U Kv fD @sdZddlmZmZmZddlZedZGdddZ GdddZ Gd d d Z e ej d ZGd d d ZdNddZdOddZdPddZGdddZdQddZedkrdddgZdZddlmZddlmZmZed d!d"ed#d$d"fZed%d&geej ej ged'Z!eZ"d(ekrZe!e!d)ke!d k@dZ!ee!e d*dd+\Z#Z$Z%Z&e#Z'e(e#e$Z)e*d,e#+e#e#+Z#e(e#e$Z,e*d-e)e,e#e,Z#dZ-e-rej.e!d.d/d0d1ej/e$e#d d2d3ej/e$e'd d4d3e0e1e&ddD]D\Z2Z3e1e&ddD](\Z4Z5e*e2e4e6d5d6d7d#dq qe&D]Z3e*e6d8d6d7d#qe$d%d&gej ej ged;Z?d#Z-e-r"e@ej.e!d.d/d0d1ej/e$e;d d2d3ej/e$e?d dd9Z8e9e!+e!:Z$e8e$Z;ej6e8fe8j<dZ=e*d:e=e"j>e$d%d&gej ej ged;Z?d#Z-e-re@ej.e!d.d/d0d1eAd?ej/e$e;d d2d3ej/e$e?d de$d%d&gej ej ged;Z?d#Z-e-re@ej.e!d.d/d0d1ej/e$e;d d2d3ej/e$e?d d 0 maybe this implies that we cannot use it for densities that are > 0 at boundary ??? or maybe there is a mistake close to the boundary, sometimes integration works. cCs2||_ddlm}|||_d|_d|_d|_dS)Nrchebyt)r)g !g !?g{Gz?)r scipy.specialr#polyr r r)r rr#r r rrps   zChebyTPoly.__init__cCsd|jdkr0t|d|ddttjS||d|ddttjtdSdS)Nrrg?)rrrr rr&rr r rrzs &zChebyTPoly.__call__Nrr r r rr!bs  r!r'c@s eZdZdZddZddZdS)HPolyzOrthonormal (for weight=1) Hermite Polynomial, uses finite bounds for current use with DensityOrthoPoly domain is defined as [-6,6] cCs,||_ddlm}|||_d|_d|_dS)Nrhermite)?)rr%r*r&r r)r rr*r r rrs   zHPoly.__init__cCsN|j}d|tdt|dt||d}t|}|||S)Nrrr')rrlogrZgammalnlogpi2expr&)r rkZlnfactZfactr r rrs0 zHPoly.__call__Nrr r r rr(sr(cs"tfddt|D}|S)Ncsg|]}|qSr r .0ipolybaserr r szpolyvander..)rZ column_stackrange)rr8rZpolyarrr r7r polyvandersr;c st|}t||f}|tjt||f}t|D]}t|dD]}||||dk rtfdd||\} } ntfdd||\} } | |||f<| |||f<||ksH| |||f<| |||f<qHq8||fS)ainner product of continuous function (with weight=1) Parameters ---------- polys : list of callables polynomial instances lower : float lower integration limit upper : float upper integration limit weight : callable or None weighting function Returns ------- innp : ndarray symmetric 2d square array with innerproduct of all function pairs err : ndarray numerical error estimate from scipy.integrate.quad, same dimension as innp Examples -------- >>> from scipy.special import chebyt >>> polys = [chebyt(i) for i in range(4)] >>> r, e = inner_cont(polys, -1, 1) >>> r array([[ 2. , 0. , -0.66666667, 0. ], [ 0. , 0.66666667, 0. , -0.4 ], [-0.66666667, 0. , 0.93333333, 0. ], [ 0. , -0.4 , 0. , 0.97142857]]) rNcs|||SNr rp1p2weightr rzinner_cont..cs||Sr<r r=r?r@r rrBrC) lenremptyfillnanzerosr:rquad) polyslowerupperrAZn_polys innerprodZinterrr6jZinnperrr r>r inner_conts(!      rQ:0yE>csrtt|D]`}t|dD]N}||||tfdd||d}tj|||k||dsdSqq dS)aZcheck whether functions are orthonormal Parameters ---------- polys : list of polynomials or function Returns ------- is_orthonormal : bool is False if the innerproducts are not close to 0 or 1 Notes ----- this stops as soon as the first deviation from orthonormality is found. Examples -------- >>> from scipy.special import chebyt >>> polys = [chebyt(i) for i in range(4)] >>> r, e = inner_cont(polys, -1, 1) >>> r array([[ 2. , 0. , -0.66666667, 0. ], [ 0. , 0.66666667, 0. , -0.4 ], [-0.66666667, 0. , 0.93333333, 0. ], [ 0. , -0.4 , 0. , 0.97142857]]) >>> is_orthonormal_cont(polys, -1, 1, atol=1e-6) False >>> polys = [ChebyTPoly(i) for i in range(4)] >>> r, e = inner_cont(polys, -1, 1) >>> r array([[ 1.00000000e+00, 0.00000000e+00, -9.31270888e-14, 0.00000000e+00], [ 0.00000000e+00, 1.00000000e+00, 0.00000000e+00, -9.47850712e-15], [ -9.31270888e-14, 0.00000000e+00, 1.00000000e+00, 0.00000000e+00], [ 0.00000000e+00, -9.47850712e-15, 0.00000000e+00, 1.00000000e+00]]) >>> is_orthonormal_cont(polys, -1, 1, atol=1e-6) True rcs||Sr<r r=rDr rrBrCz%is_orthonormal_cont..r)rtolatolFT)r:rErrJrZallclose)rKrLrMrSrTr6rOrNr rDris_orthonormal_conts, rUc@sNeZdZdZdddZdddZddd Zd d Zd d ZddZ ddZ dS)DensityOrthoPolyaUnivariate density estimation by orthonormal series expansion Uses an orthonormal polynomial basis to approximate a univariate density. currently all arguments can be given to fit, I might change it to requiring arguments in __init__ instead. Nr3cs:dk r*|_fddt|D|_}d|_d|_dS)Ncsg|] }|qSr r r4r8r rr9sz-DensityOrthoPoly.__init__..rr)r8r:rK _corfactor _corshift)r r8rrKr rWrrs zDensityOrthoPoly.__init__c sdkr|jd|}n"|_fddt|D|_}t|dsP|dj|_}}|dkr|||j|_}||||f}|_ |d|d} ||} d| |_ | | d}|||_ | |_ fd d|D} | |_||_||S) zIestimate the orthogonal polynomial approximation to the density Ncsg|] }|qSr r r4rWr rr9.sz(DensityOrthoPoly.fit.. offsetfacrr?rcsg|]}|qSr Zmeanr5pr=r rr9Cs)rKr8r:hasattrrrZminmaxoffsetlimitsshrinkshift _transformrcoeffs_verify) r rr8rrcrKZxminZxmaxrbZinterval_lengthZ xintervalrgr r7rfit&s*     zDensityOrthoPoly.fitcsV||dkrt|j}tfddtt|j|jd|D}||}|S)Nc3s|]\}}||VqdSr<r r5cr^xevalr r Nsz,DensityOrthoPoly.evaluate..)rfrErKsumlistziprg _correction)r rmrresr rlrevaluateJs   , zDensityOrthoPoly.evaluatecCs ||S)z,alias for evaluate, except no order argument)rt)r rmr r rrRszDensityOrthoPoly.__call__cCs(|j}dtj|jf|d|_|jS)zcheck for bona fide density correction currently only checks that density integrates to 1 ` non-negativity - NotImplementedYet r[r)rcrrJrtrX)r r r r rrhVs zDensityOrthoPoly._verifycCs,|jdkr||j9}|jdkr(||j7}|S)zhbona fide density correction affine shift of density to make it into a proper density rr)rXrYrr r rrrhs     zDensityOrthoPoly._correctioncCsJ|jdj}|d|d}|j|d|j|}|j|}|||S)ztransform observation to the domain of the density uses shrink and shift attribute which are set in fit to stay rr)rKr rerd)r rr Zilenrerdr r rrfvs  zDensityOrthoPoly._transform)Nr3)Nr3N)N) rrrrrrirtrrhrrrfr r r rrVs $ rVcsndkrtdfddt|D}fdd|D}tfddt||D}|||fS)N2csg|] }|qSr r r4rWr rr9sz%density_orthopoly..csg|]}|qSr r\r]r=r rr9sc3s|]\}}||VqdSr<r rjrlr rrnsz$density_orthopoly..)rlinspacer`rar:rorq)rr8rrmrKrgrsr )r8rrmrdensity_orthopolys rw__main__r#Zfourierr*i') mixture_rvsMixtureDistributionr.r-)locZscalerg?gUUUUUU?gUUUUUU?)sizedistkwargsZchebyt_)rrmz f_hat.min()fint2ruTZred)ZbinsnormedcolorZblack)ZlwrgcCst|t|Sr<)r^r@r=r r rrBrCrBr$cCs t|dS)Nr')r^r=r r rrBrC)rzdop F integral)r}r~Zgreenzusing Chebychev polynomialszusing Fourier polynomialszusing Hermite polynomialscCsg|] }t|qSr )r(r4r r rr9%sr9r+r,i)r*r#cCsg|] }t|qSr r)r4r r rr9+si cCsg|] }t|qSr r"r4r r rr9/scCsd||dS)Nrr.r r=r r rrB0rC)rA)r3)N)rrR)r3N)SrZscipyrrrZnumpyrr rrrr!r/rr0r(r;rQrUrVrwrZexamplesZnobsZmatplotlib.pyplotZpyplotZpltZ%statsmodels.distributions.mixture_rvsryrzdictZmix_kwdsZnormZobs_distZmixZf_hatZgridrgrKZf_hat0ZtrapzZfintprintr`rZdoplothistZplotshow enumerater6r^rOr@rJriZdoprvraZxfrcZdopintZpdfZmpdffiguretitlerZabsZeyer:ZhpolysZinnZastypeintr%r*r#ZhtpolysZinntZpolyscreZdiagr r r rs%    8 ;{         &      4   4