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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)rcloadr__file__parentrec fromarraysrrs r3gamlss_pdf_datazTestJFSkewT.gamlss_pdf_data~sAww N ! !$H H vv  [ 99r5za,b)ra)rr)rO c||d|k(|d|k(z}|d|d}}t|tj||j|dy)zCompare the pdf with a table of reference values. The table of reference values was produced using R, where the Jones and Faddy skew t distribution is available in the GAMLSS package as `ST5`. r1r2rXrbrr~N)rr9r~rb)rlrr1r2rrXrbs r3test_compare_with_gamlss_rz&TestJFSkewT.test_compare_with_gamlss_rs_  S !Q &?3+?1+D E cDK3U__Q266q9Fr5N) rrrrrrrrrrr;r5r3r|r|msH;  ^^ : : [[U$>? 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 2..0#EBr5cZttjjddddy)Nrzr`rur$rrqs r3test_regression_ticket_1326z$TestChi2.test_regression_ticket_1326QsEJJNN32Cd3ej:ej<ej<fgd?d@ggdAZejjdBgdCejjdDe jBjDdEej.dFd3dGdHfe jBjFdEej.dFd3dGdHfe jBjDdIej.d3dJdGdEfe jBjFdIej.d3dJdGdHfgdKZ$ejjdLgdMdNZ%ejj"ejjdLgdOdPZ&ejdQZ'dRZ(dSZ)yT)UTestLevyStableT)autousecdtj_dtj_dtj_tj j tj_y)z2Setup default parameters for levy_stable generatorS1 piecewiseN)r9rparameterizationcdf_default_methodpdf_default_method _levy_stable _QUAD_EPSquad_epsrqs r3reset_levy_stable_paramsz'TestLevyStable.reset_levy_stable_paramssH.2*/:,/:,%*%7%7%A%A"r5ctjttjdz }tj j |jd}|S)a Sample data points for pdf computed with Nolan's stablec See - http://fs2.american.edu/jpnolan/www/stable/stable.html There's a known limitation of Nolan's executable for alpha < 0.2. The data table loaded below is generated from Nolan's stablec with the following parameter space: alpha = 0.1, 0.2, ..., 2.0 beta = -1.0, -0.9, ..., 1.0 p = 0.01, 0.05, 0.1, 0.25, 0.35, 0.5, and the equivalent for the right tail Typically inputs for stablec: stablec.exe << 1 # pdf 1 # Nolan S equivalent to S0 in scipy .25,2,.25 # alpha -1,-1,0 # beta -10,10,1 # x 1,0 # gamma, delta 2 # output file z.data/levy_stable/stable-Z1-pdf-sample-data.npyx,p,alpha,beta,pctrrcrrrrrrTrrs r3nolan_pdf_sample_dataz$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.npyrrrrrs 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)rcrrrrrrs r3nolan_loc_scale_sample_dataz*TestLevyStable.nolan_loc_scale_sample_datas/ww N ! ! ? @  r5 sample_sizer6r>marksrS0rz alpha,beta))rMr)rMr)rr)rruz gamma,deltar rrbc|tj_tj||||}tj|j |d|j \}} | dkDsJy)N)rrr_r^rrr?)r9rrrBrrj) rlrrrrrkrlsrros r3rCzTestLevyStable.test_rvssb(.>*   d%U || FF $F 7 14xxr5rrur ctjjdd}d}d}tjj ||||d}tj |d||||f\}}|d kDsJy ) z3Additional test cases for rvs for alpha equal to 1.h:rMrurr r^r_rrrTrwN)rcrrr9rrrB)rlrrr^r_rXstatros r3test_rvs_alpha1zTestLevyStable.test_rvs_alpha1 sv y!    ! !%3e'+ " -,,q-%*D#u$=?a4xxr5czgd}tjj|\}}}}t|dddt |ddt |dd t |d d|gd z}tjj|\}}} } t |dt |d t | d d t | dd y)N)wgr rzrzrzrz羼u?r r r r 67',?r r r r tA}˜.?r r r r Gz?rrwr_)\(̿r`gۧ1?rg9]c?)r r r r r rgv?r )r9rrjrrr rlrXalpha1beta1loc1scale1rialpha2beta2rrs r3rozTestLevyStable.test_fits  ',&7&7&A&A!&D#tV148E4+FGQ/ '1 9 9&+&7&7&A&A"&E#tVVQUBFFA.D&!,r5zUnknown problem with fitstart.rUzalpha,beta,delta,gamma)rrr`r)rMrr`rparametrizationc|tj_tjj||||dd}tjj |}|\}} } } t ||||g|| | | gdy)z7Test that fit agrees with rvs for each parametrization.rrr9rwr~N)r9rrrrjr) rlrrrkrrrr alpha_obsbeta_obs delta_obs gamma_obss r3 test_fit_rvszTestLevyStable.test_fit_rvs3s-<)  $$ 4U%e$% ))$/471 8Y  D% ' )Y 7 r5cNtjgd}tjj |\}}}}tjj | \}}}} t |d|dk7sJt ||t || t || t | |y)N r r rrrYrYrYr3r3rZrZr r)rcrr9rrjrr) rlrXrrrrrrrrs r3test_fit_beta_flipz!TestLevyStable.test_fit_beta_flipLs 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)rcrr9rrjr) rlSHIFTrXrrrrrrrrs r3test_fit_delta_shiftz#TestLevyStable.test_fit_delta_shiftXs 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 rrrYrYrYr3r3r%r zExpected alpha < 1, got zExpected loc < z, got ) r r rrrYrYrYr3r3r&zExpected alpha > 1, got zExpected loc > )r9rrjrrrs r3test_fit_loc_extrapz"TestLevyStable.test_fit_loc_extrapcs 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)rwrur)rVr r`)rrr)rwr?rur2r)rVrur rr`)grrrwr|r ) rwr?rVreffffff?ru?rrr2rrVr`rrrcP|}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)Npctrrrrurrrrwrrwr?r2rJrVrer)rur|r}r(r))gٿ333333ӿrwrrur1rMrVrJrwr)rrrMrr)rcisinrr alpha_range beta_range pct_ranges r3rHz7TestLevyStable.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)Nr0rrrJrMrcr3r4s r3rHz7TestLevyStable.test_pdf_nolan_samples..snGGAeHi0GGAgJ 45GGAfIz23wZ#%'wZ2% 'r5c|ddk( ztj|dzdvztj|dzS)NrrMr0rr9)rr5r6 is_linux_32r7s r3rHz7TestLevyStable.test_pdf_nolan_samples..s]wZ2%$_&GGAeHi01;&(GGAfIz2 3r5g& .=ctj|dtj|dztj|dz|ddkzS)Nr0rrrJr9r4s r3rHz7TestLevyStable.test_pdf_nolan_samples..s[GGAeHi0GGAgJ 45GGAfIz23wZ3&(r5 fft-simpsonr3c|ddk\tj|dztj|dztj|dzS)Nrrr0rr9r4s r3rHz7TestLevyStable.test_pdf_nolan_samples..s[wZ3&GGAeHi01GGAgJ 45GGAfIz23r5roctj|dtj|dztj|dz|ddkDz|ddkzS)Nr0rrr rr9r4s r3rHz7TestLevyStable.test_pdf_nolan_samples.. smGGAeHi0GGAgJ 45GGAfIz23wZ!^%wZ#% 'r5Nz2Density calculations experimental for FFT method.*rXrrr rr_r^rrcalcabserrrelerrrorDz pdf test  failed with method 'z ' [platform: z]  Ferr_msgverbose)platformunamesystemmachinejoin processorr\r9rrr recordrsrbrcrrrisnanrrr)rlrr7r5r6rrK platform_desctestsixdefault_methodr filter_funcsubdatasuprosubdata2failuresmessager;s ``` @r3test_pdf_nolan_samplesz%TestLevyStable.test_pdf_nolan_samplesos6% 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 Nr0rrrM)r1rrrwrVrJrwrr9r4s r3rHz7TestLevyStable.test_cdf_nolan_samples..fsGGAeHi0GGAgJ 45GGAfIz23 wZ2-GGAfI/ABCuX-/ wZ2-GGAfI?@uX-/  r5r?cVtj|dtj|dztj|dz|ddk(tj|dgdz|ddk(zz|ddk(tj|dgdz|ddk(zzSr_r9r4s r3rHz7TestLevyStable.test_cdf_nolan_samples..{sGGAeHi0GGAgJ 45GGAfIz237r)& +=>?5T)+ 7r)& ?;<5T)+ r5r=r3ctj|dtj|dztj|dz|ddkDzS)Nr0rr333333?r9r4s r3rHz7TestLevyStable.test_cdf_nolan_samples..s[GGAeHi0GGAgJ 45GGAfIz23wZ#%'r5rctj|dtj|dztj|dz|ddkDz|ddkzS)Nr0rrrrcr9r4s r3rHz7TestLevyStable.test_cdf_nolan_samples..nGGAeHi0GGAgJ 45GGAfIz23wZ#%'wZ3& (r5rctj|dtj|dztj|dz|ddkDz|ddkzS)Nr0rrrrr9r4s r3rHz7TestLevyStable.test_cdf_nolan_samples..rer5rwctj|dtj|dztj|dz|ddkDz|ddkzS)Nr0rrrMrr9r4s r3rHz7TestLevyStable.test_cdf_nolan_samples..rer5Nz[Cumulative density calculations experimental for FFT method. Use piecewise method instead.*rXrrr rr@rrrArorDz cdf test rEz' rFFrG)r\r9rrr rPrsrjrcrrrrQrrr)rlrr7r5r6rrSrTrUrrVrWrXrorYrZr[s ``` r3test_cdf_nolan_samplesz%TestLevyStable.test_cdf_nolan_samplesHs*4%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: rrrrbrjc 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+32bitrrbz4Test unstable on some platforms; see gh-17839, 17859rrS)rbrjrXrrr`rr@r3N)rJrKrL architecturerrr9rrrrrbrjr) rlrrrrKr;rrWrr@s 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 r`r`r`r`rrrrrrrr)gtF_?gܵ|?gQ?g@?gec]?gHP?gPkw?g!rh?gTN?g ?gDioɴ?g r?gH?g+?gHPs?gX?g&?g 0*?gv/?ga2U0*?)rrerur rrerur rrerur rrerur rrerur rrzDensity calculation unstable.*)categoryr[r rr@N) rcrrr rKrsr9rrrbr)rlrdecimal_placesxsdensitybetasrXrbs 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)rrrr )r`rrYr)rYr/rrc|tjj|d|d|d|dd}t||y)Nrr r`rrr4)r9rr)rlrrQobserveds r3rzTestLevyStable.test_stats#sG$$** 1IvayfQivay+  Hh/r5r)rerurzfunction,beta,points,expectedrMrrYrzrr~cd|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)rrcrr )rlrrrrrQs r3!test_distribution_outside_supportz0TestLevyStable.test_distribution_outside_support1s@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=ryNr9rrrrbrlrXrrrQs r3test_x_equal_zetaz TestLevyStable.test_x_equal_zeta_s9Z&-"    ! !!5t ! <  r5) )rr{r|gz¶?)rr}r~gV?)rrrgRE)3\?)rrrgB?)rrrg/^?)-C6r{r|gH?)rr}r~gn?)rrrgx\?)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 rryNrrs r3test_x_near_zetazTestLevyStable.test_x_near_zetas9D&-"    ! !!5t ! <  r5c*tjSro) threadingLockrqs r3levy_stable_lockzTestLevyStable.levy_stable_locks~~r5c Btjj}tdddd}tj }t j d i|}|5d|_d|_|jd i|d|d d }|jd|d  }tj||k(rJd|_d|_|jd i|d|d d }|jd|d  } t| |t||dddy#1swYyxYw) NrrVrzrMrrr^r_rrrY rr;) rcrrr r9rrrrr rlrrr unfrozenfrozen unfrozen_afrozen_a unfrozen_bfrozen_bs r3$test_frozen_parameterization_gh20821z3TestLevyStable.test_frozen_parameterization_gh20821s ii##Ccs#>$$"",V,  /(,H %&*F #%UURc)nUJzzrIzGHvvh*45 55(,H %&*F #%UURc)nUJzzrIzGH : . X . / / /s B;DDc tjj}tdddd}tj }|5d|_t j d i|}|jd i|d|dd }|jd|d }t||d |_t j d i|}|jd i|d|dd }|jd|d } t| |dddy#1swYyxYw) NrrVrzrMrrrYrrrr;) rcrrr r9rrrrrs r3%test_frozen_parameterization_gh20821bz4TestLevyStable.test_frozen_parameterization_gh20821bsii##Ccs#>$$  /(,H %&&00F%UURc)nUJzzrIzGH : .(,H %&&00F%UURc)nUJzzrIzGH : . / / /s B=DD N)*rrrrrrrrrrrrrrCrrror|rr r#r'rcrR thread_unsafer\rhrmrurrrr9rrjrbrzrrrrrr;r5r3rrsV^^D!B"B ^^B ^^))V ^^   [[ [[ FLL lfll4v{{7G7GH   [[/$> [[C [[]VV,<=>? $ [[ [[Vc1X. / -0 [[>? 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This unit test is also a regression test for ticket 1548. The exact values are: mean: mu = a / (a + 1) variance: sigma**2 = a / ((a + 2) * (a + 1) ** 2) skewness: One formula (see https://en.wikipedia.org/wiki/Skewness) is gamma_1 = (E[X**3] - 3*mu*E[X**2] + 2*mu**3) / sigma**3 A short calculation shows that E[X**k] is a / (a + k), so gamma_1 can be implemented as n = a/(a+3) - 3*(a/(a+1))*a/(a+2) + 2*(a/(a+1))**3 d = sqrt(a/((a+2)*(a+1)**2)) ** 3 gamma_1 = n/d Either by simplifying, or by a direct calculation of mu_3 / sigma**3, one gets the more concise formula: gamma_1 = -2.0 * ((a - 1) / (a + 3)) * sqrt((a + 2) / a) kurtosis: (See https://en.wikipedia.org/wiki/Kurtosis) The excess kurtosis is gamma_2 = mu_4 / sigma**4 - 3 A bit of calculus and algebra (sympy helps) shows that mu_4 = 3*a*(3*a**2 - a + 2) / ((a+1)**4 * (a+2) * (a+3) * (a+4)) so gamma_2 = 3*(3*a**2 - a + 2) * (a+2) / (a*(a+3)*(a+4)) - 3 which can be rearranged to gamma_2 = 6 * (a**3 - a**2 - 6*a + 2) / (a*(a+3)*(a+4)) )rM)rurrzr)ry)gUUUUUU?r! g^cQg333333rrN)r9rr)rPr1 exact_mvskrs r3test_powerlaw_statsr !sG>, = ?E4 :~~##Av#6!$ 34r5c\tjjdd}t|dy)Nrr rz)r9rrrros r3test_powerlaw_edger !s" a#ACr5cvtjjdd}t|dtjj dgd}t|t j ddgtjjdgd}t|t j dt j gy)Nrr rz)rerMrrM)r9r,rrrbrcrr s r3test_exponpow_edger !s a#AC 1./ARVVS#&' a!12ARVVS266'*+r5c^tjjddd}t|dy)Nrr rM)r9gengammarbrr s r3test_gengamma_edger !s$ 1a#ACr5za, c, ref, tol))g`6Ar g2j!@r})rr ggA@r})rr rr})rr g#@r)g`FArgI~mr})gl$W}Lrg.I#Ur}c\ttjj||||yr)rr9r rA)r1rr~rMs r3test_gengamma_extreme_entropyr !s!"ENN**1a0#C@r5ctjjddd}|dk(sJtjjddd}|tj k(sJy)Nrr rrz)r9r rbrrcr)rors r3!test_gengamma_endpoint_with_neg_cr !sO 1a$A 8O8 >> Ar *D BFF7??r5ctjjddd}t|dtjjddd}t|dy)NrrJrMg>rYgqq?)r9r rrr s r3test_gengamma_munpr !sF Rb)A:& RR(A6"r5c\tjgd}tjd5t5}|j t d|j t dtjj|ddddddy#1swYxYw#1swYyxYw)N)0g>X*ȿgPix#?g=?g&^?goϿg뵏˿gO_ZQ?g\?g~?g)/?gRܶ?g핚οgH߃[g?g S"gm_ пg7k?g#BF?g KU= ?gT?gZg[Qs?g2?g6Hwƭ8?g*- ?ga]?g'?g޻?g?J_?g~sw?g0FfͿgg"@ӿg?%:˿g&zM?g?Gg F?g~,g HgW2ҿgTzy?gwm ǿg@gW=ֿg 't?g".v1glg.Hꮿg,?g*˿rrz:The maximum number of subdivisions .50. has been achieved.z-floating point number truncated to an integer) rcrrr rKrrsr9r>r)r* rXs r3test_ksone_fit_freezer !s  / 0A X &   C JJ)# $ JJ~F H KKOOA     s# B"A BB"B B""B+c Ttjttddd }gd}t t j j||dt t j j|dzj|dt j j|dzd z j}tjt j j|t j j|z }t ||d y) Nrrf r)gDB.g8_$g%`OAgjhwHRgWMB@v`g7VY}igۦx\Arg xg#g/dg3$gegd)gNq{3g يGǓgr 4g37' gfFg?dJgضg2, gm%gӅJgz;pg ɿ{gk2gij<%giVSͶgBR4g Q/Mr%ry+=y|=r) rcrr rrr9rrrealimagrr)rXrQderivderiv_expecteds r3test_norm_logcdfr "s Dq#q)* ++A0HEJJL''*H4@EJJL''F 388(NZZ  q6z *5 0 6 6EVVEJJ--a05::3D3DQ3GGHNE>6r5ctjgd}tjgd}tjj |}t ||dy)N)gMb{Gzg{Gztg~jtX)g#]"sIINN4 C!$A)U[[__aA3?)U[[__aAqcB)U[[__aAq!<)U[[__aA3cJ)U[[__b!CH)U[[__aQCsK)U[[__aQCsK)U[[..A3cJ)U[[00"aSL)U[[__aQCsK KKOOAq! 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