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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,pctrrjrrrrrrTr|s r3nolan_pdf_sample_dataz$TestLevyStable.nolan_pdf_sample_datasM6w NN ! < =  v  /C DD r5ctjttjdz }tj|jd}|S)a#Sample data points for cdf computed with Nolan's stablec See - http://fs2.american.edu/jpnolan/www/stable/stable.html There's a known limitation of Nolan's executable for alpha < 0.2. The data table loaded below is generated from Nolan's stablec with the following parameter space: alpha = 0.1, 0.2, ..., 2.0 beta = -1.0, -0.9, ..., 1.0 p = 0.01, 0.05, 0.1, 0.25, 0.35, 0.5, and the equivalent for the right tail Ideally, Nolan's output for CDF values should match the percentile from where they have been sampled from. Even more so as we extract percentile x positions from stablec too. However, we note at places Nolan's stablec will produce absolute errors in order of 1e-5. We compare against his calculations here. In future, once we less reliant on Nolan's paper we might switch to comparing directly at percentiles (those x values being produced from some alternative means). Typically inputs for stablec: stablec.exe << 2 # cdf 1 # Nolan S equivalent to S0 in scipy .25,2,.25 # alpha -1,-1,0 # beta -10,10,1 # x 1,0 # gamma, delta 2 # output file z.data/levy_stable/stable-Z1-cdf-sample-data.npyr rr r|s r3nolan_cdf_sample_dataz$TestLevyStable.nolan_cdf_sample_datasNJw NN ! < =  v  /C DD r5cbtjttjdz }|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)rjrrrrr|s r3nolan_loc_scale_sample_dataz*TestLevyStable.nolan_loc_scale_sample_datas1w NN ! ? @   r5 sample_sizer=rEmarksrS0rz alpha,beta))rTr)rTr)rr)r&r|z gamma,deltar rrvc|tj_tj||||}tj||d|j\}} | dksJdS)N)rrrfrerrrF)r9rrrIrrq) rsrrrrrrrlsrrvs r3rJzTestLevyStable.test_rvssp(.>*  d%U   | FF $F 7 7  14xxxxxxr5rr|r ctjdd}d}d}tj||||d}tj|d||||f\}}|d ksJd S) z3Additional test cases for rvs for alpha equal to 1.h:rTr|rrrerfrrr[r~N)rjrrr9rrrI)rsrrrerfr_statrvs r3test_rvs_alpha1zTestLevyStable.test_rvs_alpha1s y!!!   ! !%3e'+ " - -,q-%*D#u$=???a4xxxxxxr5cgd}tj|\}}}}t|dddt |ddt |dd t |d d|gd z}tj|\}}} } t |dt |d t | d d t | dd dS)N)wgrrrrr羼u?r r r r 67',?r!r!r!r!tA}˜.?r"r"r"r"Gz?rr~rf)\(̿rggۧ1?rg9]c?)r"r"r"r"r"rgv?r!)r9rrtrrr rsr_alpha1beta1loc1scale1rsalpha2beta2rrs r3ryzTestLevyStable.test_fits    ',&7&A&A!&D&D#tV148888E4+++FGQ/// '1   999 9&+&7&A&A"&E&E#tVVQUBFFA...D&!,,,,,r5zUnknown problem with fitstart.rjzalpha,beta,delta,gamma)rrrgr)rTrrgrparametrizationc|tj_tj||||dd}tj|}|\}} } } t ||||g|| | | gddS)z7Test that fit agrees with rvs for each parametrization.rrrAr~rN)r9rr,rrtr) rsrrrrrr,rr alpha_obsbeta_obs delta_obs gamma_obss r3 test_fit_rvszTestLevyStable.test_fit_rvss-<) $$ 4U%e$%  ))$//471 8Y  D% ' )Y 7      r5cxtjgd}tj|\}}}}tj| \}}}} t |d|dksJt ||t || t || t | |dS)N r r rrr`r`r`r:r:rarar r)rjrr9rrtrr) rsr_r&r'r(r)r*r+rrs r3test_fit_beta_flipz!TestLevyStable.test_fit_beta_flip8s H??? @ @&+&7&A&A!&D&D#tV&+&7&A&A1"&E&E#tVUAqyyyyFF+++EE6***D4%(((FF+++++r5cVd}tjgd}tj| \}}}}tj| |z\}}} } t ||t ||t | ||zt | |dS)Nr r4)rjrr9rrtr) rsSHIFTr_r&r'r(r)r*r+rrs r3test_fit_delta_shiftz#TestLevyStable.test_fit_delta_shiftDs H??? @ @&+&7&A&A1"&E&E#tV&+&7&A&A1"u*&M&M#tVFF+++E5)))D$,///FF+++++r5cgd}tj|\}}}}|dks Jd||t|ksJdt|d|gd}tj|\}}} } |dks Jd|| t |ksJdt |d| dS) N) r r rrr`r`r`r:r:r:r zExpected alpha < 1, got zExpected loc < z, got ) r r rrr`r`r`r:r:r;zExpected alpha > 1, got zExpected loc > )r9rrtrrr%s r3test_fit_loc_extrapz"TestLevyStable.test_fit_loc_extrapOs 6 6 6&+&7&A&A!&D&D#tVzzz>f>>zzzc!ff}}}DADDdDD}}} 7 7 7&+&7&A&A"&E&E#tVzzz>f>>zzzc"gg~~~FRFFFF~~~~~r5z pct_range,alpha_range,beta_range)r~r|r)r]r rg)rrr)r~rFr|r@r)r]r|r rrg)grrr~rr ) r~rFr]rlffffff?r|?rrr@rr]rgrrrct|}tj}|jdko |jdkd|j|j|jg}ddfdgddfd gddfd gdd fd gd dfdgd dfdgg}t |D]\} \} } } | tj_ | || |n|} t5}| tdtj | d| d| ddd}tjd5t!| gd|tj|| dz tj|| dz tj| dz g}dddn #1swxYwY||d| ktj|z}d| d | d!|d"|jjd#| }t+|| d| |d$%dddn #1swxYwYdS)&z2Test pdf values against Nolan's stablec.exe outputLinuxi686/dnir-c rtj|dtj|dztj|dz|ddk|ddkz|ddk|ddkz|ddkzz|ddktj|dd d gzz|dd ktj|dd d gzz|ddktj|dddgzz|ddktj|dddgztjtj|dgdzz|ddktj|ddgztjtj|ddgzz|ddktj|dddgztjtj|dgdzz|ddk|ddkz|ddkzz|ddk|ddkz|ddkzz|ddk|dd kz|ddkzz|ddktj|ddgztjtj|dgdzz|ddktj|dddgztjtj|dgdzz|ddktj|dd d gztjtj|dddgzz|ddkzzS)Npctrrrr|rrrr~rr~rFr@rQr]rlr)r|rrr=r>)gٿ333333ӿr~rr|rFrTr]rQr~r)rrrTr!r)rjisinrr alpha_range beta_range pct_ranges r3rPz7TestLevyStable.test_pdf_nolan_samples..sGAeHi00GAgJ 445GAfIz223 vY!^uX_.vY#-wZ3.0uX_. wZ3.GAeHsCj99: wZ3.GAeHsCj99:$wZ3.GAeHr2h778%,wZ3.GAeHsCj99:GBF1V9$5$5|||DDE-6wZ3.GAeHrd334GBF1V9$5$5t<<=7@wZ3.GAeHsCj99:GBF1V9$5$57M7M7MNNOA#JwZ3.vY#-/uX-/K(TwZ3.vY$.0uX-/U-^wZ3.vY#-/uX-/_2hwZ2-GAeHrd334GBF1V9$5$57G7G7GHHIi7rwZ2-GAeHsCj99:GBF1V9$5$5|||DDEs<|wZ2-GAeHsCj99:GBF1V9$5$5RyAAB}AF7s*GC,EHr5rrctj|dtj|dztj|dz|ddkz|ddkzS)NrErrrQrTrjrHrIs r3rPz7TestLevyStable.test_pdf_nolan_samples..sqGAeHi00GAgJ 445GAfIz223wZ#%'wZ2% 'r5c|ddk ztj|dzdvztj|dzS)NrrTrErrN)rrJrK is_linux_32rLs r3rPz7TestLevyStable.test_pdf_nolan_samples..s_wZ2%$_&GAeHi001;&(GAfIz22 3r5g& .=ctj|dtj|dztj|dz|ddkzS)NrErrrQrNrIs r3rPz7TestLevyStable.test_pdf_nolan_samples..s^GAeHi00GAgJ 445GAfIz223wZ3&(r5 fft-simpsonr;c|ddktj|dztj|dztj|dzS)Nrr&rErrNrIs r3rPz7TestLevyStable.test_pdf_nolan_samples..s^wZ3&GAeHi001GAgJ 445GAfIz223r5rvctj|dtj|dztj|dz|ddkz|ddkzS)NrErrr r&rNrIs r3rPz7TestLevyStable.test_pdf_nolan_samples..spGAeHi00GAgJ 445GAfIz223wZ!^%wZ#% 'r5Nz2Density calculations experimental for FFT method.*r_rrr rrfrerrcalcabserrrelerrrvrYz pdf test  failed with method 'z ' [platform: z]  Ferr_msgverbose)platformunamesystemmachinejoin processorrjr9rrr recordrzrirjrrrisnanrrr)rsr rLrJrKrr` platform_desctestsixdefault_methodr filter_funcsubdatasuprvsubdata2failuresmessagerPs ``` @r3test_pdf_nolan_samplesz%TestLevyStable.test_pdf_nolan_samples[s4%  lg-I%-62I  \5=%/ :<< tJJJJJJL ^U%%%%%% U%%%%%%% W'''''' t&&&&&& t&&&&&& kC L"+5!1!1* *  B3AE  0 + 7;;t,,=A "$$%  "H%))CLG$FO *[h///  0444F1ws|#344F1ws|#344rvgcl7K7KK  H               $H%-(1++ WWWWW"/WW4.PsGAeHi00GAgJ 445GAfIz223 wZ2-GAfI/A/A/ABBCuX-/ wZ2-GAfI??@uX-/  r5rFctj|dtj|dztj|dz|ddktj|dgdz|ddkzz|ddktj|dgdz|ddkzzSrtrNrIs r3rPz7TestLevyStable.test_cdf_nolan_samples..esGAeHi00GAgJ 445GAfIz2237r)& +=+=+=>>?5T)+ 7r)& ???;;<5T)+ r5rRr;ctj|dtj|dztj|dz|ddkzS)NrErr333333?rNrIs r3rPz7TestLevyStable.test_cdf_nolan_samples..ws^GAeHi00GAgJ 445GAfIz223wZ#%'r5rctj|dtj|dztj|dz|ddkz|ddkzS)NrErrrrxrNrIs r3rPz7TestLevyStable.test_cdf_nolan_samples..qGAeHi00GAgJ 445GAfIz223wZ#%'wZ3& (r5rctj|dtj|dztj|dz|ddkz|ddkzS)NrErrrrrNrIs r3rPz7TestLevyStable.test_cdf_nolan_samples..rzr5r~ctj|dtj|dztj|dz|ddkz|ddkzS)NrErrrTrrNrIs r3rPz7TestLevyStable.test_cdf_nolan_samples..rzr5Nz[Cumulative density calculations experimental for FFT method. Use piecewise method instead.*r_rrr rrUrrrVrvrYz cdf test rZz' r[Fr\)rjr9rrr rerzrqrjrrrrfrrr)rsrrLrJrKrrhrirjrrkrlrmrvrnrorps ``` r3test_cdf_nolan_samplesz%TestLevyStable.test_cdf_nolan_samples3s2%U%%%%%% ,T$$$$$$ &t&&&&&& t&&&&&& t&&&&&& t&&&&&& GL \"+5!1!1( (  B3AE  0 + 7;;t,,=A "$$#  "@ %))CLG$FO *[h///  0444F1ws|#344F1ws|#344rvgcl7K7KK  H               $H%-(1++ArAAAA&n2AA6>AACL#! ;# # # # # # # # # # # # # # #  ( ( s9 A'G2A(E& G&E* *G-E* .AGG G rrrrirqctj}|jdkodtjdv}|r|dkrt jd|}dt j_dt j_ ||d|k}d|t j_ |d vsJ|dkrt jj nt jj }||d |d |d d d} t| ||ddS)zGTests for pdf and cdf where loc, scale are different from 0, 1 r@32bitrriz4Test unstable on some platforms; see gh-17839, 17859rrS)rirqr_rrrgrrUr;N)r_r`ra architecturerrr9rrrrrirqr) rsrrrr`rPrrlrrNs r3test_location_scalez"TestLevyStable.test_location_scales  lg-W'X=R=T=TUV=W2W   P45== KN O O O*0;,/:,tG}-.-8[[*~%%%%%)U]]E  ! !8I8M X CL''*GFO1!    GDM400000r5zmethod,decimal_placesrCrrc tjgd}tjgd}tjgd}tjd5t5}|t d|t j_t j |d|dd }t||||d d d n #1swxYwYd d d d S#1swxYwYd S) z sample points extracted from Tables and Graphs of Stable Probability Density Functions - Donald R Holt - 1973 - p 187. )rrrrr r r r rgrgrgrgrrrrrrrr)gtF_?gܵ|?gQ?g@?gec]?gHP?gPkw?g!rh?gTN?g ?gDioɴ?g r?gH?g+?gHPs?gX?g&?g 0*?gv/?ga2U0*?)rrlr|r rrlr|r rrlr|r rrlr|r rrlr|r rrzDensity calculation unstable.*)categoryrpr rrUN) rjrrr rSrzr9rrrir)rsrdecimal_placesxsdensitybetasrmris r3'test_pdf_alpha_equals_one_beta_non_zeroz6TestLevyStable.test_pdf_alpha_equals_one_beta_non_zerosX H H H  (           [X & & &  (9(;(; s JJ'8    4:E  0#''AuA1'EEC Wnf                                    s7C/'A$C C/C C/C C//C36C3zparams,expected)r#r$rr )rgrr`r)r`r9rrctj|d|d|d|dd}t||dS)Nrr rgrrrG)r9rr)rsrrXobserveds r3rzTestLevyStable.test_stats sU$** 1IvayfQivay+   Hh/////r5r)rlr|rzfunction,beta,points,expectedrTrr`rrrcd|cxkrdksnJt||||tjt||dS)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)rrjr#r)rsrrrrrXs r3!test_distribution_outside_supportz0TestLevyStable.test_distribution_outside_supportseP5}}}}1}}}}}} HV5t 4 4 4 GCKK * *     r5zx,alpha,beta,expected))r/iZ?s0?gz5F?)r~H?|/mg>m?)r.d5 ?L ,ٿg깃\?)r>?ЀjoؿgA]4?)rpa2?Յ пgqQ?cdtj_ttj||||dS)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=rNr9rrrrirsr_rrrXs r3test_x_equal_zetaz TestLevyStable.test_x_equal_zetaIsJZ&-"   ! !!5t ! < <      r5) )rrrgz¶?)rrrgV?)rrrgRE)3\?)rrrgB?)rrrg/^?)-C6rrgH?)rrrgn?)rrrgx\?)rrrgVM%?)rrrgm?cdtj_ttj||||dS)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. 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