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The pdf has been calculated for (a,b)=(2,3), (a,b)=(8,4), and (a,b)=(12,13) for x in `np.linspace(-10, 10, 41)`. N.B. the `ST5` distribution in R uses an alternative parameterization in terms of nu and tau, where: - nu = (a - b) / (a * b * (a + b)) ** 0.5 - tau = 2 / (a + b) z"data/jf_skew_t_gamlss_pdf_data.npyz x,pdf,a,bnames)rdloadr__file__parentrec fromarraysrss r3gamlss_pdf_datazTestJFSkewT.gamlss_pdf_datalsAww N ! !$H H vv  [ 99r5za,b)r`)rr)rP c||d|k(|d|k(z}|d|d}}t|tj||j|dy)zCompare the pdf with a table of reference values. The table of reference values was produced using R, where the Jones and Faddy skew t distribution is available in the GAMLSS package as `ST5`. r1r2rYrcrrN)rr9r}rc)rmrr1r2rrYrcs r3test_compare_with_gamlss_rz&TestJFSkewT.test_compare_with_gamlss_r|s_  S !Q &?3+?1+D E cDK3U__Q266q9Fr5N) rrrrrrrrrrr;r5r3r{r{[sH;  ^^ : : [[U$>? 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 2..0#EBr5cZttjjddddy)Nr{rarvr%rrrs r3test_regression_ticket_1326z$TestChi2.test_regression_ticket_1326=sEJJNN32Cd3ej8ej:ej:fgd?d@ggdAZejjdBgdCejjdDej@jBdEej.dFd3dGdHfej@jDdEej.dFd3dGdHfej@jBdIej.d3dJdGdEfej@jDdIej.d3dJdGdHfgdKZ#ejjdLgdMdNZ$ejj"ejjdLgdOdPZ%dQZ&dRZ'yS)TTestLevyStableT)autousecdtj_dtj_dtj_tj j tj_y)z2Setup default parameters for levy_stable generatorS1 piecewiseN)r9rparameterizationcdf_default_methodpdf_default_method _levy_stable _QUAD_EPSquad_epsrrs r3reset_levy_stable_paramsz'TestLevyStable.reset_levy_stable_paramswsH.2*/:,/:,%*%7%7%A%A"r5ctjttjdz }tj j |jd}|S)a Sample data points for pdf computed with Nolan's stablec See - http://fs2.american.edu/jpnolan/www/stable/stable.html There's a known limitation of Nolan's executable for alpha < 0.2. The data table loaded below is generated from Nolan's stablec with the following parameter space: alpha = 0.1, 0.2, ..., 2.0 beta = -1.0, -0.9, ..., 1.0 p = 0.01, 0.05, 0.1, 0.25, 0.35, 0.5, and the equivalent for the right tail Typically inputs for stablec: stablec.exe << 1 # pdf 1 # Nolan S equivalent to S0 in scipy .25,2,.25 # alpha -1,-1,0 # beta -10,10,1 # x 1,0 # gamma, delta 2 # output file z.data/levy_stable/stable-Z1-pdf-sample-data.npyx,p,alpha,beta,pctrrdrrrrrrTrss r3nolan_pdf_sample_dataz$TestLevyStable.nolan_pdf_sample_datasL6ww N ! ! < = vv  /C D r5ctjttjdz }tj j |jd}|S)a#Sample data points for cdf computed with Nolan's stablec See - http://fs2.american.edu/jpnolan/www/stable/stable.html There's a known limitation of Nolan's executable for alpha < 0.2. The data table loaded below is generated from Nolan's stablec with the following parameter space: alpha = 0.1, 0.2, ..., 2.0 beta = -1.0, -0.9, ..., 1.0 p = 0.01, 0.05, 0.1, 0.25, 0.35, 0.5, and the equivalent for the right tail Ideally, Nolan's output for CDF values should match the percentile from where they have been sampled from. Even more so as we extract percentile x positions from stablec too. However, we note at places Nolan's stablec will produce absolute errors in order of 1e-5. We compare against his calculations here. In future, once we less reliant on Nolan's paper we might switch to comparing directly at percentiles (those x values being produced from some alternative means). Typically inputs for stablec: stablec.exe << 2 # cdf 1 # Nolan S equivalent to S0 in scipy .25,2,.25 # alpha -1,-1,0 # beta -10,10,1 # x 1,0 # gamma, delta 2 # output file z.data/levy_stable/stable-Z1-cdf-sample-data.npyrrrrss r3nolan_cdf_sample_dataz$TestLevyStable.nolan_cdf_sample_datasMJww N ! ! < = vv  /C D r5cdtjttjdz }|S)a&Sample data where loc, scale are different from 0, 1 Data extracted in similar way to pdf/cdf above using Nolan's stablec but set to an arbitrary location scale of (2, 3) for various important parameters alpha, beta and for parameterisations S0 and S1. z1data/levy_stable/stable-loc-scale-sample-data.npy)rdrrrrrss r3nolan_loc_scale_sample_dataz*TestLevyStable.nolan_loc_scale_sample_datas/ww N ! ! ? @  r5 sample_sizer7r?marksrS0rz alpha,beta))rNr)rNr)rr)rrvz gamma,deltar rrac|tj_tj||||}tj|j |d|j \}} | dkDsJy)N)rrr`r_rrr@)r9rrrCrrk) rmrrrrrlrlsrrps r3rDzTestLevyStable.test_rvssb(.>*   d%U || FF $F 7 14xxr5rrvr ctjjdd}d}d}tjj ||||d}tj |d||||f\}}|d kDsJy ) z3Additional test cases for rvs for alpha equal to 1.h:rNrvrr r_r`rrrUrxN)rdrrr9rrrC)rmrrr_r`rYstatrps r3test_rvs_alpha1zTestLevyStable.test_rvs_alpha1sv y!    ! !%3e'+ " -,,q-%*D#u$=?a4xxr5czgd}tjj|\}}}}t|dddt |ddt |dd t |d d|gd z}tjj|\}}} } t |dt |d t | d d t | dd y)N)wgrr{r{r{r{羼u?r r r r 67',?r r r r tA}˜.?r r r r Gz?rrxr`)\(̿ragۧ1?rg9]c?)r r r r r rgv?r )r9rrkrrr rmrYalpha1beta1loc1scale1rjalpha2beta2rrs r3rpzTestLevyStable.test_fits  ',&7&7&A&A!&D#tV148E4+FGQ/ '1 9 9&+&7&7&A&A"&E#tVVQUBFFA.D&!,r5zUnknown problem with fitstart.rTzalpha,beta,delta,gamma)rrrar)rNrrarparametrizationc|tj_tjj||||dd}tjj |}|\}} } } t ||||g|| | | gdy)z7Test that fit agrees with rvs for each parametrization.rrr:rxrN)r9rrrrkr) rmrrrlrrrr alpha_obsbeta_obs delta_obs gamma_obss r3 test_fit_rvszTestLevyStable.test_fit_rvss-<)  $$ 4U%e$% ))$/471 8Y  D% ' )Y 7 r5cNtjgd}tjj |\}}}}tjj | \}}}} t |d|dk7sJt ||t || t || t | |y)N r r rrrZrZrZr4r4r[r[r r)rdrr9rrkrr) rmrYrrrrrrrrs r3test_fit_beta_flipz!TestLevyStable.test_fit_beta_flip8s HH? @&+&7&7&A&A!&D#tV&+&7&7&A&A1"&E#tVUAqyyFF+EE6*D4%(FF+r5c6d}tjgd}tjj | \}}}}tjj | |z\}}} } t ||t ||t | ||zt | |y)Nr r)rdrr9rrkr) rmSHIFTrYrrrrrrrrs r3test_fit_delta_shiftz#TestLevyStable.test_fit_delta_shiftDs HH? @&+&7&7&A&A1"&E#tV&+&7&7&A&A1"u*&M#tVFF+E5)D$,/FF+r5cpgd}tjj|\}}}}|dks Jd||t|ksJdt|d|gd}tjj|\}}} } |dkDs Jd|| t |kDsJdt |d| y) N) r r rrrZrZrZr4r4r#r zExpected alpha < 1, got zExpected loc < z, got ) r r rrrZrZrZr4r4r$zExpected alpha > 1, got zExpected loc > )r9rrkrrrs r3test_fit_loc_extrapz"TestLevyStable.test_fit_loc_extrapOs 6&+&7&7&A&A!&D#tVz>5fX>>zc!f}DAxvdVDD} 7&+&7&7&A&A"&E#tVz>5fX>>zc"g~FR vFF~r5z pct_range,alpha_range,beta_range)rxrvr)rWr ra)rrr)rxr@rvr4r)rWrvr rra)grrrxr}r ) rxr@rWrfffffff?rv?rrr4rrWrarrrcP|}tj}|jdk(xr|jdk(dj |j|j|j g}ddfdgddfd gddfd gdd fd gd dfdgd dfdgg}t |D]Y\} \} } } | tj_ | || |n|} t5}|jtdtjj| d| d| ddd}tjd5t!| gd|tj"|| dz tj"|| dz tj"| dz g}ddd|d| k\tj$|z}d| d | d!|d"|j&j(d#| }t+|| d| |d$%ddd\y#1swYnxYw#1swYtxYw)&z2Test pdf values against Nolan's stablec.exe outputLinuxi686/dnir'c vtj|dtj|dztj|dz|ddk(|ddk(z|ddk\|ddk\z|ddk(zz|ddktj|dd d gzz|dd ktj|dd d gzz|ddktj|dddgzz|ddk(tj|dddgztjtj|dgdzz|ddk(tj|ddgztjtj|ddgzz|ddk(tj|dddgztjtj|dgdzz|ddk(|ddk(z|ddk(zz|ddk(|ddk(z|ddk(zz|ddk(|dd k(z|ddk(zz|ddk(tj|ddgztjtj|dgdzz|ddk(tj|dddgztjtj|dgdzz|ddk(tj|dd d gztjtj|dddgzz|ddk\zzS)Npctrrrrvrrrrxrrxr@r4rKrWrfr)rvr}r~r&r')gٿ333333ӿrxrrvr/rNrWrKrxr)rrrNrr)rdisinrr alpha_range beta_range pct_ranges r3rIz7TestLevyStable.test_pdf_nolan_samples..sGGAeHi0GGAgJ 45GGAfIz23 vY!^uX_.vY#-wZ3.0uX_. wZ3.GGAeHsCj9: wZ3.GGAeHsCj9:$wZ3.GGAeHr2h78%,wZ3.GGAeHsCj9:GGBFF1V9$5|DE-6wZ3.GGAeHrd34GGBFF1V9$5t<=7@wZ3.GGAeHsCj9:GGBFF1V9$57MNOA#JwZ3.vY#-/uX-/K(TwZ3.vY$.0uX-/U-^wZ3.vY#-/uX-/_2hwZ2-GGAeHrd34GGBFF1V9$57GHIi7rwZ2-GGAeHsCj9:GGBFF1V9$5|DEs<|wZ2-GGAeHsCj9:GGBFF1V9$5RyAB}AF7s*GC,EHr5rrctj|dtj|dztj|dz|ddkDz|ddk7zS)Nr.rrrKrNrdr1r2s r3rIz7TestLevyStable.test_pdf_nolan_samples..snGGAeHi0GGAgJ 45GGAfIz23wZ#%'wZ2% 'r5c|ddk( ztj|dzdvztj|dzS)NrrNr.rr7)rr3r4 is_linux_32r5s r3rIz7TestLevyStable.test_pdf_nolan_samples..s]wZ2%$_&GGAeHi01;&(GGAfIz2 3r5g& .=ctj|dtj|dztj|dz|ddkzS)Nr.rrrKr7r2s r3rIz7TestLevyStable.test_pdf_nolan_samples..s[GGAeHi0GGAgJ 45GGAfIz23wZ3&(r5 fft-simpsonr4c|ddk\tj|dztj|dztj|dzS)Nrrr.rr7r2s r3rIz7TestLevyStable.test_pdf_nolan_samples..s[wZ3&GGAeHi01GGAgJ 45GGAfIz23r5rpctj|dtj|dztj|dz|ddkDz|ddkzS)Nr.rrr rr7r2s r3rIz7TestLevyStable.test_pdf_nolan_samples..smGGAeHi0GGAgJ 45GGAfIz23wZ!^%wZ#% 'r5Nz2Density calculations experimental for FFT method.*rYrrr rr`r_rrcalcabserrrelerrrprBz pdf test  failed with method 'z ' [platform: z]  Ferr_msgverbose)platformunamesystemmachinejoin processorr^r9rrr recordrtrcrdrrrisnanrrr)rmrr5r3r4rrI platform_desctestsixdefault_methodr filter_funcsubdatasuprpsubdata2failuresmessager9s ``` @r3test_pdf_nolan_samplesz%TestLevyStable.test_pdf_nolan_samples[s4% llg-I%--62I  \\5==%// :< tJL ^U% U% W' t& t& kC L"+5!1*  B3AE   0 + 7;t,=A "$%  "H%%))CLG$FO *[[h/ 04FF1ws|#34FF1ws|#34rvvgcl7KK H $H%-((1+  t#88HI""/HNN4H4H3IH:W CL#! ?% %  * $  % % s'*AHAH"A!HH HH% cz|}ddfdgddfdgddfdgdd fd gdd fd gdd fdgg}t|D]V\}\}} } |tj_| || |n|} t 5} | j t dtjj| d| d| ddd} tjd5t| gd| tj| | dz tj| | dz tj| dz g}ddd|d| k\tj| z}d|d|d|jjd|}t| | d| |d !dddYy#1swYkxYw#1swYqxYw)"z4 Test cdf values against Nolan's stablec.exe output.rrcXtj|dtj|dztj|dz|ddk(tj|dgdz|ddk(z|ddk(tj|dgdz|ddk(zzzS Nr.rrrN)r/rrrxrWrKrxrr7r2s r3rIz7TestLevyStable.test_cdf_nolan_samples..PsGGAeHi0GGAgJ 45GGAfIz23 wZ2-GGAfI/ABCuX-/ wZ2-GGAfI?@uX-/  r5r@cVtj|dtj|dztj|dz|ddk(tj|dgdz|ddk(zz|ddk(tj|dgdz|ddk(zzSr]r7r2s r3rIz7TestLevyStable.test_cdf_nolan_samples..esGGAeHi0GGAgJ 45GGAfIz237r)& +=>?5T)+ 7r)& ?;<5T)+ r5r;r4ctj|dtj|dztj|dz|ddkDzS)Nr.rr333333?r7r2s r3rIz7TestLevyStable.test_cdf_nolan_samples..ws[GGAeHi0GGAgJ 45GGAfIz23wZ#%'r5rctj|dtj|dztj|dz|ddkDz|ddkzS)Nr.rrrrar7r2s r3rIz7TestLevyStable.test_cdf_nolan_samples..nGGAeHi0GGAgJ 45GGAfIz23wZ#%'wZ3& (r5rctj|dtj|dztj|dz|ddkDz|ddkzS)Nr.rrrrr7r2s r3rIz7TestLevyStable.test_cdf_nolan_samples..rcr5rxctj|dtj|dztj|dz|ddkDz|ddkzS)Nr.rrrNrr7r2s r3rIz7TestLevyStable.test_cdf_nolan_samples..rcr5Nz[Cumulative density calculations experimental for FFT method. Use piecewise method instead.*rYrrr rr>rrr?rprBz cdf test rCz' rDFrE)r^r9rrr rNrtrkrdrrrrOrrr)rmrr5r3r4rrQrRrSrrTrUrVrprWrXrYs ``` r3test_cdf_nolan_samplesz%TestLevyStable.test_cdf_nolan_samples3s*2%U% ,T$ &t& t& t& t& GL \"+5!1(  B3AE   0 + 7;t,=A "$#  "@ %%))CLG$FO *[[h/ 04FF1ws|#34FF1ws|#34rvvgcl7KK H $H%-((1+'rd*??Os&nn2232hZACL#! ;# #  ( $  # # s'AF0AF$:AF0$F- )F00F: rrrrcrkc tj}|jdk(xrdtjdv}|r|dk(rt j d|}dt j_dt j_ ||d|k(}d|t j_ |d vsJ|dk(rt jjnt jj}||d |d |d d d} t| ||dy)zGTests for pdf and cdf where loc, scale are different from 0, 1 r)32bitrrcz4Test unstable on some platforms; see gh-17839, 17859rrS)rcrkrYrrrarr>r4N)rHrIrJ architecturerrr9rrrrrcrkr) rmrrrrIr9rrUrrBs r3test_location_scalez"TestLevyStable.test_location_scales llg-W'X=R=R=TUV=W2W  45= KKN O*0;,/:,tG}-./0[*~%%%%)U]E   ! !8I8I8M8M  CL''*GFO1!  GDM40r5zmethod,decimal_placesr,rrc tjgd}tjgd}tjgd}tjd5t5}|j t d|t j_t jj|d|dd }t||||d d d d d d y #1swYxYw#1swYy xYw) z sample points extracted from Tables and Graphs of Stable Probability Density Functions - Donald R Holt - 1973 - p 187. )rrrrr r r r rararararrrrrrrr)gtF_?gܵ|?gQ?g@?gec]?gHP?gPkw?g!rh?gTN?g ?gDioɴ?g r?gH?g+?gHPs?gX?g&?g 0*?gv/?ga2U0*?)rrfrvr rrfrvr rrfrvr rrfrvr rrfrvr rrzDensity calculation unstable.*)categoryrYr rr>N) rdrrr rLrtr9rrrcr)rmrdecimal_placesxsdensitybetasrVrcs r3'test_pdf_alpha_equals_one_beta_non_zeroz6TestLevyStable.test_pdf_alpha_equals_one_beta_non_zerosXX H ((     [[X & (9(; s JJ'8  4:E   0##''AuA1'EC Wnf       s% C#'ACC#C C##C,zparams,expected)r r rr )rarrZr)rZr0rrc|tjj|d|d|d|dd}t||y)Nrr rarrr3)r9rr)rmrrRobserveds r3rzTestLevyStable.test_stats sG$$** 1IvayfQivay+  Hh/r5r)rfrvrzfunction,beta,points,expectedrNrrZr{rrcd|cxkrdksJJt||||tjt||y)a\Ensure the pdf/cdf routines do not return nan outside support. This distribution's support becomes truncated in a few special cases: support is [mu, infty) if alpha < 1 and beta = 1 support is (-infty, mu] if alpha < 1 and beta = -1 Otherwise, the support is all reals. Here, mu is zero by default. rr rrN)rrdrr )rmrrrrrRs r3!test_distribution_outside_supportz0TestLevyStable.test_distribution_outside_supports@P5}1}}} V5t 4 GGCK * r5zx,alpha,beta,expected))r/iZ?s0?gz5F?)r~H?|/mg>m?)r.d5 ?L ,ٿg깃\?)r>?ЀjoؿgA]4?)rpa2?Յ пgqQ?cdtj_ttjj ||||y)aTest pdf for x equal to zeta. With S1 parametrization: x0 = x + zeta if alpha != 1 So, for x = 0, x0 will be close to zeta. When case "x equal zeta" is not handled properly and quad_eps is not low enough: - pdf may be less than 0 - logpdf is nan The points from the parametrize block are found randomly so that PDF is less than 0. Reference values taken from MATLAB https://www.mathworks.com/help/stats/stable-distribution.html Afc=rwNr9rrrrcrmrYrrrRs r3test_x_equal_zetaz TestLevyStable.test_x_equal_zetaIs9Z&-"    ! !!5t ! <  r5) )rryrzgz¶?)rr{r|gV?)rr}r~gRE)3\?)rrrgB?)rrrg/^?)-C6ryrzgH?)rr{r|gn?)rr}r~gx\?)rrrgVM%?)rrrgm?cdtj_ttjj ||||y)aTest pdf for x near zeta. With S1 parametrization: x0 = x + zeta if alpha != 1 So, for x = 0, x0 will be close to zeta. When case "x near zeta" is not handled properly and quad_eps is not low enough: - pdf may be less than 0 - logpdf is nan The points from the parametrize block are found randomly so that PDF is less than 0. Reference values taken from MATLAB https://www.mathworks.com/help/stats/stable-distribution.html rrwNrrs r3test_x_near_zetazTestLevyStable.test_x_near_zeta}s9D&-"    ! !!5t ! <  r5ctjj}tdddd}tj }t j d i|}d|_d|_|jd i|d|d d }|jd|d  }tj||k(rJd|_d|_|jd i|d|d d }|jd|d  }t||t||y) NrrWr{rNrrr_r`rrrZ rr;) rdrrr r9rrrrr rmrr unfrozenfrozen unfrozen_afrozen_a unfrozen_bfrozen_bs r3$test_frozen_parameterization_gh20821z3TestLevyStable.test_frozen_parameterization_gh20821sii##Ccs#>$$"",V,$(!"&!X\\QFQ#i.Q ::2C N:C66(j0111$(!"&!X\\QFQ#i.Q ::2C N:CXz*Z*r5ctjj}tdddd}tj }d|_t j d i|}|jd i|d|dd }|jd|d }t||d |_t j d i|}|jd i|d|dd }|jd|d }t||y) NrrWr{rNrrrZrrrr;) rdrrr r9rrrrrs r3%test_frozen_parameterization_gh20821bz4TestLevyStable.test_frozen_parameterization_gh20821bsii##Ccs#>$$$(!"",V,!X\\QFQ#i.Q ::2C N:CXz*$(!"",V,!X\\QFQ#i.Q ::2C N:CXz*r5N)(rrrrrrrrrrrrrrDrrrpr{rrr!r%rdrSrZrfrkrsrrrr9rrkrcrxrrrrr;r5r3rrvsV^^D!B"B ^^B ^^))V ^^   [[ [[ FLL lfll4v{{7G7GH   [[/$> [[C [[]VV,<=>? $ [[ [[Vc1X. / -0 [[>? 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