a \:bE @s,dZddlmZmZmZddlZedZGddde Z Gddde Z Gd d d e Z e ejd ZGd d d e ZdOddZdPddZdQddZGddde ZdRddZedkr(gdZdZddlmZddlmZmZeddd ed!d"d fZ ed#d$geej!ej!ge d%Z"eZ#d&evrbe"e"d'ke"d k@dZ"ee"e d(dd)\Z$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$dZ.e.rej/e"d,d-d.d/ej0e%e$d d0d1ej0e%e(d d2d1e1e2e'ddD]D\Z3Z4e2e'ddD](\Z5Z6e+e3e5e7d3d4d5d!dqqe'D]Z4e+e7d6d4d5d!qDd7evr,ej8e"e d(d8Z9e:e",e";Z%e9e%Ze+d9e>e#j?e%d#d$gej!ej!ge d:Z@d!Z.e.r,eAej/e"d,d-d.d/ej0e%ed8Z9e:e",e";Z%e9e%Ze+d9e>e#j?e%d#d$gej!ej!ge d:Z@d!Z.e.reAej/e"d,d-d.d/eBd?ej0e%ee+d9e>e#j?e%d#d$gej!ej!ge d:Z@d!Z.e.reAej/e"d,d-d.d/ej0e%e 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')rrlogrZgammalnlogpi2Zexpr&)r rkZlnfactZfactr r rrs0 zHPoly.__call__Nrr r r rr(sr(cs"tfddt|D}|S)Ncsg|]}|qSr r .0ipolybaserr r zpolyvander..)rZ column_stackrange)rr7rZpolyarrr r6r polyvandersr;c st|}t||f}|tjt||f}t|D]}t|dD]}||||durtfdd||\} } 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 rr9zinner_cont..cs||Sr<r r=r?r@r rrBr9) lenremptyZfillnanZzerosr:rquad) polyslowerupperrAZn_polys innerprodZinterrr5jZinnperrr r>r inner_conts&!      rN: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=rCr rrBr9z%is_orthonormal_cont..r)rtolatolFT)r:rDrrGrZallclose)rHrIrJrPrQr5rLrKr rCris_orthonormal_conts, rRc@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. Nr2cs:dur*|_fddt|D|_}d|_d|_dS)Ncsg|] }|qSr r r3r7r rr8r9z-DensityOrthoPoly.__init__..rr)r7r:rH _corfactor _corshift)r r7rrHr rTrrs zDensityOrthoPoly.__init__c sdur|jd|}n"|_fddt|D|_}t|dsP|dj|_}}|dur|||j|_}||||f}|_ |d|d} ||} d| |_ | | d}|||_ | |_ fd d|D} | |_||_||S) zIestimate the orthogonal polynomial approximation to the density Ncsg|] }|qSr r r3rTr rr8.r9z(DensityOrthoPoly.fit.. offsetfacrr?rcsg|]}|qSr Zmeanr4pr=r rr8Cr9)rHr7r:hasattrrrWminmaxoffsetlimitsshrinkshift _transformrcoeffs_verify) r rr7rr`rHZxminZxmaxr_Zinterval_lengthZ xintervalrdr r6rfit&s*     zDensityOrthoPoly.fitcsV||durt|j}tfddtt|j|jd|D}||}|S)Nc3s|]\}}||VqdSr<r r4cr[xevalr r Nr9z,DensityOrthoPoly.evaluate..)rcrDrHsumlistziprd _correction)r rjrresr rirevaluateJs   , zDensityOrthoPoly.evaluatecCs ||S)z,alias for evaluate, except no order argument)rq)r rjr r rrRszDensityOrthoPoly.__call__cCs*|j}dtj|jg|Rd|_|jS)zcheck for bona fide density correction currently only checks that density integrates to 1 ` non-negativity - NotImplementedYet rXr)r`rrGrqrU)r r r r rreVs zDensityOrthoPoly._verifycCs,|jdkr||j9}|jdkr(||j7}|S)zhbona fide density correction affine shift of density to make it into a proper density rr)rUrVrr r rrohs     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)rHr rbra)r rr Zilenrbrar r rrcvs  zDensityOrthoPoly._transform)Nr2)Nr2N)N) rrrrrrfrqrrerorcr r r rrSs $ rScsndurtdfddt|D}fdd|D}tfddt||D}|||fS)N2csg|] }|qSr r r3rTr rr8r9z%density_orthopoly..csg|]}|qSr rYrZr=r rr8r9c3s|]\}}||VqdSr<r rgrir rrkr9z$density_orthopoly..)rlinspacer]r^r:rlrn)rr7rrjrHrdrpr )r7rrjrdensity_orthopolys rt__main__)r#fourierr*i') mixture_rvsMixtureDistributionr.r-)ZlocZscalerg?gUUUUUU?gUUUUUU?)sizedistkwargsZchebyt_)rrjz f_hat.min()fint2rrTZred)ZbinsZnormedcolorZblack)ZlwrgcCst|t|Sr<)r[r@r=r r rrBr9rBr$cCs t|dS)Nr')r[r=r r rrBr9r#)rzdop F integral)rzr{Zgreenzusing Chebychev polynomialsrvzusing Fourier polynomialsr*zusing Hermite polynomialscCsg|] }t|qSr )r(r3r r rr8%r9r8r+r,i)r*r#cCsg|] }t|qSr r)r3r r rr8+r9i cCsg|] }t|qSr r"r3r r rr8/r9cCsd||dS)Nrr.r r=r r rrB0r9)rA)r2)N)rrO)r2N)TrZscipyrrrZnumpyrr robjectrrr!r/rr0r(r;rNrRrSrtrZexamplesZnobsZmatplotlib.pyplotZpyplotZpltZ%statsmodels.distributions.mixture_rvsrwrxdictZmix_kwdsZnormZobs_distZmixZf_hatZgridrdrHZf_hat0ZtrapzZfintprintr]r~ZdoplotZhistZplotZshow enumerater5r[rLr@rGrfZdoprsr^Zxfr`ZdopintZpdfZmpdfZfiguretitlerWabsZeyer:ZhpolysZinnZastypeintr%r*r#ZhtpolysZinntZpolyscreZdiagr r r rs%    8 ;{        &      4   4