'YHhD dZddlmZmZmZddlZejdZGddZ GddZ Gd d Z ejejd z ZGd d ZdEdZdFdZdGdZGddZdHdZedk(rgdZdZddlmZddlmZmZeddeddfZeddgeej@ej@ge Z!eZ"d!evrBe!e!d"kDe!d kzdz Z!ee!e d#d$\Z#Z$Z%Z&e#Z'ejPe#e$Z)e*d%e#jWe#e#jWz Z#ejPe#e$Z,e*d&e)e,e#e,zZ#dZ-e-rSej\e!d'd(d)*ej^e$e#d d+,ej^e$e'd d-,ej`e1e&ddD]8\Z2Z3e1e&ddD]%\Z4Z5e*e2e4ejld.d/dd':e&D]Z3e*ejld0d/dd1evrejoe!e d#2Z8ejre!jWe!juZ$e8e$Z;ejle8ge8jxdZ=e*d3e=e"j}e$ddgej@ej@ge4Z?dZ-e-reejej\e!d'd(d)*ej^e$e;d d+,ej^e$e?d d5,ejd6d7evrkeZ8de8_Be8joe!e d82Z8ejre!jWe!juZ$e8e$Z;ejle8ge8jxdZ=e*d3e=e"j}e$ddgej@ej@ge4Z?dZ-e-reejej\e!d'd(d)*ejd9ej^e$e;d d+,ej^e$e?d d5,e*ejtejee8jLdddddejdz d:evr|eZ8de8_Be8joe!ed#2Z8ejre!jWe!juZ$e8e$Z;ejle8ge8jxdZ=e*d3e=e"j}e$ddgej@ej@ge4Z?dZ-e-rvejej\e!d'd(d)*ej^e$e;d d+,ej^e$e?d d5,ejd;ej`e*ejtejee8jLdddddejdz eEdDcgc] }e| c}ZFeeFdzjeIdd?lJmKZKmLZLeEdDcgc] }eK| c}ZMeeMd@dAdZNe*eNd>zjeIeEdBDcgc] }eL| c}ZOeeOd/ddCD\ZPZQe*ejtejePejejePz yycc}wcc}wcc}w)Iadensity estimation based on orthogonal polynomials Author: Josef Perktold Created: 2011-05017 License: BSD 2 versions work: based on Fourier, FPoly, and chebychev T, ChebyTPoly also hermite polynomials, HPoly, works other versions need normalization TODO: * check fourier case again: base is orthonormal, but needs offsetfact = 0 and does not integrate to 1, rescaled looks good * hermite: works but DensityOrthoPoly requires currently finite bounds I use it with offsettfactor 0.5 in example * not implemented methods: - add bonafide density correction - add transformation to domain of polynomial base - DONE possible problem: what is the behavior at the boundary, offsetfact requires more work, check different cases, add as option moved to polynomial class by default, as attribute * convert examples to test cases * need examples with large density on boundary, beta ? * organize poly classes in separate module, check new numpy.polynomials, polyvander * MISE measures, order selection, ... enhancements: * other polynomial bases: especially for open and half open support * wavelets * local or piecewise approximations )stats integratespecialN@ceZdZdZdZdZy)FPolyaBOrthonormal (for weight=1) Fourier Polynomial on [0,1] orthonormal polynomial but density needs corfactor that I do not see what it is analytically parameterization on [0,1] from Sam Efromovich: Orthogonal series density estimation, 2010 John Wiley & Sons, Inc. WIREs Comp Stat 2010 2 467-476 cB||_d|_|j|_y)N)r)orderdomain intdomainselfr s k/var/www/html/planif/env/lib/python3.12/site-packages/statsmodels/sandbox/nonparametric/densityorthopoly.py__init__zFPoly.__init__<s  c|jdk(rtj|Sttjtj |jz|zzSNr)r np ones_likesqr2cospirxs r__call__zFPoly.__call__AsA ::?<<? ""&&!3a!788 8rN__name__ __module__ __qualname____doc__rrrrrr.s % 9rrceZdZdZdZdZy)F2PolyaOrthogonal (for weight=1) Fourier Polynomial on [0,pi] is orthogonal but first component does not square-integrate to 1 final result seems to need a correction factor of sqrt(pi) _corfactor = sqrt(pi) from integrating the density Parameterization on [0, pi] from Peter Hall, Cross-Validation and the Smoothing of Orthogonal Series Density Estimators, JOURNAL OF MULTIVARIATE ANALYSIS 21, 189-206 (1987) cp||_dtjf|_|j|_d|_yr)r rrr r offsetfactorrs rrzF2Poly.__init__Us+ "%%j rc,|jdk(r9tj|tjtjz St tj |j|zztjtjz Sr)r rrsqrtrrrrs rrzF2Poly.__call__[sY ::?<<?RWWRUU^3 3"&&a00277255>A ArNrr"rrr$r$Gs  Brr$ceZdZdZdZdZy) ChebyTPolyaLOrthonormal (for weight=1) Chebychev Polynomial on (-1,1) Notes ----- integration requires to stay away from boundary, offsetfactor > 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. cb||_ddlm}|||_d|_d|_d|_y)Nr)chebyt)r )g !g !?g{Gz?)r scipy.specialr,polyr r r&)rr r,s rrzChebyTPoly.__init__os- (5M  * rcX|jdk(rEtj|d|dzz dzz tjtjz S|j |d|dzz dzz tjtjz tjdzS)Nrr g?)r rrr(rr/rs rrzChebyTPoly.__call__ys~ ::?<<?a1f%55rwwruu~E E99Q<1QT6T"22BGGBEENBBGGAJN NrNrr"rrr*r*as !Orr*r1ceZdZdZdZdZy)HPolyzOrthonormal (for weight=1) Hermite Polynomial, uses finite bounds for current use with DensityOrthoPoly domain is defined as [-6,6] cT||_ddlm}|||_d|_d|_y)Nr)hermite)?)r r.r5r/r r&)rr r5s rrzHPoly.__init__s& )EN  rc|j}d|tjdztj|dzzt zz||zdz z }tj |}|j||zS)Nrr r1)r rlogrgammalnlogpi2expr/)rrklnfactfacts rrzHPoly.__call__se JJ!BFF2J,1)==FG!A#a%Ovvf~yy|d""rNrr"rrr3r3s  #rr3c tjt|Dcgc]}|||c}}|Scc}wN)r column_stackrange)rpolybaser ipolyarrs r polyvanderrJs4oouU|D!{x{1~DEG NEs;c t|}tj||f}|jtjtj ||f}t |D]}t |dzD]n}|| ||  tj fd||\} } ntj fd||\} } | |||f<| |||f<||k(ra| |||f<| |||f<p||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]]) r c8||z|zSrDr")rp1p2weights rzinner_cont..sRU2a5[5Jrc&||zSrDr"rrMrNs rrPzinner_cont..sRU2a5[r) lenremptyfillnanzerosrFrquad) polyslowerupperrOn_polys innerprodinterrrHjinnperrrMrNs ` @@r inner_contrbsB%jG'7+,I NN266 XXw( )F 7^ "qs "AqBqB!%NN+J',e5 c&NN+@%O c!IacNF1Q3K6!% !A#!qs  " " f rc tt|D]\}t|dzD]I}|||| tj fd||d}t j |||k(||rHy^y)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 r c&||zSrDr"rRs rrPz%is_orthonormal_cont..sAr!urr)rtolatolFT)rFrSrrXrallclose) rYrZr[rerfrHr_r]rMrNs @@ris_orthonormal_contrhs}X3u: qs AqBqB!'$(4K4>>#A ADK&$(6M4&=#A AFT[ )fQi/4K ?* !I-"4F] __Q'')./A1Q4++-//    5"E*0s D34D8c |j|t|j}tfdt t |j |jd|D}|j|}|S)Nc3:K|]\}}||zywrDr".0crxevals r z,DensityOrthoPoly.evaluate..MsTA!AeH*T)rxrSrYsumlistziprz _correction)rrr ress ` revaluatezDensityOrthoPoly.evaluateIsc& = OETc$++tzz.J)KFU)STTs# rc$|j|S)z,alias for evaluate, except no order argument)r)rrs rrzDensityOrthoPoly.__call__Qs}}U##rc|j}dtj|jg|dz |_|jS)zcheck for bona fide density correction currently only checks that density integrates to 1 ` non-negativity - NotImplementedYet rpr)rurrXrrl)rr s rr{zDensityOrthoPoly._verifyUs<KK Y^^DMMFIFqIIrc~|jdk7r||jz}|jdk7r||jz }|S)zhbona fide density correction affine shift of density to make it into a proper density r r)rlrmrs rrzDensityOrthoPoly._correctiongs= ??a   A >>Q   Arc|jdj}|d|dz }|j|d|jz |z z }|j|z}||z |zS)ztransform observation to the domain of the density uses shrink and shift attribute which are set in fit to stay rr )rYr rwrv)rrr ilenrwrvs rrxzDensityOrthoPoly._transformusfA%%q F1I% VAYt{{2477t#E V##r)NrB)NrBNrD) rrr r!rrrrr{rrxr"rrrjrjs+"H$$ $rrjcL3tj|j|jdt |Dcgc] }|| }}|Dcgc]}||j }}t fdt||D}|||fScc}wcc}w)N2c3:K|]\}}||zywrDr"rs rrz$density_orthopoly..s 8TQa%j 8r)rlinspacerrrsrFryrr) rrGr rrHrYrrzrs ` rdensity_orthopolyrs } AEEGAEEGB/#(, /QXa[ /E /&+ +qtkkm +F + 8S%7 8 8C vu $$ 0,s BB!__main__)r,fourierr5i') mixture_rvsMixtureDistributionr:r8)locscaler g?gUUUUUU?gUUUUUU?)sizedistkwargschebyt_)r rz f_hat.min()fint2rTred)binsnormedcolorblack)lwrgc0t|t|zSrD)rrNrs rrPrPs1Q41:rr-ct|dzS)Nr1)rrs rrPrPs1Q47rr,)r zdop F integral)rrgreenzusing Chebychev polynomialsrzusing Fourier polynomialsr5zusing Hermite polynomialsr6r7i)r5r,i cd||zz dzS)Nr r:r"rs rrPrP/sq1u6Fr)rO)rBrD)rg:0yE>)rBN)Sr!scipyrrrnumpyrr(rrr$r*r;rr=r3rJrbrhrjrrexamplesnobsmatplotlib.pyplotpyplotplt%statsmodels.distributions.mixture_rvsrrdictmix_kwdsnormobs_distmixf_hatgridrzrYf_hat0trapzfintprintrrrdoplothistplotshow enumeraterHrr_rNrXrdoprrsxfrudopintpdfmpdffiguretitleroabseyerFhpolysinnastypeintr.r5r,htpolysinntpolyscrediag)rHs0rrsM$J,+rwwr{992BB4OO@ q##*5p4vw$w$v%. z/H D#R B'(<=HD;TUZZ8P"$H  C HXb[XaZ89#=&7xSU]a%b"tVUyud+ mUYY[)$ t, gtU#   CHHXBt5 A CHHT5Qg 6 CHHT6as 3 CHHJU2AY' ICAa!%), I"a.)..)=r!DQGH I I ;A .)..!2Bq9 : ; 8 $$Xz$Dr{{8<<>8<<>: Y1cjj1!4 'wwtd4[ EJJ/G"$  CJJL CHHXBt5 A CHHT2!7 3 CHHT4AW 5 CII3 4H  gghbg1r{{8<<>8<<>: Y1cjj1!4 'wwtd4[ EJJ/G"$  CJJL CHHXBt5 A CII1 2 CHHT2!7 3 CHHT4AW 5 fbffVRVVJsyy!}a;A>q IJKLH  gghRg0r{{8<<>8<<>: Y1cjj1!4 'wwtd4[ EJJ/G"$  CJJL CHHXBt5 A CHHT2!7 3 CHHT4AW 5 CII1 2 CHHJ fbffVRVVJsyy!}a;A>q IJKL !&a )1eAh )F VR #A &C &"&&fbffQi( )* 3v:  c "#-#(8,awqz,G gsB ' *D 4;  s #$!&q *AfQi *F fb!,F GDAq &"&&GBGGGBGGAJ//0 12_F* -+s3]+9]0<]5