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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,pctrrirrrrrrTr{s r3nolan_pdf_sample_dataz$TestLevyStable.nolan_pdf_sample_datasM6w NN ! < =  v  /C DD r5ctjttjdz }tj|jd}|S)a#Sample data points for cdf computed with Nolan's stablec See - http://fs2.american.edu/jpnolan/www/stable/stable.html There's a known limitation of Nolan's executable for alpha < 0.2. The data table loaded below is generated from Nolan's stablec with the following parameter space: alpha = 0.1, 0.2, ..., 2.0 beta = -1.0, -0.9, ..., 1.0 p = 0.01, 0.05, 0.1, 0.25, 0.35, 0.5, and the equivalent for the right tail Ideally, Nolan's output for CDF values should match the percentile from where they have been sampled from. Even more so as we extract percentile x positions from stablec too. However, we note at places Nolan's stablec will produce absolute errors in order of 1e-5. We compare against his calculations here. In future, once we less reliant on Nolan's paper we might switch to comparing directly at percentiles (those x values being produced from some alternative means). Typically inputs for stablec: stablec.exe << 2 # cdf 1 # Nolan S equivalent to S0 in scipy .25,2,.25 # alpha -1,-1,0 # beta -10,10,1 # x 1,0 # gamma, delta 2 # output file z.data/levy_stable/stable-Z1-cdf-sample-data.npyrrrr{s r3nolan_cdf_sample_dataz$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)rirrrrr{s r3nolan_loc_scale_sample_dataz*TestLevyStable.nolan_loc_scale_sample_datas1w NN ! ? @   r5 sample_sizer<rDmarksr S0rz alpha,beta))rSr)rSr)rr)r%r{z gamma,deltar rrxc|tj_tj||||}tj||d|j\}} | dksJdS)N)rrrerdrrrE)r9rr rHrrp) rrr rrrrqrlsrrus r3rIzTestLevyStable.test_rvssp(.>*  d%U   | FF $F 7 7  14xxxxxxr5rr{r ctjdd}d}d}tj||||d}tj|d||||f\}}|d ksJd S) z3Additional test cases for rvs for alpha equal to 1.h:rSr{rrrdrerrrZr}N)rirrr9rrrH)rrrrrdrer^statrus r3test_rvs_alpha1zTestLevyStable.test_rvs_alpha1s y!!!   ! !%3e'+ " - -,q-%*D#u$=???a4xxxxxxr5cgd}tj|\}}}}t|dddt |ddt |dd t |d d|gd z}tj|\}}} } t |dt |d t | d d t | dd dS)N)wgr'rrrr羼u?r(r(r(r(67',?r)r)r)r)tA}˜.?r*r*r*r*Gz?rr}re)\(̿rfgۧ1?rg9]c?)r*r*r*r*r*rgv?r))r9rrsrrr rrr^alpha1beta1loc1scale1rralpha2beta2rrs r3rxzTestLevyStable.test_fit(s    ',&7&A&A!&D&D#tV148888E4+++FGQ/// '1   999 9&+&7&A&A"&E&E#tVVQUBFFA...D&!,,,,,r5zUnknown problem with fitstart.rlzalpha,beta,delta,gamma)rrrfr)rSrrfrparametrizationc|tj_tj||||dd}tj|}|\}} } } t ||||g|| | | gddS)z7Test that fit agrees with rvs for each parametrization.rrr@r}rN)r9rr4rrsr) rrrrrqrr4rr alpha_obsbeta_obs delta_obs gamma_obss r3 test_fit_rvszTestLevyStable.test_fit_rvs@s-<) $$ 4U%e$%  ))$//471 8Y  D% ' )Y 7      r5cxtjgd}tj|\}}}}tj| \}}}} t |d|dksJt ||t || t || t | |dS)N r r rrr_r_r_r9r9r`r`r r)rirr9rrsrr) rrr^r.r/r0r1r2r3rrs r3test_fit_beta_flipz!TestLevyStable.test_fit_beta_flipYs 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 r<)rirr9rrsr) rrSHIFTr^r.r/r0r1r2r3rrs r3test_fit_delta_shiftz#TestLevyStable.test_fit_delta_shiftes 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_r9r9rBr zExpected alpha < 1, got zExpected loc < z, got ) r r rrr_r_r_r9r9rCzExpected alpha > 1, got zExpected loc > )r9rrsrrr-s r3test_fit_loc_extrapz"TestLevyStable.test_fit_loc_extrapps 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 rf)rrr)r}rEr{r?r)r\r{r rrf)grrr}rr ) r}rEr\rkffffff?r{?rrr?rr\rfrrrct|}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}rEr?rPr\rkr)r{rrrErF)gٿ333333ӿr}rr{rNrSr\rPr}r)rrrSr r)riisinrr alpha_range beta_range pct_ranges r3rOz7TestLevyStable.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,EHr5r rctj|dtj|dztj|dz|ddkz|ddkzS)NrMrrrPrSrirPrQs r3rOz7TestLevyStable.test_pdf_nolan_samples..sqGAeHi00GAgJ 445GAfIz223wZ#%'wZ2% 'r5c|ddk ztj|dzdvztj|dzS)NrrSrMrrV)rrRrS is_linux_32rTs r3rOz7TestLevyStable.test_pdf_nolan_samples..s_wZ2%$_&GAeHi001;&(GAfIz22 3r5g& .=ctj|dtj|dztj|dz|ddkzS)NrMrrrPrVrQs r3rOz7TestLevyStable.test_pdf_nolan_samples.. s^GAeHi00GAgJ 445GAfIz223wZ3&(r5 fft-simpsonr:c|ddktj|dztj|dztj|dzS)Nrr%rMrrVrQs r3rOz7TestLevyStable.test_pdf_nolan_samples..s^wZ3&GAeHi001GAgJ 445GAfIz223r5ructj|dtj|dztj|dz|ddkz|ddkzS)NrMrrr r%rVrQs r3rOz7TestLevyStable.test_pdf_nolan_samples..spGAeHi00GAgJ 445GAfIz223wZ!^%wZ#% 'r5Nz2Density calculations experimental for FFT method.*r^rrr rrerdrrcalcabserrrelerrruraz pdf test  failed with method 'z ' [platform: z]  Ferr_msgverbose)platformunamesystemmachinejoin processorrir9rr r recordryrhrirrrisnanrrr)rrrrTrRrSrrh platform_desctestsixdefault_methodr filter_funcsubdatasuprusubdata2failuresmessagerXs ``` @r3test_pdf_nolan_samplesz%TestLevyStable.test_pdf_nolan_samples|s6%  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.ssGAeHi00GAgJ 445GAfIz223 wZ2-GAfI/A/A/ABBCuX-/ wZ2-GAfI??@uX-/  r5rEctj|dtj|dztj|dz|ddktj|dgdz|ddkzz|ddktj|dgdz|ddkzzSr|rVrQs r3rOz7TestLevyStable.test_cdf_nolan_samples..sGAeHi00GAgJ 445GAfIz2237r)& +=+=+=>>?5T)+ 7r)& ???;;<5T)+ r5rZr:ctj|dtj|dztj|dz|ddkzS)NrMrr333333?rVrQs r3rOz7TestLevyStable.test_cdf_nolan_samples..s^GAeHi00GAgJ 445GAfIz223wZ#%'r5rctj|dtj|dztj|dz|ddkz|ddkzS)NrMrrrrrVrQs r3rOz7TestLevyStable.test_cdf_nolan_samples..qGAeHi00GAgJ 445GAfIz223wZ#%'wZ3& (r5rctj|dtj|dztj|dz|ddkz|ddkzS)NrMrrrrrVrQs r3rOz7TestLevyStable.test_cdf_nolan_samples..rr5r}ctj|dtj|dztj|dz|ddkz|ddkzS)NrMrrrSrrVrQs r3rOz7TestLevyStable.test_cdf_nolan_samples..rr5Nz[Cumulative density calculations experimental for FFT method. Use piecewise method instead.*r^rrr rr]rrr^ruraz cdf test rbz' rcFrd)rir9rr r rmryrprirrrrnrrr)rrrrTrRrSrrprqrrrrsrtrururvrwrxs ``` r3test_cdf_nolan_samplesz%TestLevyStable.test_cdf_nolan_samplesUs4%U%%%%%% ,T$$$$$$ &t&&&&&& t&&&&&& t&&&&&& t&&&&&& GL \"+5!1!1( (  B3AE  0 + 7;;t,,=A "$$#  "@ %))CLG$FO *[h///  0444F1ws|#344F1ws|#344rvgcl7K7KK  H               $H%-(1++ArAAAA&n2AA6>AACL#! ;# # # # # # # # # # # # # # #  ( ( s9 A'G2A(E& G&E* *G-E* .AGG G rrrrhrpctj}|jdkodtjdv}|r|dkrt jd|}dt j_dt j_ ||d|k}d|t j_ |d vsJ|dkrt jj nt jj }||d |d |d d d} t| ||ddS)zGTests for pdf and cdf where loc, scale are different from 0, 1 rH32bitrrhz4Test unstable on some platforms; see gh-17839, 17859r rS)rhrpr^rrrfrr]r:N)rgrhri architecturerrr9rr r r rhrpr) rrrrrrhrXrrtrrMs r3test_location_scalez"TestLevyStable.test_location_scales  lg-W'X=R=T=TUV=W2W   P45== KN O O O*0;,/:,tG}-.-8[[*~%%%%%)U]]E  ! !8I8M X CL''*GFO1!    GDM400000r5zmethod,decimal_placesrKrr c tjgd}tjgd}tjgd}tjd5t5}|t d|t j_t j |d|dd }t||||d d d n #1swxYwYd d d d S#1swxYwYd S) z sample points extracted from Tables and Graphs of Stable Probability Density Functions - Donald R Holt - 1973 - p 187. )rrrrr r r r rfrfrfrfrrrrrrrr)gtF_?gܵ|?gQ?g@?gec]?gHP?gPkw?g!rh?gTN?g ?gDioɴ?g r?gH?g+?gHPs?gX?g&?g 0*?gv/?ga2U0*?)rrkr{r rrkr{r rrkr{r rrkr{r rrkr{r rrzDensity calculation unstable.*)categoryrxr rr]N) rirrr rRryr9rr rhr)rrrdecimal_placesxsdensitybetasrurhs r3'test_pdf_alpha_equals_one_beta_non_zeroz6TestLevyStable.test_pdf_alpha_equals_one_beta_non_zero sX H H H  (           [X & & &  (9(;(; s JJ'8    4:E  0#''AuA1'EEC Wnf                                    s7C/'A$C C/C C/C C//C36C3zparams,expected)r+r,rr )rfrr_r)r_r8rrctj|d|d|d|dd}t||dS)Nrr rfrrrI)r9rr)rrrrWobserveds r3rzTestLevyStable.test_stats0sU$** 1IvayfQivay+   Hh/////r5r)rkr{rzfunction,beta,points,expectedrSrr_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)rrir"r)rrrrrrrWs r3!test_distribution_outside_supportz0TestLevyStable.test_distribution_outside_support>seP5}}}}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=rNr9rrrrhrrr^rrrWs r3test_x_equal_zetaz TestLevyStable.test_x_equal_zetalsJZ&-"   ! !!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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