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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 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.npyrrrrys r0nolan_cdf_sample_dataz$TestLevyStable.nolan_cdf_sample_datasNJw NN ! < =  v  /C DD r2cbtjttjdz }|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)rZrrrrrys r0nolan_loc_scale_sample_dataz*TestLevyStable.nolan_loc_scale_sample_datas1w NN ! ? @   r2 sample_sizer8r@marksrS0r z alpha,beta))rOr)rOr)rr)r'rz gamma,deltarrr{c|tj_tj||||}tj|||j|j\}} | dksJdS)N)rrrlrkrrA)r6rrrDrr]rv) r_rrrrrmrlsrrqs r0rEzTestLevyStable.test_rvs sr(.>*  d%U   | FF $(F ; ;RV  14xxxxxxr2rrrcd}d}d}tj||||d|j}tj|d||||f\}}|dksJd S) z3Additional test cases for rvs for alpha equal to 1.rOrrrr<rrWrN)r6rrr]rD)r_rrrkrlrestatrqs r0test_rvs_alpha1zTestLevyStable.test_rvs_alpha1*s|   ! !%3e'+$( " D D,q-%*D#u$=???a4xxxxxxr2cgd}tj|\}}}}t|dddt |ddt |dd t |d d|gd z}tj|\}}} } t |dt |d t | d d t | dd dS)N)wgr)rrrr羼u?r*r*r*r*67',?r+r+r+r+tA}˜.?r,r,r,r,Gz?rrra)\(̿rmgۧ1?rg9]c?)r,r,r,r,r,rgv?r+)r6rrqrrr r_realpha1beta1loc1scale1rpalpha2beta2rrs r0rvzTestLevyStable.test_fit7s    ',&7&A&A!&D&D#tV148888E4+++FGQ/// '1   999 9&+&7&A&A"&E&E#tVVQUBFFA...D&!,,,,,r2zUnknown problem with fitstart.rpzalpha,beta,delta,gamma)rrrmr)rOrrmrparametrizationc|tj_tj||||d|j}tj|}|\}} } } t ||||g|| | | gddS)z7Test that fit agrees with rvs for each parametrization.rr<rrN)r6rr6rr]rqr) r_rrrmrr6rr alpha_obsbeta_obs delta_obs gamma_obss r0 test_fit_rvszTestLevyStable.test_fit_rvsOs-<) $$ 4U%e$(%  ))$//471 8Y  D% ' )Y 7      r2cxtjgd}tj|\}}}}tj| \}}}} t |d|dksJt ||t || t || t | |dS)N rrrrrfrfrfr6r6rgrgrr)rZrr6rrqrr) r_rer0r1r2r3r4r5rrs r0test_fit_beta_flipz!TestLevyStable.test_fit_beta_fliphs H??? @ @&+&7&A&A!&D&D#tV&+&7&A&A1"&E&E#tVUAqyyyyFF+++EE6***D4%(((FF+++++r2cVd}tjgd}tj| \}}}}tj| |z\}}} } t ||t ||t | ||zt | |dS)Nrr>)rZrr6rrqr) r_SHIFTrer0r1r2r3r4r5rrs r0test_fit_delta_shiftz#TestLevyStable.test_fit_delta_shiftts H??? @ @&+&7&A&A1"&E&E#tV&+&7&A&A1"u*&M&M#tVFF+++E5)))D$,///FF+++++r2cgd}tj|\}}}}|dks Jd||t|ksJdt|d|gd}tj|\}}} } |dks Jd|| t |ksJdt |d| dS) N) rrrrrfrfrfr6r6rDrzExpected alpha < 1, got zExpected loc < z, got ) rrrrrfrfrfr6r6rEzExpected alpha > 1, got zExpected loc > )r6rrqrrr/s r0test_fit_loc_extrapz"TestLevyStable.test_fit_loc_extraps 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~~~~~r2z pct_range,alpha_range,beta_range)rrr)rcrrm)rrr)rrArrAr)rcrrrrm)grrryrr) rrArcrgffffff?r?rrrArrcrmrrrc|}tj}|jdko |jdkd|j|j|jg}ddfdgddfd gddfd gdd fd gd dfdgd dfdgg}t |D]\} \} } } | tj_ | || |n|} d}tj 5tj d|ttj| 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)NpctrrrrrrrrrryrArArLrcrgr)rrrrGrH)gٿ333333ӿryrrrPrOrcrLryr)rrrOr"r)rZisinrr alpha_range beta_range pct_ranges r0rLz7TestLevyStable.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,EHr2r rctj|dtj|dztj|dz|ddkz|ddkzS)NrOrrrLrOrZrRrSs r0rLz7TestLevyStable.test_pdf_nolan_samples..sqGAeHi00GAgJ 445GAfIz223wZ#%'wZ2% 'r2c|ddk ztj|dzdvztj|dzS)NrrOrOrrX)rrTrU is_linux_32rVs r0rLz7TestLevyStable.test_pdf_nolan_samples.. s_wZ2%$_&GAeHi001;&(GAfIz22 3r2g& .=ctj|dtj|dztj|dz|ddkzS)NrOrrrLrXrSs r0rLz7TestLevyStable.test_pdf_nolan_samples..s^GAeHi00GAgJ 445GAfIz223wZ3&(r2 fft-simpsonr6c|ddktj|dztj|dztj|dzS)Nrr'rOrrXrSs r0rLz7TestLevyStable.test_pdf_nolan_samples.. s^wZ3&GAeHi001GAgJ 445GAfIz223r2rqctj|dtj|dztj|dz|ddkz|ddkzS)NrOrrrr'rXrSs r0rLz7TestLevyStable.test_pdf_nolan_samples..(spGAeHi00GAgJ 445GAfIz223wZ!^%wZ#% 'r2Nz0Density calculations experimental for FFT methodrrerrrrrlrkrcalcabserrrelerrrqrcz pdf test  failed with method 'z ' [platform: z]  Ferr_msgverbose)platformunamesystemmachinejoin processorrjr6rrrrrsrrurorZrrrisnanrrr)r_rrVrTrUrrj platform_desctestsixdefault_methodr filter_funcsubdatar,rqsubdata2failuresmessagerZs ``` @r0test_pdf_nolan_samplesz%TestLevyStable.test_pdf_nolan_sampless4%  lg-I%-62I  \5=%/ :<< tJJJJJJL ^U%%%%%% U%%%%%%% W'''''' t&&&&&& t&&&&&& kC L"+5!1!1( (  B3AE  0 + 7;;t,,=A DC(**" " '#~FFF%))CLG$FO *[h///  0444F1ws|#344F1ws|#344rvgcl7K7KK  H               $H%-(1++ WWWWW"/WW4.~sGAeHi00GAgJ 445GAfIz223 wZ2-GAfI/A/A/ABBCuX-/ wZ2-GAfI??@uX-/  r2rActj|dtj|dztj|dz|ddktj|dgdz|ddkzz|ddktj|dgdz|ddkzzSr|rXrSs r0rLz7TestLevyStable.test_cdf_nolan_samples..sGAeHi00GAgJ 445GAfIz2237r)& +=+=+=>>?5T)+ 7r)& ???;;<5T)+ r2r\r6ctj|dtj|dztj|dz|ddkzS)NrOrr333333?rXrSs r0rLz7TestLevyStable.test_cdf_nolan_samples..s^GAeHi00GAgJ 445GAfIz223wZ#%'r2rctj|dtj|dztj|dz|ddkz|ddkzS)NrOrrrrrXrSs r0rLz7TestLevyStable.test_cdf_nolan_samples..qGAeHi00GAgJ 445GAfIz223wZ#%'wZ3& (r2rctj|dtj|dztj|dz|ddkz|ddkzS)NrOrrrrrXrSs r0rLz7TestLevyStable.test_cdf_nolan_samples..rr2rctj|dtj|dztj|dz|ddkz|ddkzS)NrOrrrOrrXrSs r0rLz7TestLevyStable.test_cdf_nolan_samples..rr2NrzZCumulative density calculations experimental for FFT method. Use piecewise method instead.rerrrrr_rr`rqrcz cdf test rdz' reFrf)rjr6rrrrrsrrurvrZrrrrorrr)r_rrVrTrUrrqrrrsrrtrurqrvrwrxs ``` r0test_cdf_nolan_samplesz%TestLevyStable.test_cdf_nolan_samplesas2%U%%%%%% ,T$$$$$$ &t&&&&&& t&&&&&& t&&&&&& t&&&&&& GL \"+5!1!1( (  B3AE  0 + 7;;t,,=A (**# # '>" $$$ %))CLG$FO *[h///  0444F1ws|#344F1ws|#344rvgcl7K7KK  H               $H%-(1++ArAAAA&n2AA6>AACL#! ;# # # # # # # # # # # # # # #  ( ( s9A'G7A(E+ G+E/ /G2E/ 3AGG "G rrrrorvctj}|jdkodtjdv}|r|dkrt jd|}dt j_dt j_ ||d|k}d|t j_ |d vsJ|dkrt jj nt jj }||d |d |d d d} t| ||ddS)zGTests for pdf and cdf where loc, scale are different from 0, 1 rJ32bitrroz4Test unstable on some platforms; see gh-17839, 17859r rS)rorvrerrrmrr_r6N)rirjrk architecturerrr6rrrrrorvr) r_rrrrjrZrrurrNs r0test_location_scalez"TestLevyStable.test_location_scales  lg-W'X=R=T=TUV=W2W   P45== KN O O O*0;,/:,tG}-.-8[[*~%%%%%)U]]E  ! !8I8M X CL''*GFO1!    GDM400000r2zmethod,decimal_placesrMrr c tjgd}tjgd}tjgd}tjd5tj5tjdt d|tj_ tj |d|dd }t||||d d d n #1swxYwYd d d d S#1swxYwYd S) z sample points extracted from Tables and Graphs of Stable Probability Density Functions - Donald R Holt - 1973 - p 187. )rrrrrrrrrmrmrmrmrrrrrrrr)gtF_?gܵ|?gQ?g@?gec]?gHP?gPkw?g!rh?gTN?g ?gDioɴ?g r?gH?g+?gHPs?gX?g&?g 0*?gv/?ga2U0*?)rrgrrrrgrrrrgrrrrgrrrrgrrrrzDensity calculation unstable.*)categoryrxrrr_N) rZrrrrrsrrur6rrror)r_rdecimal_placesxsdensitybetasros r0'test_pdf_alpha_equals_one_beta_non_zeroz6TestLevyStable.test_pdf_alpha_equals_one_beta_non_zerosX H H H  (           [X & & &  (?(A(A    #H'8    4:E  0#''AuA1'EEC Wnf                                    s7C4,A$C C4C C4#C $C44C8;C8zparams,expected)r-r.rr)rmrrfr)rfr5rrctj|d|d|d|dd}t||dS)NrrrmrrrL)r6rr)r_rrTobserveds r0rzTestLevyStable.test_stats;sU$** 1IvayfQivay+   Hh/////r2r)rgrrzfunction,beta,points,expectedrOrrfrrrcd|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. rrrrN)rrZrr)r_rrrrrTs r0!test_distribution_outside_supportz0TestLevyStable.test_distribution_outside_supportIseP5}}}}1}}}}}} HV5t 4 4 4 GCKK * *     r2zx,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=rNr6rrrror_rerrrTs r0test_x_equal_zetaz TestLevyStable.test_x_equal_zetawsJZ&-"   ! !!5t ! < <      r2) )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. Reference values taken from MATLAB https://www.mathworks.com/help/stats/stable-distribution.html rrNrrs r0test_x_near_zetazTestLevyStable.test_x_near_zetasJD&-"   ! !!5t ! < <      r2c(tjSrz) threadingLockr^s r0levy_stable_lockzTestLevyStable.levy_stable_locks~r2c Vtjj}tdddd}tj}t jd i|}|5d|_d|_|jd i|d|d d }|d|d  }tj||krJd|_d|_|jd i|d|d d }|d|d  } t| |t||ddddS#1swxYwYdS) Nr'rcrrOrrrkrlr!r rf rr8) rZr[r\rr6rrrrr r_rr]runfrozenfrozen unfrozen_afrozen_a unfrozen_bfrozen_bs r0$test_frozen_parameterization_gh20821z3TestLevyStable.test_frozen_parameterization_gh20821si#Ccs#>>>$",,V,,  / /(,H %&*F #%UUURcc)nnUUUUJzzrIzGGHvh*455 5 55(,H %&*F #%UUURcc)nnUUUUJzzrIzGGH : . . . 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