Sr)r#rl)r rmrs r rzquad2d..funs# OQqE " "tAwwrrrncdSrr rqs r r zquad2d..s eAhrcdSrr rss r r zquad2d..s U1Xrrt)rr-r.rros``` r quad2drsl(''''' 73a%(,>,>,>,>%%%% ' ''r__main__)assert_almost_equalassert_array_almost_equalmvn)rrr!r)rr)r!r?)rg?333333?)rrrir)rowvarr:ctjt|tjt|z Sr)r#r_bvn1r]bvnr s r r r s3 ,,rvcggajj/A/AAr)r:r:)rrg@)g&Ng&Ng&N)decimal)gue>?g7P?gTw?)rrr)gEOQם?g4x=?g9"8?ctjt|tjt|z Sr)r#r_mvn3r]mvn3br s r r r s3 ,,rveiill/C/CCri@ rgdggm0_?gX*#g!4 i)grgammalnr rrr numpy.testingrrexamplesr@arraycovxmu3cov3r"rr+rrArcrdrr!bvn2rr|rrr]zerosr(r`r_valmvn3cr)mvtmvt1mvt31r rr r@s^PPbEEEEEE%51111f!, %t4!&a6a6a6a6H...,33333333jm:m:m:m:m:m:m:m:b KKKKKKKK\l-l-l-l-l-|l-l-l-`   O f1f1f1f1f1,f1f1f1R 9G'''' zLLLLLLLLwH B 28c3Z#s, - -D ,,C 28*********, - -D  ob$''gg4g   chhqkk fbfS###$$$ cjjll cggqennr626!99--r1VRVAYY;//AA dkk# dggdmmTZZ--... dggdmmTZZ--... dggcllDJJsOO,,,hr4   1    rx.........011llxT""  L!!!MLL!!4;;t#4#4erJJJJRRR!!488D>>5BGGGG'''**D11OOO!!599T??ERHHHH!$$CC dnnS!!""" dnnSvn../// #eQ  C 828RG#4#4557I "$$$$"R 1 1224F "$$$$JJxrxB  !!#455 3ua  DKK"R!!""$566 ''&//Cs1---swBBBB CT1  E $MMM #c4  C JJJqr