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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,pctrrTrrrrrrTrps r0nolan_pdf_sample_dataz$TestLevyStable.nolan_pdf_sample_datasL6ww N ! ! < = vv  /C D r2ctjttjdz }tj j |jd}|S)a#Sample data points for cdf computed with Nolan's stablec See - http://fs2.american.edu/jpnolan/www/stable/stable.html There's a known limitation of Nolan's executable for alpha < 0.2. The data table loaded below is generated from Nolan's stablec with the following parameter space: alpha = 0.1, 0.2, ..., 2.0 beta = -1.0, -0.9, ..., 1.0 p = 0.01, 0.05, 0.1, 0.25, 0.35, 0.5, and the equivalent for the right tail Ideally, Nolan's output for CDF values should match the percentile from where they have been sampled from. Even more so as we extract percentile x positions from stablec too. However, we note at places Nolan's stablec will produce absolute errors in order of 1e-5. We compare against his calculations here. In future, once we less reliant on Nolan's paper we might switch to comparing directly at percentiles (those x values being produced from some alternative means). Typically inputs for stablec: stablec.exe << 2 # cdf 1 # Nolan S equivalent to S0 in scipy .25,2,.25 # alpha -1,-1,0 # beta -10,10,1 # x 1,0 # gamma, delta 2 # output file z.data/levy_stable/stable-Z1-cdf-sample-data.npyrrrrps r0nolan_cdf_sample_dataz$TestLevyStable.nolan_cdf_sample_datasMJww N ! ! < = vv  /C D r2cdtjttjdz }|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)rTrrrrrps r0nolan_loc_scale_sample_dataz*TestLevyStable.nolan_loc_scale_sample_datas/ww N ! ! ? @  r2 sample_sizer2r:marksrS0rz alpha,beta))rIr)rIr)rr)rryz gamma,deltarrrfc|tj_tj||||}tj|j ||j |j \}} | dkDsJy)N)rrrfrerr;)r6rrr>rrWrp) rYrrrrrgrlsrrks r0r?zTestLevyStable.test_rvs sf(.>*   d%U || FF $((F ;RVV 14xxr2rryrcd}d}d}tjj||||d|j}tj|d||||f\}}|dkDsJy ) z3Additional test cases for rvs for alpha equal to 1.rIryrrr5rrQr{N)r6rrrWr>)rYrrrerfr_statrks r0test_rvs_alpha1zTestLevyStable.test_rvs_alpha1*sm    ! !%3e'+$(( " D,,q-%*D#u$=?a4xxr2czgd}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?rrrr67',?rrrrtA}˜.?rrrrGz?rr{r[)\(̿rggۧ1?rg9]c?)rrrrrrgv?r)r6rrhrrr rYr_alpha1beta1loc1scale1rgalpha2beta2rrs r0rmzTestLevyStable.test_fit7s  ',&7&7&A&A!&D#tV148E4+FGQ/ '1 9 9&+&7&7&A&A"&E#tVVQUBFFA.D&!,r2zUnknown problem with fitstart.rYzalpha,beta,delta,gamma)rrrgr)rIrrgrparametrizationc|tj_tjj||||d|j}tjj |}|\}} } } t ||||g|| | | gdy)z7Test that fit agrees with rvs for each parametrization.rr5r{rN)r6rrrrWrhr) rYrrrgrrrr alpha_obsbeta_obs delta_obs gamma_obss r0 test_fit_rvszTestLevyStable.test_fit_rvsOs-<)  $$ 4U%e$((% ))$/471 8Y  D% ' )Y 7 r2cNtjgd}tjj |\}}}}tjj | \}}}} t |d|dk7sJt ||t || t || t | |y)N rrrrr`r`r`r0r0rararr)rTrr6rrhrr) rYr_rrrrrrrrs r0test_fit_beta_flipz!TestLevyStable.test_fit_beta_fliphs HH? @&+&7&7&A&A!&D#tV&+&7&7&A&A1"&E#tVUAqyyFF+EE6*D4%(FF+r2c6d}tjgd}tjj | \}}}}tjj | |z\}}} } t ||t ||t | ||zt | |y)Nrr&)rTrr6rrhr) rYSHIFTr_rrrrrrrrs r0test_fit_delta_shiftz#TestLevyStable.test_fit_delta_shiftts HH? @&+&7&7&A&A1"&E#tV&+&7&7&A&A1"u*&M#tVFF+E5)D$,/FF+r2cpgd}tjj|\}}}}|dks Jd||t|ksJdt|d|gd}tjj|\}}} } |dkDs Jd|| t |kDsJdt |d| y) N) rrrrr`r`r`r0r0r,rzExpected alpha < 1, got zExpected loc < z, got ) rrrrr`r`r`r0r0r-zExpected alpha > 1, got zExpected loc > )r6rrhrrrs r0test_fit_loc_extrapz"TestLevyStable.test_fit_loc_extraps 6&+&7&7&A&A!&D#tVz>5fX>>zc!f}DAxvdVDD} 7&+&7&7&A&A"&E#tVz>5fX>>zc"g~FR vFF~r2z pct_range,alpha_range,beta_range)r{ryr)r]rrg)rrr)r{r;ryr5r)r]ryrrrg)grrrsrxr) r{r;r]raffffff?ry?rrr5rr]rgr rrcr|}tj}|jdk(xr|jdk(dj |j|j|j g}ddfdgddfd gddfd gdd fd gd dfdgd dfdgg}t |D]j\} \} } } | tj_ | || |n|} d}tj5tjd|ttjj| d| d| ddd}tj d5t#| gd|tj$|| dz tj$|| dz tj$| dz g}ddd|d| k\tj&|z}d| d | d!|d"|j(j*d#| }t-|| d| |d$%dddmy#1swYnxYw#1swYxYw)&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)Npctrrrryrrrr{rrsr;r5rFr]rar)ryrxryr/r0)gٿ333333ӿrsrryr8rIr]rFrsr)rrrIrr )rTisinrr} alpha_range beta_range pct_ranges r0rEz7TestLevyStable.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,EHr2rrctj|dtj|dztj|dz|ddkDz|ddk7zS)Nr7rrrFrIrTr:r;s r0rEz7TestLevyStable.test_pdf_nolan_samples..snGGAeHi0GGAgJ 45GGAfIz23wZ#%'wZ2% 'r2c|ddk( ztj|dzdvztj|dzS)NrrIr7rr@)r}r<r= is_linux_32r>s r0rEz7TestLevyStable.test_pdf_nolan_samples.. s]wZ2%$_&GGAeHi01;&(GGAfIz2 3r2g& .=ctj|dtj|dztj|dz|ddkzS)Nr7rrrFr@r;s r0rEz7TestLevyStable.test_pdf_nolan_samples..s[GGAeHi0GGAgJ 45GGAfIz23wZ3&(r2 fft-simpsonr/c|ddk\tj|dztj|dztj|dzS)Nrrr7rr@r;s r0rEz7TestLevyStable.test_pdf_nolan_samples.. s[wZ3&GGAeHi01GGAgJ 45GGAfIz23r2rkctj|dtj|dztj|dz|ddkDz|ddkzS)Nr7rrrrr@r;s r0rEz7TestLevyStable.test_pdf_nolan_samples..(smGGAeHi0GGAgJ 45GGAfIz23wZ!^%wZ#% 'r2Nz0Density calculations experimental for FFT methodrr_rrrrrfrercalcabserrrelerrrkrKz pdf test  failed with method 'z ' [platform: z]  Ferr_msgverbose)platformunamesystemmachinejoin processorr^r6rrrlrmrrorirTrrrisnanrrr)rYrr>r<r=rrR platform_desctestsixdefault_methodr filter_funcsubdatar,rksubdata2failuresmessagerBs ``` @r0test_pdf_nolan_samplesz%TestLevyStable.test_pdf_nolan_sampless4% llg-I%--62I  \\5==%// :< tJL ^U% U% W' t& t& kC L"+5!1(  B3AE   0 + 7;t,=A DC((*" ''#~F%%))CLG$FO *[[h/ 04FF1ws|#34FF1ws|#34rvvgcl7KK H $H%-((1+  t#88HI""/HNN4H4H3IH:W CL#! 9" "  (   " " s'6AH,AH 3A!H, H) %H,,H6 c|}ddfdgddfdgddfdgdd fd gdd fd gdd fdgg}t|D]e\}\}} } |tj_| || |n|} t j 5t j ddttjj| d| d| ddd} tjd5t| gd| tj| | dz tj| | dz tj| dz g} ddd | d| k\tj| z}d|d|d|jjd|}t!| | d| |d !dddhy#1swYkxYw#1swYxYw)"z4 Test cdf values against Nolan's stablec.exe output.rrcXtj|dtj|dztj|dz|ddk(tj|dgdz|ddk(z|ddk(tj|dgdz|ddk(zzzS Nr7rrrI)r8rrr{r]rFrsrr@r;s r0rEz7TestLevyStable.test_cdf_nolan_samples..~sGGAeHi0GGAgJ 45GGAfIz23 wZ2-GGAfI/ABCuX-/ wZ2-GGAfI?@uX-/  r2r;cVtj|dtj|dztj|dz|ddk(tj|dgdz|ddk(zz|ddk(tj|dgdz|ddk(zzSrdr@r;s r0rEz7TestLevyStable.test_cdf_nolan_samples..sGGAeHi0GGAgJ 45GGAfIz237r)& +=>?5T)+ 7r)& ?;<5T)+ r2rDr/ctj|dtj|dztj|dz|ddkDzS)Nr7rr333333?r@r;s r0rEz7TestLevyStable.test_cdf_nolan_samples..s[GGAeHi0GGAgJ 45GGAfIz23wZ#%'r2rctj|dtj|dztj|dz|ddkDz|ddkzS)Nr7rrrrhr@r;s r0rEz7TestLevyStable.test_cdf_nolan_samples..nGGAeHi0GGAgJ 45GGAfIz23wZ#%'wZ3& (r2rctj|dtj|dztj|dz|ddkDz|ddkzS)Nr7rrrrr@r;s r0rEz7TestLevyStable.test_cdf_nolan_samples..rjr2r{ctj|dtj|dztj|dz|ddkDz|ddkzS)Nr7rrrIrr@r;s r0rEz7TestLevyStable.test_cdf_nolan_samples..rjr2NrzZCumulative density calculations experimental for FFT method. Use piecewise method instead.r_rrrrrGrrHrkrKz cdf test rLz' rMFrN)r^r6rrrlrmrrorprTrrrrWrrr)rYrr>r<r=rrYrZr[rr\r]rkr^r_r`s ``` r0test_cdf_nolan_samplesz%TestLevyStable.test_cdf_nolan_samplesas22%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' AF?+AF3 AF?3F< 8F??G rrrrirpc 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 r232bitrriz4Test unstable on some platforms; see gh-17839, 17859rrS)rirpr_rrrgrrGr/N)rQrRrS architecturerrr6rrrrrirpr) rYrrrrRrBrr]rrBs r0test_location_scalez"TestLevyStable.test_location_scales llg-W'X=R=R=TUV=W2W  45= KKN O*0;,/:,tG}-./0[*~%%%%)U]E   ! !8I8I8M8M  CL''*GFO1!  GDM40r2zmethod,decimal_placesr5rrc tjgd}tjgd}tjgd}tjd5tj5tj dt d|tj_ tjj|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. )rrrrrrrrrgrgrgrgrrrrrrrr)gtF_?gܵ|?gQ?g@?gec]?gHP?gPkw?g!rh?gTN?g ?gDioɴ?g r?gH?g+?gHPs?gX?g&?g 0*?gv/?ga2U0*?)rraryrrraryrrraryrrraryrrraryrrrzDensity calculation unstable.*)categoryr`rrrGN) rTrrrlrmrror6rrrir)rYrdecimal_placesxsdensitybetasris r0'test_pdf_alpha_equals_one_beta_non_zeroz6TestLevyStable.test_pdf_alpha_equals_one_beta_non_zerosXX H ((     [[X & (?(?(A   # #H'8 4:E   0##''AuA1'EC Wnf       s%C21A$C&C2&C/ +C22C;zparams,expected)rrrr)rgrr`r)r`r,rrc|tjj|d|d|d|dd}t||y)Nrrrgrrr8)r6rr)rYrrNobserveds r0rzTestLevyStable.test_stats;sG$$** 1IvayfQivay+  Hh/r2r)raryrzfunction,beta,points,expectedrIrr`r~rrzcd|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. rrrrN)rrTrr )rYrrrrrNs r0!test_distribution_outside_supportz0TestLevyStable.test_distribution_outside_supportIs@P5}1}}} V5t 4 GGCK * r2zx,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=r~Nr6rrrrirYr_rrrNs r0test_x_equal_zetaz TestLevyStable.test_x_equal_zetaws9Z&-"    ! !!5t ! <  r2) )rrrgz¶?)rrrgV?)rrrgRE)3\?)rrrgB?)rrrg/^?)-C6rrgH?)rrrgn?)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 rr~Nrrs r0test_x_near_zetazTestLevyStable.test_x_near_zetas9D&-"    ! !!5t ! <  r2c*tjSrt) threadingLockrXs r0levy_stable_lockzTestLevyStable.levy_stable_locks~~r2c 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) Nrr]r~rIrrrerfr rr` rr8) rTrUrVr r6rrrrr rYrrWr unfrozenfrozen unfrozen_afrozen_a unfrozen_bfrozen_bs r0$test_frozen_parameterization_gh20821z3TestLevyStable.test_frozen_parameterization_gh20821s ii##Ccs#>$$"",V,  /(,H %&*F #%UURc)nUJzzrIzGHvvh*45 55(,H %&*F #%UURc)nUJzzrIzGH : . X . / / /s B;DDc tjj}tdddd}tj }|5d|_t j d i|}|jd i|d|dd }|jd|d }t||d |_t j d i|}|jd i|d|dd }|jd|d } t| |dddy#1swYyxYw) Nrr]r~rIrr r`rrrr8) rTrUrVr r6rrrrrs r0%test_frozen_parameterization_gh20821bz4TestLevyStable.test_frozen_parameterization_gh20821bsii##Ccs#>$$  /(,H %&&00F%UURc)nUJzzrIzGH : .(,H %&&00F%UURc)nUJzzrIzGH : . / / /s B=DD N)*rrrrZrrrrrrrrrrr?rrrmxfailr$r'r*r.rTrNrarmrrrzrrrr6rrprirrrrrrr8r2r0rrs5V^^D!B"B ^^B ^^))V ^^   [[ [[ FLL lfll4v{{7G7GH   [[/$> [[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)) )rI)ryrr~r)r})gUUUUUU?r$ g^cQg333333rrN)r6rr)rMr. exact_mvskrs r0test_powerlaw_statsr !sG>, = ?E4 :~~##Av#6!$ 34r2c\tjjdd}t|dy)Nrrr~)r6rrrrks r0test_powerlaw_edger !s" a#ACr2cvtjjdd}t|dtjj dgd}t|t j ddgtjjdgd}t|t j dt j gy)Nrrr~)rarIrrI)r6r0rrrirTrr s r0test_exponpow_edger !s a#AC 1./ARVVS#&' a!12ARVVS266'*+r2c^tjjddd}t|dy)NrrrI)r6gengammarirr s r0test_gengamma_edger !s$ 1a#ACr2za, c, ref, tol))g`6Arg2j!@r)rrggA@r)rrrr)rrg#@r)g`FArgI~mr)gl$W}Lrg.I#Urc\ttjj||||yr)rr6r r>)r.rrzrJs r0test_gengamma_extreme_entropyr !s!"ENN**1a0#C@r2ctjjddd}|dk(sJtjjddd}|tj k(sJy)Nrrrr~)r6r rirrTr)rkrs r0!test_gengamma_endpoint_with_neg_cr "sO 1a$A 8O8 >> Ar *D BFF7??r2ctjjddd}t|dtjjddd}t|dy)NrrFrIg>r`gqq?)r6r rrr s r0test_gengamma_munpr "sF Rb)A:& RR(A6"r2ctjgd}tjd5tj5tj ddt tj ddttjj|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) rTrrrlrmrrror6r@r)r. s r0test_ksone_fit_freezer "s  / 0A X &   $ $ &   # #L"   # #?  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