a BCCf@sddlZddlmZddlmZmZddlZddlZddl m Z ddl m Z m Z mZddlmZddlmZddlmZddlmZddlmmZdd lmZmZd d lmZd d lm Z!m"Z#d d l$m%Z%m&Z&m'Z'm(Z(m)Z)m*Z*m+Z+d dl,m-Z-m.Z.m/Z/d dl0m1Z1m2Z2m3Z3m4Z4m5Z5m6Z6m7Z7d dl8m9Z9ddl:m;mZ>ddl?m@Z@ddlAm;Z;ddZBddZCdddZDGddde'ZEeEddddZFGdd d e'ZGeGdddd!d"ZHGd#d$d$e'ZIeIdd%d&ZJeKd'ejLZMeNeMZOd(d)ZPd*d+ZQd,d-ZRd.d/ZSd0d1ZTd2d3ZUd4d5ZVd6d7ZWGd8d9d9e'ZXeXd:d;ZYGdd&Z[Gd?d@d@e'Z\e\ejL dAejLdAdBdZ]GdCdDdDe'Z^e^dddEdZ_GdFdGdGe`ZaGdHdIdIe@ZbdJdKZcdLdMZdGdNdOdOe'ZeeedddPdZfGdQdRdRe'ZgegddSd&ZhGdTdUdUe'ZieidddVdZjGdWdXdXe'ZkekddYd&ZlGdZd[d[e'Zmemdd\d&ZnGd]d^d^ekZoeodd_d&ZpGd`dadae'Zqeqdbd;ZrGdcdddde'Zsesdded&ZtGdfdgdge'Zueuddhd&ZvGdidjdje'ZwewejL ejLdkdZxGdldmdme'Zyeydnd;ZzGdodpdpe'Z{e{dqd;Z|Gdrdsdse'Z}e}ddtd&Z~Gdudvdve'Zedwd;ZdxdyZGdzd{d{e'Zedd|d&ZGd}d~d~e'Zeddd&ZGddde'Zeddd&ZGddde'Zeddd&ZGddde'Zeddd&ZGddde'Zeddd&ZGddde'Zeddd&ZGddde'Zedd;ZGddde'ZedddZGddde'Zedd;ZGddde'Zeddd&ZGddde'Zeddd&ZGddde'Zedd;ZddZGddde'Zeddd&ZGdddeZeddd&ZGddde'Zeddd&ZGddde'Zeddd&ZGddde'Zedd;ZGddde'Zeddd&ZddZGddde'Zedd;ZGddde'Zedd;ZGddde'Zeddd&ZGdd„de'Zeddd&ZGddńde'Zeddd&ZGddȄde'Zedd;ZGdd˄de'ZeddddZGdd΄de'Zeddd&ZGddфde'Zeddd&ZGddԄde'Zeddd&ZGddׄde'Zedd;ZGddڄde'Zeddd&ZGdd݄de'Zedd;ZGddde'ZeddddZGddde'Zedd;ZGddde'Zedd;ZGddde'Zedd;ZddZGddde'Zeddd&ZGddde'ZedddZGddde'Zedd;ZGddde'Zedd;ZGddde'Zeddd&ZddZGddde'Zeddd&ZGddde'Zeddd&ZGddde'Zeddd&ZGddde'Zedd d&ZGd d d e'Zed d;ZGd dde'Zeddd&ZGddde'Zedd;ZGddde'Zeddd&ZddZGddde'Zeddd&ZGddde'Zeddd&ZGddde'Zed d;ZGd!d"d"e'Zed#d;ZGd$d%d%e'Zedd&d&ZGd'd(d(e'Zedd)d&ZGd*d+d+e'Zed,d;ZGd-d.d.e'Zeddd/dZGd0d1d1e'Zedd2d&ZGd3d4d4e'Zed5d;ZGd6d7d7e'Zed8dd9dZGd:d;d;e'Zeddd>e'Zed?d;Zed@d;ZGdAdBdBe'ZeddCd&ZGdDdEdEe'ZeddFd&ZGdGdHdHe'Zed8ddIdZ GdJdKdKe'Z e dLd;Z GdMdNdNe'Z e dOd;Z GdPdQdQe'ZedddRdZedddSdZej r,dTe_GdUdVdVe'ZedddWdZGdXdYdYe'ZeddZd&Zd[d\Zd]d^Zd_d`ZGdadbdbe'Zedcd ddZGdedfdfe'Zeddgd&ZGdhdidie'Zedjd;ZGdkdldleaZGdmdndne'Z e dddodZ!Gdpdqdqe'Z"e"drd;Z#e"ejL ejLdsdZ$GdtdudueZ%e%ddvd&Z&Gdwdxdxe'Z'e'dd'ejLdydZ(Gdzd{d{e'Z)e)d|d;Z*Gd}d~d~e'Z+e+ddd&Z,Gddde'Z-e-dddZ.ddZ/Gddde'Z0e0dddddZ1Gddde'Z2Gddde'Z3e3ddej4dZ5Gddde'Z6e6ddd&Z7e8e9:;Ze=e>dgZ?dS(N)Iterable)wrapscached_property Polynomial)extend_notes_in_docstringreplace_notes_in_docstringinherit_docstring_from)LowLevelCallable)optimize) integrate) _lazyselect _lazywhere)_stats)tukeylambda_variancetukeylambda_kurtosis)get_distribution_names _kurtosis rv_continuous_skew_get_fixed_fit_value _check_shape _ShapeInfo)kolmognkolmognpkolmogni)_XMIN_LOGXMIN_EULER_ZETA3_SQRT_PI_SQRT_2_OVER_PI_LOG_SQRT_2_OVER_PI) CensoredData) root_scalar)FitErrorcCsD|dd|dd|dd|dd|r@td|dS)a Remove the optimizer-related keyword arguments 'loc', 'scale' and 'optimizer' from `kwds`. Then check that `kwds` is empty, and raise `TypeError("Unknown arguments: %s." % kwds)` if it is not. This function is used in the fit method of distributions that override the default method and do not use the default optimization code. `kwds` is modified in-place. locNscale optimizermethodzUnknown arguments: %s.)pop TypeError)kwdsr.Z/var/www/html/django/DPS/env/lib/python3.9/site-packages/scipy/stats/_continuous_distns.py_remove_optimizer_parameters's    r0cstfdd}|S)Ncsz|dd}t|t}|dks2|rT|dkrTtt||j|g|Ri|S|r^|j}||g|Ri|SdS)Nr*mlemmr) getlower isinstancer$ num_censoredsupertypefit _uncensored)selfdataargsr-r*censoredfunr.r/wrapper>s "z _call_super_mom..wrapper)r)r@rAr.r?r/_call_super_mom:s rBcsV|p |d}||}fdd}|||sR|d9}||}d}t|r t|q |S)Nrcst|t|kSNnpsignlbrackrbrackr?r.r/interval_contains_rootVsz1_get_left_bracket..interval_contains_rootzVThe solver could not find a bracket containing a root to an MLE first order condition.)rEisinfFitSolverError)r@rIrHdiffrJmsgr.r?r/_get_left_bracketOs     rPc@sHeZdZdZddZddZddZdd Zd d Zd d Z ddZ dS) ksone_genaKolmogorov-Smirnov one-sided test statistic distribution. This is the distribution of the one-sided Kolmogorov-Smirnov (KS) statistics :math:`D_n^+` and :math:`D_n^-` for a finite sample size ``n >= 1`` (the shape parameter). %(before_notes)s See Also -------- kstwobign, kstwo, kstest Notes ----- :math:`D_n^+` and :math:`D_n^-` are given by .. math:: D_n^+ &= \text{sup}_x (F_n(x) - F(x)),\\ D_n^- &= \text{sup}_x (F(x) - F_n(x)),\\ where :math:`F` is a continuous CDF and :math:`F_n` is an empirical CDF. `ksone` describes the distribution under the null hypothesis of the KS test that the empirical CDF corresponds to :math:`n` i.i.d. random variates with CDF :math:`F`. %(after_notes)s References ---------- .. [1] Birnbaum, Z. W. and Tingey, F.H. "One-sided confidence contours for probability distribution functions", The Annals of Mathematical Statistics, 22(4), pp 592-596 (1951). %(example)s cCs|dk|t|k@SNrrEroundr;nr.r.r/ _argcheckszksone_gen._argcheckcCstdddtjfdgSNrVTrTFrrEinfr;r.r.r/ _shape_infoszksone_gen._shape_infocCst|| SrC)scuZ _smirnovpr;xrVr.r.r/_pdfszksone_gen._pdfcCs t||SrC)r^Z _smirnovcr_r.r.r/_cdfszksone_gen._cdfcCs t||SrC)scZsmirnovr_r.r.r/_sfsz ksone_gen._sfcCs t||SrC)r^Z _smirnovcir;qrVr.r.r/_ppfszksone_gen._ppfcCs t||SrC)rcZsmirnovirer.r.r/_isfszksone_gen._isfN) __name__ __module__ __qualname____doc__rWr]rarbrdrgrhr.r.r.r/rQfs%rQ?ksone)abnamec@sPeZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ dS) kstwo_genaKolmogorov-Smirnov two-sided test statistic distribution. This is the distribution of the two-sided Kolmogorov-Smirnov (KS) statistic :math:`D_n` for a finite sample size ``n >= 1`` (the shape parameter). %(before_notes)s See Also -------- kstwobign, ksone, kstest Notes ----- :math:`D_n` is given by .. math:: D_n = \text{sup}_x |F_n(x) - F(x)| where :math:`F` is a (continuous) CDF and :math:`F_n` is an empirical CDF. `kstwo` describes the distribution under the null hypothesis of the KS test that the empirical CDF corresponds to :math:`n` i.i.d. random variates with CDF :math:`F`. %(after_notes)s References ---------- .. [1] Simard, R., L'Ecuyer, P. "Computing the Two-Sided Kolmogorov-Smirnov Distribution", Journal of Statistical Software, Vol 39, 11, 1-18 (2011). %(example)s cCs|dk|t|k@SrRrSrUr.r.r/rWszkstwo_gen._argcheckcCstdddtjfdgSrXrZr\r.r.r/r]szkstwo_gen._shape_infocCs dt|ts|nt|dfSN?rn)r5rrEZ asanyarrayrUr.r.r/ _get_supportszkstwo_gen._get_supportcCs t||SrC)rr_r.r.r/raszkstwo_gen._pdfcCs t||SrCrr_r.r.r/rbszkstwo_gen._cdfcCst||ddSNFcdfrwr_r.r.r/rdsz kstwo_gen._sfcCst||ddS)NTryrrer.r.r/rgszkstwo_gen._ppfcCst||ddSrxr{rer.r.r/rhszkstwo_gen._isfN) rirjrkrlrWr]rvrarbrdrgrhr.r.r.r/rss$rskstwo)momtyperprqrrc@s@eZdZdZddZddZddZdd Zd d Zd d Z dS) kstwobign_genaLimiting distribution of scaled Kolmogorov-Smirnov two-sided test statistic. This is the asymptotic distribution of the two-sided Kolmogorov-Smirnov statistic :math:`\sqrt{n} D_n` that measures the maximum absolute distance of the theoretical (continuous) CDF from the empirical CDF. (see `kstest`). %(before_notes)s See Also -------- ksone, kstwo, kstest Notes ----- :math:`\sqrt{n} D_n` is given by .. math:: D_n = \text{sup}_x |F_n(x) - F(x)| where :math:`F` is a continuous CDF and :math:`F_n` is an empirical CDF. `kstwobign` describes the asymptotic distribution (i.e. the limit of :math:`\sqrt{n} D_n`) under the null hypothesis of the KS test that the empirical CDF corresponds to i.i.d. random variates with CDF :math:`F`. %(after_notes)s References ---------- .. [1] Feller, W. "On the Kolmogorov-Smirnov Limit Theorems for Empirical Distributions", Ann. Math. Statist. Vol 19, 177-189 (1948). %(example)s cCsgSrCr.r\r.r.r/r] szkstwobign_gen._shape_infocCs t| SrC)r^Z_kolmogpr;r`r.r.r/raszkstwobign_gen._pdfcCs t|SrC)r^Z_kolmogcrr.r.r/rbszkstwobign_gen._cdfcCs t|SrC)rcZ kolmogorovrr.r.r/rdszkstwobign_gen._sfcCs t|SrC)r^Z _kolmogcir;rfr.r.r/rgszkstwobign_gen._ppfcCs t|SrC)rcZkolmogirr.r.r/rhszkstwobign_gen._isfN) rirjrkrlr]rarbrdrgrhr.r.r.r/r~s$r~ kstwobign)rprrrKcCst|d dtSNrK@)rEexp _norm_pdf_Cr`r.r.r/ _norm_pdf,srcCs|d dtSr)_norm_pdf_logCrr.r.r/ _norm_logpdf0srcCs t|SrC)rcZndtrrr.r.r/ _norm_cdf4srcCs t|SrC)rcZlog_ndtrrr.r.r/ _norm_logcdf8srcCs t|SrC)rcZndtrirfr.r.r/ _norm_ppf<srcCs t| SrCrrr.r.r/_norm_sf@srcCs t| SrCrrr.r.r/ _norm_logsfDsrcCs t| SrCrrr.r.r/ _norm_isfHsrc@seZdZdZddZd!ddZddZd d Zd d Zd dZ ddZ ddZ ddZ ddZ ddZddZeeeddddZdd ZdS)"norm_genaA normal continuous random variable. The location (``loc``) keyword specifies the mean. The scale (``scale``) keyword specifies the standard deviation. %(before_notes)s Notes ----- The probability density function for `norm` is: .. math:: f(x) = \frac{\exp(-x^2/2)}{\sqrt{2\pi}} for a real number :math:`x`. %(after_notes)s %(example)s cCsgSrCr.r\r.r.r/r]csznorm_gen._shape_infoNcCs ||SrC)standard_normalr;size random_stater.r.r/_rvsfsz norm_gen._rvscCst|SrCrrr.r.r/raisz norm_gen._pdfcCst|SrCrrr.r.r/_logpdfmsznorm_gen._logpdfcCst|SrCrrr.r.r/rbpsz norm_gen._cdfcCst|SrCrrr.r.r/_logcdfssznorm_gen._logcdfcCst|SrCrrr.r.r/rdvsz norm_gen._sfcCst|SrC)rrr.r.r/_logsfysznorm_gen._logsfcCst|SrCrrr.r.r/rg|sz norm_gen._ppfcCst|SrCrrr.r.r/rhsz norm_gen._isfcCsdS)N)rmrnrmrmr.r\r.r.r/rsznorm_gen._statscCsdtdtjdSNrurKrrElogpir\r.r.r/_entropysznorm_gen._entropya} For the normal distribution, method of moments and maximum likelihood estimation give identical fits, and explicit formulas for the estimates are available. This function uses these explicit formulas for the maximum likelihood estimation of the normal distribution parameters, so the `optimizer` and `method` arguments are ignored. notescKs|dd}|dd}t||dur8|dur8tdt|}t|sXtd|durj|}n|}|durt||d}n|}||fS)Nflocfscale3All parameters fixed. There is nothing to optimize.$The data contains non-finite values.rK) r+r0 ValueErrorrEasarrayisfiniteallmeansqrt)r;r<r-rrr'r(r.r.r/r9s    z norm_gen.fitcCs"|ddkrt|dSdSdS)z @returns Moments of standard normal distribution for integer n >= 0 See eq. 16 of https://arxiv.org/abs/1209.4340v2 rKrrrmN)rcZ factorial2rUr.r.r/_munps znorm_gen._munp)NN)rirjrkrlr]rrarrbrrdrrgrhrrrBrrr9rr.r.r.r/rLs"   rnorm)rrc@sFeZdZdZejZddZddZddZ dd Z d d Z d d Z dS) alpha_gena&An alpha continuous random variable. %(before_notes)s Notes ----- The probability density function for `alpha` ([1]_, [2]_) is: .. math:: f(x, a) = \frac{1}{x^2 \Phi(a) \sqrt{2\pi}} * \exp(-\frac{1}{2} (a-1/x)^2) where :math:`\Phi` is the normal CDF, :math:`x > 0`, and :math:`a > 0`. `alpha` takes ``a`` as a shape parameter. %(after_notes)s References ---------- .. [1] Johnson, Kotz, and Balakrishnan, "Continuous Univariate Distributions, Volume 1", Second Edition, John Wiley and Sons, p. 173 (1994). .. [2] Anthony A. Salvia, "Reliability applications of the Alpha Distribution", IEEE Transactions on Reliability, Vol. R-34, No. 3, pp. 251-252 (1985). %(example)s cCstdddtjfdgSNrpFrFFrZr\r.r.r/r]szalpha_gen._shape_infocCs$d|dt|t|d|SNrnrK)rrr;r`rpr.r.r/raszalpha_gen._pdfcCs,dt|t|d|tt|S)Nrn)rErrrrr.r.r/rszalpha_gen._logpdfcCst|d|t|SNrnrrr.r.r/rbszalpha_gen._cdfcCsdt|t|t|Sr)rErrrr;rfrpr.r.r/rgszalpha_gen._ppfcCstjgdtjgdSNrKrEr[nanr;rpr.r.r/rszalpha_gen._statsN) rirjrkrlr_open_support_mask _support_maskr]rarrbrgrr.r.r.r/rsralphac@sHeZdZdZddZddZddZdd Zd d Zd d Z ddZ dS) anglit_genaAn anglit continuous random variable. %(before_notes)s Notes ----- The probability density function for `anglit` is: .. math:: f(x) = \sin(2x + \pi/2) = \cos(2x) for :math:`-\pi/4 \le x \le \pi/4`. %(after_notes)s %(example)s cCsgSrCr.r\r.r.r/r] szanglit_gen._shape_infocCstd|Sr)rEcosrr.r.r/ra szanglit_gen._pdfcCst|tjddSNrrEsinrrr.r.r/rbszanglit_gen._cdfcCst|tjddSr)rErrrr.r.r/rdszanglit_gen._sfcCstt|tjdSNr)rEarcsinrrrr.r.r/rgszanglit_gen._ppfcCs>dtjtjddddtjddtjtjddfS) Nrmrurr`rKrErr\r.r.r/rszanglit_gen._statscCsdtdSNrrKrErr\r.r.r/rszanglit_gen._entropyN) rirjrkrlr]rarbrdrgrrr.r.r.r/rsrranglitc@s@eZdZdZddZddZddZdd Zd d Zd d Z dS) arcsine_gena An arcsine continuous random variable. %(before_notes)s Notes ----- The probability density function for `arcsine` is: .. math:: f(x) = \frac{1}{\pi \sqrt{x (1-x)}} for :math:`0 < x < 1`. %(after_notes)s %(example)s cCsgSrCr.r\r.r.r/r]8szarcsine_gen._shape_infocCsLtjdd,dtjt|d|WdS1s>0YdS)Nignoredividernr)rEerrstaterrrr.r.r/ra;szarcsine_gen._pdfcCsdtjtt|SNr)rErrrrr.r.r/rb@szarcsine_gen._cdfcCsttjd|dSrrrr.r.r/rgCszarcsine_gen._ppfcCsd}d}d}d}||||fS)Nrug?rr.r;mumu2g1g2r.r.r/rFs zarcsine_gen._statscCsdS)Ngοr.r\r.r.r/rMszarcsine_gen._entropyN rirjrkrlr]rarbrgrrr.r.r.r/r$srarcsinec@seZdZdZddZdS) FitDataErrorz=Raised when input data is inconsistent with fixed parameters.cCs d|d|d|df|_dS)Nz>Invalid values in `data`. Maximum likelihood estimation with z requires that z < (x - loc)/scale < z for each x in `data`.r=)r;Zdistrr4upperr.r.r/__init__YszFitDataError.__init__Nrirjrkrlrr.r.r.r/rTsrc@seZdZdZddZdS)rMzN Raised when a solver fails to converge while fitting a distribution. cCs d}||dd7}|f|_dS)Nz1Solver for the MLE equations failed to converge:  )replacer=)r;mesgZemsgr.r.r/rgszFitSolverError.__init__Nrr.r.r.r/rMasrMcCs*t||}||| t|}|SrCrcpsi)rprqrVs1psiabfuncr.r.r/ _beta_mle_amsrcCsJ|\}}t||}||| t|||| t|g}|SrCr)thetarVrs2rprqrrr.r.r/ _beta_mle_abvs rcseZdZdZddZdddZddZd d Zd d Zd dZ ddZ ddZ ddZ fddZ eeeddfddZddZZS)beta_genaA beta continuous random variable. %(before_notes)s Notes ----- The probability density function for `beta` is: .. math:: f(x, a, b) = \frac{\Gamma(a+b) x^{a-1} (1-x)^{b-1}} {\Gamma(a) \Gamma(b)} for :math:`0 <= x <= 1`, :math:`a > 0`, :math:`b > 0`, where :math:`\Gamma` is the gamma function (`scipy.special.gamma`). `beta` takes :math:`a` and :math:`b` as shape parameters. %(after_notes)s %(example)s cCs0tdddtjfd}tdddtjfd}||gSNrpFrrrqrZr;iaibr.r.r/r]szbeta_gen._shape_infoNcCs||||SrCbeta)r;rprqrrr.r.r/rsz beta_gen._rvscCs>tjddt|||WdS1s00YdSNrover)rEr_boostZ _beta_pdfr;r`rprqr.r.r/rasz beta_gen._pdfcCs6t|d| t|d|}|t||8}|Sr)rcxlog1pyxlogybetaln)r;r`rprqlPxr.r.r/rs"zbeta_gen._logpdfcCst|||SrC)rZ _beta_cdfrr.r.r/rbsz beta_gen._cdfcCst|||SrC)rZ_beta_sfrr.r.r/rdsz beta_gen._sfcCs>tjddt|||WdS1s00YdSr)rErrZ _beta_isfrr.r.r/rhsz beta_gen._isfcCs>tjddt|||WdS1s00YdSr)rErrZ _beta_ppfr;rfrprqr.r.r/rgsz beta_gen._ppfcCs,t||t||t||t||fSrC)rZ _beta_meanZ_beta_varianceZ_beta_skewnessZ_beta_kurtosis_excessr;rprqr.r.r/rs     zbeta_gen._statscsTt|tr|}t|t|fdd}t|d\}}tj|||fdS)Ncs|\}}d||t||d||dt||}|d|dd|d|d|dd|||d}|||||d||d}|d9}||gS)NrKrrEr)r`rprqskkurrr.r/rs 8@$z beta_gen._fitstart..func)rnrnr) r5r$ _uncensorrrr fsolver7 _fitstart)r;r<rrprq __class__r r/r s zbeta_gen._fitstartz In the special case where `method="MLE"` and both `floc` and `fscale` are given, a `ValueError` is raised if any value `x` in `data` does not satisfy `floc < x < floc + fscale`. rcs:|dd}|dd}|dus(|durBtj|g|Ri|S|dd|ddt|gd}t|gd}t||dur|durtdt| stdt |||}t |dkst |dkrt d |||d | }|dus|dur|dur(|} d|}d|}n|} | |d|} tjt| | t|t|fd d \} } } }| dkr~t|d | d} |dur.| | } } nt|}t| }|d||jddd}||} d||} tjt| | gt|||fd d \} } } }| dkr&t|d | \} } | | ||fS)Nrrf0faZfix_a)f1fbZfix_brrrrrr4rT)r= full_output)r)Zddof)r3r7r9r+rr0rrErrZravelanyrrr r rlenrsumrMrclog1pvarr)r;r<r=r-rrrrxbarrqrprinfoierrrrfacr r.r/r9s`               z beta_gen.fitcCsdd}dd}dd}dd}|d kr:|d kr:|||S|d krd||d krd|||krd|||S|d kr||d kr|||kr|||S|||SdS) NcSsJt|||dt||dt|||dt||Sr)rcrrrprqr.r.r/regularBs z"beta_gen._entropy..regularcSs||}dtdtjt|t|dt|d}d|d|d|dd|d }d |d |d|d|d }d |d |d|d|d }||||d S) NrurKrrni xr)rprqsum_ablog_termt1t2t3r.r.r/asymptotic_ab_largeFs4($$z.beta_gen._entropy..asymptotic_ab_largecSs||}t||dt|}dd|dd||dd|dd|dd|d d |d d d|dd||dd |dd|dd |d d |d d}|t||t|dt|}|||S)NrrK r#r$r'r%r<~)rcgammalnrrErr)rprqr(r*r+r)r.r.r/asymptotic_b_largePs0:        *z-beta_gen._entropy..asymptotic_b_largecSsB|dkr dSt|}t|}t|d|d}|dd|S)Nrnir&rK)rElog10int)vjdigitsdr.r.r/threshold_large\s  z*beta_gen._entropy..threshold_largegRAg(RAg.Ar.)r;rprqr r-r6r>r.r.r/rAs        zbeta_gen._entropy)NN)rirjrkrlr]rrarrbrdrhrgrr rBrrr9r __classcell__r.r.r r/rs   errc@sXeZdZdZejZddZdddZddZ d d Z d d Z d dZ ddZ ddZdS) betaprime_genaA beta prime continuous random variable. %(before_notes)s Notes ----- The probability density function for `betaprime` is: .. math:: f(x, a, b) = \frac{x^{a-1} (1+x)^{-a-b}}{\beta(a, b)} for :math:`x >= 0`, :math:`a > 0`, :math:`b > 0`, where :math:`\beta(a, b)` is the beta function (see `scipy.special.beta`). `betaprime` takes ``a`` and ``b`` as shape parameters. The distribution is related to the `beta` distribution as follows: If :math:`X` follows a beta distribution with parameters :math:`a, b`, then :math:`Y = X/(1-X)` has a beta prime distribution with parameters :math:`a, b` ([1]_). The beta prime distribution is a reparametrized version of the F distribution. The beta prime distribution with shape parameters ``a`` and ``b`` and ``scale = s`` is equivalent to the F distribution with parameters ``d1 = 2*a``, ``d2 = 2*b`` and ``scale = (a/b)*s``. For example, >>> from scipy.stats import betaprime, f >>> x = [1, 2, 5, 10] >>> a = 12 >>> b = 5 >>> betaprime.pdf(x, a, b, scale=2) array([0.00541179, 0.08331299, 0.14669185, 0.03150079]) >>> f.pdf(x, 2*a, 2*b, scale=(a/b)*2) array([0.00541179, 0.08331299, 0.14669185, 0.03150079]) %(after_notes)s References ---------- .. [1] Beta prime distribution, Wikipedia, https://en.wikipedia.org/wiki/Beta_prime_distribution %(example)s cCs0tdddtjfd}tdddtjfd}||gSrrZrr.r.r/r]szbetaprime_gen._shape_infoNcCs(tj|||d}tj|||d}||SNrr)gammarvs)r;rprqrru1u2r.r.r/rszbetaprime_gen._rvscCst||||SrCrErrrr.r.r/raszbetaprime_gen._pdfcCs,t|d|t|||t||Sr)rcrrrrr.r.r/rszbetaprime_gen._logpdfcCs"t|dk|||gdddddS)NrcSstdd|||SrRrrdx_Za_b_r.r.r/z$betaprime_gen._cdf..cSst|d|||SrRrrbrIr.r.r/rLrMf2rrr.r.r/rbs zbetaprime_gen._cdfcCs"t|dk|||gdddddS)NrcSstdd|||SrRrNrIr.r.r/rLrMz#betaprime_gen._sf..cSst|d|||SrRrHrIr.r.r/rLrMrOrQrr.r.r/rds zbetaprime_gen._sfcCst|||\}}}tj|||}tjdd|d|}Wdn1sR0Y|dk}dtj||||||d||<|S)NrrrgH.?)rEbroadcast_arraysstatsrrgrrh)r;prprqroutir.r.r/rgs*(zbetaprime_gen._ppfcs"t|k||ffddtjdS)Ncs(tjfddtddDddS)Ncs g|]}|d|qSrr..0rWrr.r/ rMz9betaprime_gen._munp....rrZaxis)rEprodrangerrVrr/rLrMz%betaprime_gen._munp.. fillvaluerrEr[)r;rVrprqr.r_r/rs   zbetaprime_gen._munp)NN)rirjrkrlrrrr]rrarrbrdrgrr.r.r.r/r@rs/  r@ betaprimec@sBeZdZdZddZddZddZdd Zdd d Zd dZ dS) bradford_genabA Bradford continuous random variable. %(before_notes)s Notes ----- The probability density function for `bradford` is: .. math:: f(x, c) = \frac{c}{\log(1+c) (1+cx)} for :math:`0 <= x <= 1` and :math:`c > 0`. `bradford` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s %(example)s cCstdddtjfdgSNcFrrrZr\r.r.r/r]szbradford_gen._shape_infocCs|||dt|Srrcrr;r`rfr.r.r/raszbradford_gen._pdfcCst||t|SrCrgrhr.r.r/rbszbradford_gen._cdfcCst|t||SrCrcexpm1rr;rfrfr.r.r/rgszbradford_gen._ppfmvcCsxtd|}||||}|d|d|d|||}d}d}d|vrtdd||d|||dd||||dd}|t|||dd|d||dd|}d |vrl|d|d|d|d d d||||d |dd|||d|d d|d}|d|||dd|d}||||fS)NrnrrKsr/ rrkrr)rErr)r;rfmomentsrorrrrr.r.r/rs $F: B $zbradford_gen._statscCs$td|}|dt||SNrrr)r;rfror.r.r/rszbradford_gen._entropyN)rlrr.r.r.r/rds rdbradfordc@sheZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ ddZdS)burr_genaA Burr (Type III) continuous random variable. %(before_notes)s See Also -------- fisk : a special case of either `burr` or `burr12` with ``d=1`` burr12 : Burr Type XII distribution mielke : Mielke Beta-Kappa / Dagum distribution Notes ----- The probability density function for `burr` is: .. math:: f(x; c, d) = c d \frac{x^{-c - 1}} {{(1 + x^{-c})}^{d + 1}} for :math:`x >= 0` and :math:`c, d > 0`. `burr` takes ``c`` and ``d`` as shape parameters for :math:`c` and :math:`d`. This is the PDF corresponding to the third CDF given in Burr's list; specifically, it is equation (11) in Burr's paper [1]_. The distribution is also commonly referred to as the Dagum distribution [2]_. If the parameter :math:`c < 1` then the mean of the distribution does not exist and if :math:`c < 2` the variance does not exist [2]_. The PDF is finite at the left endpoint :math:`x = 0` if :math:`c * d >= 1`. %(after_notes)s References ---------- .. [1] Burr, I. W. "Cumulative frequency functions", Annals of Mathematical Statistics, 13(2), pp 215-232 (1942). .. [2] https://en.wikipedia.org/wiki/Dagum_distribution .. [3] Kleiber, Christian. "A guide to the Dagum distributions." Modeling Income Distributions and Lorenz Curves pp 97-117 (2008). %(example)s cCs0tdddtjfd}tdddtjfd}||gSNrfFrrr=rZr;icidr.r.r/r]Lszburr_gen._shape_infocCs8t|dk|||gddddd}|jdkr4|dS|S)NrcSs$|||||dd||SrRr.rJZc_Zd_r.r.r/rLUrMzburr_gen._pdf..cSs,|||| dd|| |dSNrnrr.rzr.r.r/rLVsrOr.rndimr;r`rfr=outputr.r.r/raQs z burr_gen._pdfcCs8t|dk|||gddddd}|jdkr4|dS|S)NrcSs>t|t|t||d||dt||SrR)rErrcrrrzr.r.r/rL_s&z"burr_gen._logpdf..cSs<t|t|t| d|t|d|| SrRrErrcrrrzr.r.r/rLasrOr.r|r~r.r.r/r\s zburr_gen._logpdfcCsd|| | SrRr.r;r`rfr=r.r.r/rbhsz burr_gen._cdfcCst|| | SrCrgrr.r.r/rkszburr_gen._logcdfcCst||||SrCrErrrr.r.r/rdnsz burr_gen._sfcCstd|| |  SrRrErrr.r.r/rqszburr_gen._logsfcCs|d|dd|SNrr.r;rfrfr=r.r.r/rgtsz burr_gen._ppfcCs$td|| }t|d|SNrrcrrj)r;rfrfr=Z_qr.r.r/rhwsz burr_gen._isfc Cstdddd|}t||d||\}}}}t|dk|tj}||d} t|dk| tj} t|dk||||| fdd tjd } t|d k|||||| fd d tjd } t|d kr| | | | fS|| | | fS)NrrrnrKr@cSs*|d||d|dt|dS)NrrKr)rfe1e2e3mu2_if_cr.r.r/rLs z!burr_gen._stats..r`@cSs8|d||d||dd|d|ddS)NrrrKrr.)rfrrre4rr.r.r/rLsr) rEarangereshapercrwhererrr}item) r;rfr=ncrrrrrrrrrr.r.r/r{s(   zburr_gen._statscs\ddt|t|t|}}}t||k||k@||k@|||ffddtjS)NcSs$d||}|td|||SrrcrrVrfr=rr.r.r/__munps zburr_gen._munp..__munpcs |||SrCr.)rfr=rVZ_burr_gen__munpr.r/rLrMz burr_gen._munp..)rErrr)r;rVrfr=r.rr/rs "  zburr_gen._munpN)rirjrkrlr]rarrbrrdrrgrhrrr.r.r.r/rus0  ruburrc@sXeZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ dS) burr12_gena}A Burr (Type XII) continuous random variable. %(before_notes)s See Also -------- fisk : a special case of either `burr` or `burr12` with ``d=1`` burr : Burr Type III distribution Notes ----- The probability density function for `burr12` is: .. math:: f(x; c, d) = c d \frac{x^{c-1}} {(1 + x^c)^{d + 1}} for :math:`x >= 0` and :math:`c, d > 0`. `burr12` takes ``c`` and ``d`` as shape parameters for :math:`c` and :math:`d`. This is the PDF corresponding to the twelfth CDF given in Burr's list; specifically, it is equation (20) in Burr's paper [1]_. %(after_notes)s The Burr type 12 distribution is also sometimes referred to as the Singh-Maddala distribution from NIST [2]_. References ---------- .. [1] Burr, I. W. "Cumulative frequency functions", Annals of Mathematical Statistics, 13(2), pp 215-232 (1942). .. [2] https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/b12pdf.htm .. [3] "Burr distribution", https://en.wikipedia.org/wiki/Burr_distribution %(example)s cCs0tdddtjfd}tdddtjfd}||gSrvrZrwr.r.r/r]szburr12_gen._shape_infocCst||||SrCrGrr.r.r/raszburr12_gen._pdfcCs:t|t|t|d|t| d||SrRrrr.r.r/rszburr12_gen._logpdfcCst|||| SrCrcrjrrr.r.r/rbszburr12_gen._cdfcCstd|||  SrRrgrr.r.r/rszburr12_gen._logcdfcCst||||SrCrrr.r.r/rdszburr12_gen._sfcCst| ||SrCrcrrr.r.r/rszburr12_gen._logsfcCs"td|t| d|S)Nr.rrirr.r.r/rgszburr12_gen._ppfcCs(dd}t|||k|||f|tjdS)NcSs$d||}|td|||Srrrr.r.r/moment_if_existss z*burr12_gen._munp..moment_if_existsr`)rrEr)r;rVrfr=rr.r.r/rszburr12_gen._munpN) rirjrkrlr]rarrbrrdrrgrr.r.r.r/rs,rburr12c@speZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ ddZddZdS)fisk_genaA Fisk continuous random variable. The Fisk distribution is also known as the log-logistic distribution. %(before_notes)s See Also -------- burr Notes ----- The probability density function for `fisk` is: .. math:: f(x, c) = \frac{c x^{c-1}} {(1 + x^c)^2} for :math:`x >= 0` and :math:`c > 0`. Please note that the above expression can be transformed into the following one, which is also commonly used: .. math:: f(x, c) = \frac{c x^{-c-1}} {(1 + x^{-c})^2} `fisk` takes ``c`` as a shape parameter for :math:`c`. `fisk` is a special case of `burr` or `burr12` with ``d=1``. Suppose ``X`` is a logistic random variable with location ``l`` and scale ``s``. Then ``Y = exp(X)`` is a Fisk (log-logistic) random variable with ``scale = exp(l)`` and shape ``c = 1/s``. %(after_notes)s %(example)s cCstdddtjfdgSrerZr\r.r.r/r]!szfisk_gen._shape_infocCst||dSr)rrarhr.r.r/ra$sz fisk_gen._pdfcCst||dSr)rrbrhr.r.r/rb(sz fisk_gen._cdfcCst||dSr)rrdrhr.r.r/rd+sz fisk_gen._sfcCst||dSr)rrrhr.r.r/r.szfisk_gen._logpdfcCst||dSr)rrrhr.r.r/r2szfisk_gen._logcdfcCst||dSr)rrrhr.r.r/r5szfisk_gen._logsfcCst||dSr)rrgrhr.r.r/rg8sz fisk_gen._ppfcCst||dSr)rrhrkr.r.r/rh;sz fisk_gen._isfcCst||dSr)rrr;rVrfr.r.r/r>szfisk_gen._munpcCs t|dSr)rrr;rfr.r.r/rAszfisk_gen._statscCsdt|Srrrr.r.r/rDszfisk_gen._entropyN)rirjrkrlr]rarbrdrrrrgrhrrrr.r.r.r/rs*rfiskc@sZeZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ dddZ dS) cauchy_gena A Cauchy continuous random variable. %(before_notes)s Notes ----- The probability density function for `cauchy` is .. math:: f(x) = \frac{1}{\pi (1 + x^2)} for a real number :math:`x`. %(after_notes)s %(example)s cCsgSrCr.r\r.r.r/r]_szcauchy_gen._shape_infocCsdtjd||Srrrr.r.r/rabszcauchy_gen._pdfcCsddtjt|SrtrErarctanrr.r.r/rbfszcauchy_gen._cdfcCsttj|tjdSrrEtanrrr.r.r/rgiszcauchy_gen._ppfcCsddtjt|Srtrrr.r.r/rdlszcauchy_gen._sfcCsttjdtj|Srrrr.r.r/rhoszcauchy_gen._isfcCstjtjtjtjfSrCrErr\r.r.r/rrszcauchy_gen._statscCstdtjSrrr\r.r.r/ruszcauchy_gen._entropyNcCs8t|tr|}t|gd\}}}|||dfS)N2KrKr5r$r rEZ percentile)r;r<r=p25p50p75r.r.r/r xs zcauchy_gen._fitstart)N) rirjrkrlr]rarbrgrdrhrrr r.r.r.r/rKsrcauchyc@sbeZdZdZddZdddZddZd d Zd d Zd dZ ddZ ddZ ddZ ddZ dS)chi_genaA chi continuous random variable. %(before_notes)s Notes ----- The probability density function for `chi` is: .. math:: f(x, k) = \frac{1}{2^{k/2-1} \Gamma \left( k/2 \right)} x^{k-1} \exp \left( -x^2/2 \right) for :math:`x >= 0` and :math:`k > 0` (degrees of freedom, denoted ``df`` in the implementation). :math:`\Gamma` is the gamma function (`scipy.special.gamma`). Special cases of `chi` are: - ``chi(1, loc, scale)`` is equivalent to `halfnorm` - ``chi(2, 0, scale)`` is equivalent to `rayleigh` - ``chi(3, 0, scale)`` is equivalent to `maxwell` `chi` takes ``df`` as a shape parameter. %(after_notes)s %(example)s cCstdddtjfdgSNdfFrrrZr\r.r.r/r]szchi_gen._shape_infoNcCsttj|||dSrA)rErchi2rDr;rrrr.r.r/rsz chi_gen._rvscCst|||SrCrGr;r`rr.r.r/rasz chi_gen._pdfcCsJtddtd|td|}|t|d|d|dS)NrKrurn)rErrcr5r)r;r`rlr.r.r/rs*zchi_gen._logpdfcCstd|d|dSNrurKrcgammaincrr.r.r/rbsz chi_gen._cdfcCstd|d|dSrrc gammainccrr.r.r/rdsz chi_gen._sfcCstdtd||SNrKrurErrc gammaincinvr;rfrr.r.r/rgsz chi_gen._ppfcCstdtd||SrrErrc gammainccinvrr.r.r/rhsz chi_gen._isfcCstdtd|d}|||}d|d|dd|tt|d}d|d|d|dd|dd|d}|t|d }||||fS) NrKrurr?rnrrr)rErrcpochrpowerr;rrrrrr.r.r/rs  .4zchi_gen._statscCs&dd}dd}t|dk|f||dS)NcSs6td|d|td|dtd|Sr)rcr5rErdigammarr.r.r/regular_formulas &z)chi_gen._entropy..regular_formulacSsDdttjd|dd|ddd|d|dd S) NrurKr.rrgll?rrr.r.r/asymptotic_formulas *  z,chi_gen._entropy..asymptotic_formulagr@rOrQ)r;rrrr.r.r/rs zchi_gen._entropy)NNrirjrkrlr]rrarrbrdrgrhrrr.r.r.r/rs  rchic@sbeZdZdZddZdddZddZd d Zd d Zd dZ ddZ ddZ ddZ ddZ dS)chi2_genaA chi-squared continuous random variable. For the noncentral chi-square distribution, see `ncx2`. %(before_notes)s See Also -------- ncx2 Notes ----- The probability density function for `chi2` is: .. math:: f(x, k) = \frac{1}{2^{k/2} \Gamma \left( k/2 \right)} x^{k/2-1} \exp \left( -x/2 \right) for :math:`x > 0` and :math:`k > 0` (degrees of freedom, denoted ``df`` in the implementation). `chi2` takes ``df`` as a shape parameter. The chi-squared distribution is a special case of the gamma distribution, with gamma parameters ``a = df/2``, ``loc = 0`` and ``scale = 2``. %(after_notes)s %(example)s cCstdddtjfdgSrrZr\r.r.r/r]szchi2_gen._shape_infoNcCs |||SrC)Z chisquarerr.r.r/rsz chi2_gen._rvscCst|||SrCrGrr.r.r/rasz chi2_gen._pdfcCs<t|dd||dt|dtd|dS)NrrrK)rcrr5rErrr.r.r/rszchi2_gen._logpdfcCs t||SrC)rcZchdtrrr.r.r/rbsz chi2_gen._cdfcCs t||SrC)rcZchdtrcrr.r.r/rd sz chi2_gen._sfcCs t||SrC)rcZchdtrir;rTrr.r.r/rh sz chi2_gen._isfcCsdt|d|Srrcrrr.r.r/rgsz chi2_gen._ppfcCs2|}d|}dtd|}d|}||||fS)NrKr(@rrr.r.r/rs zchi2_gen._statscCs.d|}dd}dd}t|dk|f||dS)NrucSs*|tdt|d|t|SNrKr)rErrcr5r)half_dfr.r.r/rsz*chi2_gen._entropy..regular_formulacSs\tdddtdtj}d|}|d|d|d|ddt||S)NrKrurgUUUUUUUUUUUUտgllg@r)rrfhr.r.r/r!s" z-chi2_gen._entropy..asymptotic_formula}rOrQ)r;rrrrr.r.r/rs zchi2_gen._entropy)NN)rirjrkrlr]rrarrbrdrhrgrrr.r.r.r/rs! rrc@sXeZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ dS) cosine_gena\A cosine continuous random variable. %(before_notes)s Notes ----- The cosine distribution is an approximation to the normal distribution. The probability density function for `cosine` is: .. math:: f(x) = \frac{1}{2\pi} (1+\cos(x)) for :math:`-\pi \le x \le \pi`. %(after_notes)s %(example)s cCsgSrCr.r\r.r.r/r]Iszcosine_gen._shape_infocCsdtjdt|SNrurrErrrr.r.r/raLszcosine_gen._pdfcCs(t|}t|dk|fddtj dS)Nr.cSst|tdtjSr)rErrrrfr.r.r/rLSrMz$cosine_gen._logpdf..r`)rErrr[rhr.r.r/rPs   zcosine_gen._logpdfcCs t|SrCr^Z _cosine_cdfrr.r.r/rbVszcosine_gen._cdfcCs t| SrCrrr.r.r/rdYszcosine_gen._sfcCs t|SrCr^Z_cosine_invcdfr;rTr.r.r/rg\szcosine_gen._ppfcCs t| SrCrrr.r.r/rh_szcosine_gen._isfcCsJtjtjdd}dtjdddtjtjdd}d |d |fS) Nrrr2rZ@rrKrmr)r;r:ror.r.r/rbs*zcosine_gen._statscCstdtjdS)Nrrnrr\r.r.r/rgszcosine_gen._entropyN rirjrkrlr]rarrbrdrgrhrrr.r.r.r/r4srcosinec@sbeZdZdZddZdddZddZd d Zd d Zd dZ ddZ ddZ ddZ ddZ dS) dgamma_genaA double gamma continuous random variable. The double gamma distribution is also known as the reflected gamma distribution [1]_. %(before_notes)s Notes ----- The probability density function for `dgamma` is: .. math:: f(x, a) = \frac{1}{2\Gamma(a)} |x|^{a-1} \exp(-|x|) for a real number :math:`x` and :math:`a > 0`. :math:`\Gamma` is the gamma function (`scipy.special.gamma`). `dgamma` takes ``a`` as a shape parameter for :math:`a`. %(after_notes)s References ---------- .. [1] Johnson, Kotz, and Balakrishnan, "Continuous Univariate Distributions, Volume 1", Second Edition, John Wiley and Sons (1994). %(example)s cCstdddtjfdgSrrZr\r.r.r/r]szdgamma_gen._shape_infoNcCs2|j|d}tj|||d}|t|dkddSNrrBrurr.)uniformrCrDrEr)r;rprruZgmr.r.r/rs zdgamma_gen._rvscCs2t|}ddt|||dt| Sr)absrcrCrErr;r`rpaxr.r.r/raszdgamma_gen._pdfcCs0t|}t|d||tdt|Sr)rrcrrErr5rr.r.r/rszdgamma_gen._logpdfc Cs0t|dkddt||dt|| SNrru)rErrcrrrr.r.r/rbs zdgamma_gen._cdfc Cs0t|dkdt||ddt|| Sr)rErrcrrrr.r.r/rds zdgamma_gen._sfcCstj|tdSNru)rSrCrrErrr.r.r/rszdgamma_gen._entropyc Cs0t|dkt|d|dt|d| SrrErrcrrrr.r.r/rgs zdgamma_gen._ppfc Cs0t|dkt|d|d t|d|Srrrr.r.r/rhs zdgamma_gen._isfcCs,||d}d|d|d|d|dfS)Nrnrmrrr.)r;rprr.r.r/rs zdgamma_gen._stats)NN)rirjrkrlr]rrarrbrdrrgrhrr.r.r.r/rns rdgammac@sjeZdZdZddZdddZddZd d Zd d Zd dZ ddZ ddZ ddZ ddZ ddZdS) dweibull_genavA double Weibull continuous random variable. %(before_notes)s Notes ----- The probability density function for `dweibull` is given by .. math:: f(x, c) = c / 2 |x|^{c-1} \exp(-|x|^c) for a real number :math:`x` and :math:`c > 0`. `dweibull` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s %(example)s cCstdddtjfdgSrerZr\r.r.r/r]szdweibull_gen._shape_infoNcCs2|j|d}tj|||d}|t|dkddSr)r weibull_minrDrEr)r;rfrrrwr.r.r/rs zdweibull_gen._rvscCs0t|}|d||dt|| }|SNrrn)rrEr)r;r`rfrPxr.r.r/ras$zdweibull_gen._pdfcCs4t|}t|tdt|d|||Sr)rrErrcr)r;r`rfrr.r.r/rszdweibull_gen._logpdfcCs.dtt|| }t|dkd||SNrurr)rErrr)r;r`rfZCx1r.r.r/rbszdweibull_gen._cdfcCsFdt|dk|d|}tt| d|}t|dk|| SNrrurn)rErrr)r;rfrfrr.r.r/rgszdweibull_gen._ppfcCs.dtjt||}t|dk|d|Sr)rSrrdrErr)r;r`rfZhalf_weibull_min_sfr.r.r/rdszdweibull_gen._sfcCs<dt|dk|d|}tj||}t|dk| |Sr)rErrSrrh)r;rfrfZdouble_qZweibull_min_isfr.r.r/rhszdweibull_gen._isfcCs"d|dtdd||S)NrrKrnrcrCrr.r.r/rszdweibull_gen._munpcCsdSN)rNrNr.rr.r.r/rszdweibull_gen._statscCstj|td}|Sr)rSrrrEr)r;rfrr.r.r/rszdweibull_gen._entropy)NN)rirjrkrlr]rrarrbrgrdrhrrrr.r.r.r/rs rdweibullc@seZdZdZddZdddZddZd d Zd d Zd dZ ddZ ddZ ddZ ddZ ddZeeeddddZdS) expon_genaEAn exponential continuous random variable. %(before_notes)s Notes ----- The probability density function for `expon` is: .. math:: f(x) = \exp(-x) for :math:`x \ge 0`. %(after_notes)s A common parameterization for `expon` is in terms of the rate parameter ``lambda``, such that ``pdf = lambda * exp(-lambda * x)``. This parameterization corresponds to using ``scale = 1 / lambda``. The exponential distribution is a special case of the gamma distributions, with gamma shape parameter ``a = 1``. %(example)s cCsgSrCr.r\r.r.r/r]$szexpon_gen._shape_infoNcCs ||SrC)standard_exponentialrr.r.r/r'szexpon_gen._rvscCs t| SrCrErrr.r.r/ra*szexpon_gen._pdfcCs| SrCr.rr.r.r/r.szexpon_gen._logpdfcCst|  SrCrcrjrr.r.r/rb1szexpon_gen._cdfcCst|  SrCrgrr.r.r/rg4szexpon_gen._ppfcCs t| SrCrrr.r.r/rd7sz expon_gen._sfcCs| SrCr.rr.r.r/r:szexpon_gen._logsfcCs t| SrCrrr.r.r/rh=szexpon_gen._isfcCsdS)N)rnrnr@r.r\r.r.r/r@szexpon_gen._statscCsdSrr.r\r.r.r/rCszexpon_gen._entropyz When `method='MLE'`, this function uses explicit formulas for the maximum likelihood estimation of the exponential distribution parameters, so the `optimizer`, `loc` and `scale` keyword arguments are ignored. rc Ost|dkrtd|dd}|dd}t||durL|durLtdt|}t|sltd| }|dur|}n|}||krt d|tj d|dur| |}n|}t |t |fS) NrToo many arguments.rrrrexponr)rr,r+r0rrErrrminrr[rfloat r;r<r=r-rrdata_minr'r(r.r.r/r9Fs(    z expon_gen.fit)NN)rirjrkrlr]rrarrbrgrdrrhrrrBrrr9r.r.r.r/r s  rrc@sJeZdZdZddZdddZddZd d Zd d Zd dZ ddZ dS) exponnorm_genaAn exponentially modified Normal continuous random variable. Also known as the exponentially modified Gaussian distribution [1]_. %(before_notes)s Notes ----- The probability density function for `exponnorm` is: .. math:: f(x, K) = \frac{1}{2K} \exp\left(\frac{1}{2 K^2} - x / K \right) \text{erfc}\left(-\frac{x - 1/K}{\sqrt{2}}\right) where :math:`x` is a real number and :math:`K > 0`. It can be thought of as the sum of a standard normal random variable and an independent exponentially distributed random variable with rate ``1/K``. %(after_notes)s An alternative parameterization of this distribution (for example, in the Wikipedia article [1]_) involves three parameters, :math:`\mu`, :math:`\lambda` and :math:`\sigma`. In the present parameterization this corresponds to having ``loc`` and ``scale`` equal to :math:`\mu` and :math:`\sigma`, respectively, and shape parameter :math:`K = 1/(\sigma\lambda)`. .. versionadded:: 0.16.0 References ---------- .. [1] Exponentially modified Gaussian distribution, Wikipedia, https://en.wikipedia.org/wiki/Exponentially_modified_Gaussian_distribution %(example)s cCstdddtjfdgS)NKFrrrZr\r.r.r/r]szexponnorm_gen._shape_infoNcCs |||}||}||SrC)rr)r;rrrexpvalZgvalr.r.r/rs zexponnorm_gen._rvscCst|||SrCrG)r;r`rr.r.r/raszexponnorm_gen._pdfcCs2d|}|d||}|t||t|SNrnrurrEr)r;r`rinvKZexpargr.r.r/rszexponnorm_gen._logpdfcCs:d|}|d||}|t||}t|t|Sr rrrErr;r`rr r Zlogprodr.r.r/rbszexponnorm_gen._cdfcCs<d|}|d||}|t||}t| t|Sr r rr.r.r/rdszexponnorm_gen._sfcCsD||}d|}d|d|d}d|||d}||||fS)NrnrKrrrrr.)r;rZK2ZopK2ZskwZkrtr.r.r/rs zexponnorm_gen._stats)NN) rirjrkrlr]rrarrbrdrr.r.r.r/rys) r exponnormcCstt||S)a' Compute (1 + x)**y - 1. Uses expm1 and xlog1py to avoid loss of precision when (1 + x)**y is close to 1. Note that the inverse of this function with respect to x is ``_pow1pm1(x, 1/y)``. That is, if t = _pow1pm1(x, y) then x = _pow1pm1(t, 1/y) )rErjrcrr`yr.r.r/_pow1pm1src@sHeZdZdZddZddZddZdd Zd d Zd d Z ddZ dS) exponweib_genaAn exponentiated Weibull continuous random variable. %(before_notes)s See Also -------- weibull_min, numpy.random.Generator.weibull Notes ----- The probability density function for `exponweib` is: .. math:: f(x, a, c) = a c [1-\exp(-x^c)]^{a-1} \exp(-x^c) x^{c-1} and its cumulative distribution function is: .. math:: F(x, a, c) = [1-\exp(-x^c)]^a for :math:`x > 0`, :math:`a > 0`, :math:`c > 0`. `exponweib` takes :math:`a` and :math:`c` as shape parameters: * :math:`a` is the exponentiation parameter, with the special case :math:`a=1` corresponding to the (non-exponentiated) Weibull distribution `weibull_min`. * :math:`c` is the shape parameter of the non-exponentiated Weibull law. %(after_notes)s References ---------- https://en.wikipedia.org/wiki/Exponentiated_Weibull_distribution %(example)s cCs0tdddtjfd}tdddtjfd}||gSNrpFrrrfrZr;rrxr.r.r/r]szexponweib_gen._shape_infocCst||||SrCrGr;r`rprfr.r.r/ra szexponweib_gen._pdfcCsR|| }t| }t|t|t|d||t|d|}|Sr)rcrjrErr)r;r`rprfZnegxcexm1cZlogpr.r.r/rs  "zexponweib_gen._logpdfcCst||  }||SrCr)r;r`rprfrr.r.r/rbszexponweib_gen._cdfcCs$t|d|  td|Sr)rcrrEr)r;rfrprfr.r.r/rgszexponweib_gen._ppfcCstt||  | SrC)rrErrr.r.r/rdszexponweib_gen._sfcCs"tt| d|  d|SrR)rErr)r;rTrprfr.r.r/rh!szexponweib_gen._isfN rirjrkrlr]rarrbrgrdrhr.r.r.r/rs(r exponweibc@sHeZdZdZddZddZddZdd Zd d Zd d Z ddZ dS) exponpow_genaAn exponential power continuous random variable. %(before_notes)s Notes ----- The probability density function for `exponpow` is: .. math:: f(x, b) = b x^{b-1} \exp(1 + x^b - \exp(x^b)) for :math:`x \ge 0`, :math:`b > 0`. Note that this is a different distribution from the exponential power distribution that is also known under the names "generalized normal" or "generalized Gaussian". `exponpow` takes ``b`` as a shape parameter for :math:`b`. %(after_notes)s References ---------- http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Exponentialpower.pdf %(example)s cCstdddtjfdgSNrqFrrrZr\r.r.r/r]Dszexponpow_gen._shape_infocCst|||SrCrGr;r`rqr.r.r/raGszexponpow_gen._pdfcCs8||}dt|t|d||t|}|SNrrn)rErrcrr)r;r`rqxbfr.r.r/rKs,zexponpow_gen._logpdfcCstt||  SrCrrr.r.r/rbPszexponpow_gen._cdfcCstt|| SrCrErrcrjrr.r.r/rdSszexponpow_gen._sfcCstt| d|SrrcrrErrr.r.r/rhVszexponpow_gen._isfcCsttt|  d|Srpowrcrr;rfrqr.r.r/rgYszexponpow_gen._ppfN) rirjrkrlr]rarrbrdrhrgr.r.r.r/r(srexponpowc@s`eZdZdZejZddZdddZddZ d d Z d d Z d dZ ddZ ddZddZdS)fatiguelife_gena0A fatigue-life (Birnbaum-Saunders) continuous random variable. %(before_notes)s Notes ----- The probability density function for `fatiguelife` is: .. math:: f(x, c) = \frac{x+1}{2c\sqrt{2\pi x^3}} \exp(-\frac{(x-1)^2}{2x c^2}) for :math:`x >= 0` and :math:`c > 0`. `fatiguelife` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s References ---------- .. [1] "Birnbaum-Saunders distribution", https://en.wikipedia.org/wiki/Birnbaum-Saunders_distribution %(example)s cCstdddtjfdgSrerZr\r.r.r/r]}szfatiguelife_gen._shape_infoNcCsD||}d||}||}dd|d|td|}|S)NrurnrKr)rrEr)r;rfrrzr`x2tr.r.r/rs   "zfatiguelife_gen._rvscCst|||SrCrGrhr.r.r/raszfatiguelife_gen._pdfcCsZt|d|ddd||dtd|dtdtjdt|S)NrrKrrurrrhr.r.r/rs6 zfatiguelife_gen._logpdfcCs$td|t|dt|Sr)rrErrhr.r.r/rbszfatiguelife_gen._cdfcCs*|t|}d|t|dddSN?rKrrrErr;rfrftmpr.r.r/rgs zfatiguelife_gen._ppfcCs$td|t|dt|Sr)rrErrhr.r.r/rdszfatiguelife_gen._sfcCs,| t|}d|t|dddSr*r,r-r.r.r/rhszfatiguelife_gen._isfcCst||}|dd}d|d}||d}d|d|dt|d}d |d |d |d}||||fS) Nrrnrrr rrr]gD@rEr)r;rfc2rdenrrrr.r.r/rs    zfatiguelife_gen._stats)NN)rirjrkrlrrrr]rrarrbrgrdrhrr.r.r.r/r&`s r& fatiguelifec@sJeZdZdZddZddZdddZd d Zd d Zd dZ ddZ dS)foldcauchy_genaoA folded Cauchy continuous random variable. %(before_notes)s Notes ----- The probability density function for `foldcauchy` is: .. math:: f(x, c) = \frac{1}{\pi (1+(x-c)^2)} + \frac{1}{\pi (1+(x+c)^2)} for :math:`x \ge 0` and :math:`c \ge 0`. `foldcauchy` takes ``c`` as a shape parameter for :math:`c`. %(example)s cCs|dkSNrr.rr.r.r/rWszfoldcauchy_gen._argcheckcCstdddtjfdgSNrfFrrYrZr\r.r.r/r]szfoldcauchy_gen._shape_infoNcCsttj|||dS)Nr'rr)rrrDr;rfrrr.r.r/rs zfoldcauchy_gen._rvscCs2dtjdd||ddd||dSNrnrrKrrhr.r.r/raszfoldcauchy_gen._pdfcCs&dtjt||t||Srrrhr.r.r/rbszfoldcauchy_gen._cdfcCs&td||td||tjSrR)rEarctan2rrhr.r.r/rdszfoldcauchy_gen._sfcCstjtjtjtjfSrCrrr.r.r/rszfoldcauchy_gen._stats)NN rirjrkrlrWr]rrarbrdrr.r.r.r/r5s r5 foldcauchyc@sZeZdZdZddZdddZddZd d Zd d Zd dZ ddZ ddZ ddZ dS)f_genaAn F continuous random variable. For the noncentral F distribution, see `ncf`. %(before_notes)s See Also -------- ncf Notes ----- The F distribution with :math:`df_1 > 0` and :math:`df_2 > 0` degrees of freedom is the distribution of the ratio of two independent chi-squared distributions with :math:`df_1` and :math:`df_2` degrees of freedom, after rescaling by :math:`df_2 / df_1`. The probability density function for `f` is: .. math:: f(x, df_1, df_2) = \frac{df_2^{df_2/2} df_1^{df_1/2} x^{df_1 / 2-1}} {(df_2+df_1 x)^{(df_1+df_2)/2} B(df_1/2, df_2/2)} for :math:`x > 0`. `f` accepts shape parameters ``dfn`` and ``dfd`` for :math:`df_1`, the degrees of freedom of the chi-squared distribution in the numerator, and :math:`df_2`, the degrees of freedom of the chi-squared distribution in the denominator, respectively. %(after_notes)s %(example)s cCs0tdddtjfd}tdddtjfd}||gS)NdfnFrrdfdrZ)r;ZidfnZidfdr.r.r/r] szf_gen._shape_infoNcCs||||SrC)r)r;r?r@rrr.r.r/r sz f_gen._rvscCst||||SrCrGr;r`r?r@r.r.r/ra sz f_gen._pdfcCs~d|}d|}|dt||dt|t|dd|||dt|||t|d|d}|SNrnrKr)rErrcrr)r;r`r?r@rVmrr.r.r/r s 60z f_gen._logpdfcCst|||SrC)rcZfdtrrAr.r.r/rb sz f_gen._cdfcCst|||SrC)rcZfdtrcrAr.r.r/rd sz f_gen._sfcCst|||SrC)rcZfdtri)r;rfr?r@r.r.r/rg# sz f_gen._ppfc Csd|d|}}|d|d|d|df\}}}}t|dk||fddtj} t|d k||||fd dtj} t|d k||||fd dtj} | td9} t|d k| ||fddtj} | d9} | | | | fS)Nrnrrr @rKcSs||SrCr.)v2v2_2r.r.r/rL, rMzf_gen._stats..rcSs$d||||||d|Srr.)v1rErFv2_4r.r.r/rL1 srcSs&d|||t||||Srr)rGrFrHv2_6r.r.r/rL7 srcSsd||||S)Nrr.)rrIv2_8r.r.r/rL> rMr)rrEr[rr) r;r?r@rGrErFrHrIrJrrrrr.r.r/r& s2$ z f_gen._statscCsnd|}d|}d||}t|t|t||d|t|d|t||t|Sr)rErrcrr)r;r?r@Zhalf_dfnZhalf_dfdZhalf_sumr.r.r/rD s  zf_gen._entropy)NN) rirjrkrlr]rrarrbrdrgrrr.r.r.r/r>s$ r>rc@sJeZdZdZddZddZdddZd d Zd d Zd dZ ddZ dS) foldnorm_genazA folded normal continuous random variable. %(before_notes)s Notes ----- The probability density function for `foldnorm` is: .. math:: f(x, c) = \sqrt{2/\pi} cosh(c x) \exp(-\frac{x^2+c^2}{2}) for :math:`x \ge 0` and :math:`c \ge 0`. `foldnorm` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s %(example)s cCs|dkSr6r.rr.r.r/rWr szfoldnorm_gen._argcheckcCstdddtjfdgSr7rZr\r.r.r/r]u szfoldnorm_gen._shape_infoNcCst|||SrCrrr9r.r.r/rx szfoldnorm_gen._rvscCst||t||SrCrrhr.r.r/ra{ szfoldnorm_gen._pdfcCs2td}dt|||t|||Sr)rErrcerf)r;r`rfZsqrt_twor.r.r/rb s zfoldnorm_gen._cdfcCst||t||SrCrrhr.r.r/rd szfoldnorm_gen._sfcCs||}td|tdtj}d||t|td}|d||}d||||||}|t|d}||ddd||}|d|d d |d|d7}||dd }||||fS) NrrKrrrrrDr)rErrrrcrMr)r;rfr2Zexpfacrrrrr.r.r/r s $zfoldnorm_gen._stats)NNr<r.r.r.r/rK\ s rKfoldnormcs|eZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ eeddfddZZS)weibull_min_genaWeibull minimum continuous random variable. The Weibull Minimum Extreme Value distribution, from extreme value theory (Fisher-Gnedenko theorem), is also often simply called the Weibull distribution. It arises as the limiting distribution of the rescaled minimum of iid random variables. %(before_notes)s See Also -------- weibull_max, numpy.random.Generator.weibull, exponweib Notes ----- The probability density function for `weibull_min` is: .. math:: f(x, c) = c x^{c-1} \exp(-x^c) for :math:`x > 0`, :math:`c > 0`. `weibull_min` takes ``c`` as a shape parameter for :math:`c`. (named :math:`k` in Wikipedia article and :math:`a` in ``numpy.random.weibull``). Special shape values are :math:`c=1` and :math:`c=2` where Weibull distribution reduces to the `expon` and `rayleigh` distributions respectively. Suppose ``X`` is an exponentially distributed random variable with scale ``s``. Then ``Y = X**k`` is `weibull_min` distributed with shape ``c = 1/k`` and scale ``s**k``. %(after_notes)s References ---------- https://en.wikipedia.org/wiki/Weibull_distribution https://en.wikipedia.org/wiki/Fisher-Tippett-Gnedenko_theorem %(example)s cCstdddtjfdgSrerZr\r.r.r/r] szweibull_min_gen._shape_infocCs$|t||dtt|| SrRr#rErrhr.r.r/ra szweibull_min_gen._pdfcCs$t|t|d|t||SrRrErrcrr#rhr.r.r/r szweibull_min_gen._logpdfcCstt||  SrCrcrjr#rhr.r.r/rb szweibull_min_gen._cdfcCstt|  d|Srr"rkr.r.r/rg szweibull_min_gen._ppfcCst|||SrCrrhr.r.r/rd szweibull_min_gen._sfcCs t|| SrCr#rhr.r.r/r szweibull_min_gen._logsfcCst| d|SrRrrkr.r.r/rh szweibull_min_gen._isfcCstd|d|Srrrr.r.r/r szweibull_min_gen._munpcCst |t|tdSrRrrErrr.r.r/r szweibull_min_gen._entropya If ``method='mm'``, parameters fixed by the user are respected, and the remaining parameters are used to match distribution and sample moments where possible. For example, if the user fixes the location with ``floc``, the parameters will only match the distribution skewness and variance to the sample skewness and variance; no attempt will be made to match the means or minimize a norm of the errors. rc s8t|tr:|dkr |}ntj|g|Ri|S|ddr`tj|g|Ri|St||||\}}}}|dd }ddt |d}|} | kr|d kr|dur|stj|g|Ri|S|d krd \} } } n,t |r|dnd} |d d} |d d} |durN| durNt fd dd|gddj} n|dur\|} |dur| durt|} t| tdd| tdd| d} n|dur|} |dur| durt|}|| tdd| } n|dur|} |d kr| | | fStj|| f| | d|SdS)NrsuperfitFr*r1cSsjtdd|}tdd|}tdd|}d|dd|||}||dd}||S)NrrKrrr)rfZgamma1Zgamma2Zgamma3numr3r.r.r/skew s z!weibull_min_gen.fit..skewg@r2NNNr'r(cs |SrCr.rrmrXr.r/rL% rMz%weibull_min_gen.fit..g{Gz?bisect)bracketr*rrKr'r()r5r$r6r r7r9r+_check_fit_input_parametersr3r4rSrXrr%rootrErrrcrCr)r;r<r=r-fcrrr*Zmax_cZs_minrfr'r(r:rCr rZr/r9 sP            4     zweibull_min_gen.fit)rirjrkrlr]rarrbrgrdrrhrrrrr9r?r.r.r r/rP s, rPrcseZdZdZddZddZfddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ ddZddZddZZS)truncweibull_min_gena9A doubly truncated Weibull minimum continuous random variable. %(before_notes)s See Also -------- weibull_min, truncexpon Notes ----- The probability density function for `truncweibull_min` is: .. math:: f(x, a, b, c) = \frac{c x^{c-1} \exp(-x^c)}{\exp(-a^c) - \exp(-b^c)} for :math:`a < x <= b`, :math:`0 \le a < b` and :math:`c > 0`. `truncweibull_min` takes :math:`a`, :math:`b`, and :math:`c` as shape parameters. Notice that the truncation values, :math:`a` and :math:`b`, are defined in standardized form: .. math:: a = (u_l - loc)/scale b = (u_r - loc)/scale where :math:`u_l` and :math:`u_r` are the specific left and right truncation values, respectively. In other words, the support of the distribution becomes :math:`(a*scale + loc) < x <= (b*scale + loc)` when :math:`loc` and/or :math:`scale` are provided. %(after_notes)s References ---------- .. [1] Rinne, H. "The Weibull Distribution: A Handbook". CRC Press (2009). %(example)s cCs|dk||k@|dk@SNrmr.r;rfrprqr.r.r/rWn sztruncweibull_min_gen._argcheckcCsFtdddtjfd}tdddtjfd}tdddtjfd}|||gS)NrfFrrrprYrqrZ)r;rxrrr.r.r/r]q sz truncweibull_min_gen._shape_infocstj|ddS)N)rrrrr7r r;r<r r.r/r w sztruncweibull_min_gen._fitstartcCs||fSrCr.rcr.r.r/rv{ sz!truncweibull_min_gen._get_supportcCsLtt|| tt|| }|t||dtt|| |SrRrErr#)r;r`rfrprqdenumr.r.r/ra~ s$ztruncweibull_min_gen._pdfcCsRttt|| tt|| }t|t|d|t|||SrR)rErrr#rcr)r;r`rfrprqlogdenumr.r.r/r s*ztruncweibull_min_gen._logpdfcCsPtt|| tt|| }tt|| tt|| }||SrCrfr;r`rfrprqrWrgr.r.r/rb s$$ztruncweibull_min_gen._cdfcCs\ttt|| tt|| }ttt|| tt|| }||SrCrErrr#r;r`rfrprqZlognumrhr.r.r/r s**ztruncweibull_min_gen._logcdfcCsPtt|| tt|| }tt|| tt|| }||SrCrfrir.r.r/rd s$$ztruncweibull_min_gen._sfcCs\ttt|| tt|| }ttt|| tt|| }||SrCrjrkr.r.r/r s**ztruncweibull_min_gen._logsfc CsBttd|tt|| |tt||  d|SrRr#rErrr;rfrfrprqr.r.r/rh s 0`. `weibull_max` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s References ---------- https://en.wikipedia.org/wiki/Weibull_distribution https://en.wikipedia.org/wiki/Fisher-Tippett-Gnedenko_theorem %(example)s cCstdddtjfdgSrerZr\r.r.r/r] szweibull_max_gen._shape_infocCs(|t| |dtt| | SrRrQrhr.r.r/ra szweibull_max_gen._pdfcCs(t|t|d| t| |SrRrRrhr.r.r/r szweibull_max_gen._logpdfcCstt| | SrCrfrhr.r.r/rb szweibull_max_gen._cdfcCst| | SrCrTrhr.r.r/r szweibull_max_gen._logcdfcCstt| |  SrCrSrhr.r.r/rd szweibull_max_gen._sfcCstt| d| Sr)r#rErrkr.r.r/rg szweibull_max_gen._ppfcCs4td|d|}t|dr(d}nd}||S)NrnrKr.r)rcrCr9)r;rVrfvalsgnr.r.r/r s  zweibull_max_gen._munpcCst |t|tdSrRrUrr.r.r/r szweibull_max_gen._entropyN) rirjrkrlr]rarrbrrdrgrrr.r.r.r/ro s$ro weibull_max)rqrrc@s`eZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ dS)genlogistic_genaA generalized logistic continuous random variable. %(before_notes)s Notes ----- The probability density function for `genlogistic` is: .. math:: f(x, c) = c \frac{\exp(-x)} {(1 + \exp(-x))^{c+1}} for real :math:`x` and :math:`c > 0`. In literature, different generalizations of the logistic distribution can be found. This is the type 1 generalized logistic distribution according to [1]_. It is also referred to as the skew-logistic distribution [2]_. `genlogistic` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s References ---------- .. [1] Johnson et al. "Continuous Univariate Distributions", Volume 2, Wiley. 1995. .. [2] "Generalized Logistic Distribution", Wikipedia, https://en.wikipedia.org/wiki/Generalized_logistic_distribution %(example)s cCstdddtjfdgSrerZr\r.r.r/r] szgenlogistic_gen._shape_infocCst|||SrCrGrhr.r.r/ra szgenlogistic_gen._pdfcCsL|d |dkd}t|}t||||dtt| SNrr)rErrrcrr)r;r`rfZmultZabsxr.r.r/r! s zgenlogistic_gen._logpdfcCsdt| | }|SrRr)r;r`rfZCxr.r.r/rb) szgenlogistic_gen._cdfcCs| tt| SrC)rErrrhr.r.r/r- szgenlogistic_gen._logcdfcCstt|d| Sr)rErrcpowm1rkr.r.r/rg0 szgenlogistic_gen._ppfcCst||| SrCrcrjrrhr.r.r/rd3 szgenlogistic_gen._sfcCs|d||SrRrgrkr.r.r/rh6 szgenlogistic_gen._isfcCstt|}tjtjdtd|}dtd|dt}|t|d}tjdddtd|}||d }||||fS) NrrKrrrr.@rr)rrcrrErzetar rr;rfrrrrr.r.r/r9 s zgenlogistic_gen._statscCst|dk|fdddddS)Ng^AcSs"t| t|dtdSrR)rErrcrrrr.r.r/rLD rMz*genlogistic_gen._entropy..cSsdd|tdSrrrr.r.r/rLJ rMrOrQrr.r.r/rB s zgenlogistic_gen._entropyN)rirjrkrlr]rarrbrrgrdrhrrr.r.r.r/rs s  rs genlogisticc@szeZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ dddZddZddZdS) genpareto_genaA generalized Pareto continuous random variable. %(before_notes)s Notes ----- The probability density function for `genpareto` is: .. math:: f(x, c) = (1 + c x)^{-1 - 1/c} defined for :math:`x \ge 0` if :math:`c \ge 0`, and for :math:`0 \le x \le -1/c` if :math:`c < 0`. `genpareto` takes ``c`` as a shape parameter for :math:`c`. For :math:`c=0`, `genpareto` reduces to the exponential distribution, `expon`: .. math:: f(x, 0) = \exp(-x) For :math:`c=-1`, `genpareto` is uniform on ``[0, 1]``: .. math:: f(x, -1) = 1 %(after_notes)s %(example)s cCs t|SrCrErrr.r.r/rWt szgenpareto_gen._argcheckcCstddtj tjfdgSNrfFrrZr\r.r.r/r]w szgenpareto_gen._shape_infocCsBt|}t|dk|fddtj}t|dk|j|j}||fS)NrcSsd|Srr.rr.r.r/rL} rMz,genpareto_gen._get_support..)rErrr[rrp)r;rfrqrpr.r.r/rvz s  zgenpareto_gen._get_supportcCst|||SrCrGrhr.r.r/ra szgenpareto_gen._pdfcCs$t||k|dk@||fdd| S)NrcSst|d|| |Srrr`rfr.r.r/rL rMz'genpareto_gen._logpdf..rQrhr.r.r/r szgenpareto_gen._logpdfcCst| |  SrC)rcZ inv_boxcox1prhr.r.r/rb szgenpareto_gen._cdfcCst| | SrC)rcZ inv_boxcoxrhr.r.r/rd szgenpareto_gen._sfcCs$t||k|dk@||fdd| S)NrcSst|| |SrCrgrr.r.r/rL rMz&genpareto_gen._logsf..rQrhr.r.r/r szgenpareto_gen._logsfcCst| |  SrC)rcboxcox1prkr.r.r/rg szgenpareto_gen._ppfcCst||  SrC)rcboxcoxrkr.r.r/rh szgenpareto_gen._isfrlcCsd|vrd}nt|dk|fddtj}d|vr6d}nt|dk|fddtj}d|vr^d}nt|d k|fd dtj}d |vrd}nt|d k|fd dtj}||||fS)NrCrcSs dd|SrRr.xir.r.r/rL rMz&genpareto_gen._stats..r:rucSsdd|ddd|Srr.rr.r.r/rL rMrmgUUUUUU?cSs*dd|tdd|dd|S)NrKrrrrr.r.r/rL s ror+cSs@ddd|d|d|ddd|dd|dS)NrrrKrr.rr.r.r/rL s "  rrEr[r)r;rfrrrCr:rmror.r.r/r s2    zgenpareto_gen._statscs0ddt|dk|ffddtdS)NcSspd}td|d}t|t||D]$\}}||d|d||}q&t||dk|d||tjS)Nrmrrr.rnr)rErziprccombrr[)rVrfrproZkiZcnkr.r.r/r s z#genpareto_gen._munp..__munprcs |SrCr.rZ_genpareto_gen__munprVr.r/rL rMz%genpareto_gen._munp..r)rrcrCrr.rr/r s    zgenpareto_gen._munpcCsd|Srr.rr.r.r/r szgenpareto_gen._entropyN)rl)rirjrkrlrWr]rvrarrbrdrrgrhrrrr.r.r.r/r}P s#  r} genparetoc@sHeZdZdZddZddZddZdd Zd d Zd d Z ddZ dS) genexpon_gena!A generalized exponential continuous random variable. %(before_notes)s Notes ----- The probability density function for `genexpon` is: .. math:: f(x, a, b, c) = (a + b (1 - \exp(-c x))) \exp(-a x - b x + \frac{b}{c} (1-\exp(-c x))) for :math:`x \ge 0`, :math:`a, b, c > 0`. `genexpon` takes :math:`a`, :math:`b` and :math:`c` as shape parameters. %(after_notes)s References ---------- H.K. Ryu, "An Extension of Marshall and Olkin's Bivariate Exponential Distribution", Journal of the American Statistical Association, 1993. N. Balakrishnan, Asit P. Basu (editors), *The Exponential Distribution: Theory, Methods and Applications*, Gordon and Breach, 1995. ISBN 10: 2884491929 %(example)s cCsFtdddtjfd}tdddtjfd}tdddtjfd}|||gS)NrpFrrrqrfrZ)r;rrrxr.r.r/r] szgenexpon_gen._shape_infoc CsH||t| | t| |||t| | |SrCrcrjrErr;r`rprqrfr.r.r/ra s(zgenexpon_gen._pdfcCsHt||t| | | |||t| | |SrCrErrcrjrr.r.r/r szgenexpon_gen._logpdfcCs0t| |||t| | | SrCrrr.r.r/rb szgenexpon_gen._cdfcCsF||}||t| |}|t| |t| j|SrC)rErrclambertwrrealr;rTrprqrfrmr)r.r.r/rg szgenexpon_gen._ppfcCs.t| |||t| | |SrCr rr.r.r/rd szgenexpon_gen._sfcCsD||}||t||}|t| |t| j|SrC)rErrcrrrrr.r.r/rh szgenexpon_gen._isfNrr.r.r.r/r srgenexponcseZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ ddZddZfddZddZddZZS) genextreme_genaBA generalized extreme value continuous random variable. %(before_notes)s See Also -------- gumbel_r Notes ----- For :math:`c=0`, `genextreme` is equal to `gumbel_r` with probability density function .. math:: f(x) = \exp(-\exp(-x)) \exp(-x), where :math:`-\infty < x < \infty`. For :math:`c \ne 0`, the probability density function for `genextreme` is: .. math:: f(x, c) = \exp(-(1-c x)^{1/c}) (1-c x)^{1/c-1}, where :math:`-\infty < x \le 1/c` if :math:`c > 0` and :math:`1/c \le x < \infty` if :math:`c < 0`. Note that several sources and software packages use the opposite convention for the sign of the shape parameter :math:`c`. `genextreme` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s %(example)s cCs t|SrCr~rr.r.r/rW5 szgenextreme_gen._argcheckcCstddtj tjfdgSrrZr\r.r.r/r]8 szgenextreme_gen._shape_infocCsLt|dkdt|ttj}t|dkdt|t tj }||fSNrrn)rErmaximumrr[minimum)r;rf_b_ar.r.r/rv; s $zgenextreme_gen._get_supportcCs$t||k|dk@||fdd| S)NrcSst| ||SrCrgrr.r.r/rLC rMz+genextreme_gen._loglogcdf..rQrhr.r.r/ _loglogcdf@ s zgenextreme_gen._loglogcdfcCst|||SrCrGrhr.r.r/raE szgenextreme_gen._pdfcCst||k|dk@||fddd}t| }|||}t|}t||dk|tj k@dt|dk|tj kB|||fddtj d}t||dk|dk@d|S)NrcSs||SrCr.rr.r.r/rLL rMz(genextreme_gen._logpdf..rmrcSs| ||SrCr.)pex2Zlpex2Zlex2r.r.r/rLT rMr`)rrcrrrErZputmaskr[)r;r`rfZcxZlogex2Zlogpex2rlogpdfr.r.r/rK s"   zgenextreme_gen._logpdfcCst||| SrC)rErrrhr.r.r/rY szgenextreme_gen._logcdfcCst|||SrCrErrrhr.r.r/rb\ szgenextreme_gen._cdfcCst||| SrCrvrhr.r.r/rd_ szgenextreme_gen._sfcCs6tt|  }t||k|dk@||fdd|S)NrcSst| | |SrCrrr.r.r/rLe rMz%genextreme_gen._ppf..)rErrr;rfrfr`r.r.r/rgb szgenextreme_gen._ppfcCs8tt|   }t||k|dk@||fdd|S)NrcSst| | |SrCrrr.r.r/rLj rMz%genextreme_gen._isf..)rErrcrrrr.r.r/rhg szgenextreme_gen._isfc sfdd}|d}|d}|d}|d}ttdktjdd ||d}ttdktjdd ttdd dtd d}d } tt| kt ttd} td ktj| } td ktj|d|} t dk||||fddtjd} tt| dkdt dt tjd| }t dk|||||fddtjd}tt| dkd|d}| | ||fS)Ncst|dSrRrr_rr.r/gm sz genextreme_gen._stats..grrKrrgHz>rrrn+=rrNrcSs(t|| |d|||dSNrKrrD)rfrrg3g2mg12r.r.r/rL sz'genextreme_gen._stats..r`g(\?r/rgпcSs(|d|d|||||dS)NrrrKr.)rrrg4rr.r.r/rL sgq= ףp?333333@r) rErrrrcrjr5rrrrr )r;rfrrrrrrZgam2kZepsZgamkrCr:Zsk1rZku1rr.rr/rl s4 ,0, 2 zgenextreme_gen._statscs>t|tr|}t|}|dkr(d}nd}tj||fdS)NrrurNrr5r$r rr7r )r;r<rrpr r.r/r  s zgenextreme_gen._fitstartcCsdtd|d}d||tjt||d|t||ddd}t||dk|tjS)Nrrrnr.r\)rErrrcrrCrr[)r;rVrfrovalsr.r.r/r s $zgenextreme_gen._munpcCstd|dSrRr{rr.r.r/r szgenextreme_gen._entropy)rirjrkrlrWr]rvrrarrrbrdrgrhrr rrr?r.r.r r/r s &! r genextremecsd}fdd}dkrDtd}dkrntj||dd}|Sn*d kr`td d }nd |}tj||d dd\}}}}|dkrtd|dS)afInverse of the digamma function (real positive arguments only). This function is used in the `fit` method of `gamma_gen`. The function uses either optimize.fsolve or optimize.newton to solve `sc.digamma(x) - y = 0`. There is probably room for improvement, but currently it works over a wide range of y: >>> import numpy as np >>> rng = np.random.default_rng() >>> y = 64*rng.standard_normal(1000000) >>> y.min(), y.max() (-311.43592651416662, 351.77388222276869) >>> x = [_digammainv(t) for t in y] >>> np.abs(sc.digamma(x) - y).max() 1.1368683772161603e-13 gox?cst|SrC)rcrrrr.r/r sz_digammainv..funcgrur&绽|=)Ztolrg-@g뭁,?rndy=T)xtolrrz"_digammainv: fsolve failed, y = %rr)rErr Znewtonr  RuntimeError)rZ_emrx0valuerrrr.rr/ _digammainv s    rcseZdZdZddZdddZddZd d Zd d Zd dZ ddZ ddZ ddZ ddZ fddZeeddfddZZS) gamma_genaA gamma continuous random variable. %(before_notes)s See Also -------- erlang, expon Notes ----- The probability density function for `gamma` is: .. math:: f(x, a) = \frac{x^{a-1} e^{-x}}{\Gamma(a)} for :math:`x \ge 0`, :math:`a > 0`. Here :math:`\Gamma(a)` refers to the gamma function. `gamma` takes ``a`` as a shape parameter for :math:`a`. When :math:`a` is an integer, `gamma` reduces to the Erlang distribution, and when :math:`a=1` to the exponential distribution. Gamma distributions are sometimes parameterized with two variables, with a probability density function of: .. math:: f(x, \alpha, \beta) = \frac{\beta^\alpha x^{\alpha - 1} e^{-\beta x }}{\Gamma(\alpha)} Note that this parameterization is equivalent to the above, with ``scale = 1 / beta``. %(after_notes)s %(example)s cCstdddtjfdgSrrZr\r.r.r/r] szgamma_gen._shape_infoNcCs |||SrCstandard_gamma)r;rprrr.r.r/r szgamma_gen._rvscCst|||SrCrGrr.r.r/ra szgamma_gen._pdfcCst|d||t|Sr)rcrr5rr.r.r/r szgamma_gen._logpdfcCs t||SrCrrr.r.r/rb szgamma_gen._cdfcCs t||SrCrrr.r.r/rd sz gamma_gen._sfcCs t||SrCrrr.r.r/rg szgamma_gen._ppfcCs t||SrCrcrrr.r.r/rh szgamma_gen._isfcCs||dt|d|fS)Nrrrrr.r.r/r szgamma_gen._statscCs&dd}dd}t|dk|f||dS)NcSs t|d||t|SrRrcrr5rpr.r.r/r sz+gamma_gen._entropy..regular_formulacSsRddtdtjt|dd||dd|dd |d d S) NrurnrKrrr#r/r$rr%r'rrr.r.r/r" s,   z.gamma_gen._entropy..asymptotic_formularOrQ)r;rprrr.r.r/r s zgamma_gen._entropycs<t|tr|}t|}dd|d}tj||fdS)Nr:0yE>rKrr)r;r<rrpr r.r/r - s  zgamma_gen._fitstarta< When the location is fixed by using the argument `floc` and `method='MLE'`, this function uses explicit formulas or solves a simpler numerical problem than the full ML optimization problem. So in that case, the `optimizer`, `loc` and `scale` arguments are ignored. rcs|dd}|dd}t|ts6|durP|dkrPtj|g|Ri|S|ddt|gd}|dd}t||dur|dur|durt dt |}t | st d|dkrt |}t |} t ||d } |||} } } | dur,| dur,| dur,| d | } | durN| durNt | | } | durn| durn| || } | dur| dur| | d } | dur|| | } | dur|| | } | dur|| | } | | | fSt ||krtd |t jd |d kr ||}|}|dur|dur.|} nlt |t |d t d d dd}|d}|d}tjfdd||d d} || } n$t |t |}t|} |} | || fS)Nrr*r1r2rrrrrrKrCrrrpr/g333333?gffffff?cst|t|SrC)rErrcrrrmr.r/rL rMzgamma_gen.fit..)Zdisp)r3r5r$r4r7r9r+rr0rrErrrrrrrrr[rr brentqr)r;r<r=r-rr*rrm1m2m3rpr'r(rZaestZxarrfr rr/r99 sr                      * z gamma_gen.fit)NN)rirjrkrlr]rrarrbrdrgrhrrr rrr9r?r.r.r r/r s(  rrCcsHeZdZdZddZddZfddZeedd fd d Z Z S) erlang_genaAn Erlang continuous random variable. %(before_notes)s See Also -------- gamma Notes ----- The Erlang distribution is a special case of the Gamma distribution, with the shape parameter `a` an integer. Note that this restriction is not enforced by `erlang`. It will, however, generate a warning the first time a non-integer value is used for the shape parameter. Refer to `gamma` for examples. cCs<tt||k}|s4d|d}tj|tdd|dkS)NzRThe shape parameter of the erlang distribution has been given a non-integer value .r stacklevelr)rErfloorwarningswarnRuntimeWarning)r;rpZallintmessager.r.r/rW szerlang_gen._argcheckcCstdddtjfdgS)NrpTrrYrZr\r.r.r/r] szerlang_gen._shape_infocs@t|tr|}tddt|d}tt|j||fdS)NrrrKr)r5r$r r9rr7rr )r;r<rpr r.r/r  s zerlang_gen._fitstarta The Erlang distribution is generally defined to have integer values for the shape parameter. This is not enforced by the `erlang` class. When fitting the distribution, it will generally return a non-integer value for the shape parameter. By using the keyword argument `f0=`, the fit method can be constrained to fit the data to a specific integer shape parameter.rcstj|g|Ri|SrC)r7r9r;r<r=r-r r.r/r9 szerlang_gen.fit) rirjrkrlrWr]r rrr9r?r.r.r r/r s   rerlangc@sjeZdZdZddZddZddZdd Zd d Zdd dZ ddZ ddZ ddZ ddZ ddZd S) gengamma_genaA generalized gamma continuous random variable. %(before_notes)s See Also -------- gamma, invgamma, weibull_min Notes ----- The probability density function for `gengamma` is ([1]_): .. math:: f(x, a, c) = \frac{|c| x^{c a-1} \exp(-x^c)}{\Gamma(a)} for :math:`x \ge 0`, :math:`a > 0`, and :math:`c \ne 0`. :math:`\Gamma` is the gamma function (`scipy.special.gamma`). `gengamma` takes :math:`a` and :math:`c` as shape parameters. %(after_notes)s References ---------- .. [1] E.W. Stacy, "A Generalization of the Gamma Distribution", Annals of Mathematical Statistics, Vol 33(3), pp. 1187--1192. %(example)s cCs|dk|dk@Sr6r.)r;rprfr.r.r/rWszgengamma_gen._argcheckcCs4tdddtjfd}tddtj tjfd}||gSrrZrr.r.r/r]szgengamma_gen._shape_infocCst||||SrCrGrr.r.r/ra szgengamma_gen._pdfcs,t|dk|dkB||ffddtj dS)Nrcs4tt|t|d|||tSrR)rErrrcrr5rrr.r/rLs z&gengamma_gen._logpdf..r`rbrr.rr/rs zgengamma_gen._logpdfcCs2||}t||}t||}t|dk||Sr6rcrrrErr;r`rprfZxcval1val2r.r.r/rbs  zgengamma_gen._cdfNcCs|j||d}|d|S)Nrrnr)r;rprfrrrUr.r.r/rszgengamma_gen._rvscCs2||}t||}t||}t|dk||Sr6rrr.r.r/rd s  zgengamma_gen._sfcCs2t||}t||}t|dk||d|SrrcrrrErr;rfrprfrrr.r.r/rg&s  zgengamma_gen._ppfcCs2t||}t||}t|dk||d|Srrrr.r.r/rh+s  zgengamma_gen._isfcCst||d|Sr)rcr)r;rVrprfr.r.r/r0szgengamma_gen._munpcCs,dd}dd}t|dk||f||d}|S)NcSsBt|}|d|||}t|tt|}||}|SrR)rcrr5rErr)rprfrpABrr.r.r/r 5s  z&gengamma_gen._entropy..regularcSsptt|dtt||dd|ddt||dd|dd|dd |S) NrKrrr$rr#r/r%r')rrrErr)rprfr.r.r/ asymptotic<s  0z)gengamma_gen._entropy..asymptotici@rrPrQ)r;rprfr rrr.r.r/r4szgengamma_gen._entropy)NN)rirjrkrlrWr]rarrbrrdrgrhrrr.r.r.r/r s rgengammac@s@eZdZdZddZddZddZdd Zd d Zd d Z dS)genhalflogistic_genaA generalized half-logistic continuous random variable. %(before_notes)s Notes ----- The probability density function for `genhalflogistic` is: .. math:: f(x, c) = \frac{2 (1 - c x)^{1/(c-1)}}{[1 + (1 - c x)^{1/c}]^2} for :math:`0 \le x \le 1/c`, and :math:`c > 0`. `genhalflogistic` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s %(example)s cCstdddtjfdgSrerZr\r.r.r/r]_szgenhalflogistic_gen._shape_infocCs|jd|fSrrrr.r.r/rvbsz genhalflogistic_gen._get_supportcCsBd|}td||}||d}||}d|d|dSr:rEr)r;r`rflimitr.Ztmp0tmp2r.r.r/raes  zgenhalflogistic_gen._pdfcCs2d|}td||}||}d|d|Sr{r)r;r`rfrr.rr.r.r/rbnszgenhalflogistic_gen._cdfcCs d|dd|d||Sr{r.rkr.r.r/rgtszgenhalflogistic_gen._ppfcCsdd|dtdSrrrr.r.r/rwszgenhalflogistic_gen._entropyN) rirjrkrlr]rvrarbrgrr.r.r.r/rIs rgenhalflogisticcsveZdZdZddZddZfddZdd Zd d Zd d e ddZ ddZ ddZ dddZ ddZZS)genhyperbolic_genu A generalized hyperbolic continuous random variable. %(before_notes)s See Also -------- t, norminvgauss, geninvgauss, laplace, cauchy Notes ----- The probability density function for `genhyperbolic` is: .. math:: f(x, p, a, b) = \frac{(a^2 - b^2)^{p/2}} {\sqrt{2\pi}a^{p-1/2} K_p\Big(\sqrt{a^2 - b^2}\Big)} e^{bx} \times \frac{K_{p - 1/2} (a \sqrt{1 + x^2})} {(\sqrt{1 + x^2})^{1/2 - p}} for :math:`x, p \in ( - \infty; \infty)`, :math:`|b| < a` if :math:`p \ge 0`, :math:`|b| \le a` if :math:`p < 0`. :math:`K_{p}(.)` denotes the modified Bessel function of the second kind and order :math:`p` (`scipy.special.kv`) `genhyperbolic` takes ``p`` as a tail parameter, ``a`` as a shape parameter, ``b`` as a skewness parameter. %(after_notes)s The original parameterization of the Generalized Hyperbolic Distribution is found in [1]_ as follows .. math:: f(x, \lambda, \alpha, \beta, \delta, \mu) = \frac{(\gamma/\delta)^\lambda}{\sqrt{2\pi}K_\lambda(\delta \gamma)} e^{\beta (x - \mu)} \times \frac{K_{\lambda - 1/2} (\alpha \sqrt{\delta^2 + (x - \mu)^2})} {(\sqrt{\delta^2 + (x - \mu)^2} / \alpha)^{1/2 - \lambda}} for :math:`x \in ( - \infty; \infty)`, :math:`\gamma := \sqrt{\alpha^2 - \beta^2}`, :math:`\lambda, \mu \in ( - \infty; \infty)`, :math:`\delta \ge 0, |\beta| < \alpha` if :math:`\lambda \ge 0`, :math:`\delta > 0, |\beta| \le \alpha` if :math:`\lambda < 0`. The location-scale-based parameterization implemented in SciPy is based on [2]_, where :math:`a = \alpha\delta`, :math:`b = \beta\delta`, :math:`p = \lambda`, :math:`scale=\delta` and :math:`loc=\mu` Moments are implemented based on [3]_ and [4]_. For the distributions that are a special case such as Student's t, it is not recommended to rely on the implementation of genhyperbolic. To avoid potential numerical problems and for performance reasons, the methods of the specific distributions should be used. References ---------- .. [1] O. Barndorff-Nielsen, "Hyperbolic Distributions and Distributions on Hyperbolae", Scandinavian Journal of Statistics, Vol. 5(3), pp. 151-157, 1978. https://www.jstor.org/stable/4615705 .. [2] Eberlein E., Prause K. (2002) The Generalized Hyperbolic Model: Financial Derivatives and Risk Measures. In: Geman H., Madan D., Pliska S.R., Vorst T. (eds) Mathematical Finance - Bachelier Congress 2000. Springer Finance. Springer, Berlin, Heidelberg. :doi:`10.1007/978-3-662-12429-1_12` .. [3] Scott, David J, Würtz, Diethelm, Dong, Christine and Tran, Thanh Tam, (2009), Moments of the generalized hyperbolic distribution, MPRA Paper, University Library of Munich, Germany, https://EconPapers.repec.org/RePEc:pra:mprapa:19081. .. [4] E. Eberlein and E. A. von Hammerstein. Generalized hyperbolic and inverse Gaussian distributions: Limiting cases and approximation of processes. FDM Preprint 80, April 2003. University of Freiburg. https://freidok.uni-freiburg.de/fedora/objects/freidok:7974/datastreams/FILE1/content %(example)s cCs4tt||k|dktt||k|dkBSr6)rE logical_andr)r;rTrprqr.r.r/rWszgenhyperbolic_gen._argcheckcCsNtddtj tjfd}tdddtjfd}tddtj tjfd}|||gS)NrTFrrprrYrqrZ)r;iprrr.r.r/r]szgenhyperbolic_gen._shape_infocstj|ddS)N)rrrurrdrer r.r/r szgenhyperbolic_gen._fitstartcCstjdd}|||||S)NcSst||||SrC)rZgenhyperbolic_logpdfr`rTrprqr.r.r/_logpdf_singlesz1genhyperbolic_gen._logpdf.._logpdf_singlerE vectorize)r;r`rTrprqrr.r.r/rs zgenhyperbolic_gen._logpdfcCstjdd}|||||S)NcSst||||SrC)rZgenhyperbolic_pdfrr.r.r/ _pdf_singlesz+genhyperbolic_gen._pdf.._pdf_singler)r;r`rTrprqrr.r.r/ras zgenhyperbolic_gen._pdfcCstj|ttjgdS)NZotypes)rEr__get__objectfloat64rr.r.r/rLrMzgenhyperbolic_gen.c Cst|||gtjtj}ttd|}t ||||}||t |d|t ||}d} d} ||kr|krnn2t j |||| | ddt j |||| | dd} nt j |||| | dd} t| rd} tj| tddtd td | S) z Integrate the pdf of the genhyberbolic distribution from x0 to x1. This is a private function used by _cdf() and _sf() only; either x0 will be -inf or x1 will be inf. Z_genhyperbolic_pdfrrr)epsrelepsabszdInfinite values encountered in scipy.special.kve. Values replaced by NaN to avoid incorrect results.rrrmrn)rEarrayrctypesdata_asc_void_pr from_cythonrrrckvr quadisnanrrrmaxr) rx1rTrprq user_datallcr=rrrZintgrlrOr.r.r/_integrate_pdfs:$    z genhyperbolic_gen._integrate_pdfcCs|tj ||||SrCrrEr[r;r`rTrprqr.r.r/rb!szgenhyperbolic_gen._cdfcCs||tj|||SrCrrr.r.r/rd$szgenhyperbolic_gen._sfNc Csht|dt|d}t|d}t|d}tj|||||d} tj||d} || t| | S)NrKrurN)rTrqr(rrrB)rE float_power geninvgaussrDrr) r;rTrprqrrr*r+r,ZgigZnormstr.r.r/r's  zgenhyperbolic_gen._rvscsDt|||\}}}t|dt|d}t|d}tddt|d}tddd}||jd|j}t|||\}}} } fd d ||| | fD\} } } }||| }|| t|dt|d| t| d}t|d t|d | d ||td dt| d d |t|d| t| d}|t|d }t|dt|d|d| |td d|t|dtdd t| dt|dt|d d| d||td dt| d d t|d| }|t|d d }||||fS)NrKrurr.rrrrXc3s|]}|VqdSrCr.)rZrqb0r.r/ HrMz+genhyperbolic_gen._stats..rrrrrr/) rErRrZlinspacershaper}rcr)r;rTrprqr*r+Zintegersb1b2Zb3Zb4r1r2Zr3Zr4rCr:Zm3ermZm4eror.rr/r<sT "  zgenhyperbolic_gen._stats)NN)rirjrkrlrWr]r rra staticmethodrrbrdrrr?r.r.r r/r~sY    r genhyperbolicc@sPeZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ dS) gompertz_genaqA Gompertz (or truncated Gumbel) continuous random variable. %(before_notes)s Notes ----- The probability density function for `gompertz` is: .. math:: f(x, c) = c \exp(x) \exp(-c (e^x-1)) for :math:`x \ge 0`, :math:`c > 0`. `gompertz` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s %(example)s cCstdddtjfdgSrerZr\r.r.r/r]szgompertz_gen._shape_infocCst|||SrCrGrhr.r.r/raszgompertz_gen._pdfcCst|||t|SrCrrhr.r.r/rszgompertz_gen._logpdfcCst| t| SrCrrhr.r.r/rbszgompertz_gen._cdfcCstd|t| Srrgrkr.r.r/rgszgompertz_gen._ppfcCst| t|SrCr rhr.r.r/rdszgompertz_gen._sfcCstt| |SrCr!r;rTrfr.r.r/rhszgompertz_gen._isfcCsdt|tj||Sr)rErrc_ufuncs _scaled_exp1rr.r.r/rszgompertz_gen._entropyN rirjrkrlr]rarrbrgrdrhrr.r.r.r/risrgompertzcCs8t|}t|}|}t||}tj||dS)N)weights)rErrrZaverage)r` logweightsZmaxlogwr r.r.r/_average_with_log_weightss   r c@steZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ eeeddZdS) gumbel_r_genaA right-skewed Gumbel continuous random variable. %(before_notes)s See Also -------- gumbel_l, gompertz, genextreme Notes ----- The probability density function for `gumbel_r` is: .. math:: f(x) = \exp(-(x + e^{-x})) The Gumbel distribution is sometimes referred to as a type I Fisher-Tippett distribution. It is also related to the extreme value distribution, log-Weibull and Gompertz distributions. %(after_notes)s %(example)s cCsgSrCr.r\r.r.r/r]szgumbel_r_gen._shape_infocCst||SrCrGrr.r.r/raszgumbel_r_gen._pdfcCs| t| SrCrrr.r.r/rszgumbel_r_gen._logpdfcCstt|  SrCrrr.r.r/rbszgumbel_r_gen._cdfcCst|  SrCrrr.r.r/rszgumbel_r_gen._logcdfcCstt|  SrCrrr.r.r/rgszgumbel_r_gen._ppfcCstt|   SrCrrr.r.r/rdszgumbel_r_gen._sfcCstt|   SrCrErrrr.r.r/rhszgumbel_r_gen._isfcCs0ttjtjddtdtjdtdfS)Nrr/rrrrrErrr r\r.r.r/rszgumbel_r_gen._statscCstdSrr{r\r.r.r/rszgumbel_r_gen._entropyc st|||\}}fdd}|dur6|}||n|durR|fddn fdd|dd}|d|d} } fd d } | | | s| d ks| tjkr| d} | d9} qtj| | fd d d } | j}|dur|n|||fS)Ncs$| t |ttSrC)rc logsumexprErr)r(r<r.r/get_loc_from_scalesz,gumbel_r_gen.fit..get_loc_from_scalecs:t|}t|}||SrC)rErrr)r(Zterm1Zterm2r<r'r.r/rszgumbel_r_gen.fit..funccs& |}t|d}||S)N)r )r r)r(ZsdataZwavgrr.r/rs  r(rrKcst|t|kSrCrDrGrr.r/rJs  z0gumbel_r_gen.fit..interval_contains_rootrr)r\rtolr)r^r3rEr[r r%r_) r;r<r=r-rrrr( brack_startrHrIrJresr.)r<rr'r/r9s6        zgumbel_r_gen.fitN)rirjrkrlr]rarrbrrgrdrhrrrBr rr9r.r.r.r/r sr gumbel_rc@steZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ eeeddZdS) gumbel_l_genaA left-skewed Gumbel continuous random variable. %(before_notes)s See Also -------- gumbel_r, gompertz, genextreme Notes ----- The probability density function for `gumbel_l` is: .. math:: f(x) = \exp(x - e^x) The Gumbel distribution is sometimes referred to as a type I Fisher-Tippett distribution. It is also related to the extreme value distribution, log-Weibull and Gompertz distributions. %(after_notes)s %(example)s cCsgSrCr.r\r.r.r/r]Aszgumbel_l_gen._shape_infocCst||SrCrGrr.r.r/raDszgumbel_l_gen._pdfcCs|t|SrCrrr.r.r/rHszgumbel_l_gen._logpdfcCstt|  SrCrrr.r.r/rbKszgumbel_l_gen._cdfcCstt|  SrCrErrcrrr.r.r/rgNszgumbel_l_gen._ppfcCs t| SrCrrr.r.r/rQszgumbel_l_gen._logsfcCstt| SrCrrr.r.r/rdTszgumbel_l_gen._sfcCstt| SrCrrr.r.r/rhWszgumbel_l_gen._isfcCs2t tjtjddtdtjdtdfS)Nrrrrrr\r.r.r/rZszgumbel_l_gen._statscCstdSrr{r\r.r.r/r^szgumbel_l_gen._entropycOsJ|ddur|d |d<tjt| g|Ri|\}}| |fS)Nr)r3rr9rEr)r;r<r=r-Zloc_rZscale_rr.r.r/r9as $zgumbel_l_gen.fitN)rirjrkrlr]rarrbrgrrdrhrrrBr rr9r.r.r.r/r&srgumbel_lcsteZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ e eefddZZS)halfcauchy_gena A Half-Cauchy continuous random variable. %(before_notes)s Notes ----- The probability density function for `halfcauchy` is: .. math:: f(x) = \frac{2}{\pi (1 + x^2)} for :math:`x \ge 0`. %(after_notes)s %(example)s cCsgSrCr.r\r.r.r/r]szhalfcauchy_gen._shape_infocCsdtjd||Srrrr.r.r/raszhalfcauchy_gen._pdfcCstdtjt||SrrErrrcrrr.r.r/rszhalfcauchy_gen._logpdfcCsdtjt|Srrrr.r.r/rbszhalfcauchy_gen._cdfcCsttjd|Srrrr.r.r/rgszhalfcauchy_gen._ppfcCsdtjtd|SNrr)rErr;rr.r.r/rdszhalfcauchy_gen._sfcCsdttj|dSrrrr.r.r/rhszhalfcauchy_gen._isfcCstjtjtjtjfSrCrr\r.r.r/rszhalfcauchy_gen._statscCstdtjSrrr\r.r.r/rszhalfcauchy_gen._entropyc s|ddr&tj|g|Ri|St||||\}}}t|}|durj||krdtd|tjd|}n|}dd}|dur|} n |||} || fS)NrVF halfcauchyrcsR||}|jt|fdd}tdjd}t||t|fd}|jS)Ncs"|d}dt|SrrEr)r( denominatorrVZshifted_data_squaredr.r/ fun_to_solves z.find_scale..fun_to_solvernrur\)rrEsquarefinfotinyr%rr_)r'r<Z shifted_datar"Zsmallrr.r!r/ find_scales z&halfcauchy_gen.fit..find_scaler+r7r9r^rErrr[) r;r<r=r-rrrr'r'r(r r.r/r9s      zhalfcauchy_gen.fit)rirjrkrlr]rarrbrgrdrhrrrBr rr9r?r.r.r r/rusrrcsteZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ e eefddZZS)halflogistic_genaA half-logistic continuous random variable. %(before_notes)s Notes ----- The probability density function for `halflogistic` is: .. math:: f(x) = \frac{ 2 e^{-x} }{ (1+e^{-x})^2 } = \frac{1}{2} \text{sech}(x/2)^2 for :math:`x \ge 0`. %(after_notes)s References ---------- .. [1] Asgharzadeh et al (2011). "Comparisons of Methods of Estimation for the Half-Logistic Distribution". Selcuk J. Appl. Math. 93-108. %(example)s cCsgSrCr.r\r.r.r/r]szhalflogistic_gen._shape_infocCst||SrCrGrr.r.r/raszhalflogistic_gen._pdfcCs$td|dtt| Sr)rErrcrrrr.r.r/rszhalflogistic_gen._logpdfcCst|dSr)rEtanhrr.r.r/rbszhalflogistic_gen._cdfcCsdt|SrrEZarctanhrr.r.r/rgszhalflogistic_gen._ppfcCsdt| Srrcexpitrr.r.r/rdszhalflogistic_gen._sfcCst|dk|fdddddS)NrucSstd| Srrclogitrr.r.r/rLrMz'halflogistic_gen._isf..cSsdtd|Srr+rr.r.r/rLrMrOrQrr.r.r/rhs zhalflogistic_gen._isfcCs|dkrdtdS|dkr.tjtjdS|dkr>dtS|dkrXdtjddSddtd d|t|dt|dS) NrrKrrrnrr7rxr)rErrr r#rcrCryrUr.r.r/rszhalflogistic_gen._munpcCsdtdSrrr\r.r.r/rszhalflogistic_gen._entropyc s|ddr&tj|g|Ri|St||||\}}}dd}t|}|durr||krltd|tjd|}n|}|dur|n|||} || fS)NrVFcSs<|jd}tj|dd}td|d|d}d|}d|}|d||t||}d||}||}dt|dd|dd} dt|dd|ddd} | t| dd|| d|} d} d} |}| | kr8|t | | }|d||}t | || } |} q| S) Nrr\rrurKrrr) rrEsortrrrrrrcr-r)r<r'Zn_observationsZ sorted_datarTrfZpp1rrrCr(rZrelative_residualZ shifted_meanZsum_termZ scale_newr.r.r/r's,  "& z(halflogistic_gen.fit..find_scale halflogisticrr() r;r<r=r-rrr'rr'r(r r.r/r9s  " zhalflogistic_gen.fit)rirjrkrlr]rarrbrgrdrhrrrBr rr9r?r.r.r r/r)s r)r2cs~eZdZdZddZdddZddZd d Zd d Zd dZ ddZ ddZ ddZ ddZ eeefddZZS) halfnorm_genaFA half-normal continuous random variable. %(before_notes)s Notes ----- The probability density function for `halfnorm` is: .. math:: f(x) = \sqrt{2/\pi} \exp(-x^2 / 2) for :math:`x >= 0`. `halfnorm` is a special case of `chi` with ``df=1``. %(after_notes)s %(example)s cCsgSrCr.r\r.r.r/r]gszhalfnorm_gen._shape_infoNcCst|j|dSNrrLrr.r.r/rjszhalfnorm_gen._rvscCs$tdtjt| |dSrrErrrrr.r.r/ramszhalfnorm_gen._pdfcCs dtdtj||dSNrurrrr.r.r/rqszhalfnorm_gen._logpdfcCst|tdSrrcrMrErrr.r.r/rbtszhalfnorm_gen._cdfcCstd|dSrsrrr.r.r/rgwszhalfnorm_gen._ppfcCs dt|Srrrr.r.r/rdzszhalfnorm_gen._sfcCs t|dSrrrr.r.r/rh}szhalfnorm_gen._isfcCsXtdtjddtjtddtjtjdddtjdtjddfS)NrrrKrrrrrErrr\r.r.r/rs   zhalfnorm_gen._statscCsdttjddSr6rr\r.r.r/rszhalfnorm_gen._entropyc s|ddr&tj|g|Ri|St||||\}}}t|}|durj||krdtd|tjd|}n|}|dur||}ntj |d|dd}||fS)NrVFhalfnormrrK)ordercenterru) r+r7r9r^rErrr[rSZmomentrr r.r/r9s   zhalfnorm_gen.fit)NN)rirjrkrlr]rrarrbrgrdrhrrrBr rr9r?r.r.r r/r3Qs r3r9c@sPeZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ dS) hypsecant_genaA hyperbolic secant continuous random variable. %(before_notes)s Notes ----- The probability density function for `hypsecant` is: .. math:: f(x) = \frac{1}{\pi} \text{sech}(x) for a real number :math:`x`. %(after_notes)s %(example)s cCsgSrCr.r\r.r.r/r]szhypsecant_gen._shape_infocCsdtjt|Sr)rErcoshrr.r.r/raszhypsecant_gen._pdfcCsdtjtt|SrrErrrrr.r.r/rbszhypsecant_gen._cdfcCstttj|dSrrErrrrr.r.r/rgszhypsecant_gen._ppfcCsdtjtt| Srr>rr.r.r/rdszhypsecant_gen._sfcCstttj|d Srr?rr.r.r/rhszhypsecant_gen._isfcCsdtjtjdddfS)NrrrKrr\r.r.r/rszhypsecant_gen._statscCstdtjSrrr\r.r.r/rszhypsecant_gen._entropyN) rirjrkrlr]rarbrgrdrhrrr.r.r.r/r<sr< hypsecantc@s0eZdZdZddZddZddZdd Zd S) gausshyper_gena_A Gauss hypergeometric continuous random variable. %(before_notes)s Notes ----- The probability density function for `gausshyper` is: .. math:: f(x, a, b, c, z) = C x^{a-1} (1-x)^{b-1} (1+zx)^{-c} for :math:`0 \le x \le 1`, :math:`a,b > 0`, :math:`c` a real number, :math:`z > -1`, and :math:`C = \frac{1}{B(a, b) F[2, 1](c, a; a+b; -z)}`. :math:`F[2, 1]` is the Gauss hypergeometric function `scipy.special.hyp2f1`. `gausshyper` takes :math:`a`, :math:`b`, :math:`c` and :math:`z` as shape parameters. %(after_notes)s References ---------- .. [1] Armero, C., and M. J. Bayarri. "Prior Assessments for Prediction in Queues." *Journal of the Royal Statistical Society*. Series D (The Statistician) 43, no. 1 (1994): 139-53. doi:10.2307/2348939 %(example)s cCs |dk|dk@||k@|dk@S)Nrr.r.)r;rprqrfr'r.r.r/rWszgausshyper_gen._argcheckcCs`tdddtjfd}tdddtjfd}tddtj tjfd}tdddtjfd}||||gS) NrpFrrrqrfr'r.rZ)r;rrrxZizr.r.r/r]s zgausshyper_gen._shape_infocCsVt||t||||| }d|||dd||dd|||SrrcrZhyp2f1)r;r`rprqrfr'Znormalization_constantr.r.r/ras""zgausshyper_gen._pdfc Cs\t|||t||}t||||||| }t||||| }|||SrCrB) r;rVrprqrfr'rrWr3r.r.r/r szgausshyper_gen._munpN)rirjrkrlrWr]rarr.r.r.r/rAs  rA gausshyperc@s`eZdZdZejZddZddZddZ dd Z d d Z d d Z ddZ dddZddZdS) invgamma_gena_An inverted gamma continuous random variable. %(before_notes)s Notes ----- The probability density function for `invgamma` is: .. math:: f(x, a) = \frac{x^{-a-1}}{\Gamma(a)} \exp(-\frac{1}{x}) for :math:`x >= 0`, :math:`a > 0`. :math:`\Gamma` is the gamma function (`scipy.special.gamma`). `invgamma` takes ``a`` as a shape parameter for :math:`a`. `invgamma` is a special case of `gengamma` with ``c=-1``, and it is a different parameterization of the scaled inverse chi-squared distribution. Specifically, if the scaled inverse chi-squared distribution is parameterized with degrees of freedom :math:`\nu` and scaling parameter :math:`\tau^2`, then it can be modeled using `invgamma` with ``a=`` :math:`\nu/2` and ``scale=`` :math:`\nu \tau^2/2`. %(after_notes)s %(example)s cCstdddtjfdgSrerZr\r.r.r/r]3szinvgamma_gen._shape_infocCst|||SrCrGrr.r.r/ra6szinvgamma_gen._pdfcCs&|d t|t|d|SrrErrcr5rr.r.r/r:szinvgamma_gen._logpdfcCst|d|Srrrr.r.r/rb=szinvgamma_gen._cdfcCsdt||Srrrr.r.r/rg@szinvgamma_gen._ppfcCst|d|Srrrr.r.r/rdCszinvgamma_gen._sfcCsdt||Srrrr.r.r/rhFszinvgamma_gen._isfmvskcCst|dk|fddtj}t|dk|fddtj}d\}}d|vr^t|dk|fd dtj}d |vrt|d k|fd dtj}||||fS) NrcSs d|dSrr.rr.r.r/rLJrMz%invgamma_gen._stats..rKcSsd|dd|dS)NrnrKrr.rr.r.r/rLKrM)NNrmrcSsdt|d|dS)Nrrrrrr.r.r/rLRrMrorcSs dd|d|d|dS)Nrrg&@rrr.rr.r.r/rLVrMr)r;rprrrrrrr.r.r/rIs     zinvgamma_gen._statscCs*dd}dd}t|dk|f||d}|S)NcSs$||dt|t|}|Srrrprr.r.r/r Zs z&invgamma_gen._entropy..regularcSs`ddt|tdttjdd|d|dd|dd |d d }|S) NrrrKUUUUUU?rr#r/r$rr%r'rrGr.r.r/r^s*    z)invgamma_gen._entropy..asymptoticrrrQ)r;rpr rrr.r.r/rYszinvgamma_gen._entropyN)rF)rirjrkrlrrrr]rarrbrgrdrhrrr.r.r.r/rDs rDinvgammacseZdZdZejZddZdddZddZ d d Z d d Z d dZ ddZ ddZfddZfddZddZeefddZddZZS) invgauss_gena`An inverse Gaussian continuous random variable. %(before_notes)s Notes ----- The probability density function for `invgauss` is: .. math:: f(x; \mu) = \frac{1}{\sqrt{2 \pi x^3}} \exp\left(-\frac{(x-\mu)^2}{2 \mu^2 x}\right) for :math:`x \ge 0` and :math:`\mu > 0`. `invgauss` takes ``mu`` as a shape parameter for :math:`\mu`. %(after_notes)s A common shape-scale parameterization of the inverse Gaussian distribution has density .. math:: f(x; \nu, \lambda) = \sqrt{\frac{\lambda}{2 \pi x^3}} \exp\left( -\frac{\lambda(x-\nu)^2}{2 \nu^2 x}\right) Using ``nu`` for :math:`\nu` and ``lam`` for :math:`\lambda`, this parameterization is equivalent to the one above with ``mu = nu/lam``, ``loc = 0``, and ``scale = lam``. %(example)s cCstdddtjfdgSNrFrrrZr\r.r.r/r]szinvgauss_gen._shape_infoNcCs|j|d|dSNrnrwaldr;rrrr.r.r/rszinvgauss_gen._rvscCs>dtdtj|dtdd||||dS)NrnrKrrr5r;r`rr.r.r/raszinvgauss_gen._pdfcCs:dtdtjdt||||dd|S)NrNrKrrrPr.r.r/rszinvgauss_gen._logpdfcCsXdt|}t|||d}d|t| ||d}|tt||Sr)rErrrrr;r`rrrprqr.r.r/rszinvgauss_gen._logcdfcCsZdt|}t|||d}d|t| |||}|tt|| Sr)rErrrrrrQr.r.r/rszinvgauss_gen._logsfcCst|||SrCrrPr.r.r/rdszinvgauss_gen._sfcCst|||SrCrrPr.r.r/rbszinvgauss_gen._cdfcstjddddxt||\}}t||d}|dk}td||||d||<t|}t||||||<Wdn1s0Y|SNr)rrinvalidrru) rErrRr _invgauss_ppf _invgauss_isfrr7rg)r;r`rppfi_wti_nanr r.r/rgs 8zinvgauss_gen._ppfcstjddddxt||\}}t||d}|dk}td||||d||<t|}t||||||<Wdn1s0Y|SrR) rErrRrrUrTrr7rh)r;r`rZisfrWrXr r.r/rhs 8zinvgauss_gen._isfcCs||ddt|d|fS)Nrrrr)r;rr.r.r/rszinvgauss_gen._statsc s|dd}t|ts.t|tks.|dkrHtj|g|Ri|St||||\}}}}|dusn|durtj|g|Ri|St ||dkrt ddt j dn@||}t |}|durt|t |d|d}||}|||fS)Nr*r1r2rinvgaussrr.)r3r5r$r8wald_genr4r7r9r^rErrr[rrr) r;r<r=r-r*Zfshape_srrZfshape_nr r.r/r9s$    zinvgauss_gen.fitcCsJdtdtjdt|}d|}tj||}d|d|S)zV Ref.: https://moser-isi.ethz.ch/docs/papers/smos-2012-10.pdf (eq. 9) rnrKrrur)rErrrcrr)r;rrprUrqr.r.r/rs"zinvgauss_gen._entropy)NN)rirjrkrlrrrr]rrarrrrdrbrgrhrr r9rr?r.r.r r/rJls "  !rJrYc@sbeZdZdZddZddZddZdd Zd d Zd d Z dddZ ddZ ddZ ddZ dS)geninvgauss_genaXA Generalized Inverse Gaussian continuous random variable. %(before_notes)s Notes ----- The probability density function for `geninvgauss` is: .. math:: f(x, p, b) = x^{p-1} \exp(-b (x + 1/x) / 2) / (2 K_p(b)) where `x > 0`, `p` is a real number and `b > 0`\([1]_). :math:`K_p` is the modified Bessel function of second kind of order `p` (`scipy.special.kv`). %(after_notes)s The inverse Gaussian distribution `stats.invgauss(mu)` is a special case of `geninvgauss` with `p = -1/2`, `b = 1 / mu` and `scale = mu`. Generating random variates is challenging for this distribution. The implementation is based on [2]_. References ---------- .. [1] O. Barndorff-Nielsen, P. Blaesild, C. Halgreen, "First hitting time models for the generalized inverse gaussian distribution", Stochastic Processes and their Applications 7, pp. 49--54, 1978. .. [2] W. Hoermann and J. Leydold, "Generating generalized inverse Gaussian random variates", Statistics and Computing, 24(4), p. 547--557, 2014. %(example)s cCs||k|dk@Sr6r.r;rTrqr.r.r/rW$szgeninvgauss_gen._argcheckcCs4tddtj tjfd}tdddtjfd}||gS)NrTFrrqrrZ)r;rrr.r.r/r]'szgeninvgauss_gen._shape_infocCsLdd}tj|tjgd}||||}t|rHd}tj|tdd|S)NcSst|||SrC)rZgeninvgauss_logpdfr`rTrqr.r.r/ logpdf_single0sz.geninvgauss_gen._logpdf..logpdf_singlerzjInfinite values encountered in scipy.special.kve(p, b). Values replaced by NaN to avoid incorrect results.rr)rErrrrrrr)r;r`rTrqr^r'rOr.r.r/r,s zgeninvgauss_gen._logpdfcCst||||SrCrGr;r`rTrqr.r.r/ra<szgeninvgauss_gen._pdfcs<|j|\}fdd}tj|tjgd}||g|RS)NcsB|\}}t||gtjtj}ttd|}t ||dS)NZ_geninvgauss_pdfr) rErrrrrr rrr r)r`r=rTrqrrrr.r/ _cdf_singleCs z)geninvgauss_gen._cdf.._cdf_singler)rvrErr)r;r`r=rrar.r`r/rb@s zgeninvgauss_gen._cdfcCs t|dk|||fddtj S)NrcSs&|dt|||d|dSrrr]r.r.r/rLRrMz.geninvgauss_gen._logquasipdf..rbr_r.r.r/ _logquasipdfOszgeninvgauss_gen._logquasipdfNc st|r&t|r&|||||}n|jdkrT|jdkrT|||||}nt||\}}t|j|\}tt |}t |}tj ||gdgdgdggdj st fddtt| dD}|dd|||||<q|dkr|}|S) Nr multi_indexreadonlyflagsZop_flagsc3s(|] }|sj|ntdVqdSrCrcslicerZr;bcitr.r/rsz'geninvgauss_gen._rvs..rr.)rEisscalar _rvs_scalarrrrRrrr9r]emptynditerfinishedtupler^rriternext) r;rTrqrrrVshp numsamplesidxr.rjr/rUs2       zgeninvgauss_gen._rvsc3 sd}|s d}dkr d}}d}dks>dkrDd}n*tddtddkrjd}nd}tt|} t| } t| } d} |rV|rdd|} d|dd}|| dd}d| dd | |d|}t| td |dd}td |d }|t |dtj d| d}| t |d| d} | |} |}||t d|}||t d|}d}fd d }|}nrt d |}dtddd}d}|t d |}d}fdd }||krpt d|dkrt dd}| | kr| | }||j|d}|j|d} |||| } | ||}!dt|||!k}"t|"}#|#dkr|!|"| | | |#<| |#7} | dkrF|| dkrFd|| d}$t|$|d7}qnd}%t|%df}&t  |}'|'|%}(|%dkrt })dkr|)d|%}*n|)tdd}*nd\})}*|&d}+d|+t |& d},|(|*|,}-| | kr| | }t|t|}.}!|j|d}|-|j|d} | |(k}/t|/| |(|*k@}0t|/|0B}1|%| |/|(|!|/<|'|.|/<dkr|%| |0|(|)d|!|0<n$t | |0|(t |!|0<|)|!|0d|.|0<t |& d| |1|(|*d|+}2dt|2|!|1<|+t |!|1 d|.|1<t||. |!k}"t|"}#|#dkr*|!|"| | | |#<| |#7} q*t| | }!|rd|!}!|!S)NFrrTrurKrrircs|SrCrbrrqZlmrTr;r.r/logqpdfsz,geninvgauss_gen._rvs_scalar..logqpdfcs|SrCrxr)rqrTr;r.r/rzszvmin must be smaller than vmax.zumax must be positive.riPz2Not a single random variate could be generated in zH attempts. Sampling does not appear to work for the provided parameters.)rr)_moderrErrr atleast_1dr]zerosZarccosrrrbrrrrrrrZ logical_notr)3r;rTrqrurZ invert_resrCZ ratio_unifZ mode_shiftsize1dNr` simulatedZa2Za1p1q1phirZroot1root2d1d2ZvminZvmaxZumaxrzrfZxplusrWrorr:rDaccept num_acceptrOrxsZk1A1Zk2A2Zk3ZA3rrZcond1Zcond2Zcond3r'r.ryr/rns     "$&                *$0    zgeninvgauss_gen._rvs_scalarcCsX|dkr.|t|dd|dd|Std|d|dd||SdSrrr\r.r.r/r{%s&zgeninvgauss_gen._modec Cst|||}t||}t|t|B}|r|d}tj|tddtj|tj tj d}||||||<n||}|S)NzInfinite values encountered in the moment calculation involving scipy.special.kve. Values replaced by NaN to avoid incorrect results.rrdtype) rcZkverErLrrrr full_likerr) r;rVrTrqrWdenomZinf_valsrOrCr.r.r/r,s zgeninvgauss_gen._munp)NN)rirjrkrlrWr]rrarbrbrrnr{rr.r.r.r/r[s$ 7r[rcs`eZdZdZejZddZddZfddZ dd Z d d Z d d Z dddZ ddZZS)norminvgauss_genaA Normal Inverse Gaussian continuous random variable. %(before_notes)s Notes ----- The probability density function for `norminvgauss` is: .. math:: f(x, a, b) = \frac{a \, K_1(a \sqrt{1 + x^2})}{\pi \sqrt{1 + x^2}} \, \exp(\sqrt{a^2 - b^2} + b x) where :math:`x` is a real number, the parameter :math:`a` is the tail heaviness and :math:`b` is the asymmetry parameter satisfying :math:`a > 0` and :math:`|b| <= a`. :math:`K_1` is the modified Bessel function of second kind (`scipy.special.k1`). %(after_notes)s A normal inverse Gaussian random variable `Y` with parameters `a` and `b` can be expressed as a normal mean-variance mixture: `Y = b * V + sqrt(V) * X` where `X` is `norm(0,1)` and `V` is `invgauss(mu=1/sqrt(a**2 - b**2))`. This representation is used to generate random variates. Another common parametrization of the distribution (see Equation 2.1 in [2]_) is given by the following expression of the pdf: .. math:: g(x, \alpha, \beta, \delta, \mu) = \frac{\alpha\delta K_1\left(\alpha\sqrt{\delta^2 + (x - \mu)^2}\right)} {\pi \sqrt{\delta^2 + (x - \mu)^2}} \, e^{\delta \sqrt{\alpha^2 - \beta^2} + \beta (x - \mu)} In SciPy, this corresponds to `a = alpha * delta, b = beta * delta, loc = mu, scale=delta`. References ---------- .. [1] O. Barndorff-Nielsen, "Hyperbolic Distributions and Distributions on Hyperbolae", Scandinavian Journal of Statistics, Vol. 5(3), pp. 151-157, 1978. .. [2] O. Barndorff-Nielsen, "Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling", Scandinavian Journal of Statistics, Vol. 24, pp. 1-13, 1997. %(example)s cCs|dkt||k@Sr6)rEabsoluterr.r.r/rWwsznorminvgauss_gen._argcheckcCs4tdddtjfd}tddtj tjfd}||gSrrZrr.r.r/r]zsznorminvgauss_gen._shape_infocstj|ddS)N)rrurrdrer r.r/r sznorminvgauss_gen._fitstartcCs\t|d|d}|tj}td|}|t||t||||||Sr)rErrhypotrcZk1er)r;r`rprqrCfac1sqr.r.r/ras  znorminvgauss_gen._pdfc Cst|r(tj|j|tj||fddSt|}t|}g}t|||D].\}}}|tj|j|tj||fddqLt |SdS)Nrr) rErmr rrar[r|rappendr)r;r`rprqresultra0rr.r.r/rds   znorminvgauss_gen._sfc s^fdd}t|r"||||Sg}t|||D]\}}}|||||q2t|SdS)Nc sfdd}||}|||||}|dkr2|S|dkrpd}|}||}|||||dkrd|}||}qJn4d}|}||}|||||dkrd|}||}qtj||||||fjd} | S)Ncs||||SrCrd)r`rprqrfr\r.r/eqsz6norminvgauss_gen._isf.._isf_scalar..eqrrrK)r=r)rr rr) rfrprqrxmZemdeltaleftrightrr\r.r/ _isf_scalars,    z*norminvgauss_gen._isf.._isf_scalar)rErmrrr) r;rfrprqrrq0rrr.r\r/rhs !  znorminvgauss_gen._isfNcCsJt|d|d}tjd|||d}||t|tj||dS)NrKr)rrrrB)rErrYrDr)r;rprqrrrCZigr.r.r/rs znorminvgauss_gen._rvscCspt|d|d}||}|d|d}d||t|}ddd|d|d|}||||fS)NrKrrrrr)r;rprqrCrZvarianceskewnessZkurtosisr.r.r/rs  znorminvgauss_gen._stats)NN)rirjrkrlrrrrWr]r rardrhrrr?r.r.r r/r?s5  * r norminvgausscsheZdZdZejZddZddZddZ dd Z d d Z d d Z ddZ ddZdfdd ZZS)invweibull_genuAn inverted Weibull continuous random variable. This distribution is also known as the Fréchet distribution or the type II extreme value distribution. %(before_notes)s Notes ----- The probability density function for `invweibull` is: .. math:: f(x, c) = c x^{-c-1} \exp(-x^{-c}) for :math:`x > 0`, :math:`c > 0`. `invweibull` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s References ---------- F.R.S. de Gusmao, E.M.M Ortega and G.M. Cordeiro, "The generalized inverse Weibull distribution", Stat. Papers, vol. 52, pp. 591-619, 2011. %(example)s cCstdddtjfdgSrerZr\r.r.r/r]szinvweibull_gen._shape_infocCs8t|| d}t|| }t| }|||SrrErr)r;r`rfxc1Zxc2r.r.r/ras zinvweibull_gen._pdfcCst|| }t| SrCr)r;r`rfrr.r.r/rbszinvweibull_gen._cdfcCst||   SrC)rErjrhr.r.r/rdszinvweibull_gen._sfcCstt| d|Sr)rErrrkr.r.r/rgszinvweibull_gen._ppfcCst|  d|SNr.rrr.r.r/rh szinvweibull_gen._isfcCstd||SrRrrr.r.r/r szinvweibull_gen._munpcCsdtt|t|SrRrUrr.r.r/rszinvweibull_gen._entropyNcs |dur dn|}tj||dS)N)rrrd)r;r<r=r r.r/r szinvweibull_gen._fitstart)N)rirjrkrlrrrr]rarbrdrgrhrrr r?r.r.r r/rsr invweibullc@sBeZdZdZddZddZdddZd d Zd d Zd dZ dS) jf_skew_t_genaJones and Faddy skew-t distribution. %(before_notes)s Notes ----- The probability density function for `jf_skew_t` is: .. math:: f(x; a, b) = C_{a,b}^{-1} \left(1+\frac{x}{\left(a+b+x^2\right)^{1/2}}\right)^{a+1/2} \left(1-\frac{x}{\left(a+b+x^2\right)^{1/2}}\right)^{b+1/2} for real numbers :math:`a>0` and :math:`b>0`, where :math:`C_{a,b} = 2^{a+b-1}B(a,b)(a+b)^{1/2}`, and :math:`B` denotes the beta function (`scipy.special.beta`). When :math:`ab`, the distribution is positively skewed. If :math:`a=b`, then we recover the `t` distribution with :math:`2a` degrees of freedom. `jf_skew_t` takes :math:`a` and :math:`b` as shape parameters. %(after_notes)s References ---------- .. [1] M.C. Jones and M.J. Faddy. "A skew extension of the t distribution, with applications" *Journal of the Royal Statistical Society*. Series B (Statistical Methodology) 65, no. 1 (2003): 159-174. :doi:`10.1111/1467-9868.00378` %(example)s cCs0tdddtjfd}tdddtjfd}||gSrrZrr.r.r/r]@szjf_skew_t_gen._shape_infocCsd||dt||t||}d|t|||d|d}d|t|||d|d}|||SNrKrru)rcrrEr)r;r`rprqrfrrr.r.r/raEs*&&zjf_skew_t_gen._pdfNcCsF||||}d|dt||}dt|d|}||Sr)rrEr)r;rprqrrrrd3r.r.r/rKszjf_skew_t_gen._rvscCs0d|t|||dd}t|||SNrrKru)rErrcZbetainc)r;r`rprqrr.r.r/rbQs"zjf_skew_t_gen._cdfcCsFt|||}d|dt||}dt|d|}||Sr)rrVrEr)r;rfrprqrrrr.r.r/rgUszjf_skew_t_gen._ppfcCsLdd}|d|k|d|k@|dk@}t||||ftj|tjgdtjS)zReturns the n-th moment(s) where all the following hold: - n >= 0 - a > n / 2 - b > n / 2 The result is np.nan in all other cases. c Ss||d|}d|t||}t|d}t|ddkdd}t|d|||d||}t||||}|||S)zgComputes E[T^(n_k)] where T is skew-t distributed with parameters a_k and b_k. rurKrrr.)rcrrErrrr) Zn_kZa_kZb_krWrindicesrqr=Z sum_termsr.r.r/ nth_momentds$z'jf_skew_t_gen._munp..nth_momentrurrrrErrr)r;rVrprqrZnth_moment_validr.r.r/r[s  zjf_skew_t_gen._munp)NN) rirjrkrlr]rarrbrgrr.r.r.r/rs$ r jf_skew_tc@sNeZdZdZejZddZddZddZ dd Z d d Z d d Z ddZ dS) johnsonsb_gena!A Johnson SB continuous random variable. %(before_notes)s See Also -------- johnsonsu Notes ----- The probability density function for `johnsonsb` is: .. math:: f(x, a, b) = \frac{b}{x(1-x)} \phi(a + b \log \frac{x}{1-x} ) where :math:`x`, :math:`a`, and :math:`b` are real scalars; :math:`b > 0` and :math:`x \in [0,1]`. :math:`\phi` is the pdf of the normal distribution. `johnsonsb` takes :math:`a` and :math:`b` as shape parameters. %(after_notes)s %(example)s cCs|dk||k@Sr6r.rr.r.r/rWszjohnsonsb_gen._argcheckcCs4tddtj tjfd}tdddtjfd}||gSNrpFrrqrrZrr.r.r/r]szjohnsonsb_gen._shape_infocCs.t||t|}|d|d||Sr{)rrcr/)r;r`rprqtrmr.r.r/raszjohnsonsb_gen._pdfcCst||t|SrC)rrcr/rr.r.r/rbszjohnsonsb_gen._cdfcCstd|t||Sr)rcr-rrr.r.r/rgszjohnsonsb_gen._ppfcCst||t|SrC)rrcr/rr.r.r/rdszjohnsonsb_gen._sfcCstd|t||Sr)rcr-rrr.r.r/rhszjohnsonsb_gen._isfN)rirjrkrlrrrrWr]rarbrgrdrhr.r.r.r/r~sr johnsonsbc@sReZdZdZddZddZddZdd Zd d Zd d Z ddZ dddZ dS) johnsonsu_gena-A Johnson SU continuous random variable. %(before_notes)s See Also -------- johnsonsb Notes ----- The probability density function for `johnsonsu` is: .. math:: f(x, a, b) = \frac{b}{\sqrt{x^2 + 1}} \phi(a + b \log(x + \sqrt{x^2 + 1})) where :math:`x`, :math:`a`, and :math:`b` are real scalars; :math:`b > 0`. :math:`\phi` is the pdf of the normal distribution. `johnsonsu` takes :math:`a` and :math:`b` as shape parameters. The first four central moments are calculated according to the formulas in [1]_. %(after_notes)s References ---------- .. [1] Taylor Enterprises. "Johnson Family of Distributions". https://variation.com/wp-content/distribution_analyzer_help/hs126.htm %(example)s cCs|dk||k@Sr6r.rr.r.r/rWszjohnsonsu_gen._argcheckcCs4tddtj tjfd}tdddtjfd}||gSrrZrr.r.r/r]szjohnsonsu_gen._shape_infocCs8||}t||t|}|dt|d|Sr)rrEarcsinhr)r;r`rprqr(rr.r.r/raszjohnsonsu_gen._pdfcCst||t|SrC)rrErrr.r.r/rbszjohnsonsu_gen._cdfcCstt|||SrC)rEsinhrrr.r.r/rgszjohnsonsu_gen._ppfcCst||t|SrC)rrErrr.r.r/rdszjohnsonsu_gen._sfcCstt|||SrC)rErrrr.r.r/rhszjohnsonsu_gen._isfrlcCsd\}}}}|d}t|} ||} d|vrB| d t| }d|vrndt|| td| d}d|vr| dt|d} d t| } | | dtd | } tdd| td| d }| | | |}d |vrd d | } d | d| dtd| } | dtd | } dd | dd| d | d }dd| td| d}| | | ||d }||||fS)NNNNNr#rCrur:rKrrmrrrorrr)rErrrcrjr=r)r;rprqrrrrrrZbn2Zexpbn2Za_br*r+r,rZt4r.r.r/rs,  $$  "$zjohnsonsu_gen._statsN)rl) rirjrkrlrWr]rarbrgrdrhrr.r.r.r/rs#r johnsonsuc@sreZdZdZddZdddZddZd d Zd d Zd dZ ddZ ddZ ddZ e eeddddZdS) laplace_gena A Laplace continuous random variable. %(before_notes)s Notes ----- The probability density function for `laplace` is .. math:: f(x) = \frac{1}{2} \exp(-|x|) for a real number :math:`x`. %(after_notes)s %(example)s cCsgSrCr.r\r.r.r/r]-szlaplace_gen._shape_infoNcCs|jdd|dS)Nrrr)laplacerr.r.r/r0szlaplace_gen._rvscCsdtt| Sr)rErrrr.r.r/ra3szlaplace_gen._pdfc Cs\tjdd<t|dkddt| dt|WdS1sN0YdS)Nrrrrnru)rErrrrr.r.r/rb7szlaplace_gen._cdfcCs || SrCrbrr.r.r/rd;szlaplace_gen._sfcCs,t|dktdd| td|SrrErrrr.r.r/rg?szlaplace_gen._ppfcCs || SrCrwrr.r.r/rhBszlaplace_gen._isfcCsdS)N)rrKrrr.r\r.r.r/rFszlaplace_gen._statscCstddSrrr\r.r.r/rIszlaplace_gen._entropyz This function uses explicit formulas for the maximum likelihood estimation of the Laplace distribution parameters, so the keyword arguments `loc`, `scale`, and `optimizer` are ignored. rcOsRt||||\}}}|dur&t|}|durJtt||t|}||fSrC)r^rEmedianrrr)r;r<r=r-rrr.r.r/r9Ls  zlaplace_gen.fit)NN)rirjrkrlr]rrarbrdrgrhrrrBrrr9r.r.r.r/rs  rrc@sXeZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ dS)laplace_asymmetric_genuAn asymmetric Laplace continuous random variable. %(before_notes)s See Also -------- laplace : Laplace distribution Notes ----- The probability density function for `laplace_asymmetric` is .. math:: f(x, \kappa) &= \frac{1}{\kappa+\kappa^{-1}}\exp(-x\kappa),\quad x\ge0\\ &= \frac{1}{\kappa+\kappa^{-1}}\exp(x/\kappa),\quad x<0\\ for :math:`-\infty < x < \infty`, :math:`\kappa > 0`. `laplace_asymmetric` takes ``kappa`` as a shape parameter for :math:`\kappa`. For :math:`\kappa = 1`, it is identical to a Laplace distribution. %(after_notes)s Note that the scale parameter of some references is the reciprocal of SciPy's ``scale``. For example, :math:`\lambda = 1/2` in the parameterization of [1]_ is equivalent to ``scale = 2`` with `laplace_asymmetric`. References ---------- .. [1] "Asymmetric Laplace distribution", Wikipedia https://en.wikipedia.org/wiki/Asymmetric_Laplace_distribution .. [2] Kozubowski TJ and Podgórski K. A Multivariate and Asymmetric Generalization of Laplace Distribution, Computational Statistics 15, 531--540 (2000). :doi:`10.1007/PL00022717` %(example)s cCstdddtjfdgS)NkappaFrrrZr\r.r.r/r]sz"laplace_asymmetric_gen._shape_infocCst|||SrCrGr;r`rr.r.r/raszlaplace_asymmetric_gen._pdfcCs6d|}|t|dk| |}|t||8}|Srtr)r;r`rkapinvrr.r.r/rszlaplace_asymmetric_gen._logpdfcCsLd|}||}t|dkdt| |||t||||SrtrErrr;r`rr kappkapinvr.r.r/rbs  zlaplace_asymmetric_gen._cdfc CsLd|}||}t|dkt| |||dt||||Srtrrr.r.r/rds  zlaplace_asymmetric_gen._sfcCsPd|}||}t|||ktd||| |t||||SrRrr;rfrrrr.r.r/rgs zlaplace_asymmetric_gen._ppfcCsPd|}||}t|||kt||| |td||||SrRrrr.r.r/rhs zlaplace_asymmetric_gen._isfcCsd|}||}||||}ddt|dtdt|dd}ddt|dtdt|dd}||||fS) NrrrrrrrrKr1)r;rrmnrrrr.r.r/rs ,,zlaplace_asymmetric_gen._statscCsdt|d|SrRrr;rr.r.r/rszlaplace_asymmetric_gen._entropyNrr.r.r.r/rds+rlaplace_asymmetriccCsnt|tst|}|dd}|dd}|jrBt|jdnd}g}g}|jr|jdd} t | D]T\} } dt | } | d| d| g} t || }| | | ||durn||| <qndd d d ddh|}t ||}|rtd |d t||krtdd||h|vr*tdt|tr>|n|}t|sZtd|g|||RS)Nrr,r rZfix_r'r(r)r*zUnknown keyword arguments: rzToo many positional arguments.rr)r5r$rErr3shapesrsplitr enumeratestrrrset differencer,rr rrr)distr<r=r-rrZ num_shapesZ fshape_keysZfshapesrr;rmkeynamesrpZ known_keysZ unknown_keysZ uncensoredr.r.r/r^sB         r^c@sNeZdZdZejZddZddZddZ dd Z d d Z d d Z ddZ dS)levy_genagA Levy continuous random variable. %(before_notes)s See Also -------- levy_stable, levy_l Notes ----- The probability density function for `levy` is: .. math:: f(x) = \frac{1}{\sqrt{2\pi x^3}} \exp\left(-\frac{1}{2x}\right) for :math:`x > 0`. This is the same as the Levy-stable distribution with :math:`a=1/2` and :math:`b=1`. %(after_notes)s Examples -------- >>> import numpy as np >>> from scipy.stats import levy >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1) Calculate the first four moments: >>> mean, var, skew, kurt = levy.stats(moments='mvsk') Display the probability density function (``pdf``): >>> # `levy` is very heavy-tailed. >>> # To show a nice plot, let's cut off the upper 40 percent. >>> a, b = levy.ppf(0), levy.ppf(0.6) >>> x = np.linspace(a, b, 100) >>> ax.plot(x, levy.pdf(x), ... 'r-', lw=5, alpha=0.6, label='levy pdf') Alternatively, the distribution object can be called (as a function) to fix the shape, location and scale parameters. This returns a "frozen" RV object holding the given parameters fixed. Freeze the distribution and display the frozen ``pdf``: >>> rv = levy() >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf') Check accuracy of ``cdf`` and ``ppf``: >>> vals = levy.ppf([0.001, 0.5, 0.999]) >>> np.allclose([0.001, 0.5, 0.999], levy.cdf(vals)) True Generate random numbers: >>> r = levy.rvs(size=1000) And compare the histogram: >>> # manual binning to ignore the tail >>> bins = np.concatenate((np.linspace(a, b, 20), [np.max(r)])) >>> ax.hist(r, bins=bins, density=True, histtype='stepfilled', alpha=0.2) >>> ax.set_xlim([x[0], x[-1]]) >>> ax.legend(loc='best', frameon=False) >>> plt.show() cCsgSrCr.r\r.r.r/r]Cszlevy_gen._shape_infocCs.dtdtj||tdd|SNrrKr.r5rr.r.r/raFsz levy_gen._pdfcCsttd|Sr)rcerfcrErrr.r.r/rbJsz levy_gen._cdfcCsttd|Srr7rr.r.r/rdNsz levy_gen._sfcCst|d}d||SNrKrnrr;rfrpr.r.r/rgQs z levy_gen._ppfcCsddt|dSr)rcZerfinvrr.r.r/rhVsz levy_gen._isfcCstjtjtjtjfSrCrr\r.r.r/rYszlevy_gen._statsNrirjrkrlrrrr]rarbrdrgrhrr.r.r.r/rsHrlevyc@sNeZdZdZejZddZddZddZ dd Z d d Z d d Z ddZ dS) levy_l_genaA left-skewed Levy continuous random variable. %(before_notes)s See Also -------- levy, levy_stable Notes ----- The probability density function for `levy_l` is: .. math:: f(x) = \frac{1}{|x| \sqrt{2\pi |x|}} \exp{ \left(-\frac{1}{2|x|} \right)} for :math:`x < 0`. This is the same as the Levy-stable distribution with :math:`a=1/2` and :math:`b=-1`. %(after_notes)s Examples -------- >>> import numpy as np >>> from scipy.stats import levy_l >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1) Calculate the first four moments: >>> mean, var, skew, kurt = levy_l.stats(moments='mvsk') Display the probability density function (``pdf``): >>> # `levy_l` is very heavy-tailed. >>> # To show a nice plot, let's cut off the lower 40 percent. >>> a, b = levy_l.ppf(0.4), levy_l.ppf(1) >>> x = np.linspace(a, b, 100) >>> ax.plot(x, levy_l.pdf(x), ... 'r-', lw=5, alpha=0.6, label='levy_l pdf') Alternatively, the distribution object can be called (as a function) to fix the shape, location and scale parameters. This returns a "frozen" RV object holding the given parameters fixed. Freeze the distribution and display the frozen ``pdf``: >>> rv = levy_l() >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf') Check accuracy of ``cdf`` and ``ppf``: >>> vals = levy_l.ppf([0.001, 0.5, 0.999]) >>> np.allclose([0.001, 0.5, 0.999], levy_l.cdf(vals)) True Generate random numbers: >>> r = levy_l.rvs(size=1000) And compare the histogram: >>> # manual binning to ignore the tail >>> bins = np.concatenate(([np.min(r)], np.linspace(a, b, 20))) >>> ax.hist(r, bins=bins, density=True, histtype='stepfilled', alpha=0.2) >>> ax.set_xlim([x[0], x[-1]]) >>> ax.legend(loc='best', frameon=False) >>> plt.show() cCsgSrCr.r\r.r.r/r]szlevy_l_gen._shape_infocCs6t|}dtdtj||tdd|Sr)rrErrrr;r`rr.r.r/raszlevy_l_gen._pdfcCs"t|}dtdt|dSr)rrrErrr.r.r/rbszlevy_l_gen._cdfcCst|}dtdt|Sr)rrrErrr.r.r/rdszlevy_l_gen._sfcCst|dd}d||S)NrnrKrrrr.r.r/rgszlevy_l_gen._ppfcCsdt|ddS)Nr.rKrrr.r.r/rhszlevy_l_gen._isfcCstjtjtjtjfSrCrr\r.r.r/rszlevy_l_gen._statsNrr.r.r.r/r`sGrlevy_lcseZdZdZddZdddZddZd d Zd d Zd dZ ddZ ddZ ddZ ddZ ddZddZeeefddZZS) logistic_genaA logistic (or Sech-squared) continuous random variable. %(before_notes)s Notes ----- The probability density function for `logistic` is: .. math:: f(x) = \frac{\exp(-x)} {(1+\exp(-x))^2} `logistic` is a special case of `genlogistic` with ``c=1``. Remark that the survival function (``logistic.sf``) is equal to the Fermi-Dirac distribution describing fermionic statistics. %(after_notes)s %(example)s cCsgSrCr.r\r.r.r/r]szlogistic_gen._shape_infoNcCs |j|dSr4)logisticrr.r.r/rszlogistic_gen._rvscCst||SrCrGrr.r.r/raszlogistic_gen._pdfcCs$t| }|dtt|Sr)rErrcrr)r;r`rr.r.r/rs zlogistic_gen._logpdfcCs t|SrCr,rr.r.r/rbszlogistic_gen._cdfcCs t|SrCrcZ log_expitrr.r.r/rszlogistic_gen._logcdfcCs t|SrCr.rr.r.r/rgszlogistic_gen._ppfcCs t| SrCr,rr.r.r/rdszlogistic_gen._sfcCs t| SrCrrr.r.r/rszlogistic_gen._logsfcCs t| SrCr.rr.r.r/rhszlogistic_gen._isfcCsdtjtjdddfS)Nrrg333333?rr\r.r.r/rszlogistic_gen._statscCsdSrr.r\r.r.r/rszlogistic_gen._entropyc sD|ddr&tjg|Ri|St|||\}}t|\}}|d||d|}}|ffdd |ffdd fd d }|dur|durt|f} | j d }|}nH|dur|durt|f} | j d }|}nt|||f} | j \}}t |}| j r*||fStjg|Ri|S) NrVFr'r(cs$||}tt|dSr)rErrcr-)r'r(rfr<rVr.r/dl_dlocs z!logistic_gen.fit..dl_dloccs(||}t|t|dSr)rErr*)r(r'rfrr.r/ dl_dscales z#logistic_gen.fit..dl_dscalecs|\}}||||fSrCr.)paramsr'r()rrr.r/r#szlogistic_gen.fit..funcr) r+r7r9r^rr r3r r_r`rsuccess) r;r<r=r-rrr'r(rrr )r<rrrVr/r9s2     zlogistic_gen.fit)NN)rirjrkrlr]rrarrbrrgrdrrhrrrBr rr9r?r.r.r r/rs  rrc@sbeZdZdZddZdddZddZd d Zd d Zd dZ ddZ ddZ ddZ ddZ dS) loggamma_genaA log gamma continuous random variable. %(before_notes)s Notes ----- The probability density function for `loggamma` is: .. math:: f(x, c) = \frac{\exp(c x - \exp(x))} {\Gamma(c)} for all :math:`x, c > 0`. Here, :math:`\Gamma` is the gamma function (`scipy.special.gamma`). `loggamma` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s %(example)s cCstdddtjfdgSrerZr\r.r.r/r]Xszloggamma_gen._shape_infoNcCs.t|j|d|dt|j|d|S)Nrr)rErrCrr9r.r.r/r[s zloggamma_gen._rvscCs"t||t|t|SrCrErrcr5rhr.r.r/raiszloggamma_gen._pdfcCs||t|t|SrCrrhr.r.r/rmszloggamma_gen._logpdfcCs t|tk||fdddddS)NcSst||t|dSrRrrr.r.r/rLrMz#loggamma_gen._cdf..cSst|t|SrC)rcrrErrr.r.r/rLrMrOrrrhr.r.r/rbpszloggamma_gen._cdfcCs.t||}t|tk|||fdddddS)NcSst|t|d|SrRrErrfrfr.r.r/rLrMz#loggamma_gen._ppf..cSs t|SrCrrr.r.r/rLrMrO)rcrrrr;rfrfrr.r.r/rgs  zloggamma_gen._ppfcCs t|tk||fdddddS)NcSst||t|d SrR)rErjrcr5rr.r.r/rLrMz"loggamma_gen._sf..cSst|t|SrC)rcrrErrr.r.r/rLrMrOrrhr.r.r/rdszloggamma_gen._sfcCs.t||}t|tk|||fdddddS)NcSst| t|d|SrR)rErrcr5rr.r.r/rLrMz#loggamma_gen._isf..cSs t|SrCrrr.r.r/rLrMrO)rcrrrrr.r.r/rhs  zloggamma_gen._isfcCsNt|}td|}td|t|d}td|||}||||fS)NrrKrr)rcrZ polygammarEr)r;rfrrrZexcess_kurtosisr.r.r/rs   zloggamma_gen._statscCs*dd}dd}t|dk|f||d}|S)NcSs t||t||}|SrC)rcr5r)rfrr.r.r/r sz&loggamma_gen._entropy..regularcSsBdt||dd|dd|dd}t|}|S)NrNrrr$rr0)rErrr)rftermrr.r.r/rs2 z)loggamma_gen._entropy..asymptotic-rrQ)r;rfr rrr.r.r/rszloggamma_gen._entropy)NNrirjrkrlr]rrarrbrgrdrhrrr.r.r.r/r?s  rloggammacsleZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ e e efddZZS)loglaplace_genaTA log-Laplace continuous random variable. %(before_notes)s Notes ----- The probability density function for `loglaplace` is: .. math:: f(x, c) = \begin{cases}\frac{c}{2} x^{ c-1} &\text{for } 0 < x < 1\\ \frac{c}{2} x^{-c-1} &\text{for } x \ge 1 \end{cases} for :math:`c > 0`. `loglaplace` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s Suppose a random variable ``X`` follows the Laplace distribution with location ``a`` and scale ``b``. Then ``Y = exp(X)`` follows the log-Laplace distribution with ``c = 1 / b`` and ``scale = exp(a)``. References ---------- T.J. Kozubowski and K. Podgorski, "A log-Laplace growth rate model", The Mathematical Scientist, vol. 28, pp. 49-60, 2003. %(example)s cCstdddtjfdgSrerZr\r.r.r/r]szloglaplace_gen._shape_infocCs,|d}t|dk|| }|||dSrrEr)r;r`rfZcd2r.r.r/raszloglaplace_gen._pdfcCs(t|dkd||dd|| SNrrurrhr.r.r/rbszloglaplace_gen._cdfcCs(t|dkdd||d|| Srrrhr.r.r/rdszloglaplace_gen._sfcCs.t|dkd|d|dd|d|SNrurrnrKrrrkr.r.r/rgszloglaplace_gen._ppfcCs.t|dkdd|d|d|d|Srrrkr.r.r/rhszloglaplace_gen._isfcCs^tjdd>|d|d}}t||k|||tjWdS1sP0YdS)NrrrK)rErrr[)r;rVrfr2Zn2r.r.r/rszloglaplace_gen._munpcCstd|dSrrrr.r.r/rszloglaplace_gen._entropyc st||||\}}}}|dur@tt||j|g|Ri|St||kr^td|tjd|dkrn||}tjt ||durt |nd|durd|nddd\}}|} |durt |n|} |durd|n|} | | | fS)N loglaplacerrrr1)rrr*) r^r7r8r9rErrr[rrr) r;r<r=r-r`rrrprqr'r(rfr r.r/r9s$ "  zloglaplace_gen.fit)rirjrkrlr]rarbrdrgrhrrrBr rr9r?r.r.r r/rs rrcCst|dk||fddtj S)NrcSs:t|d d|dt||tdtjSr)rErrrr`rmr.r.r/rLsz!_lognorm_logpdf..rbrr.r.r/_lognorm_logpdfsrcseZdZdZejZddZdddZddZ d d Z d d Z d dZ ddZ ddZddZddZddZddZeeeddfddZZS) lognorm_genaA lognormal continuous random variable. %(before_notes)s Notes ----- The probability density function for `lognorm` is: .. math:: f(x, s) = \frac{1}{s x \sqrt{2\pi}} \exp\left(-\frac{\log^2(x)}{2s^2}\right) for :math:`x > 0`, :math:`s > 0`. `lognorm` takes ``s`` as a shape parameter for :math:`s`. %(after_notes)s Suppose a normally distributed random variable ``X`` has mean ``mu`` and standard deviation ``sigma``. Then ``Y = exp(X)`` is lognormally distributed with ``s = sigma`` and ``scale = exp(mu)``. %(example)s The logarithm of a log-normally distributed random variable is normally distributed: >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy import stats >>> fig, ax = plt.subplots(1, 1) >>> mu, sigma = 2, 0.5 >>> X = stats.norm(loc=mu, scale=sigma) >>> Y = stats.lognorm(s=sigma, scale=np.exp(mu)) >>> x = np.linspace(*X.interval(0.999)) >>> y = Y.rvs(size=10000) >>> ax.plot(x, X.pdf(x), label='X (pdf)') >>> ax.hist(np.log(y), density=True, bins=x, label='log(Y) (histogram)') >>> ax.legend() >>> plt.show() cCstdddtjfdgS)NrmFrrrZr\r.r.r/r]Lszlognorm_gen._shape_infoNcCst|||SrCrErr)r;rmrrr.r.r/rOszlognorm_gen._rvscCst|||SrCrGr;r`rmr.r.r/raRszlognorm_gen._pdfcCs t||SrCrrr.r.r/rVszlognorm_gen._logpdfcCstt||SrCrrErrr.r.r/rbYszlognorm_gen._cdfcCstt||SrCr rr.r.r/r\szlognorm_gen._logcdfcCst|t|SrCrErrr;rfrmr.r.r/rg_szlognorm_gen._ppfcCstt||SrCrrErrr.r.r/rdbszlognorm_gen._sfcCstt||SrC)rrErrr.r.r/reszlognorm_gen._logsfcCst|t|SrCrErrrr.r.r/rhhszlognorm_gen._isfcCsVt||}t|}||d}t|dd|}tgd|}||||fSNrrK)rrKrrr2)rErrpolyval)r;rmrTrrrrr.r.r/rks   zlognorm_gen._statscCs&ddtdtjdt|SNrurrKr)r;rmr.r.r/rsszlognorm_gen._entropyaF When `method='MLE'` and the location parameter is fixed by using the `floc` argument, this function uses explicit formulas for the maximum likelihood estimation of the log-normal shape and scale parameters, so the `optimizer`, `loc` and `scale` keyword arguments are ignored. If the location is free, a likelihood maximum is found by setting its partial derivative wrt to location to 0, and solving by substituting the analytical expressions of shape and scale (or provided parameters). See, e.g., equation 3.1 in A. Clifford Cohen & Betty Jones Whitten (1980) Estimation in the Three-Parameter Lognormal Distribution, Journal of the American Statistical Association, 75:370, 399-404 https://doi.org/10.2307/2287466 rcsb|ddr&tjg|Ri|St||}|\}t}fddfdd}fdd}|durt|} || } || } || } d | } | d kr|| } || } | d 9} qt| rt| stjg|Ri|Stt | tj | d }||}d | |} t|rvt|rvt |t | krv| | }||}| d 9} q,t|rt|stjg|Ri|St ||| fd }|j stjg|Ri|S||j}|| kr|jn|| }n||krtd dtj d|}|\}}|r>|dksXtjg|Ri|S|||fS)NrVFcsZdusdurt|}p.t|}pPtt|t|d}||fSr)rErrrr)r'Zlndatar(r)r<rfshaper.r/get_shape_scales "z(lognorm_gen.fit..get_shape_scalecs8|\}}|}tdt|||d|SrrErr)r'rr(Zshifted)r<rr.r/dL_dLocs z lognorm_gen.fit..dL_dLoccs |\}}|||f SrC)nnlf)r'rr()r<rr;r.r/lls zlognorm_gen.fit..llrKgưrr#lognormrmrr)r+r7r9r^rErspacingrr nextafterr[rFr% convergedr_rrW)r;r<r=r- parametersrrrrr rIZdL_dLoc_rbrackZ ll_rbrackrrHZdL_dLoc_lbrackrZll_rootr'rr(r )r<rrrr;r/r9vsV            zlognorm_gen.fit)NN)rirjrkrlrrrr]rrarrbrrgrdrrhrrrBrr9r?r.r.r r/rs"+  rrc@sheZdZdZejZddZdddZddZ d d Z d d Z d dZ ddZ ddZddZddZdS) gibrat_genaBA Gibrat continuous random variable. %(before_notes)s Notes ----- The probability density function for `gibrat` is: .. math:: f(x) = \frac{1}{x \sqrt{2\pi}} \exp(-\frac{1}{2} (\log(x))^2) `gibrat` is a special case of `lognorm` with ``s=1``. %(after_notes)s %(example)s cCsgSrCr.r\r.r.r/r]szgibrat_gen._shape_infoNcCst||SrCrrr.r.r/rszgibrat_gen._rvscCst||SrCrGrr.r.r/raszgibrat_gen._pdfcCs t|dSrrrr.r.r/rszgibrat_gen._logpdfcCstt|SrCrrr.r.r/rb szgibrat_gen._cdfcCstt|SrCrrr.r.r/rg szgibrat_gen._ppfcCstt|SrCrrr.r.r/rdszgibrat_gen._sfcCstt|SrCrrr.r.r/rhszgibrat_gen._isfcCsNtj}t|}||d}t|dd|}tgd|}||||fSr)rEerr)r;rTrrrrr.r.r/rs   zgibrat_gen._statscCsdtdtjdSrrr\r.r.r/rszgibrat_gen._entropy)NN)rirjrkrlrrrr]rrarrbrgrdrhrrr.r.r.r/r s r gibratc@sbeZdZdZddZdddZddZd d Zd d Zd dZ ddZ ddZ ddZ ddZ dS) maxwell_genaA Maxwell continuous random variable. %(before_notes)s Notes ----- A special case of a `chi` distribution, with ``df=3``, ``loc=0.0``, and given ``scale = a``, where ``a`` is the parameter used in the Mathworld description [1]_. The probability density function for `maxwell` is: .. math:: f(x) = \sqrt{2/\pi}x^2 \exp(-x^2/2) for :math:`x >= 0`. %(after_notes)s References ---------- .. [1] http://mathworld.wolfram.com/MaxwellDistribution.html %(example)s cCsgSrCr.r\r.r.r/r]@szmaxwell_gen._shape_infoNcCstjd||dS)NrrBrrDrr.r.r/rCszmaxwell_gen._rvscCs t||t| |dSr)r"rErrr.r.r/raFszmaxwell_gen._pdfcCsNtjdd.tdt|d||WdS1s@0YdS)NrrrKru)rErr#rrr.r.r/rJszmaxwell_gen._logpdfcCstd||dSNrrrrr.r.r/rbOszmaxwell_gen._cdfcCstdtd|Srrrr.r.r/rgRszmaxwell_gen._ppfcCstd||dSrrrr.r.r/rdUszmaxwell_gen._sfcCstdtd|Srrrr.r.r/rhXszmaxwell_gen._isfcCsrdtjd}dtdtjddtjtdddtj|ddtjtjd tjd |dfS) NrrrKr r&rrirErrr;rpr.r.r/r[s  $zmaxwell_gen._statscCstdtdtjdSr)rrErrr\r.r.r/rbszmaxwell_gen._entropy)NNrr.r.r.r/r%s rmaxwellc@s@eZdZdZddZddZddZdd Zd d Zd d Z dS) mielke_genaA Mielke Beta-Kappa / Dagum continuous random variable. %(before_notes)s Notes ----- The probability density function for `mielke` is: .. math:: f(x, k, s) = \frac{k x^{k-1}}{(1+x^s)^{1+k/s}} for :math:`x > 0` and :math:`k, s > 0`. The distribution is sometimes called Dagum distribution ([2]_). It was already defined in [3]_, called a Burr Type III distribution (`burr` with parameters ``c=s`` and ``d=k/s``). `mielke` takes ``k`` and ``s`` as shape parameters. %(after_notes)s References ---------- .. [1] Mielke, P.W., 1973 "Another Family of Distributions for Describing and Analyzing Precipitation Data." J. Appl. Meteor., 12, 275-280 .. [2] Dagum, C., 1977 "A new model for personal income distribution." Economie Appliquee, 33, 327-367. .. [3] Burr, I. W. "Cumulative frequency functions", Annals of Mathematical Statistics, 13(2), pp 215-232 (1942). %(example)s cCs0tdddtjfd}tdddtjfd}||gS)NroFrrrmrZ)r;iki_sr.r.r/r]szmielke_gen._shape_infocCs,|||dd||d|d|Srr.r;r`rormr.r.r/raszmielke_gen._pdfcCsftjddFt|t||dt||d||WdS1sX0YdS)Nrrr)rErrrrr.r.r/rszmielke_gen._logpdfcCs ||d|||d|Srr.rr.r.r/rbszmielke_gen._cdfcCs(t||d|}t|d|d|SrrT)r;rfrormZqskr.r.r/rgszmielke_gen._ppfcCs"dd}t||k|||f|tjS)NcSs2t|||td||t||SrRr)rVrormr.r.r/rsz$mielke_gen._munp..nth_momentrb)r;rVrormrr.r.r/rszmielke_gen._munpN) rirjrkrlr]rarrbrgrr.r.r.r/ris!rmielkec@sheZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ ddZdS) kappa4_genap Kappa 4 parameter distribution. %(before_notes)s Notes ----- The probability density function for kappa4 is: .. math:: f(x, h, k) = (1 - k x)^{1/k - 1} (1 - h (1 - k x)^{1/k})^{1/h-1} if :math:`h` and :math:`k` are not equal to 0. If :math:`h` or :math:`k` are zero then the pdf can be simplified: h = 0 and k != 0:: kappa4.pdf(x, h, k) = (1.0 - k*x)**(1.0/k - 1.0)* exp(-(1.0 - k*x)**(1.0/k)) h != 0 and k = 0:: kappa4.pdf(x, h, k) = exp(-x)*(1.0 - h*exp(-x))**(1.0/h - 1.0) h = 0 and k = 0:: kappa4.pdf(x, h, k) = exp(-x)*exp(-exp(-x)) kappa4 takes :math:`h` and :math:`k` as shape parameters. The kappa4 distribution returns other distributions when certain :math:`h` and :math:`k` values are used. +------+-------------+----------------+------------------+ | h | k=0.0 | k=1.0 | -inf<=k<=inf | +======+=============+================+==================+ | -1.0 | Logistic | | Generalized | | | | | Logistic(1) | | | | | | | | logistic(x) | | | +------+-------------+----------------+------------------+ | 0.0 | Gumbel | Reverse | Generalized | | | | Exponential(2) | Extreme Value | | | | | | | | gumbel_r(x) | | genextreme(x, k) | +------+-------------+----------------+------------------+ | 1.0 | Exponential | Uniform | Generalized | | | | | Pareto | | | | | | | | expon(x) | uniform(x) | genpareto(x, -k) | +------+-------------+----------------+------------------+ (1) There are at least five generalized logistic distributions. Four are described here: https://en.wikipedia.org/wiki/Generalized_logistic_distribution The "fifth" one is the one kappa4 should match which currently isn't implemented in scipy: https://en.wikipedia.org/wiki/Talk:Generalized_logistic_distribution https://www.mathwave.com/help/easyfit/html/analyses/distributions/gen_logistic.html (2) This distribution is currently not in scipy. References ---------- J.C. Finney, "Optimization of a Skewed Logistic Distribution With Respect to the Kolmogorov-Smirnov Test", A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College, (August, 2004), https://digitalcommons.lsu.edu/gradschool_dissertations/3672 J.R.M. Hosking, "The four-parameter kappa distribution". IBM J. Res. Develop. 38 (3), 25 1-258 (1994). B. Kumphon, A. Kaew-Man, P. Seenoi, "A Rainfall Distribution for the Lampao Site in the Chi River Basin, Thailand", Journal of Water Resource and Protection, vol. 4, 866-869, (2012). :doi:`10.4236/jwarp.2012.410101` C. Winchester, "On Estimation of the Four-Parameter Kappa Distribution", A Thesis Submitted to Dalhousie University, Halifax, Nova Scotia, (March 2000). http://www.nlc-bnc.ca/obj/s4/f2/dsk2/ftp01/MQ57336.pdf %(after_notes)s %(example)s cCs t||dj}tj|ddS)NrT fill_value)rErRrfull)r;rrorr.r.r/rWszkappa4_gen._argcheckcCs8tddtj tjfd}tddtj tjfd}||gS)NrFrrorZ)r;Zihrr.r.r/r]szkappa4_gen._shape_infoc Cst|dk|dkt|dk|dkt|dk|dkt|dk|dkt|dk|dkt|dk|dkg}dd}dd}dd}dd }t|||||||g||gtjd }d d}d d}t|||||||g||gtjd } || fS) NrcSsdt|| |Sr)rErrror.r.r/rsz#kappa4_gen._get_support..f0cSs t|SrCrr!r.r.r/rsz#kappa4_gen._get_support..f1cSs$tt|}tj |dd<|SrCrErorr[rrorpr.r.r/f3sz#kappa4_gen._get_support..f3cSsd|Srr.r!r.r.r/f5sz#kappa4_gen._get_support..f5defaultcSsd|Srr.r!r.r.r/r'scSs"tt|}tj|dd<|SrCr"r#r.r.r/r*srErr r) r;rrocondlistrrr$r%rrr.r.r/rv s0zkappa4_gen._get_supportcCst||||SrCrGr;r`rror.r.r/ra5szkappa4_gen._pdfc Cst|dk|dkt|dk|dkt|dk|dkt|dk|dkg}dd}dd}dd}dd }t|||||g|||gtjd S) NrcSsDtd|d| |td|d| d||d|S)zpdf = (1.0 - k*x)**(1.0/k - 1.0)*( 1.0 - h*(1.0 - k*x)**(1.0/k))**(1.0/h-1.0) logpdf = ... rnrr`rror.r.r/r@s(zkappa4_gen._logpdf..f0cSs.td|d| |d||d|S)z~pdf = (1.0 - k*x)**(1.0/k - 1.0)*np.exp(-( 1.0 - k*x)**(1.0/k)) logpdf = ... rnrr+r.r.r/rHszkappa4_gen._logpdf..f1cSs(| td|d| t| S)z]pdf = np.exp(-x)*(1.0 - h*np.exp(-x))**(1.0/h - 1.0) logpdf = ... rn)rcrrErr+r.r.r/rPOszkappa4_gen._logpdf..f2cSs| t| S)zDpdf = np.exp(-x-np.exp(-x)) logpdf = ... rr+r.r.r/r$Uszkappa4_gen._logpdf..f3r&r( r;r`rror)rrrPr$r.r.r/r:s zkappa4_gen._logpdfcCst||||SrCrr*r.r.r/rb`szkappa4_gen._cdfc Cst|dk|dkt|dk|dkt|dk|dkt|dk|dkg}dd}dd}dd}dd }t|||||g|||gtjd S) NrcSs(d|t| d||d|S)zVcdf = (1.0 - h*(1.0 - k*x)**(1.0/k))**(1.0/h) logcdf = ... rnrgr+r.r.r/riszkappa4_gen._logcdf..f0cSsd||d| S)zLcdf = np.exp(-(1.0 - k*x)**(1.0/k)) logcdf = ... rnr.r+r.r.r/roszkappa4_gen._logcdf..f1cSs d|t| t| S)zLcdf = (1.0 - h*np.exp(-x))**(1.0/h) logcdf = ... rn)rcrrErr+r.r.r/rPuszkappa4_gen._logcdf..f2cSst|  S)zBcdf = np.exp(-np.exp(-x)) logcdf = ... rr+r.r.r/r${szkappa4_gen._logcdf..f3r&r(r,r.r.r/rcs zkappa4_gen._logcdfc Cst|dk|dkt|dk|dkt|dk|dkt|dk|dkg}dd}dd}dd}dd }t|||||g|||gtjd S) NrcSs d|dd||||Srr.rfrror.r.r/rszkappa4_gen._ppf..f0cSsd|dt| |Srrr-r.r.r/rszkappa4_gen._ppf..f1cSst||  t|S)z,ppf = -np.log((1.0 - (q**h))/h) r!r-r.r.r/rPszkappa4_gen._ppf..f2cSstt|  SrCrr-r.r.r/r$szkappa4_gen._ppf..f3r&r() r;rfrror)rrrPr$r.r.r/rgs zkappa4_gen._ppfcCsDt|dk|dk|dkg}dd}dd}t|||g||gddS)NrcSsd||tSrZastyper9r!r.r.r/rsz&kappa4_gen._get_stats_info..f0cSsd|tSrr.r!r.r.r/rsz&kappa4_gen._get_stats_info..f1rr&)rErr )r;rror)rrr.r.r/_get_stats_infos zkappa4_gen._get_stats_infocs0|||fddtddD}|ddS)Ncs$g|]}t|krdntjqSrCrErr)rZrUmaxrr.r/r[rMz%kappa4_gen._stats..rr)r/r^)r;rrooutputsr.r1r/rs zkappa4_gen._statscGs@||d|d}||kr"tjStj|jdd|f|ddSNrrr)r/rErr r _mom_integ1)r;rCr=r2r.r.r/_mom1_scszkappa4_gen._mom1_scN)rirjrkrlrWr]rvrarrbrrgr/rr6r.r.r.r/rsX)&#rkappa4csXeZdZdZddZddZddZfdd Zd d Zd d Z ddZ ddZ Z S) kappa3_gena*Kappa 3 parameter distribution. %(before_notes)s Notes ----- The probability density function for `kappa3` is: .. math:: f(x, a) = a (a + x^a)^{-(a + 1)/a} for :math:`x > 0` and :math:`a > 0`. `kappa3` takes ``a`` as a shape parameter for :math:`a`. References ---------- P.W. Mielke and E.S. Johnson, "Three-Parameter Kappa Distribution Maximum Likelihood and Likelihood Ratio Tests", Methods in Weather Research, 701-707, (September, 1973), :doi:`10.1175/1520-0493(1973)101<0701:TKDMLE>2.3.CO;2` B. Kumphon, "Maximum Entropy and Maximum Likelihood Estimation for the Three-Parameter Kappa Distribution", Open Journal of Statistics, vol 2, 415-419 (2012), :doi:`10.4236/ojs.2012.24050` %(after_notes)s %(example)s cCstdddtjfdgSrrZr\r.r.r/r]szkappa3_gen._shape_infocCs||||d|dSrr.rr.r.r/raszkappa3_gen._pdfcCs||||d|Srr.rr.r.r/rbszkappa3_gen._cdfc st||\}}t||}d}||k}ttd||||||||  }||k}|||||<|||<|S)Ng{Gz?r)rErRr7rdrcrjr)r;r`rpsfcutoffrWZsf2i2r r.r/rds2zkappa3_gen._sfcCs||| dd|Srr.rr.r.r/rgszkappa3_gen._ppfcCs*t| | }t|}||d|Srr)r;rfrpZlgrr.r.r/rhs zkappa3_gen._isfcs$fddtddD}|ddS)Ncs$g|]}t|krdntjqSrCr0rYrr.r/r[rMz%kappa3_gen._stats..rr)r^)r;rpr3r.rr/rszkappa3_gen._statscGs6t||dkrtjStj|jdd|f|ddSr4)rErrr rr5)r;rCr=r.r.r/r6szkappa3_gen._mom1_sc) rirjrkrlr]rarbrdrgrhrr6r?r.r.r r/r8s  r8kappa3c@sReZdZdZddZdddZddZd d Zd d Zd dZ ddZ ddZ dS) moyal_genaA Moyal continuous random variable. %(before_notes)s Notes ----- The probability density function for `moyal` is: .. math:: f(x) = \exp(-(x + \exp(-x))/2) / \sqrt{2\pi} for a real number :math:`x`. %(after_notes)s This distribution has utility in high-energy physics and radiation detection. It describes the energy loss of a charged relativistic particle due to ionization of the medium [1]_. It also provides an approximation for the Landau distribution. For an in depth description see [2]_. For additional description, see [3]_. References ---------- .. [1] J.E. Moyal, "XXX. Theory of ionization fluctuations", The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, vol 46, 263-280, (1955). :doi:`10.1080/14786440308521076` (gated) .. [2] G. Cordeiro et al., "The beta Moyal: a useful skew distribution", International Journal of Research and Reviews in Applied Sciences, vol 10, 171-192, (2012). http://www.arpapress.com/Volumes/Vol10Issue2/IJRRAS_10_2_02.pdf .. [3] C. Walck, "Handbook on Statistical Distributions for Experimentalists; International Report SUF-PFY/96-01", Chapter 26, University of Stockholm: Stockholm, Sweden, (2007). http://www.stat.rice.edu/~dobelman/textfiles/DistributionsHandbook.pdf .. versionadded:: 1.1.0 %(example)s cCsgSrCr.r\r.r.r/r]7szmoyal_gen._shape_infoNcCstjdd||d}t| S)NrurK)rpr(rr)rCrDrEr)r;rrrEr.r.r/r:s zmoyal_gen._rvscCs*td|t| tdtjSNrNrK)rErrrrr.r.r/ra?szmoyal_gen._pdfcCsttd|tdSr>)rcrrErrrr.r.r/rbBszmoyal_gen._cdfcCsttd|tdSr>)rcrMrErrrr.r.r/rdEsz moyal_gen._sfcCstdt|d Sr)rErrcZerfcinvrr.r.r/rgHszmoyal_gen._ppfcCsPtdtj}tjdd}dtdtdtjd}d}||||fS)NrKrr)rEr euler_gammarrrcryrr.r.r/rKs "zmoyal_gen._statscCs$|dkrtdtjS|dkrBtjddtdtjdS|dkrdtjdtdtj}tdtjd}dtd}|||S|dkrd tdtdtj}dtjdtdtjd}tdtjd }d tjd d }||||S||SdS) NrnrKrrrrrqr8rr7)rErr@rrcryr6)r;rVZtmp1rZtmp3Ztmp4r.r.r/rRs "  "zmoyal_gen._munp)NN) rirjrkrlr]rrarbrdrgrrr.r.r.r/r= s* r=moyalc@steZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ dddZdddZdS) nakagami_gena`A Nakagami continuous random variable. %(before_notes)s Notes ----- The probability density function for `nakagami` is: .. math:: f(x, \nu) = \frac{2 \nu^\nu}{\Gamma(\nu)} x^{2\nu-1} \exp(-\nu x^2) for :math:`x >= 0`, :math:`\nu > 0`. The distribution was introduced in [2]_, see also [1]_ for further information. `nakagami` takes ``nu`` as a shape parameter for :math:`\nu`. %(after_notes)s References ---------- .. [1] "Nakagami distribution", Wikipedia https://en.wikipedia.org/wiki/Nakagami_distribution .. [2] M. Nakagami, "The m-distribution - A general formula of intensity distribution of rapid fading", Statistical methods in radio wave propagation, Pergamon Press, 1960, 3-36. :doi:`10.1016/B978-0-08-009306-2.50005-4` %(example)s cCs|dkSr6r.)r;nur.r.r/rWsznakagami_gen._argcheckcCstdddtjfdgS)NrDFrrrZr\r.r.r/r]sznakagami_gen._shape_infocCst|||SrCrGr;r`rDr.r.r/rasznakagami_gen._pdfcCs@tdt||t|td|d|||dSr)rErrcrr5rEr.r.r/rs  znakagami_gen._logpdfcCst||||SrCrrEr.r.r/rbsznakagami_gen._cdfcCstd|t||Srr)r;rfrDr.r.r/rgsznakagami_gen._ppfcCst||||SrCrrEr.r.r/rdsznakagami_gen._sfcCstd|t||SrRr)r;rTrDr.r.r/rhsznakagami_gen._isfcCst|dt|}d||}|dd||d|t|d}d|d|d|d |d d |d}|||d}||||fS) NrurnrrrrrrK)rcrrErr)r;rDrrrrr.r.r/rs  (0znakagami_gen._statsc Cst|}t|}t|}||dt|}dt|td}|||}tj }|dk}|||dd||||<| |dS)NrurNrKgj@rr/r.) rErr|rcr5rrrSrrr) r;rDrrrr1rZ norm_entropyrWr.r.r/rs      znakagami_gen._entropyNcCst|j||d|Sr4)rErr)r;rDrrr.r.r/rsznakagami_gen._rvscCsZt|tr|}|dur$d|j}t|}tt||dt|}|||fS)N)rnrK) r5r$r ZnumargsrErrrr)r;r<r=r'r(r.r.r/r s    znakagami_gen._fitstart)NN)N)rirjrkrlrWr]rarrbrgrdrhrrrr r.r.r.r/rCks rCnakagamicCsz|dd}t|t|}}t|d||d||d}t|||d}t|dk||fddtj dS) NrrnrurKrcSs|t|SrCr)rUrfr.r.r/rLrMz_ncx2_log_pdf..)rra)rErrcriverr[)r`rrdf2rnsrZcorrr.r.r/ _ncx2_log_pdfs $rKc@sbeZdZdZddZddZdddZd d Zd d Zd dZ ddZ ddZ ddZ ddZ dS)ncx2_genaA non-central chi-squared continuous random variable. %(before_notes)s Notes ----- The probability density function for `ncx2` is: .. math:: f(x, k, \lambda) = \frac{1}{2} \exp(-(\lambda+x)/2) (x/\lambda)^{(k-2)/4} I_{(k-2)/2}(\sqrt{\lambda x}) for :math:`x >= 0`, :math:`k > 0` and :math:`\lambda \ge 0`. :math:`k` specifies the degrees of freedom (denoted ``df`` in the implementation) and :math:`\lambda` is the non-centrality parameter (denoted ``nc`` in the implementation). :math:`I_\nu` denotes the modified Bessel function of first order of degree :math:`\nu` (`scipy.special.iv`). `ncx2` takes ``df`` and ``nc`` as shape parameters. %(after_notes)s %(example)s cCs|dkt|@|dk@Sr6r~r;rrr.r.r/rWszncx2_gen._argcheckcCs0tdddtjfd}tdddtjfd}||gS)NrFrrrrYrZr;idfincr.r.r/r]szncx2_gen._shape_infoNcCs||||SrC)Znoncentral_chisquare)r;rrrrr.r.r/rsz ncx2_gen._rvscCs0tj|td|dk@}t||||ftdddS)NrrcSs t||SrC)rrr`r_r.r.r/rL rMz"ncx2_gen._logpdf..r)rE ones_likeboolrrKr;r`rrcondr.r.r/r szncx2_gen._logpdfcCsbtj|td|dk@}tjdd,t||||ftjdddWdS1sT0YdS)NrrrrcSs t||SrC)rrarQr.r.r/rLrMzncx2_gen._pdf..r)rErSrTrrrZ _ncx2_pdfrUr.r.r/ras z ncx2_gen._pdfcCsbtj|td|dk@}tjdd,t||||ftjdddWdS1sT0YdS)NrrrrcSs t||SrC)rrbrQr.r.r/rLrMzncx2_gen._cdf..r)rErSrTrrrZ _ncx2_cdfrUr.r.r/rbs z ncx2_gen._cdfcCsbtj|td|dk@}tjdd,t||||ftjdddWdS1sT0YdS)NrrrrcSs t||SrC)rrgrQr.r.r/rLrMzncx2_gen._ppf..r)rErSrTrrrZ _ncx2_ppf)r;rfrrrVr.r.r/rgs z ncx2_gen._ppfcCsbtj|td|dk@}tjdd,t||||ftjdddWdS1sT0YdS)NrrrrcSs t||SrC)rrdrQr.r.r/rL%rMzncx2_gen._sf..r)rErSrTrrrZ_ncx2_sfrUr.r.r/rd!s z ncx2_gen._sfcCsbtj|td|dk@}tjdd,t||||ftjdddWdS1sT0YdS)NrrrrcSs t||SrC)rrhrQr.r.r/rL+rMzncx2_gen._isf..r)rErSrTrrrZ _ncx2_isfrUr.r.r/rh's z ncx2_gen._isfcCs,t||t||t||t||fSrC)rZ _ncx2_meanZ_ncx2_varianceZ_ncx2_skewnessZ_ncx2_kurtosis_excessrMr.r.r/r-s     zncx2_gen._stats)NN)rirjrkrlrWr]rrrarbrgrdrhrr.r.r.r/rLs rLncx2c@sdeZdZdZddZddZdddZd d Zd d Zd dZ ddZ ddZ ddZ dddZ dS)ncf_genaA non-central F distribution continuous random variable. %(before_notes)s See Also -------- scipy.stats.f : Fisher distribution Notes ----- The probability density function for `ncf` is: .. math:: f(x, n_1, n_2, \lambda) = \exp\left(\frac{\lambda}{2} + \lambda n_1 \frac{x}{2(n_1 x + n_2)} \right) n_1^{n_1/2} n_2^{n_2/2} x^{n_1/2 - 1} \\ (n_2 + n_1 x)^{-(n_1 + n_2)/2} \gamma(n_1/2) \gamma(1 + n_2/2) \\ \frac{L^{\frac{n_1}{2}-1}_{n_2/2} \left(-\lambda n_1 \frac{x}{2(n_1 x + n_2)}\right)} {B(n_1/2, n_2/2) \gamma\left(\frac{n_1 + n_2}{2}\right)} for :math:`n_1, n_2 > 0`, :math:`\lambda \ge 0`. Here :math:`n_1` is the degrees of freedom in the numerator, :math:`n_2` the degrees of freedom in the denominator, :math:`\lambda` the non-centrality parameter, :math:`\gamma` is the logarithm of the Gamma function, :math:`L_n^k` is a generalized Laguerre polynomial and :math:`B` is the beta function. `ncf` takes ``df1``, ``df2`` and ``nc`` as shape parameters. If ``nc=0``, the distribution becomes equivalent to the Fisher distribution. %(after_notes)s %(example)s cCs|dk|dk@|dk@Sr6r.)r;df1rIrr.r.r/rWbszncf_gen._argcheckcCsFtdddtjfd}tdddtjfd}tdddtjfd}|||gS)NrYFrrrIrrYrZ)r;Zidf1Zidf2rPr.r.r/r]eszncf_gen._shape_infoNcCs|||||SrC)Z noncentral_f)r;r?r@rrrr.r.r/rksz ncf_gen._rvscCst||||SrC)rZ_ncf_pdfr;r`r?r@rr.r.r/ransz ncf_gen._pdfcCst||||SrC)rZ_ncf_cdfrZr.r.r/rbwsz ncf_gen._cdfcCs@tjdd t||||WdS1s20YdSr)rErrZ_ncf_ppf)r;rfr?r@rr.r.r/rgzsz ncf_gen._ppfcCst||||SrC)rZ_ncf_sfrZr.r.r/rd~sz ncf_gen._sfcCs@tjdd t||||WdS1s20YdSr)rErrZ_ncf_isfrZr.r.r/rhsz ncf_gen._isfcCs|d||}t|d|td||t|d}|t| d|9}|t|d|d|d|9}|S)Nrnrur)rcr5rErhyp1f1)r;rVr?r@rrprr.r.r/rs 2"z ncf_gen._munprlc Cs\t|||}t|||}d|vr2t|||nd}d|vrLt|||nd}||||fSNrmro)rZ _ncf_meanZ _ncf_varianceZ _ncf_skewnessZ_ncf_kurtosis_excess) r;r?r@rrrrrrrr.r.r/rszncf_gen._stats)NN)rl)rirjrkrlrWr]rrarbrgrdrhrrr.r.r.r/rX9s(  rXncfc@sbeZdZdZddZdddZddZd d Zd d Zd dZ ddZ ddZ ddZ ddZ dS)t_genaA Student's t continuous random variable. For the noncentral t distribution, see `nct`. %(before_notes)s See Also -------- nct Notes ----- The probability density function for `t` is: .. math:: f(x, \nu) = \frac{\Gamma((\nu+1)/2)} {\sqrt{\pi \nu} \Gamma(\nu/2)} (1+x^2/\nu)^{-(\nu+1)/2} where :math:`x` is a real number and the degrees of freedom parameter :math:`\nu` (denoted ``df`` in the implementation) satisfies :math:`\nu > 0`. :math:`\Gamma` is the gamma function (`scipy.special.gamma`). %(after_notes)s %(example)s cCstdddtjfdgSrrZr\r.r.r/r]szt_gen._shape_infoNcCs|j||dSr4)Z standard_trr.r.r/rsz t_gen._rvscs&t|tjk||fddfdddS)NcSs t|SrC)rrar`rr.r.r/rLrMzt_gen._pdf..cst||SrCrGr_r\r.r/rLsrrbrr.r\r/ras  z t_gen._pdfcCs*dd}dd}t|tjk||f||dS)NcSsNttd|ddt|ttj|ddt|||Sr)rErrcrrrr_r.r.r/t_logpdfs zt_gen._logpdf..t_logpdfcSs t|SrC)rrr_r.r.r/ norm_logpdfsz"t_gen._logpdf..norm_logpdfrrb)r;r`rr`rar.r.r/rsz t_gen._logpdfcCs t||SrCrcZstdtrrr.r.r/rbsz t_gen._cdfcCst|| SrCrbrr.r.r/rdsz t_gen._sfcCs t||SrCrcZstdtritrr.r.r/rgsz t_gen._ppfcCst|| SrCrcrr.r.r/rhsz t_gen._isfc Cst|}t|dkdtj}|dk|dk@|dkt|@|f}ddddddf}t|||ftj}t|dkdtj}|dk|d k@|d kt|@|f}d dd dd df}t|||ftj}||||fS) NrrmrKcSsttj|jSrCrE broadcast_tor[rrr.r.r/rLrMzt_gen._stats..cSs ||dSrr.rr.r.r/rLrMcSstd|jSrRrErerrr.r.r/rLrMrrcSsttj|jSrCrdrr.r.r/rLrMcSs d|dS)Nrrr.rr.r.r/rLrMcSstd|jSr6rfrr.r.r/rLrM)rEZisposinfrr[rr r) r;rZ infinite_dfrr) choicelistrrrr.r.r/rs, z t_gen._statscCs<|tjkrtSdd}dd}t|dk|f||d}|S)NcSsH|d}|dd}|t|t|tt|t|dSr)rcrrErrr)rZhalfZhalf1r.r.r/r s  zt_gen._entropy..regularcSsPtd||dd|dd|ddd|d |d d}|S) Nrr#rr$rr%rg333333?r0r2)rr)rrr.r.r/rs&   z"t_gen._entropy..asymptoticdr)rEr[rrr)r;rr rrr.r.r/rs  zt_gen._entropy)NNrr.r.r.r/r^s   r^r)c@s\eZdZdZddZddZdddZd d Zd d Zd dZ ddZ ddZ dddZ dS)nct_genaA non-central Student's t continuous random variable. %(before_notes)s Notes ----- If :math:`Y` is a standard normal random variable and :math:`V` is an independent chi-square random variable (`chi2`) with :math:`k` degrees of freedom, then .. math:: X = \frac{Y + c}{\sqrt{V/k}} has a non-central Student's t distribution on the real line. The degrees of freedom parameter :math:`k` (denoted ``df`` in the implementation) satisfies :math:`k > 0` and the noncentrality parameter :math:`c` (denoted ``nc`` in the implementation) is a real number. %(after_notes)s %(example)s cCs|dk||k@Sr6r.rMr.r.r/rW*sznct_gen._argcheckcCs4tdddtjfd}tddtj tjfd}||gS)NrFrrrrZrNr.r.r/r]-sznct_gen._shape_infoNcCs8tj|||d}tj|||d}|t|t|S)Nr8rB)rrDrrEr)r;rrrrrVr2r.r.r/r2sz nct_gen._rvsc Cs2|d}|d}||}|||}||}|dt|t|d|td||d|dt|t|d}t|} |d|} td||t|ddd| t|t|dd}t|ddd| tt|t|dd} | || 9} t | ddS)NrnrrrKrrur) rErrcr5rrr[rrCclip) r;r`rrrVr(rWrtrm1rZvalFtrm2r.r.r/ra7s( *   &  z nct_gen._pdfcCsHtjdd(tt|||ddWdS1s:0YdSNrrrr)rErrjrZ_nct_cdfr;r`rrr.r.r/rbJsz nct_gen._cdfcCs>tjddt|||WdS1s00YdSr)rErrZ_nct_ppf)r;rfrrr.r.r/rgNsz nct_gen._ppfcCsHtjdd(tt|||ddWdS1s:0YdSrm)rErrjrZ_nct_sfrnr.r.r/rdRsz nct_gen._sfcCs>tjddt|||WdS1s00YdSr)rErrZ_nct_isfrnr.r.r/rhVsz nct_gen._isfrlcCsTt||}t||}d|vr,t||nd}d|vrDt||nd}||||fSr\)rZ _nct_meanZ _nct_varianceZ _nct_skewnessZ_nct_kurtosis_excess)r;rrrrrrrrr.r.r/rZs   znct_gen._stats)NN)rl) rirjrkrlrWr]rrarbrgrdrhrr.r.r.r/ris rinctcsneZdZdZddZddZddZdd Zd d Zd d Z dddZ ddZ e e efddZZS) pareto_genaLA Pareto continuous random variable. %(before_notes)s Notes ----- The probability density function for `pareto` is: .. math:: f(x, b) = \frac{b}{x^{b+1}} for :math:`x \ge 1`, :math:`b > 0`. `pareto` takes ``b`` as a shape parameter for :math:`b`. %(after_notes)s %(example)s cCstdddtjfdgSrrZr\r.r.r/r]{szpareto_gen._shape_infocCs||| dSrRr.rr.r.r/ra~szpareto_gen._pdfcCsd|| SrRr.rr.r.r/rbszpareto_gen._cdfcCstd|d|S)NrrrTr$r.r.r/rgszpareto_gen._ppfcCs || SrCr.rr.r.r/rdszpareto_gen._sfcCst|d|Srr1r$r.r.r/rhszpareto_gen._isfrlc Csd\}}}}d|vrT|dk}t||}tjt|tjd}t||||dd|vr|dk}t||}tjt|tjd}t||||d|ddd |vr|d k}t||}tjt|tjd}d|dt|d|d t|} t||| d |vr||d k}t||}tjt|tjd}dtgd|tgd|} t||| ||||fS)NrrCrrrnr:rKrrmrrrorr)rnrnrFr)rngrrm) rEextractr rr[Zplacerrr) r;rqrrrrrrmaskZbtrr.r.r/rs4   "  ,  zpareto_gen._statscCsdd|t|Srrrr.r.r/rszpareto_gen._entropycst|||}|\}}|durHt||p4dkrHtddtjdjdfdd||urvdurnnfddfd d fd d fd d}t|dd}|d|d} } || | s| dks| tjkr| d} | d9} qt| | gd} | j r|| j } t| } pB| | }| | tksrt| } t | d} || | fSt j fi|Sn|durt|} n|} |pt| } p҈| | }|| | fS)Nrparetorrcstt||SrCr)r(location)r<ndatar.r/ get_shapesz!pareto_gen.fit..get_shapecs ||SrCr.)rr()rur.r/ dL_dScalesz!pareto_gen.fit..dL_dScalecs|dtd|SrRr)rrtrr.r/ dL_dLocationsz$pareto_gen.fit..dL_dLocationcs0t|}p||}||||SrC)rEr)r(rtr)rxrwr<rrvr.r/r"sz$pareto_gen.fit..fun_to_solvecst|t|kSrCrDrGr"r.r/rJs  z.pareto_gen.fit..interval_contains_rootr(rKr#)r^rErrr[rrr3r%r r_r r7r9)r;r<r=r-r rrrJrrHrIrr(r'rr )rxrwr<rr"rvrur/r9sH          zpareto_gen.fit)rl)rirjrkrlr]rarbrgrdrhrrrBr rr9r?r.r.r r/rpes rprsc@s`eZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ dS) lomax_genaA Lomax (Pareto of the second kind) continuous random variable. %(before_notes)s Notes ----- The probability density function for `lomax` is: .. math:: f(x, c) = \frac{c}{(1+x)^{c+1}} for :math:`x \ge 0`, :math:`c > 0`. `lomax` takes ``c`` as a shape parameter for :math:`c`. `lomax` is a special case of `pareto` with ``loc=-1.0``. %(after_notes)s %(example)s cCstdddtjfdgSrerZr\r.r.r/r]szlomax_gen._shape_infocCs|dd||dSrr.rhr.r.r/ra!szlomax_gen._pdfcCst||dt|SrRrrhr.r.r/r%szlomax_gen._logpdfcCst| t| SrCrirhr.r.r/rb(szlomax_gen._cdfcCst| t|SrC)rErrcrrhr.r.r/rd+sz lomax_gen._sfcCs| t|SrCrgrhr.r.r/r.szlomax_gen._logsfcCstt|  |SrCrirkr.r.r/rg1szlomax_gen._ppfcCs|d|dSrr.rkr.r.r/rh4szlomax_gen._isfcCs$tj|ddd\}}}}||||fS)NrrF)r'rr)rsrSrzr.r.r/r7szlomax_gen._statscCsdd|t|Srrrr.r.r/r;szlomax_gen._entropyN)rirjrkrlr]rarrbrdrrgrhrrr.r.r.r/rzsrzlomaxcseZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ dddZ ddZ eeeddfddZZS) pearson3_genaA pearson type III continuous random variable. %(before_notes)s Notes ----- The probability density function for `pearson3` is: .. math:: f(x, \kappa) = \frac{|\beta|}{\Gamma(\alpha)} (\beta (x - \zeta))^{\alpha - 1} \exp(-\beta (x - \zeta)) where: .. math:: \beta = \frac{2}{\kappa} \alpha = \beta^2 = \frac{4}{\kappa^2} \zeta = -\frac{\alpha}{\beta} = -\beta :math:`\Gamma` is the gamma function (`scipy.special.gamma`). Pass the skew :math:`\kappa` into `pearson3` as the shape parameter ``skew``. %(after_notes)s %(example)s References ---------- R.W. Vogel and D.E. McMartin, "Probability Plot Goodness-of-Fit and Skewness Estimation Procedures for the Pearson Type 3 Distribution", Water Resources Research, Vol.27, 3149-3158 (1991). L.R. Salvosa, "Tables of Pearson's Type III Function", Ann. Math. Statist., Vol.1, 191-198 (1930). "Using Modern Computing Tools to Fit the Pearson Type III Distribution to Aviation Loads Data", Office of Aviation Research (2003). c Csd}d}d}td||\}}}|}t||k}|}d|||} || d} || | } | ||| } ||| ||| | | fS)Nrmrng>rrK)rErRcopyr) r;r`rXr'r(Znorm2pearson_transitionansrrinvmaskrrrytransxr.r.r/ _preprocessps  zpearson3_gen._preprocesscCs t|SrCr~)r;rXr.r.r/rWszpearson3_gen._argcheckcCstddtj tjfdgS)NrXFrrZr\r.r.r/r]szpearson3_gen._shape_infocCs$d}d}|}d|d}||||fS)NrmrnrrKr.)r;rXrCr:rmror.r.r/rs  zpearson3_gen._statscCs@t|||}|jdkr.t|r*dS|Sd|t|<|S)Nrrm)rErrr}r)r;r`rXr~r.r.r/ras  zpearson3_gen._pdfc CsT|||\}}}}}}}} tt||||<tt|t||||<|SrC)rrErrrrCr) r;r`rXr~rrrrrrrRr.r.r/rs  zpearson3_gen._logpdfc Cs|||\}}}}}}}}t||||<t||j}t||dk} ||dk} t|| || || <t||dk} ||dk} t|| || || <|Sr6) rrrErerrrCrzr9 r;r`rXr~rrrrrRrZ invmask1aZ invmask1bZ invmask2aZ invmask2br.r.r/rbs   zpearson3_gen._cdfc Cs|||\}}}}}}}}t||||<t||j}t||dk} ||dk} t|| || || <t||dk} ||dk} t|| || || <|Sr6) rrrErerrrCr9rzrr.r.r/rds   zpearson3_gen._sfNc Csvt||}|dg|\}}}}}}} } |} |j| } || ||<|| | || ||<|dkrr|d}|S)Nrr.)rErerrrrr) r;rXrrr~rRrrrrrryZnsmallZnbigr.r.r/rs   zpearson3_gen._rvsc Csh|||\}}}}}}}} t||||<||}d||dk||dk<t|||| ||<|Srt)rrrcr) r;rfrXr~rRrrrrrryr.r.r/rgs zpearson3_gen._ppfze Note that method of moments (`method='MM'`) is not available for this distribution. rcs@|dddkrtdn"tt||j|g|Ri|SdS)Nr*ZMMzhFit `method='MM'` is not available for the Pearson3 distribution. Please try the default `method='MLE'`.)r3NotImplementedErrorr7r8r9rr r.r/r9s zpearson3_gen.fit)NN)rirjrkrlrrWr]rrarrbrdrrgrBrrr9r?r.r.r r/r|Bs-    r|pearson3cseZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ fddZeeeddfddZZS) powerlaw_genaA power-function continuous random variable. %(before_notes)s See Also -------- pareto Notes ----- The probability density function for `powerlaw` is: .. math:: f(x, a) = a x^{a-1} for :math:`0 \le x \le 1`, :math:`a > 0`. `powerlaw` takes ``a`` as a shape parameter for :math:`a`. %(after_notes)s For example, the support of `powerlaw` can be adjusted from the default interval ``[0, 1]`` to the interval ``[c, c+d]`` by setting ``loc=c`` and ``scale=d``. For a power-law distribution with infinite support, see `pareto`. `powerlaw` is a special case of `beta` with ``b=1``. %(example)s cCstdddtjfdgSrrZr\r.r.r/r]* szpowerlaw_gen._shape_infocCs|||dSrr.rr.r.r/ra- szpowerlaw_gen._pdfcCst|t|d|SrR)rErrcrrr.r.r/r1 szpowerlaw_gen._logpdfcCs ||dSrr.rr.r.r/rb4 szpowerlaw_gen._cdfcCs|t|SrCrrr.r.r/r7 szpowerlaw_gen._logcdfcCst|d|SrrTrr.r.r/rg: szpowerlaw_gen._ppfcCst|| SrC)rcru)r;rTrpr.r.r/rd= szpowerlaw_gen._sfcCs |||SrCr.)r;rVrpr.r.r/r@ szpowerlaw_gen._munpcCsn||d||d|ddd|d|dt|d|dtgd|||d|dfS) NrnrrKr#rr)rr.rFrKr)rErrrr.r.r/rD s  $&zpowerlaw_gen._statscCsdd|t|Srrrr.r.r/rJ szpowerlaw_gen._entropycst|||dk|dkB@SNrr)r7rrr r.r/rM s zpowerlaw_gen._support_maska: Notes specifically for ``powerlaw.fit``: If the location is a free parameter and the value returned for the shape parameter is less than one, the true maximum likelihood approaches infinity. This causes numerical difficulties, and the resulting estimates are approximate. rcs |ddr&tjg|Ri|SttdkrRtjg|Ri|St|||\}}|fg}||id}|dur̈ |kst ddd|dur̈ ||kst ddd|dur|dkrt d|t krd}t |dd d d |dur4|dur4||||fS|durt tj } pb| |} || | |f} t |tj} p| |} || | |f}| |kr| | |fS| | |fS|dur|}p||}|||fSfd d }ddddfddfddfdd}durvdkrv|Sdurdkr|S|}||} |}||}| |kr|ddkr|S| |kr|ddkr|Stjg|Ri|SdS)NrVFrpowerlawrzKNegative or zero `fscale` is outside the range allowed by the distribution.z0`fscale` must be greater than the range of data.cSs0t|}| tt|||t|SrC)rrErr)r<r'r(rr.r.r/rv sz#powerlaw_gen.fit..get_shapecSs ||SrC)rrr.r.r/ get_scale sz#powerlaw_gen.fit..get_scalecsrttj }t|t|jjkrDt|t|jj}t|tj}pf||}|||fSrC) rEr rr[rr%rr&rFr'r(r)r<rrrvr.r/fit_loc_scale_w_shape_lt_1 s z4powerlaw_gen.fit..fit_loc_scale_w_shape_lt_1cSs|jd ||Sr6)r)r<rr(r.r.r/rw sz#powerlaw_gen.fit..dL_dScalecSs|dtd||SrRr)r<rr'r.r.r/rx sz&powerlaw_gen.fit..dL_dLocationcs2t|tj }p$||}||SrCrEr r[r)rxr<rrrvr.r/dL_dLocation_star sz+powerlaw_gen.fit..dL_dLocation_starcs>t|tj }p$||}||||SrCrr)rxrwr<rrrvr.r/r" s   z&powerlaw_gen.fit..fun_to_solvec sttj }|}|dkrB|}|d9}q fdd}|d}d}|||s|tj kr|}|d9}qZtj||fd}t|jtj }t|tj}p̈||}|||fS)NrrKcst|t|kSrCrDrGryr.r/rJ s  zTpowerlaw_gen.fit..fit_loc_scale_w_shape_gt_1..interval_contains_rootrrnr#)rEr rr[r r%r_) rIrrJrHrWr_r'r(r)rr<rr"rrvr.r/fit_loc_scale_w_shape_gt_1 s$         z4powerlaw_gen.fit..fit_loc_scale_w_shape_gt_1)r+r7r9rrEuniquer^r Z _reduce_funcrrrrptpr r[r)r;r<r=r-rrZpenalized_nllf_argsZpenalized_nllfrOZloc_lt1Z shape_lt1Zll_lt1Zloc_gt1Z shape_gt1Zll_gt1r(rrrZ fit_shape_lt1Z fit_shape_gt1r )rxrrwr<rr"rrvr/r9Q sr(            $  zpowerlaw_gen.fit)rirjrkrlr]rarrbrrgrdrrrrrBrrr9r?r.r.r r/r s   rrc@sVeZdZdZejZddZddZddZ dd Z d d Z d d Z ddZ ddZdS)powerlognorm_genaA power log-normal continuous random variable. %(before_notes)s Notes ----- The probability density function for `powerlognorm` is: .. math:: f(x, c, s) = \frac{c}{x s} \phi(\log(x)/s) (\Phi(-\log(x)/s))^{c-1} where :math:`\phi` is the normal pdf, and :math:`\Phi` is the normal cdf, and :math:`x > 0`, :math:`s, c > 0`. `powerlognorm` takes :math:`c` and :math:`s` as shape parameters. %(after_notes)s %(example)s cCs0tdddtjfd}tdddtjfd}||gS)NrfFrrrmrZ)r;rxrr.r.r/r]@!szpowerlognorm_gen._shape_infocCst||||SrCrGr;r`rfrmr.r.r/raE!szpowerlognorm_gen._pdfcCsLt|t|t|tt||tt| ||dSrrErrrrr.r.r/rH!s zpowerlognorm_gen._logpdfcCst|||| SrCrrr.r.r/rbM!szpowerlognorm_gen._cdfcCs|d|||SrR)rhr;rfrfrmr.r.r/rgP!szpowerlognorm_gen._ppfcCst||||SrCrrr.r.r/rdS!szpowerlognorm_gen._sfcCstt| ||SrCr rr.r.r/rV!szpowerlognorm_gen._logsfcCstt|d| |SrRrrr.r.r/rhY!szpowerlognorm_gen._isfN)rirjrkrlrrrr]rarrbrgrdrrhr.r.r.r/r&!sr powerlognormc@sPeZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ dS) powernorm_genahA power normal continuous random variable. %(before_notes)s Notes ----- The probability density function for `powernorm` is: .. math:: f(x, c) = c \phi(x) (\Phi(-x))^{c-1} where :math:`\phi` is the normal pdf, :math:`\Phi` is the normal cdf, :math:`x` is any real, and :math:`c > 0` [1]_. `powernorm` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s References ---------- .. [1] NIST Engineering Statistics Handbook, Section 1.3.6.6.13, https://www.itl.nist.gov/div898/handbook//eda/section3/eda366d.htm %(example)s cCstdddtjfdgSrerZr\r.r.r/r]|!szpowernorm_gen._shape_infocCs|t|t| |dSrrrrhr.r.r/ra!szpowernorm_gen._pdfcCs$t|t||dt| SrRrrhr.r.r/r!szpowernorm_gen._logpdfcCst||| SrCrrhr.r.r/rb!szpowernorm_gen._cdfcCsttd|d| Sr)rr#rkr.r.r/rg!szpowernorm_gen._ppfcCst|||SrCrrhr.r.r/rd!szpowernorm_gen._sfcCs|t| SrCrrhr.r.r/r!szpowernorm_gen._logsfcCsttt|| SrC)rrErrrkr.r.r/rh!szpowernorm_gen._isfN) rirjrkrlr]rarrbrgrdrrhr.r.r.r/r`!sr powernormc@sReZdZdZddZddZddZdd Zd d Zd d Z dddZ ddZ dS) rdist_gena/An R-distributed (symmetric beta) continuous random variable. %(before_notes)s Notes ----- The probability density function for `rdist` is: .. math:: f(x, c) = \frac{(1-x^2)^{c/2-1}}{B(1/2, c/2)} for :math:`-1 \le x \le 1`, :math:`c > 0`. `rdist` is also called the symmetric beta distribution: if B has a `beta` distribution with parameters (c/2, c/2), then X = 2*B - 1 follows a R-distribution with parameter c. `rdist` takes ``c`` as a shape parameter for :math:`c`. This distribution includes the following distribution kernels as special cases:: c = 2: uniform c = 3: `semicircular` c = 4: Epanechnikov (parabolic) c = 6: quartic (biweight) c = 8: triweight %(after_notes)s %(example)s cCstdddtjfdgSrerZr\r.r.r/r]!szrdist_gen._shape_infocCst|||SrCrGrhr.r.r/ra!szrdist_gen._pdfcCs*td t|dd|d|dSr)rErrrrhr.r.r/r!szrdist_gen._logpdfcCst|dd|d|dSrrNrhr.r.r/rb!szrdist_gen._cdfcCst|dd|d|dSrrHrhr.r.r/rd!sz rdist_gen._sfcCsdt||d|ddSr)rrgrkr.r.r/rg!szrdist_gen._ppfNcCsd||d|d|dSrrr9r.r.r/r!szrdist_gen._rvscCs8d|dt|dd|d}|td|dS)NrrKrnrrur)r;rVrf numeratorr.r.r/r!s$zrdist_gen._munp)NN) rirjrkrlr]rarrbrdrgrrr.r.r.r/r!s! rrrdistcseZdZdZejZddZdddZddZ d d Z d d Z d dZ ddZ ddZddZddZddZeeeddfddZZS) rayleigh_gena7A Rayleigh continuous random variable. %(before_notes)s Notes ----- The probability density function for `rayleigh` is: .. math:: f(x) = x \exp(-x^2/2) for :math:`x \ge 0`. `rayleigh` is a special case of `chi` with ``df=2``. %(after_notes)s %(example)s cCsgSrCr.r\r.r.r/r]!szrayleigh_gen._shape_infoNcCstjd||dS)NrKrBrrr.r.r/r!szrayleigh_gen._rvscCst||SrCrGr;rUr.r.r/ra!szrayleigh_gen._pdfcCst|d||Srrrr.r.r/r!szrayleigh_gen._logpdfcCstd|d Sr>rrr.r.r/rb!szrayleigh_gen._cdfcCstdt| SNr)rErrcrrr.r.r/rg"szrayleigh_gen._ppfcCst||SrCrrr.r.r/rd"szrayleigh_gen._sfcCs d||S)NrNr.rr.r.r/r"szrayleigh_gen._logsfcCstdt|Sr)rErrrr.r.r/rh "szrayleigh_gen._isfcCsZdtj}ttjd|ddtjdttj|ddtj|d|dfS)NrrKrrrrrrr.r.r/r "s   zrayleigh_gen._statscCstdddtdS)NrrrurKrUr\r.r.r/r"szrayleigh_gen._entropya Notes specifically for ``rayleigh.fit``: If the location is fixed with the `floc` parameter, this method uses an analytical formula to find the scale. Otherwise, this function uses a numerical root finder on the first order conditions of the log-likelihood function to find the MLE. Only the (optional) `loc` parameter is used as the initial guess for the root finder; the `scale` parameter and any other parameters for the optimizer are ignored. rcs*|ddr&tjg|Ri|St|||\}}fdd}fdd}|ffdd }|durt|d krtd d tjd n |||fS|d } | dur| d } |dur|n|} t t tj } t | | } t j| | | fd} | jst| j| j}|p ||}||fS)NrVFcs"t|ddtdSr)rErrr'rr.r/ scale_mle&"sz#rayleigh_gen.fit..scale_mlecs@|}|}|d}d|}||dt|Sr)rr)r'rrrZs3rr.r/loc_mle+"s   z!rayleigh_gen.fit..loc_mlecs$|}||dd|Sr)r)r'r(rrr.r/loc_mle_scale_fixed4"sz-rayleigh_gen.fit..loc_mle_scale_fixedrrayleighrrr'r#)r+r7r9r^rErrr[r3r r rrPr r%r rMflagr_)r;r<r=r-rrrrrZloc0r@rIrHrr'r(r rr/r9"s0        zrayleigh_gen.fit)NN)rirjrkrlrrrr]rrarrbrgrdrrhrrrBrr9r?r.r.r r/r!s   rrcseZdZdZddZddZfddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ dZeeedfddZZS)reciprocal_gena,A loguniform or reciprocal continuous random variable. %(before_notes)s Notes ----- The probability density function for this class is: .. math:: f(x, a, b) = \frac{1}{x \log(b/a)} for :math:`a \le x \le b`, :math:`b > a > 0`. This class takes :math:`a` and :math:`b` as shape parameters. %(after_notes)s %(example)s This doesn't show the equal probability of ``0.01``, ``0.1`` and ``1``. This is best when the x-axis is log-scaled: >>> import numpy as np >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1) >>> ax.hist(np.log10(r)) >>> ax.set_ylabel("Frequency") >>> ax.set_xlabel("Value of random variable") >>> ax.xaxis.set_major_locator(plt.FixedLocator([-2, -1, 0])) >>> ticks = ["$10^{{ {} }}$".format(i) for i in [-2, -1, 0]] >>> ax.set_xticklabels(ticks) # doctest: +SKIP >>> plt.show() This random variable will be log-uniform regardless of the base chosen for ``a`` and ``b``. Let's specify with base ``2`` instead: >>> rvs = %(name)s(2**-2, 2**0).rvs(size=1000) Values of ``1/4``, ``1/2`` and ``1`` are equally likely with this random variable. Here's the histogram: >>> fig, ax = plt.subplots(1, 1) >>> ax.hist(np.log2(rvs)) >>> ax.set_ylabel("Frequency") >>> ax.set_xlabel("Value of random variable") >>> ax.xaxis.set_major_locator(plt.FixedLocator([-2, -1, 0])) >>> ticks = ["$2^{{ {} }}$".format(i) for i in [-2, -1, 0]] >>> ax.set_xticklabels(ticks) # doctest: +SKIP >>> plt.show() cCs|dk||k@Sr6r.rr.r.r/rW"szreciprocal_gen._argcheckcCs0tdddtjfd}tdddtjfd}||gSrrZrr.r.r/r]"szreciprocal_gen._shape_infocs2t|tr|}tj|t|t|fdSNrr5r$r r7r rErrrer r.r/r "s zreciprocal_gen._fitstartcCs||fSrCr.rr.r.r/rv"szreciprocal_gen._get_supportcCst||||SrCrGrr.r.r/ra"szreciprocal_gen._pdfcCs&t| tt|t|SrCrrr.r.r/r"szreciprocal_gen._logpdfcCs(t|t|t|t|SrCrrr.r.r/rb"szreciprocal_gen._cdfcCs(tt||t|t|SrCrErrrr.r.r/rg"szreciprocal_gen._ppfc CsNdt|t||}ttt|t||t|}||SrR)rErrr _log_diff)r;rVrprqr*r+r.r.r/r"s*zreciprocal_gen._munpcCs2dt|t|tt|t|Srrrr.r.r/r"szreciprocal_gen._entropyz `loguniform`/`reciprocal` is over-parameterized. `fit` automatically fixes `scale` to 1 unless `fscale` is provided by the user. rcs*|dd}tj|g|Rd|i|S)Nrr)r+r7r9)r;r<r=r-rr r.r/r9"s zreciprocal_gen.fit)rirjrkrlrWr]r rvrarrbrgrrZfit_noterrr9r?r.r.r r/rU"s3  r loguniform reciprocalc@sJeZdZdZddZddZdddZd d Zd d Zd dZ ddZ dS)rice_genaA Rice continuous random variable. %(before_notes)s Notes ----- The probability density function for `rice` is: .. math:: f(x, b) = x \exp(- \frac{x^2 + b^2}{2}) I_0(x b) for :math:`x >= 0`, :math:`b > 0`. :math:`I_0` is the modified Bessel function of order zero (`scipy.special.i0`). `rice` takes ``b`` as a shape parameter for :math:`b`. %(after_notes)s The Rice distribution describes the length, :math:`r`, of a 2-D vector with components :math:`(U+u, V+v)`, where :math:`U, V` are constant, :math:`u, v` are independent Gaussian random variables with standard deviation :math:`s`. Let :math:`R = \sqrt{U^2 + V^2}`. Then the pdf of :math:`r` is ``rice.pdf(x, R/s, scale=s)``. %(example)s cCs|dkSr6r.r;rqr.r.r/rW"szrice_gen._argcheckcCstdddtjfdgS)NrqFrrYrZr\r.r.r/r]"szrice_gen._shape_infoNcCs4|td|jd|d}t||jddS)NrK)rKrrr\)rErrr)r;rqrrr)r.r.r/r"sz rice_gen._rvscCstt|dt|Sr)rcZchndtrrEr$rr.r.r/rb"sz rice_gen._cdfc Cstt|dt|Sr)rErrcZchndtrixr$r$r.r.r/rg"sz rice_gen._ppfcCs.|t|| ||dt||Sr)rErrci0err.r.r/ra"sz rice_gen._pdfcCsH|d}d|}||d}d|t| t|t|d|Sr)rErrcrCr[)r;rVrqZnd2Zn1rr.r.r/r"s   zrice_gen._munp)NN) rirjrkrlrWr]rrbrgrarr.r.r.r/r"s  rricec@sBeZdZdZddZddZddZdd Zd d Zdd dZ d S)recipinvgauss_genaA reciprocal inverse Gaussian continuous random variable. %(before_notes)s Notes ----- The probability density function for `recipinvgauss` is: .. math:: f(x, \mu) = \frac{1}{\sqrt{2\pi x}} \exp\left(\frac{-(1-\mu x)^2}{2\mu^2x}\right) for :math:`x \ge 0`. `recipinvgauss` takes ``mu`` as a shape parameter for :math:`\mu`. %(after_notes)s %(example)s cCstdddtjfdgSrKrZr\r.r.r/r]#szrecipinvgauss_gen._shape_infocCst|||SrCrGrPr.r.r/ra#szrecipinvgauss_gen._pdfcCs t|dk||fddtj dS)NrcSs:d||d d||ddtdtj|S)NrrrKrur)r`rr.r.r/rL$#s z+recipinvgauss_gen._logpdf..r`rbrPr.r.r/r"#szrecipinvgauss_gen._logpdfcCsPd||}d||}dt|}t| |td|t| |SNrnrrErrrr;r`rrkrlZisqxr.r.r/rb(#s  zrecipinvgauss_gen._cdfcCsNd||}d||}dt|}t||td|t| |Srrrr.r.r/rd.#s  zrecipinvgauss_gen._sfNcCsd|j|d|dSrLrMrOr.r.r/r4#szrecipinvgauss_gen._rvs)NN) rirjrkrlr]rarrbrdrr.r.r.r/r#sr recipinvgaussc@sReZdZdZddZddZddZdd Zd d Zdd dZ ddZ ddZ d S)semicircular_genaA semicircular continuous random variable. %(before_notes)s See Also -------- rdist Notes ----- The probability density function for `semicircular` is: .. math:: f(x) = \frac{2}{\pi} \sqrt{1-x^2} for :math:`-1 \le x \le 1`. The distribution is a special case of `rdist` with `c = 3`. %(after_notes)s References ---------- .. [1] "Wigner semicircle distribution", https://en.wikipedia.org/wiki/Wigner_semicircle_distribution %(example)s cCsgSrCr.r\r.r.r/r]Z#szsemicircular_gen._shape_infocCsdtjtd||Srrrr.r.r/ra]#szsemicircular_gen._pdfcCs$tdtjdt| |Srrrr.r.r/r`#szsemicircular_gen._logpdfcCs.ddtj|td||t|S)Nrurnr)rErrrrr.r.r/rbc#szsemicircular_gen._cdfcCs t|dSNr)rrgrr.r.r/rgf#szsemicircular_gen._ppfNcCs2t|j|d}ttj|j|d}||Sr4)rErrrr)r;rrrUrpr.r.r/ri#szsemicircular_gen._rvscCsdS)N)rr+rrr.r\r.r.r/rp#szsemicircular_gen._statscCsdS)NgzCϑ?r.r\r.r.r/rs#szsemicircular_gen._entropy)NN) rirjrkrlr]rarrbrgrrrr.r.r.r/r;#s r semicircularc@sJeZdZdZddZddZddZdd Zd d Zdd dZ ddZ dS)skewcauchy_genaA skewed Cauchy random variable. %(before_notes)s See Also -------- cauchy : Cauchy distribution Notes ----- The probability density function for `skewcauchy` is: .. math:: f(x) = \frac{1}{\pi \left(\frac{x^2}{\left(a\, \text{sign}(x) + 1 \right)^2} + 1 \right)} for a real number :math:`x` and skewness parameter :math:`-1 < a < 1`. When :math:`a=0`, the distribution reduces to the usual Cauchy distribution. %(after_notes)s References ---------- .. [1] "Skewed generalized *t* distribution", Wikipedia https://en.wikipedia.org/wiki/Skewed_generalized_t_distribution#Skewed_Cauchy_distribution %(example)s cCst|dkSrR)rErrr.r.r/rW#szskewcauchy_gen._argcheckcCstddddgS)NrpF)rrnrrr\r.r.r/r]#szskewcauchy_gen._shape_infocCs,dtj|d|t|dddSr)rErrFrr.r.r/ra#szskewcauchy_gen._pdfc Csbt|dkd|dd|tjt|d|d|dd|tjt|d|SNrrrK)rErrrrr.r.r/rb#s **zskewcauchy_gen._cdfc Csn||d|k}t|ttjd||d|dd|ttjd||d|dd|Sr)rbrErrr)r;r`rprWr.r.r/rg#s **zskewcauchy_gen._ppfrFcCstjtjtjtjfSrCr)r;rprrr.r.r/r#szskewcauchy_gen._statscCs:t|tr|}t|gd\}}}d|||dfS)NrrmrKr)r;r<rrrr.r.r/r #s zskewcauchy_gen._fitstartN)rF) rirjrkrlrWr]rarbrgrr r.r.r.r/rz#s! r skewcauchycseZdZdZddZddZddZdd Zfd d Zd d Z ddZ ddZ d ddZ d!ddZ eddZddZeeddfddZZS)" skewnorm_genafA skew-normal random variable. %(before_notes)s Notes ----- The pdf is:: skewnorm.pdf(x, a) = 2 * norm.pdf(x) * norm.cdf(a*x) `skewnorm` takes a real number :math:`a` as a skewness parameter When ``a = 0`` the distribution is identical to a normal distribution (`norm`). `rvs` implements the method of [1]_. %(after_notes)s %(example)s References ---------- .. [1] A. Azzalini and A. Capitanio (1999). Statistical applications of the multivariate skew-normal distribution. J. Roy. Statist. Soc., B 61, 579-602. :arxiv:`0911.2093` cCs t|SrCr~rr.r.r/rW#szskewnorm_gen._argcheckcCstddtj tjfdgS)NrpFrrZr\r.r.r/r]#szskewnorm_gen._shape_infocCs t|dk||fdddddS)NrcSst|SrCrr`rpr.r.r/rL#rMz#skewnorm_gen._pdf..cSsdt|t||Srrrr.r.r/rL#rMrOrQrr.r.r/ra#szskewnorm_gen._pdfcCs t|dk||fdddddS)NrcSst|SrCrrr.r.r/rL#rMz&skewnorm_gen._logpdf..cSstdt|t||Srrrr.r.r/rL#rMrOrQrr.r.r/r#szskewnorm_gen._logpdfcs`t|}t|dd|}t||j}|dk|dk@}t||||||<t|ddS)Nrrgư>) rEr|rZ _skewnorm_cdfrerr7rbrj)r;r`rprzZ i_small_cdfr r.r/rb#s  zskewnorm_gen._cdfcCst|dd|Sr)rZ _skewnorm_ppfrr.r.r/rg#szskewnorm_gen._ppfcCs|| | SrCrrr.r.r/rd#szskewnorm_gen._sfcCst|dd|Sr)rZ _skewnorm_isfrr.r.r/rh#szskewnorm_gen._isfNcCs`|j|d}|j|d}|td|d}|||td|d}t|dk|| S)NrrrKr)normalrErr)r;rprrZu0r:r=rEr.r.r/r$s   zskewnorm_gen._rvsrFcCsgd}tdtj|td|d}d|vr>||d<d|vrVd|d|d<d|vrdtjd|td|dd |d<d |vrdtjd |dd|dd|d <|S) NrrKrrCrr:rmrrror8)r;rprrrconstr.r.r/r$s&,*zskewnorm_gen._statsc Csltdgtddgtgdtgdtgdtgdtgdtgd tgd tgd d }|S) Nrrr.)rir)ii?i)iiinir)(iSi6QiiiO)iiBi/iioir)iԷiiYei{Hxiii!) i! iׅi쇀 iiViX'i lir) is_'il.skew_dcSsBt|d}t|ttjd||dtjddS)NrHrKr)rErrFrr)rXZs_23r.r.r/d_skewg$s "z skewnorm_gen.fit..d_skewr2rYr'r(gGzgGz?rrrrKr])r+r7r9r5r$r6r r^r3r4rrSrXrErjrrrrFrrr)r;r<r=r-rrrr*rrrpr'r(rmZs_maxr=r:rCr r.r/r9G$sR          J "     zskewnorm_gen.fit)NN)rF)rirjrkrlrWr]rarrbrgrdrhrrrrrrrr9r?r.r.r r/r#s      rskewnormc@sHeZdZdZddZddZddZdd Zd d Zd d Z ddZ dS) trapezoid_genaA trapezoidal continuous random variable. %(before_notes)s Notes ----- The trapezoidal distribution can be represented with an up-sloping line from ``loc`` to ``(loc + c*scale)``, then constant to ``(loc + d*scale)`` and then downsloping from ``(loc + d*scale)`` to ``(loc+scale)``. This defines the trapezoid base from ``loc`` to ``(loc+scale)`` and the flat top from ``c`` to ``d`` proportional to the position along the base with ``0 <= c <= d <= 1``. When ``c=d``, this is equivalent to `triang` with the same values for `loc`, `scale` and `c`. The method of [1]_ is used for computing moments. `trapezoid` takes :math:`c` and :math:`d` as shape parameters. %(after_notes)s The standard form is in the range [0, 1] with c the mode. The location parameter shifts the start to `loc`. The scale parameter changes the width from 1 to `scale`. %(example)s References ---------- .. [1] Kacker, R.N. and Lawrence, J.F. (2007). Trapezoidal and triangular distributions for Type B evaluation of standard uncertainty. Metrologia 44, 117-127. :doi:`10.1088/0026-1394/44/2/003` cCs(|dk|dk@|dk@|dk@||k@Srr.r;rfr=r.r.r/rW$sztrapezoid_gen._argcheckcCs$tdddd}tdddd}||gS)NrfFrrnTTr=rrwr.r.r/r]$sztrapezoid_gen._shape_infocCsPd||d}t||k||k||k@||kgddddddg||||fS)NrKrcSs |||SrCr.r`rfr=rr.r.r/rL$rMz$trapezoid_gen._pdf..cSs|SrCr.rr.r.r/rL$rMcSs|d|d|SrRr.rr.r.r/rL$rMr )r;r`rfr=rr.r.r/ra$s ztrapezoid_gen._pdfcCs>t||k||k||k@||kgddddddg|||fS)NcSs|d|||dSrr.r`rfr=r.r.r/rL$rMz$trapezoid_gen._cdf..cSs|d||||dSrr.rr.r.r/rL$rMcSs$dd|d||dd|Srr.rr.r.r/rL$s  rrr.r.r/rb$sztrapezoid_gen._cdfcCs||||||||}}||k||k||kg}t||d||d|d||d|dtd|||dd|g}t||Sr)rbrErselect)r;rfrfr=ZqcZqdr)rgr.r.r/rg$s$ztrapezoid_gen._ppfcsz|d}t|dkd|k|dk@|dkgddfddfddg|g}dd||||dd }|S) NrrmrncSsdSrr.rr.r.r/rL$rMz%trapezoid_gen._munp..cs tdt||dSr)rErjrrr_r.r/rL$rMcsdSrr.rr_r.r/rL$rMrrKr)r;rVrfr=Zab_termZdc_termrpr.r_r/r$s   (ztrapezoid_gen._munpcCs2dd||d||tdd||Srtrrr.r.r/r$sztrapezoid_gen._entropyN) rirjrkrlrWr]rarbrgrrr.r.r.r/r$s!  r trapezoidtrapzz!trapz is an alias for `trapezoid`c@sReZdZdZdddZddZddZd d Zd d Zd dZ ddZ ddZ dS) triang_gena5A triangular continuous random variable. %(before_notes)s Notes ----- The triangular distribution can be represented with an up-sloping line from ``loc`` to ``(loc + c*scale)`` and then downsloping for ``(loc + c*scale)`` to ``(loc + scale)``. `triang` takes ``c`` as a shape parameter for :math:`0 \le c \le 1`. %(after_notes)s The standard form is in the range [0, 1] with c the mode. The location parameter shifts the start to `loc`. The scale parameter changes the width from 1 to `scale`. %(example)s NcCs|d|d|Sr) triangularr9r.r.r/r'%sztriang_gen._rvscCs|dk|dk@Srr.rr.r.r/rW*%sztriang_gen._argcheckcCstddddgS)NrfFrrrr\r.r.r/r]-%sztriang_gen._shape_infocCsLt|dk||k||k|dk@|dkgddddddddg||f}|S)NrrcSs dd|Srr.rr.r.r/rL:%rMz!triang_gen._pdf..cSs d||Srr.rr.r.r/rL;%rMcSsdd|d|Srr.rr.r.r/rL<%rMcSsd|Srr.rr.r.r/rL=%rMrr;r`rfrUr.r.r/ra0%s ztriang_gen._pdfcCsLt|dk||k||k|dk@|dkgddddddddg||f}|S)NrrcSsd|||Srr.rr.r.r/rLF%rMz!triang_gen._cdf..cSs |||SrCr.rr.r.r/rLG%rMcSs||d|||dSrr.rr.r.r/rLH%rMcSs||SrCr.rr.r.r/rLI%rMrrr.r.r/rbA%s ztriang_gen._cdfc Cs2t||kt||dtd|d|SrR)rErrrkr.r.r/rgM%sztriang_gen._ppfc Csb|ddd|||dtdd|d|d|ddtd|||ddfS) NrnrrKrrrg333333)rErrrr.r.r/rP%s  @ztriang_gen._statscCsdtdSrrrr.r.r/rV%sztriang_gen._entropy)NN) rirjrkrlrrWr]rarbrgrrr.r.r.r/r%s  rtriangcsheZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ fddZ ddZ ZS)truncexpon_genadA truncated exponential continuous random variable. %(before_notes)s Notes ----- The probability density function for `truncexpon` is: .. math:: f(x, b) = \frac{\exp(-x)}{1 - \exp(-b)} for :math:`0 <= x <= b`. `truncexpon` takes ``b`` as a shape parameter for :math:`b`. %(after_notes)s %(example)s cCstdddtjfdgSrrZr\r.r.r/r]s%sztruncexpon_gen._shape_infocCs |j|fSrCrrr.r.r/rvv%sztruncexpon_gen._get_supportcCst| t|  SrCr rr.r.r/ray%sztruncexpon_gen._pdfcCs| tt|  SrCrrr.r.r/r}%sztruncexpon_gen._logpdfcCst| t| SrCrrr.r.r/rb%sztruncexpon_gen._cdfcCst|t|  SrC)rcrrjr$r.r.r/rg%sztruncexpon_gen._ppfcCs$t| t| t| SrCr rr.r.r/rd%sztruncexpon_gen._sfcCs$tt| |t|  SrC)rErrrcrjr$r.r.r/rh%sztruncexpon_gen._isfcs|dkr.d|dt| t|  S|dkrpddd||d|dt| t|  St||SdSr)rErrcrjr7r)r;rVrqr r.r/r%s &:ztruncexpon_gen._munpcCs0t|}t|dd||dd|Srr)r;rqZeBr.r.r/r%s ztruncexpon_gen._entropy)rirjrkrlr]rvrarrbrgrdrhrrr?r.r.r r/r]%s r truncexponcCstj||gddS)Nrr\)rcrZlog_pZlog_qr.r.r/_log_sum%srcCstj||tjdgddS)N?rr\)rcrrErrr.r.r/r%srcst||\}}|dk}|dk}||B}ddfdd}dd}tj|tjtjd}||jrz||||||<||jr|||||||<||jr|||||||<t|S) z3Log of Gaussian probability mass within an intervalrcSstt|t|SrC)rrrr.r.r/mass_case_left%sz'_log_gauss_mass..mass_case_leftcs| | SrCr.rrr.r/mass_case_right%sz(_log_gauss_mass..mass_case_rightcSstt| t| SrC)rcrrrr.r.r/mass_case_central%s z*_log_gauss_mass..mass_case_central)rr)rErRrrZ complex128rr)rprq case_left case_rightZ case_centralrrrVr.rr/_log_gauss_mass%s     rcseZdZdZddZddZfddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ ddZddZddZddZd!dd ZZS)" truncnorm_genaw A truncated normal continuous random variable. %(before_notes)s Notes ----- This distribution is the normal distribution centered on ``loc`` (default 0), with standard deviation ``scale`` (default 1), and truncated at ``a`` and ``b`` *standard deviations* from ``loc``. For arbitrary ``loc`` and ``scale``, ``a`` and ``b`` are *not* the abscissae at which the shifted and scaled distribution is truncated. .. note:: If ``a_trunc`` and ``b_trunc`` are the abscissae at which we wish to truncate the distribution (as opposed to the number of standard deviations from ``loc``), then we can calculate the distribution parameters ``a`` and ``b`` as follows:: a, b = (a_trunc - loc) / scale, (b_trunc - loc) / scale This is a common point of confusion. For additional clarification, please see the example below. %(example)s In the examples above, ``loc=0`` and ``scale=1``, so the plot is truncated at ``a`` on the left and ``b`` on the right. However, suppose we were to produce the same histogram with ``loc = 1`` and ``scale=0.5``. >>> loc, scale = 1, 0.5 >>> rv = truncnorm(a, b, loc=loc, scale=scale) >>> x = np.linspace(truncnorm.ppf(0.01, a, b), ... truncnorm.ppf(0.99, a, b), 100) >>> r = rv.rvs(size=1000) >>> fig, ax = plt.subplots(1, 1) >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf') >>> ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2) >>> ax.set_xlim(a, b) >>> ax.legend(loc='best', frameon=False) >>> plt.show() Note that the distribution is no longer appears to be truncated at abscissae ``a`` and ``b``. That is because the *standard* normal distribution is first truncated at ``a`` and ``b``, *then* the resulting distribution is scaled by ``scale`` and shifted by ``loc``. If we instead want the shifted and scaled distribution to be truncated at ``a`` and ``b``, we need to transform these values before passing them as the distribution parameters. >>> a_transformed, b_transformed = (a - loc) / scale, (b - loc) / scale >>> rv = truncnorm(a_transformed, b_transformed, loc=loc, scale=scale) >>> x = np.linspace(truncnorm.ppf(0.01, a, b), ... truncnorm.ppf(0.99, a, b), 100) >>> r = rv.rvs(size=10000) >>> fig, ax = plt.subplots(1, 1) >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf') >>> ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2) >>> ax.set_xlim(a-0.1, b+0.1) >>> ax.legend(loc='best', frameon=False) >>> plt.show() cCs||kSrCr.rr.r.r/rW&sztruncnorm_gen._argcheckcCs8tddtj tjfd}tddtj tjfd}||gS)NrpFrYrq)FTrZrr.r.r/r]&sztruncnorm_gen._shape_infocs2t|tr|}tj|t|t|fdSrrrer r.r/r &s ztruncnorm_gen._fitstartcCs||fSrCr.rr.r.r/rv &sztruncnorm_gen._get_supportcCst||||SrCrGrr.r.r/ra#&sztruncnorm_gen._pdfcCst|t||SrC)rrrr.r.r/r&&sztruncnorm_gen._logpdfcCst||||SrCrrr.r.r/rb)&sztruncnorm_gen._cdfc Cspt|||\}}}tt||t||}|dk}t|rltt||||||| ||<|SNg)rErRrrrrrr)r;r`rprqZlogcdfrWr.r.r/r,&s  ,ztruncnorm_gen._logcdfcCst||||SrCrrr.r.r/rd4&sztruncnorm_gen._sfc Cspt|||\}}}tt||t||}|dk}t|rltt||||||| ||<|Sr)rErRrrrrrr)r;r`rprqZlogsfrWr.r.r/r7&s  ,ztruncnorm_gen._logsfc Csdt|}t|}||}ttdtjtj|}|t||t|d|}||}|Sr)rrErrrrr) r;rprqrrZr1Drr.r.r/r?&s  ztruncnorm_gen._entropyc Cst|||\}}}|dk}|}dd}dd}t|}||} ||} | jrj|| ||||||<| jr|| ||||||<|S)NrcSs(tt|t|t||}t|SrC)rrrErrrc ndtri_exprfrprqZ log_Phi_xr.r.r/ppf_leftN&sz$truncnorm_gen._ppf..ppf_leftcSs.tt| t| t||}t| SrC)rrrErrrcrrr.r.r/ ppf_rightS&s z%truncnorm_gen._ppf..ppf_rightrErR empty_liker) r;rfrprqrrrrrVq_leftq_rightr.r.r/rgH&s ztruncnorm_gen._ppfc Cst|||\}}}|dk}|}dd}dd}t|}||} ||} | jrj|| ||||||<| jr|| ||||||<|S)NrcSs.tt|t|t||}tt|SrC)rrrErrrcrrrr.r.r/isf_leftk&sz$truncnorm_gen._isf..isf_leftcSs4tt| t| t||}tt| SrC)rrrErrrcrrrr.r.r/ isf_rightp&s z%truncnorm_gen._isf..isf_rightr) r;rfrprqrrrrrVrrr.r.r/rhd&s ztruncnorm_gen._isfcsDfdd}t|dk||k@||k@|||ftj|tjgdtjS)Nc st||g||\}}|| g}ddg}td|dD]Ht||||ggfdddd}t|d|d}||q<|dS)z Returns n-th moment. 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Function cannot broadcast due to the loop over n rrcs||dSrRr.rror.r/rL&rMz:truncnorm_gen._munp..n_th_moment..r`rr.)rarErr^rrr) rVrprqpApBprobsrrrmkr\rr/ n_th_moment&s   z(truncnorm_gen._munp..n_th_momentrrr)r;rVrprqrr.r\r/r&s   ztruncnorm_gen._munprlcCsB|t||g||\}}dd}tj|dd}||||||S)NcSs>||}|}|| g}t||||ggdddd}dt|} t||||||ggdddd}dt|} t||||ggdddd}d|t|} t||||ggd ddd}d | t|} | |d | d|d} | t| d }| |d | d |d| |d}|| dd }|| ||fS)NcSs||SrCr.rr.r.r/rL&rMzGtruncnorm_gen._stats.._truncnorm_stats_scalar..rr`rcSs||SrCr.rr.r.r/rL&rMcSs ||dSrr.rr.r.r/rL&rMrKcSs ||dSrr.rr.r.r/rL&rMrrrr)rrErr)rprqrrrrrrrrrrrZm4Zmu3rZmu4rr.r.r/_truncnorm_stats_scalar&s0 (z5truncnorm_gen._stats.._truncnorm_stats_scalar)rr)excluded)ZpdfrErr)r;rprqrrrrrZ_truncnorm_statsr.r.r/r&s ztruncnorm_gen._stats)rl)rirjrkrlrWr]r rvrarrbrrdrrrgrhrrr?r.r.r r/r%s @  r truncnorm)rrr}cseZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ ddZddZddZddZeeefddZZS) truncpareto_genacAn upper truncated Pareto continuous random variable. %(before_notes)s See Also -------- pareto : Pareto distribution Notes ----- The probability density function for `truncpareto` is: .. math:: f(x, b, c) = \frac{b}{1 - c^{-b}} \frac{1}{x^{b+1}} for :math:`b > 0`, :math:`c > 1` and :math:`1 \le x \le c`. `truncpareto` takes `b` and `c` as shape parameters for :math:`b` and :math:`c`. Notice that the upper truncation value :math:`c` is defined in standardized form so that random values of an unscaled, unshifted variable are within the range ``[1, c]``. If ``u_r`` is the upper bound to a scaled and/or shifted variable, then ``c = (u_r - loc) / scale``. In other words, the support of the distribution becomes ``(scale + loc) <= x <= (c*scale + loc)`` when `scale` and/or `loc` are provided. %(after_notes)s References ---------- .. [1] Burroughs, S. M., and Tebbens S. F. "Upper-truncated power laws in natural systems." Pure and Applied Geophysics 158.4 (2001): 741-757. %(example)s cCs0tdddtjfd}tdddtjfd}||gS)NrqFrmrrfrnrZ)r;rrxr.r.r/r]&sztruncpareto_gen._shape_infocCs|dk|dk@SNrmrnr.r;rqrfr.r.r/rW&sztruncpareto_gen._argcheckcCs |j|fSrCrrr.r.r/rv&sztruncpareto_gen._get_supportcCs"|||d dd||SrRr.r;r`rqrfr.r.r/ra&sztruncpareto_gen._pdfc Cs:t|tt| t| |dt|SrR)rErrjrr.r.r/r&sztruncpareto_gen._logpdfcCsd|| dd||SrRr.rr.r.r/rb&sztruncpareto_gen._cdfcCs$t||  td||Srrrr.r.r/r&sztruncpareto_gen._logcdfcCs"tddd|||d|SNrr.rTr;rfrqrfr.r.r/rg&sztruncpareto_gen._ppfcCs&|| d||dd||SrRr.rr.r.r/rd'sztruncpareto_gen._sfcCs.t|| d||td||Srr rr.r.r/r'sztruncpareto_gen._logsfcCs*td||dd|||d|SrrTrr.r.r/rh'sztruncpareto_gen._isfcCsBt|dd|||dt|||dd| SrRrrr.r.r/r 's$ztruncpareto_gen._entropycCsV||kr*|t|dd||S|||||||||dSdSrR)rrEr)r;rVrqrfr.r.r/r's ztruncpareto_gen._munpcCs>t|tr|}t|\}}}t|||}||||fSrC)r5r$r rsr9r)r;r<rqr'r(rfr.r.r/r 's  ztruncpareto_gen._fitstartcs<|ddr&tjg|Ri|Sddddfddfd d fd d }fd dd fdd }dd} fdd}t||}|\} } } } ttj } | dur | dur | dur | dur t dn:| dur| dur| dur| durfdd}ttj } | }d}|d}|tj kr||||dkr|d7}|t d|}qd|tj ks|g|Ri|St |||fd}|j s|g|Ri|S|j d}|d}d}|tj krL||||dkrL|d7}|t d|}q |tj ksp|g|Ri|St |||fd}|j s|g|Ri|S|j }|}||}|||}||}td|d|d}||ks|g|Ri|Sn| }| d}d}|tj kr^||| ||| dkr^|d7}|d|}q|tj ks|g|Ri|St || f||fd}|j s|g|Ri|S|j }|}||}| }np| dur| n|| | }| p|}| p||}| dur4| dkr4tdd|d| rt| durt| rt| | | krttdd||d| dur@||}|}t|}d||ks|g|Ri|Sd|d||}td|d}z@t |||f||fd}|j s|g|Ri|WS|j }Wnt y<|}Yn0n| }||ks~| rjt|tj }n|}t|d}|||ks||}t|tj}t||r|dks|g|Ri|S||||f}| dur8| dur8|g|Ri|}|}|}||kr8|S|S)!NrVFcSstt|SrC)rErrrr.r.r/log_mean"'sz%truncpareto_gen.fit..log_meancSsdtd|SrR)rErrr.r.r/ harm_mean%'sz&truncpareto_gen.fit..harm_meancsV||}|}|}|d|}d|d|dd||t||SrRr)rfr'r(rharm_mZlog_mquot)r<rrr.r/get_b('s   z"truncpareto_gen.fit..get_bcs ||SrCr.r])mxr.r/get_c/'sz"truncpareto_gen.fit..get_ccs0|r|}|S|r,||d}|SdSrRr.)r`rr')rrr.r/get_loc2's z$truncpareto_gen.fit..get_loccs|SrCr.r)rr.r/r:'sz&truncpareto_gen.fit..get_scalecsn|}||}|dur&|||n|}||}dd|d||d|dd|d|SrRr.)r'rKr(rfrqr)r<rr rrr.r/r@'s  z$truncpareto_gen.fit..dL_dLoccSs"|t||d|||SrRr)rqlogclogmr.r.r/dL_dBI'sz"truncpareto_gen.fit..dL_dBcsttj|g|Ri|SrC)r7rr9)r<r=kwargs)rr;r.r/fallbackO'sz%truncpareto_gen.fit..fallbackz2All parameters fixed.There is nothing to optimize.csD|}||}||}dd|dt||dSrRr)r'r(rfr)r<r rrr.r/cond_b`'s z#truncpareto_gen.fit..cond_brrrr#gMbP?rK truncparetor)N)r+r7r9r^rrrEr r[rrr%r r_rrrrWr)r;r<r=r-r rr rr rr`rrZmn_infrrIrWrHrr'r(rfrqZstd_dataZ up_bound_br r Zparams_overrideZ params_superZ nllf_overrideZ nllf_superr ) r<rr rrrrrr;r/r9's                           ztruncpareto_gen.fit)rirjrkrlr]rWrvrarrbrrgrdrrhrrr rBr rr9r?r.r.r r/r&s$)rrc@sHeZdZdZddZddZddZdd Zd d Zd d Z ddZ dS)tukeylambda_gena*A Tukey-Lamdba continuous random variable. %(before_notes)s Notes ----- A flexible distribution, able to represent and interpolate between the following distributions: - Cauchy (:math:`lambda = -1`) - logistic (:math:`lambda = 0`) - approx Normal (:math:`lambda = 0.14`) - uniform from -1 to 1 (:math:`lambda = 1`) `tukeylambda` takes a real number :math:`lambda` (denoted ``lam`` in the implementation) as a shape parameter. %(after_notes)s %(example)s cCs t|SrCr~r;lamr.r.r/rW(sztukeylambda_gen._argcheckcCstddtj tjfdgS)NrFrrZr\r.r.r/r](sztukeylambda_gen._shape_infocCsjtt||}||dtd||d}dt|}t|dkt|dt|kB|dS)Nrnrrrm)rErrctklmbdarr)r;r`rZFxrr.r.r/ra(s"ztukeylambda_gen._pdfcCs t||SrC)rcr)r;r`rr.r.r/rb!(sztukeylambda_gen._cdfcCst||t| |SrC)rcrr)r;rfrr.r.r/rg$(sztukeylambda_gen._ppfcCsdt|dt|fSr6)_tlvar_tlkurtrr.r.r/r'(sztukeylambda_gen._statscsfdd}t|dddS)Ncs&tt|dtd|dSrR)rErr#)rTrr.r/integ+(sz'tukeylambda_gen._entropy..integrr)r r)r;rrr.rr/r*(s ztukeylambda_gen._entropyN) rirjrkrlrWr]rarbrgrrr.r.r.r/r'sr tukeylambdac@seZdZddZdS)FitUniformFixedScaleDataErrorcCsd|d|d|_dS)NzInvalid values in `data`. Maximum likelihood estimation with the uniform distribution and fixed scale requires that np.ptp(data) <= fscale, but np.ptp(data) = z and fscale = rr)r;rrr.r.r/r4(s z&FitUniformFixedScaleDataError.__init__N)rirjrkrr.r.r.r/r3(src@sVeZdZdZddZdddZddZd d Zd d Zd dZ ddZ e ddZ dS) uniform_gena A uniform continuous random variable. In the standard form, the distribution is uniform on ``[0, 1]``. Using the parameters ``loc`` and ``scale``, one obtains the uniform distribution on ``[loc, loc + scale]``. %(before_notes)s %(example)s cCsgSrCr.r\r.r.r/r]I(szuniform_gen._shape_infoNcCs|dd|Sr)rrr.r.r/rL(szuniform_gen._rvscCs d||kSrr.rr.r.r/raO(szuniform_gen._pdfcCs|SrCr.rr.r.r/rbR(szuniform_gen._cdfcCs|SrCr.rr.r.r/rgU(szuniform_gen._ppfcCsdS)N)rugUUUUUU?rg333333r.r\r.r.r/rX(szuniform_gen._statscCsdSrbr.r\r.r.r/r[(szuniform_gen._entropyc Ost|dkrtd|dd}|dd}t||durL|durLtdt|}t|sltd|dur|dur| }t |}q|}| |}| |krt d|||d n6t |}||krt ||d | d ||}|}t|t|fS) a Maximum likelihood estimate for the location and scale parameters. `uniform.fit` uses only the following parameters. Because exact formulas are used, the parameters related to optimization that are available in the `fit` method of other distributions are ignored here. The only positional argument accepted is `data`. Parameters ---------- data : array_like Data to use in calculating the maximum likelihood estimate. floc : float, optional Hold the location parameter fixed to the specified value. fscale : float, optional Hold the scale parameter fixed to the specified value. Returns ------- loc, scale : float Maximum likelihood estimates for the location and scale. Notes ----- An error is raised if `floc` is given and any values in `data` are less than `floc`, or if `fscale` is given and `fscale` is less than ``data.max() - data.min()``. An error is also raised if both `floc` and `fscale` are given. Examples -------- >>> import numpy as np >>> from scipy.stats import uniform We'll fit the uniform distribution to `x`: >>> x = np.array([2, 2.5, 3.1, 9.5, 13.0]) For a uniform distribution MLE, the location is the minimum of the data, and the scale is the maximum minus the minimum. >>> loc, scale = uniform.fit(x) >>> loc 2.0 >>> scale 11.0 If we know the data comes from a uniform distribution where the support starts at 0, we can use `floc=0`: >>> loc, scale = uniform.fit(x, floc=0) >>> loc 0.0 >>> scale 13.0 Alternatively, if we know the length of the support is 12, we can use `fscale=12`: >>> loc, scale = uniform.fit(x, fscale=12) >>> loc 1.5 >>> scale 12.0 In that last example, the support interval is [1.5, 13.5]. This solution is not unique. For example, the distribution with ``loc=2`` and ``scale=12`` has the same likelihood as the one above. When `fscale` is given and it is larger than ``data.max() - data.min()``, the parameters returned by the `fit` method center the support over the interval ``[data.min(), data.max()]``. rrrNrrrrr)rrru)rr,r+r0rrErrrrrrrrr) r;r<r=r-rrr'r(rr.r.r/r9^(s0K         zuniform_gen.fit)NN) rirjrkrlr]rrarbrgrrrBr9r.r.r.r/r=(s  rrcseZdZdZddZddZd ddZeefd d Z d d Z d dZ ddZ ddZ ddZeeddd!fdd ZeeeddfddZZS)" vonmises_genaU A Von Mises continuous random variable. %(before_notes)s See Also -------- scipy.stats.vonmises_fisher : Von-Mises Fisher distribution on a hypersphere Notes ----- The probability density function for `vonmises` and `vonmises_line` is: .. math:: f(x, \kappa) = \frac{ \exp(\kappa \cos(x)) }{ 2 \pi I_0(\kappa) } for :math:`-\pi \le x \le \pi`, :math:`\kappa \ge 0`. :math:`I_0` is the modified Bessel function of order zero (`scipy.special.i0`). `vonmises` is a circular distribution which does not restrict the distribution to a fixed interval. Currently, there is no circular distribution framework in SciPy. The ``cdf`` is implemented such that ``cdf(x + 2*np.pi) == cdf(x) + 1``. `vonmises_line` is the same distribution, defined on :math:`[-\pi, \pi]` on the real line. This is a regular (i.e. non-circular) distribution. Note about distribution parameters: `vonmises` and `vonmises_line` take ``kappa`` as a shape parameter (concentration) and ``loc`` as the location (circular mean). A ``scale`` parameter is accepted but does not have any effect. Examples -------- Import the necessary modules. >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.stats import vonmises Define distribution parameters. >>> loc = 0.5 * np.pi # circular mean >>> kappa = 1 # concentration Compute the probability density at ``x=0`` via the ``pdf`` method. >>> vonmises.pdf(0, loc=loc, kappa=kappa) 0.12570826359722018 Verify that the percentile function ``ppf`` inverts the cumulative distribution function ``cdf`` up to floating point accuracy. >>> x = 1 >>> cdf_value = vonmises.cdf(x, loc=loc, kappa=kappa) >>> ppf_value = vonmises.ppf(cdf_value, loc=loc, kappa=kappa) >>> x, cdf_value, ppf_value (1, 0.31489339900904967, 1.0000000000000004) Draw 1000 random variates by calling the ``rvs`` method. >>> sample_size = 1000 >>> sample = vonmises(loc=loc, kappa=kappa).rvs(sample_size) Plot the von Mises density on a Cartesian and polar grid to emphasize that it is a circular distribution. >>> fig = plt.figure(figsize=(12, 6)) >>> left = plt.subplot(121) >>> right = plt.subplot(122, projection='polar') >>> x = np.linspace(-np.pi, np.pi, 500) >>> vonmises_pdf = vonmises.pdf(x, loc=loc, kappa=kappa) >>> ticks = [0, 0.15, 0.3] The left image contains the Cartesian plot. >>> left.plot(x, vonmises_pdf) >>> left.set_yticks(ticks) >>> number_of_bins = int(np.sqrt(sample_size)) >>> left.hist(sample, density=True, bins=number_of_bins) >>> left.set_title("Cartesian plot") >>> left.set_xlim(-np.pi, np.pi) >>> left.grid(True) The right image contains the polar plot. >>> right.plot(x, vonmises_pdf, label="PDF") >>> right.set_yticks(ticks) >>> right.hist(sample, density=True, bins=number_of_bins, ... label="Histogram") >>> right.set_title("Polar plot") >>> right.legend(bbox_to_anchor=(0.15, 1.06)) cCstdddtjfdgS)NrFrrYrZr\r.r.r/r]W)szvonmises_gen._shape_infocCs|dkSr6r.rr.r.r/rWZ)szvonmises_gen._argcheckNcCs|jd||dS)Nrmr)vonmises)r;rrrr.r.r/r])szvonmises_gen._rvscs0tj|i|}t|tjdtjtjSr)r7rDrEmodr)r;r=r-rDr r.r/rD`)szvonmises_gen.rvscCs(t|t|dtjt|Sr)rErrccosm1rrrr.r.r/rae)szvonmises_gen._pdfcCs.|t|tdtjtt|Sr)rcr rErrrrr.r.r/rl)szvonmises_gen._logpdfcCs t||SrC)rZ von_mises_cdfrr.r.r/rbp)szvonmises_gen._cdfcCsdSrr.rr.r.r/ _stats_skips)szvonmises_gen._stats_skipcCs8| t|t|tdtjt||Sr)rci1errErrrr.r.r/rv)s zvonmises_gen._entropyz The default limits of integration are endpoints of the interval of width ``2*pi`` centered at `loc` (e.g. ``[-pi, pi]`` when ``loc=0``). rr.rrFc  sPtj tj} } |dur || }|dur0|| }tj|||||||fi|SrC)rErr7expect) r;rr=r'r(ZlbZubZ conditionalr-rrr r.r/r#)s zvonmises_gen.expecta Fit data is assumed to represent angles and will be wrapped onto the unit circle. `f0` and `fscale` are ignored; the returned shape is always the maximum likelihood estimate and the scale is always 1. Initial guesses are ignored. c s|ddr&tj|g|Ri|St||||\}}}}|jtj krdtj|g|Ri|St|dtj}dd}dd}|dur|n||} |dur|n||| } t| tjdtjtj} | | dfS) NrVFrKcSs t|SrC)rSZcircmeanrr.r.r/find_mu)sz!vonmises_gen.fit..find_mucstt||t|dkr(dSdkrfdd}dd}d|}||dkrh|S||dkrx|St|d||fd}|jSn ttjSdS) Nrg7yACrcst|t|SrC)rcr"r)rrUr.r/solve_for_kappa)sz=vonmises_gen.fit..find_kappa..solve_for_kapparKr)r*r\) rErrrr%r_r%rr&)r<r'r& lower_bound upper_boundZroot_resr.r%r/ find_kappa)s     z$vonmises_gen.fit..find_kappar)r+r7r9r^rprErr) r;r<r=r-rrrr$r)r'rr r.r/r9)s  9zvonmises_gen.fit)NN)Nr.rrNNF)rirjrkrlr]rWrr rrDrarrbr!rrr#rBr9r?r.r.r r/r(s$_    rr vonmises_linec@sxeZdZdZejZddZdddZddZ d d Z d d Z d dZ ddZ ddZddZddZddZddZdS)rZaXA Wald continuous random variable. %(before_notes)s Notes ----- The probability density function for `wald` is: .. math:: f(x) = \frac{1}{\sqrt{2\pi x^3}} \exp(- \frac{ (x-1)^2 }{ 2x }) for :math:`x >= 0`. `wald` is a special case of `invgauss` with ``mu=1``. %(after_notes)s %(example)s cCsgSrCr.r\r.r.r/r]*szwald_gen._shape_infoNcCs|jdd|dSrLrMrr.r.r/r*sz wald_gen._rvscCs t|dSr)rYrarr.r.r/ra *sz wald_gen._pdfcCs t|dSr)rYrbrr.r.r/rb*sz wald_gen._cdfcCs t|dSr)rYrdrr.r.r/rd*sz wald_gen._sfcCs t|dSr)rYrgrr.r.r/rg*sz wald_gen._ppfcCs t|dSr)rYrhrr.r.r/rh*sz wald_gen._isfcCs t|dSr)rYrrr.r.r/r*szwald_gen._logpdfcCs t|dSr)rYrrr.r.r/r*szwald_gen._logcdfcCs t|dSr)rYrrr.r.r/r *szwald_gen._logsfcCsdS)N)rnrnrrxr.r\r.r.r/r#*szwald_gen._statscCs tdSr)rYrr\r.r.r/r&*szwald_gen._entropy)NN)rirjrkrlrrrr]rrarbrdrgrhrrrrrr.r.r.r/rZ)s rZrNc@sHeZdZdZddZddZddZdd Zd d Zd d Z ddZ dS)wrapcauchy_genaA wrapped Cauchy continuous random variable. %(before_notes)s Notes ----- The probability density function for `wrapcauchy` is: .. math:: f(x, c) = \frac{1-c^2}{2\pi (1+c^2 - 2c \cos(x))} for :math:`0 \le x \le 2\pi`, :math:`0 < c < 1`. `wrapcauchy` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s %(example)s cCs|dk|dk@Srr.rr.r.r/rWC*szwrapcauchy_gen._argcheckcCstddddgS)NrfF)rrrrr\r.r.r/r]F*szwrapcauchy_gen._shape_infocCs4d||dtjd||d|t|SrBrrhr.r.r/raI*szwrapcauchy_gen._pdfcCs:dd}dd}d|d|}t|tjk||f||dS)NcSs"dtjt|t|dSrrErrrr`crr.r.r/rO*szwrapcauchy_gen._cdf..f1c Ss0ddtjt|tdtj|dSrr,r-r.r.r/rPS*szwrapcauchy_gen._cdf..f2rr)rrEr)r;r`rfrrPr.r.r.r/rbM*szwrapcauchy_gen._cdfc Csld|d|}dt|ttj|}dtjdt|ttjd|}t|dk||S)NrnrKrru)rErrrr)r;rfrfrpZrcqZrcmqr.r.r/rgZ*s,zwrapcauchy_gen._ppfcCstdtjd||Srrrr.r.r/r`*szwrapcauchy_gen._entropycCs2t|tr|}dt|t|dtjfSr)r5r$r rErrrrer.r.r/r c*s zwrapcauchy_gen._fitstartN) rirjrkrlrWr]rarbrgrr r.r.r.r/r+-*s r+ wrapcauchyc@sbeZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ dddZ dS) gennorm_gena0A generalized normal continuous random variable. %(before_notes)s See Also -------- laplace : Laplace distribution norm : normal distribution Notes ----- The probability density function for `gennorm` is [1]_: .. math:: f(x, \beta) = \frac{\beta}{2 \Gamma(1/\beta)} \exp(-|x|^\beta), where :math:`x` is a real number, :math:`\beta > 0` and :math:`\Gamma` is the gamma function (`scipy.special.gamma`). `gennorm` takes ``beta`` as a shape parameter for :math:`\beta`. For :math:`\beta = 1`, it is identical to a Laplace distribution. For :math:`\beta = 2`, it is identical to a normal distribution (with ``scale=1/sqrt(2)``). References ---------- .. [1] "Generalized normal distribution, Version 1", https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1 .. [2] Nardon, Martina, and Paolo Pianca. "Simulation techniques for generalized Gaussian densities." Journal of Statistical Computation and Simulation 79.11 (2009): 1317-1329 .. [3] Wicklin, Rick. "Simulate data from a generalized Gaussian distribution" in The DO Loop blog, September 21, 2016, https://blogs.sas.com/content/iml/2016/09/21/simulate-generalized-gaussian-sas.html %(example)s cCstdddtjfdgSNrFrrrZr\r.r.r/r]*szgennorm_gen._shape_infocCst|||SrCrGr;r`rr.r.r/ra*szgennorm_gen._pdfcCs(td|td|t||Srt)rErrcr5rr2r.r.r/r*szgennorm_gen._logpdfcCs2dt|}d||td|t||Srt)rErFrcrrr;r`rrfr.r.r/rb*szgennorm_gen._cdfcCs:t|d}|td|d|d||d|S)Nrurnr)rErFrcrr3r.r.r/rg*szgennorm_gen._ppfcCs|| |SrCrr2r.r.r/rd*szgennorm_gen._sfcCs||| SrCrwr2r.r.r/rh*szgennorm_gen._isfcCsNtd|d|d|g\}}}dt||dt||d|dfS)Nrnrrrmr)rcr5rEr)r;rc1c3Zc5r.r.r/r*s"zgennorm_gen._statscCs$d|td|td|Sr rEr;rr.r.r/r*szgennorm_gen._entropyNcCsL|jd||d}|d|}t|}|j|jddk}|| ||<|S)Nrrru)rCrErrandomr)r;rrrr'rrrr.r.r/r*s   zgennorm_gen._rvs)NN)rirjrkrlr]rarrbrgrdrhrrrr.r.r.r/r0o*s*r0gennormc@sPeZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ dS)halfgennorm_genaThe upper half of a generalized normal continuous random variable. %(before_notes)s See Also -------- gennorm : generalized normal distribution expon : exponential distribution halfnorm : half normal distribution Notes ----- The probability density function for `halfgennorm` is: .. math:: f(x, \beta) = \frac{\beta}{\Gamma(1/\beta)} \exp(-|x|^\beta) for :math:`x, \beta > 0`. :math:`\Gamma` is the gamma function (`scipy.special.gamma`). `halfgennorm` takes ``beta`` as a shape parameter for :math:`\beta`. For :math:`\beta = 1`, it is identical to an exponential distribution. For :math:`\beta = 2`, it is identical to a half normal distribution (with ``scale=1/sqrt(2)``). References ---------- .. [1] "Generalized normal distribution, Version 1", https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1 %(example)s cCstdddtjfdgSr1rZr\r.r.r/r]*szhalfgennorm_gen._shape_infocCst|||SrCrGr2r.r.r/ra*szhalfgennorm_gen._pdfcCs t|td|||SrrEr2r.r.r/r*szhalfgennorm_gen._logpdfcCstd|||Srrr2r.r.r/rb*szhalfgennorm_gen._cdfcCstd||d|Srrr2r.r.r/rg*szhalfgennorm_gen._ppfcCstd|||Srrr2r.r.r/rd*szhalfgennorm_gen._sfcCstd||d|Srrr2r.r.r/rh+szhalfgennorm_gen._isfcCs d|t|td|SrrEr6r.r.r/r+szhalfgennorm_gen._entropyNrr.r.r.r/r9*s#r9 halfgennormcsXeZdZdZddZddZfddZdd Zd d Zd d Z ddZ ddZ Z S)crystalball_gena Crystalball distribution %(before_notes)s Notes ----- The probability density function for `crystalball` is: .. math:: f(x, \beta, m) = \begin{cases} N \exp(-x^2 / 2), &\text{for } x > -\beta\\ N A (B - x)^{-m} &\text{for } x \le -\beta \end{cases} where :math:`A = (m / |\beta|)^m \exp(-\beta^2 / 2)`, :math:`B = m/|\beta| - |\beta|` and :math:`N` is a normalisation constant. `crystalball` takes :math:`\beta > 0` and :math:`m > 1` as shape parameters. :math:`\beta` defines the point where the pdf changes from a power-law to a Gaussian distribution. :math:`m` is the power of the power-law tail. %(after_notes)s .. versionadded:: 0.19.0 References ---------- .. [1] "Crystal Ball Function", https://en.wikipedia.org/wiki/Crystal_Ball_function %(example)s cCs|dk|dk@S)z@ Shape parameter bounds are m > 1 and beta > 0. rrr.)r;rrCr.r.r/rW0+szcrystalball_gen._argcheckcCs0tdddtjfd}tdddtjfd}||gS)NrFrrrCrrZ)r;ZibetaZimr.r.r/r]6+szcrystalball_gen._shape_infocstj|ddS)N)rrrrdrer r.r/r ;+szcrystalball_gen._fitstartcCsdd|||dt|d dtt|}dd}dd}|t|| k|||f||d S) a` Return PDF of the crystalball function. -- | exp(-x**2 / 2), for x > -beta crystalball.pdf(x, beta, m) = N * | | A * (B - x)**(-m), for x <= -beta -- rnrrKrcSst|d dSrrr`rrCr.r.r/rhsL+sz!crystalball_gen._pdf..rhscSs6|||t|d d||||| Srrr<r.r.r/lhsO+sz!crystalball_gen._pdf..lhsrrErrrrr;r`rrCrr=r>r.r.r/ra?+s $ zcrystalball_gen._pdfcCsjd|||dt|d dtt|}dd}dd}t|t|| k|||f||d S) zH Return the log of the PDF of the crystalball function. rnrrKrcSs|d dSrr.r<r.r.r/r=\+sz$crystalball_gen._logpdf..rhscSs8|t|||dd|t||||Srrr<r.r.r/r>_+sz$crystalball_gen._logpdf..lhsr)rErrrrrr@r.r.r/rU+s $ zcrystalball_gen._logpdfcCsdd|||dt|d dtt|}dd}dd}|t|| k|||f||d S) z8 Return CDF of the crystalball function rnrrKrcSs:||t|d d|dtt|t| SNrKrrrErrrr<r.r.r/r=k+s"z!crystalball_gen._cdf..rhscSsB|||t|d d||||| d|dSrArr<r.r.r/r>o+s z!crystalball_gen._cdf..lhsrr?r@r.r.r/rbd+s $ zcrystalball_gen._cdfcCsd|||dt|d dtt|}|||t|d d|d}dd}dd}t||k|||f||d S) NrnrrKrcSsvt|d d}||||d}d|tt|}||||d||| |||dd|SrrBrTrrCZeb2r1rr.r.r/ppf_lessz+s  ,z&crystalball_gen._ppf..ppf_lesscSs^t|d d}||||d}d|tt|}tt| dt|||Sr)rErrrrrCr.r.r/ ppf_greater+sz)crystalball_gen._ppf..ppf_greaterrr?)r;rTrrCrZpbetarDrEr.r.r/rgu+s$ (zcrystalball_gen._ppfcCsld|||dt|d dtt|}dd}|t|d|k|||ftj|tjgdtjS)zR Returns the n-th non-central moment of the crystalball function. rnrrKrcSs|||t|d d}|||}d|ddt|dddd|t|dd|dd}t|j}t|dD]J}|t|||||d|||d||| |d7}q|||S)z Returns n-th moment. Defined only if n+1 < m Function cannot broadcast due to the loop over n rKrrrnr.) rErrcrCrr}rr^Zbinom)rVrrCrrr=r>ror.r.r/r+s   & ,z*crystalball_gen._munp..n_th_momentr)rErrrrrrr[)r;rVrrCrrr.r.r/r+s$ zcrystalball_gen._munp) rirjrkrlrWr]r rarrbrgrr?r.r.r r/r; +s# r; crystalballzA Crystalball Function)rrlongnamecCstd|dddS)a Utility function for the argus distribution used in the pdf, sf and moment calculation. Note that for all x > 0: gammainc(1.5, x**2/2) = 2 * (_norm_cdf(x) - x * _norm_pdf(x) - 0.5). This can be verified directly by noting that the cdf of Gamma(1.5) can be written as erf(sqrt(x)) - 2*sqrt(x)*exp(-x)/sqrt(Pi). We use gammainc instead of the usual definition because it is more precise for small chi. rrKr)rr.r.r/ _argus_phi+s rHc@sTeZdZdZddZddZddZdd Zd d Zdd dZ dddZ ddZ d S) argus_gena Argus distribution %(before_notes)s Notes ----- The probability density function for `argus` is: .. math:: f(x, \chi) = \frac{\chi^3}{\sqrt{2\pi} \Psi(\chi)} x \sqrt{1-x^2} \exp(-\chi^2 (1 - x^2)/2) for :math:`0 < x < 1` and :math:`\chi > 0`, where .. math:: \Psi(\chi) = \Phi(\chi) - \chi \phi(\chi) - 1/2 with :math:`\Phi` and :math:`\phi` being the CDF and PDF of a standard normal distribution, respectively. `argus` takes :math:`\chi` as shape a parameter. Details about sampling from the ARGUS distribution can be found in [2]_. %(after_notes)s References ---------- .. [1] "ARGUS distribution", https://en.wikipedia.org/wiki/ARGUS_distribution .. [2] Christoph Baumgarten "Random variate generation by fast numerical inversion in the varying parameter case." Research in Statistics, vol. 1, 2023, doi:10.1080/27684520.2023.2279060. .. versionadded:: 0.19.0 %(example)s cCstdddtjfdgS)NrFrrrZr\r.r.r/r]+szargus_gen._shape_infocCstjddnd||}dt|ttt|}|t|dt| ||d|dWdS1s0YdS)NrrrnrrurK)rErrrrHr)r;r`rrrr.r.r/r+s  zargus_gen._logpdfcCst|||SrCrGr;r`rr.r.r/ra+szargus_gen._pdfcCsd|||SrrrJr.r.r/rb+szargus_gen._cdfcCs"t|td|dt|Sr)rHrErrJr.r.r/rd+sz argus_gen._sfNc st|}|jdkr&|j|||d}nt|j|\}tt|}t|}tj |gdgdggdj st fddt t | dD}|jd||d}||||<qf|d kr|d }|S) Nr)rurrcrdrec3s(|] }|sj|ntdVqdSrCrgrirjr.r/r+sz!argus_gen._rvs..rr.)rErrrnrrr9r]rorprqrrr^rrrs) r;rrrrVrtrurvrUr.rjr/r+s0     zargus_gen._rvscCstt|}tt|}t|}d}||}|dkr| d} ||kr||} |j| d} |j| d} | d} t| | | k}t|}|dkrDt d| |}|||||<||7}qDn(|dkrzt | d}||kr||} |j| d} |j| d} dt|d| | |} | d| dk}t|}|dkrt d| |}|||||<||7}qnv||kr||} |j d| d}||dk}t|}|dkrz||||||<||7}qzt dd||}t ||S) NrrurKrrHrg?r) rrrEr|r9r]r}rrrrrrr)r;rrurr~rr`rrr=rorr:r'rrrDZechirr.r.r/rn ,sR4                zargus_gen._rvs_scalarcCstj|td}t|}ttjd|td|dd|}t|}|dk}||}dd|d|t |||||<||}gd}t ||||<|||dddfS) NrrrrKrg?r) g_1g־rgWBargp|RH?rgE'卡?rg?) rErrrHrrrcrHrrr)r;rrrCrrrrfZcoefr.r.r/rn,s, ( zargus_gen._stats)NN)NN) rirjrkrlr]rrarbrdrrnrr.r.r.r/rI+s(  erIarguszAn Argus Function)rrrGrprqcs`eZdZdZejZddfdd ZddZdd Zd d Z d d Z ddZ fddZ Z S) rv_histograma3 Generates a distribution given by a histogram. This is useful to generate a template distribution from a binned datasample. As a subclass of the `rv_continuous` class, `rv_histogram` inherits from it a collection of generic methods (see `rv_continuous` for the full list), and implements them based on the properties of the provided binned datasample. Parameters ---------- histogram : tuple of array_like Tuple containing two array_like objects. The first containing the content of n bins, the second containing the (n+1) bin boundaries. In particular, the return value of `numpy.histogram` is accepted. density : bool, optional If False, assumes the histogram is proportional to counts per bin; otherwise, assumes it is proportional to a density. For constant bin widths, these are equivalent, but the distinction is important when bin widths vary (see Notes). If None (default), sets ``density=True`` for backwards compatibility, but warns if the bin widths are variable. Set `density` explicitly to silence the warning. .. versionadded:: 1.10.0 Notes ----- When a histogram has unequal bin widths, there is a distinction between histograms that are proportional to counts per bin and histograms that are proportional to probability density over a bin. If `numpy.histogram` is called with its default ``density=False``, the resulting histogram is the number of counts per bin, so ``density=False`` should be passed to `rv_histogram`. If `numpy.histogram` is called with ``density=True``, the resulting histogram is in terms of probability density, so ``density=True`` should be passed to `rv_histogram`. To avoid warnings, always pass ``density`` explicitly when the input histogram has unequal bin widths. There are no additional shape parameters except for the loc and scale. The pdf is defined as a stepwise function from the provided histogram. The cdf is a linear interpolation of the pdf. .. versionadded:: 0.19.0 Examples -------- Create a scipy.stats distribution from a numpy histogram >>> import scipy.stats >>> import numpy as np >>> data = scipy.stats.norm.rvs(size=100000, loc=0, scale=1.5, ... random_state=123) >>> hist = np.histogram(data, bins=100) >>> hist_dist = scipy.stats.rv_histogram(hist, density=False) Behaves like an ordinary scipy rv_continuous distribution >>> hist_dist.pdf(1.0) 0.20538577847618705 >>> hist_dist.cdf(2.0) 0.90818568543056499 PDF is zero above (below) the highest (lowest) bin of the histogram, defined by the max (min) of the original dataset >>> hist_dist.pdf(np.max(data)) 0.0 >>> hist_dist.cdf(np.max(data)) 1.0 >>> hist_dist.pdf(np.min(data)) 7.7591907244498314e-05 >>> hist_dist.cdf(np.min(data)) 0.0 PDF and CDF follow the histogram >>> import matplotlib.pyplot as plt >>> X = np.linspace(-5.0, 5.0, 100) >>> fig, ax = plt.subplots() >>> ax.set_title("PDF from Template") >>> ax.hist(data, density=True, bins=100) >>> ax.plot(X, hist_dist.pdf(X), label='PDF') >>> ax.plot(X, hist_dist.cdf(X), label='CDF') >>> ax.legend() >>> fig.show() N)densitycsb||_||_t|dkr tdt|d|_t|d|_t|jdt|jkr`td|jdd|jdd|_t |j|jd }|dur|rd}t j |t dd d }n|s|j|j|_|jt t|j|j|_t|j|j|_td |jd g|_td |jg|_|jd|d <|_|jd|d <|_tj|i|dS)a5 Create a new distribution using the given histogram Parameters ---------- histogram : tuple of array_like Tuple containing two array_like objects. The first containing the content of n bins, the second containing the (n+1) bin boundaries. In particular, the return value of np.histogram is accepted. density : bool, optional If False, assumes the histogram is proportional to counts per bin; otherwise, assumes it is proportional to a density. For constant bin widths, these are equivalent. If None (default), sets ``density=True`` for backward compatibility, but warns if the bin widths are variable. Set `density` explicitly to silence the warning. rKz)Expected length 2 for parameter histogramrrzbNumber of elements in histogram content and histogram boundaries do not match, expected n and n+1.Nr.zjBin widths are not constant. Assuming `density=True`.Specify `density` explicitly to silence this warning.rTrmrprq) _histogram_densityrrrEr_hpdf_hbins _hbin_widthsZallcloserrrrrZcumsum_hcdfZhstackrprqr7r)r; histogramrMr=rZ bins_varyrr r.r/r,s.  zrv_histogram.__init__cCs|jtj|j|ddS)z& PDF of the histogram r)Zside)rPrEZ searchsortedrQrr.r.r/ra-szrv_histogram._pdfcCst||j|jS)z3 CDF calculated from the histogram )rEinterprQrSrr.r.r/rb-szrv_histogram._cdfcCst||j|jS)zC Percentile function calculated from the histogram )rErUrSrQrr.r.r/rg-szrv_histogram._ppfcCsL|jdd|d|jdd|d|d}t|jdd|S)z$Compute the n-th non-central moment.rNr.)rQrErrP)r;rVZ integralsr.r.r/r"-s4zrv_histogram._munpcCsJt|jdddk|jddftjd}t|jdd||j S)zCompute entropy of distributionrr.rm)rrPrErrrR)r;rr.r.r/r'-s zrv_histogram._entropycs"t}|j|d<|j|d<|S)zF Set the histogram as additional constructor argument rTrM)r7_updated_ctor_paramrNrO)r;dctr r.r/rV/-s   z rv_histogram._updated_ctor_param)rirjrkrlrrrrarbrgrrrVr?r.r.r r/rL,s[0rLcsHeZdZdZddZddZfddZdd Zd d Zd d Z Z S)studentized_range_genuA studentized range continuous random variable. %(before_notes)s See Also -------- t: Student's t distribution Notes ----- The probability density function for `studentized_range` is: .. math:: f(x; k, \nu) = \frac{k(k-1)\nu^{\nu/2}}{\Gamma(\nu/2) 2^{\nu/2-1}} \int_{0}^{\infty} \int_{-\infty}^{\infty} s^{\nu} e^{-\nu s^2/2} \phi(z) \phi(sx + z) [\Phi(sx + z) - \Phi(z)]^{k-2} \,dz \,ds for :math:`x ≥ 0`, :math:`k > 1`, and :math:`\nu > 0`. `studentized_range` takes ``k`` for :math:`k` and ``df`` for :math:`\nu` as shape parameters. When :math:`\nu` exceeds 100,000, an asymptotic approximation (infinite degrees of freedom) is used to compute the cumulative distribution function [4]_ and probability distribution function. %(after_notes)s References ---------- .. [1] "Studentized range distribution", https://en.wikipedia.org/wiki/Studentized_range_distribution .. [2] Batista, Ben Dêivide, et al. "Externally Studentized Normal Midrange Distribution." Ciência e Agrotecnologia, vol. 41, no. 4, 2017, pp. 378-389., doi:10.1590/1413-70542017414047716. .. [3] Harter, H. Leon. "Tables of Range and Studentized Range." The Annals of Mathematical Statistics, vol. 31, no. 4, 1960, pp. 1122-1147. JSTOR, www.jstor.org/stable/2237810. Accessed 18 Feb. 2021. .. [4] Lund, R. E., and J. R. Lund. "Algorithm AS 190: Probabilities and Upper Quantiles for the Studentized Range." Journal of the Royal Statistical Society. Series C (Applied Statistics), vol. 32, no. 2, 1983, pp. 204-210. JSTOR, www.jstor.org/stable/2347300. Accessed 18 Feb. 2021. Examples -------- >>> import numpy as np >>> from scipy.stats import studentized_range >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1) Calculate the first four moments: >>> k, df = 3, 10 >>> mean, var, skew, kurt = studentized_range.stats(k, df, moments='mvsk') Display the probability density function (``pdf``): >>> x = np.linspace(studentized_range.ppf(0.01, k, df), ... studentized_range.ppf(0.99, k, df), 100) >>> ax.plot(x, studentized_range.pdf(x, k, df), ... 'r-', lw=5, alpha=0.6, label='studentized_range pdf') Alternatively, the distribution object can be called (as a function) to fix the shape, location and scale parameters. This returns a "frozen" RV object holding the given parameters fixed. Freeze the distribution and display the frozen ``pdf``: >>> rv = studentized_range(k, df) >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf') Check accuracy of ``cdf`` and ``ppf``: >>> vals = studentized_range.ppf([0.001, 0.5, 0.999], k, df) >>> np.allclose([0.001, 0.5, 0.999], studentized_range.cdf(vals, k, df)) True Rather than using (``studentized_range.rvs``) to generate random variates, which is very slow for this distribution, we can approximate the inverse CDF using an interpolator, and then perform inverse transform sampling with this approximate inverse CDF. This distribution has an infinite but thin right tail, so we focus our attention on the leftmost 99.9 percent. >>> a, b = studentized_range.ppf([0, .999], k, df) >>> a, b 0, 7.41058083802274 >>> from scipy.interpolate import interp1d >>> rng = np.random.default_rng() >>> xs = np.linspace(a, b, 50) >>> cdf = studentized_range.cdf(xs, k, df) # Create an interpolant of the inverse CDF >>> ppf = interp1d(cdf, xs, fill_value='extrapolate') # Perform inverse transform sampling using the interpolant >>> r = ppf(rng.uniform(size=1000)) And compare the histogram: >>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2) >>> ax.legend(loc='best', frameon=False) >>> plt.show() cCs|dk|dk@Srtr.)r;rorr.r.r/rW-szstudentized_range_gen._argcheckcCs0tdddtjfd}tdddtjfd}||gS)NroFrrrrrZ)r;rrOr.r.r/r]-sz!studentized_range_gen._shape_infocstj|ddS)N)rKrrrdrer r.r/r -szstudentized_range_gen._fitstartcsJd|\fdd}t|dd}tj||||tjddS)NZ_studentized_range_momentc szt||}||||g}t|tjtj}t t|}tj tj fdtj ffg}t ddd}t j |||ddS)Nrr-q=rrrangesopts)r_studentized_range_pdf_logconstrErrrrrr rr[dictr nquad) rror log_constargusr_datarr\r]rr cython_symbolr.r/_single_moment-s   z3studentized_range_gen._munp.._single_momentrrrr.)rvrE frompyfuncrr)r;rrorrfufuncr.rdr/r-s   zstudentized_range_gen._munpcCs2dd}t|dd}tj||||tjddS)Nc Ss|dkrTd}t||}||||g}t|tjtj}tj tjfdtjfg}n2d}||g}t|tjtj}tj tjfg}t t||}t ddd} t j ||| ddS) N順Z_studentized_range_pdfrZ!_studentized_range_pdf_asymptoticrrYrZr[)rr^rErrrrrr[r rr_r r` rfrorrerarbrcr\rr]r.r.r/ _single_pdf-s   z/studentized_range_gen._pdf.._single_pdfrrrr.)rErgrr)r;r`rorrkrhr.r.r/ra-szstudentized_range_gen._pdfcCs<dd}t|dd}ttj||||tjddddS)Nc Ss|dkrTd}t||}||||g}t|tjtj}tj tjfdtjfg}n2d}||g}t|tjtj}tj tjfg}t t||}t ddd} t j ||| ddS) NriZ_studentized_range_cdfrZ!_studentized_range_cdf_asymptoticrrYrZr[)rZ_studentized_range_cdf_logconstrErrrrrr[r rr_r r`rjr.r.r/ _single_cdf-s   z/studentized_range_gen._cdf.._single_cdfrrrr.r)rErgrjrr)r;r`rorrlrhr.r.r/rb-szstudentized_range_gen._cdf) rirjrkrlrWr]r rrarbr?r.r.r r/rX9-sn rXstudentized_range)rrrprqcsXeZdZdZddZddZddZdd Zd d Zd d Z e e fddZ Z S)rel_breitwigner_genaA relativistic Breit-Wigner random variable. %(before_notes)s See Also -------- cauchy: Cauchy distribution, also known as the Breit-Wigner distribution. Notes ----- The probability density function for `rel_breitwigner` is .. math:: f(x, \rho) = \frac{k}{(x^2 - \rho^2)^2 + \rho^2} where .. math:: k = \frac{2\sqrt{2}\rho^2\sqrt{\rho^2 + 1}} {\pi\sqrt{\rho^2 + \rho\sqrt{\rho^2 + 1}}} The relativistic Breit-Wigner distribution is used in high energy physics to model resonances [1]_. It gives the uncertainty in the invariant mass, :math:`M` [2]_, of a resonance with characteristic mass :math:`M_0` and decay-width :math:`\Gamma`, where :math:`M`, :math:`M_0` and :math:`\Gamma` are expressed in natural units. In SciPy's parametrization, the shape parameter :math:`\rho` is equal to :math:`M_0/\Gamma` and takes values in :math:`(0, \infty)`. Equivalently, the relativistic Breit-Wigner distribution is said to give the uncertainty in the center-of-mass energy :math:`E_{\text{cm}}`. In natural units, the speed of light :math:`c` is equal to 1 and the invariant mass :math:`M` is equal to the rest energy :math:`Mc^2`. In the center-of-mass frame, the rest energy is equal to the total energy [3]_. %(after_notes)s :math:`\rho = M/\Gamma` and :math:`\Gamma` is the scale parameter. For example, if one seeks to model the :math:`Z^0` boson with :math:`M_0 \approx 91.1876 \text{ GeV}` and :math:`\Gamma \approx 2.4952\text{ GeV}` [4]_ one can set ``rho=91.1876/2.4952`` and ``scale=2.4952``. To ensure a physically meaningful result when using the `fit` method, one should set ``floc=0`` to fix the location parameter to 0. References ---------- .. [1] Relativistic Breit-Wigner distribution, Wikipedia, https://en.wikipedia.org/wiki/Relativistic_Breit-Wigner_distribution .. [2] Invariant mass, Wikipedia, https://en.wikipedia.org/wiki/Invariant_mass .. [3] Center-of-momentum frame, Wikipedia, https://en.wikipedia.org/wiki/Center-of-momentum_frame .. [4] M. Tanabashi et al. (Particle Data Group) Phys. Rev. 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