U Gv fG@sddlZddlmZddlmZmZddlZddlZddl m Z ddl m Z m Z mZddlmZddlmZddlmZddlmZddlmmZdd lmZmZd d lmZd d lm Z!m"Z#d d l$m%Z%m&Z&m'Z'm(Z(m)Z)m*Z*m+Z+d dl,m-Z-m.Z.m/Z/d dl0m1Z1m2Z2m3Z3m4Z4m5Z5m6Z6ddl7m8m9Z9ddl:m;Z;ddlm8Z8ddZ?ddZ@dddZAGddde'ZBeBddddZCGddde'ZDeDdddd d!ZEGd"d#d#e'ZFeFdd$d%ZGeHd&ejIZJeKeJZLd'd(ZMd)d*ZNd+d,ZOd-d.ZPd/d0ZQd1d2ZRd3d4ZSd5d6ZTGd7d8d8e'ZUeUd9d:ZVGd;d<dd?d?e'ZYeYejI d@ejId@dAdZZGdBdCdCe'Z[e[dddDdZ\GdEdFdFe]Z^GdGdHdHe=Z_dIdJZ`dKdLZaGdMdNdNe'ZbebdddOdZcGdPdQdQe'ZdedddRd%ZeGdSdTdTe'ZfefdddUdZgGdVdWdWe'ZhehddXd%ZiGdYdZdZe'Zjejdd[d%ZkGd\d]d]ehZleldd^d%ZmGd_d`d`e'Znendad:ZoGdbdcdce'Zpepdddd%ZqGdedfdfe'Zrerddgd%ZsGdhdidie'ZtetejI ejIdjdZuGdkdldle'Zvevdmd:ZwGdndodoe'Zxexdpd:ZyGdqdrdre'Zzezddsd%Z{Gdtdudue'Z|e|dvd:Z}Gdwdxdxe'Z~e~ddyd%ZGdzd{d{e'Zedd|d%ZGd}d~d~e'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%ZGddde'Zeddd%ZGdd„de'Zeddd%ZGddńde'Zedd:ZGddȄde'ZeddddZGdd˄de'Zeddd%ZGdd΄de'Zeddd%ZGddфde'Zeddd%ZGddԄde'Zedd:ZGddׄde'Zeddd%ZGddڄde'ZeddddZGdd݄de'Zedd:ZGddde'Zedd:ZGddde'Zedd:ZddZGddde'Zeddd%ZGddde'ZedddZGddde'Zedd:ZGddde'Zedd:ZGddde'Zeddd%ZddZGddde'Zeddd%ZGddde'ZdZGdddeԃZeddd%Zeddd%Zؐdddddddd d d d d dddddddgZeD]8ZeeeeڃdeڛdeڛeաZeeee݃ qvGddde'Zeddd%ZGddde'Zeddd%ZGddde'Zedd:ZGd d!d!e'Zedd"d%ZGd#d$d$e'Zed%d:ZGd&d'd'e'Zedd(d%Zd)d*ZGd+d,d,e'Zedd-d%ZGd.d/d/e'Zedd0d%ZGd1d2d2e'Zed3d:ZGd4d5d5e'Zed6d:ZGd7d8d8e'Zedd9d%ZGd:d;d;e'Zeddd>e'Zed?d:ZGd@dAdAe'ZedddBdZGdCdDdDe'ZeddEd%ZGdFdGdGe'ZedHd:ZGdIdJdJe'ZedKddLdZGdMdNdNe'ZeddOd%ZGdPdQdQe'ZedRd:ZedSd:ZGdTdUdUe'ZeddVd%ZGdWdXdXe'Z e ddYd%Z GdZd[d[e'Z e dKdd\dZ Gd]d^d^e'Z e d_d:ZGd`dadae'Zedbd:ZGdcdddde'ZedddedZedddfdZej rdge_Gdhdidie'ZedddjdZGdkdldle'Zeddmd%ZdndoZdpdqZdrdsZGdtdudue'Zedvd dwZGdxdydye'Zeddzd%ZGd{d|d|e'Z e d}d:Z!Gd~dde^Z"Gddde'Z#e#ddddZ$Gddde'Z%e%dd:Z&e%ejI ejIddZ'GdddeZ(e(ddd%Z)Gddde'Z*e*dd&ejIddZ+Gddde'Z,e,dd:Z-Gddde'Z.e.ddd%Z/Gddde'Z0e0dddZ1ddZ2Gddde'Z3e3dddddZ4Gddde'Z5Gddde'Z6e6ddej7dZ8e9e:;<Z=e%e=e'\Z>Z?e>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_EULER_ZETA3_SQRT_PI_SQRT_2_OVER_PI_LOG_SQRT_2_OVER_PI) 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 optimizermethodUnknown arguments: %s.)pop TypeError)kwdsr-B/tmp/pip-unpacked-wheel-96ln3f52/scipy/stats/_continuous_distns.py_remove_optimizer_parameters&s    r/cstfdd}|S)NcsB|dd}|dkr.tt||j||S|f||SdS)Nr(mle)getlowersupertypefit)selfargsr,r(funr-r.wrapper<sz _call_super_mom..wrapper)r)r9r:r-r8r._call_super_mom9sr;csV|p |d}||}fdd}|||sR|d9}||}d}t|r t|q |S)Nrcst|t|kSNnpsignlbrackrbrackr8r-r.interval_contains_rootMsz1_get_left_bracket..interval_contains_rootzVThe solver could not find a bracket containing a root to an MLE first order condition.)r>isinfFitSolverError)r9rBrAZdiffrCmsgr-r8r._get_left_bracketFs     rHc@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@SNrr>roundr6nr-r-r. _argcheckszksone_gen._argcheckcCstdddtjfdgSNrNTrTFrr>infr6r-r-r. _shape_infoszksone_gen._shape_infocCst|| Sr<)scuZ _smirnovpr6xrNr-r-r._pdfszksone_gen._pdfcCs t||Sr<)rVZ _smirnovcrWr-r-r._cdfszksone_gen._cdfcCs t||Sr<)scZsmirnovrWr-r-r._sfsz ksone_gen._sfcCs t||Sr<)rVZ _smirnovcir6qrNr-r-r._ppfszksone_gen._ppfcCs t||Sr<)r[Zsmirnovir]r-r-r._isfszksone_gen._isfN) __name__ __module__ __qualname____doc__rOrUrYrZr\r_r`r-r-r-r.rI]s%rI?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@SrJrKrMr-r-r.rOszkstwo_gen._argcheckcCstdddtjfdgSrPrRrTr-r-r.rUszkstwo_gen._shape_infocCs dt|ts|nt|dfSN?rf) isinstancerr>Z asanyarrayrMr-r-r. _get_supportszkstwo_gen._get_supportcCs t||Sr<)rrWr-r-r.rYszkstwo_gen._pdfcCs t||Sr<rrWr-r-r.rZszkstwo_gen._cdfcCst||ddSNFcdfrprWr-r-r.r\sz kstwo_gen._sfcCst||ddS)NTrrrr]r-r-r.r_szkstwo_gen._ppfcCst||ddSrqrtr]r-r-r.r`szkstwo_gen._isfN) rarbrcrdrOrUrorYrZr\r_r`r-r-r-r.rks$rkkstwo)momtyperhrirjc@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 cCsgSr<r-rTr-r-r.rUszkstwobign_gen._shape_infocCs t| Sr<)rVZ_kolmogpr6rXr-r-r.rYszkstwobign_gen._pdfcCs t|Sr<)rVZ_kolmogcrxr-r-r.rZ szkstwobign_gen._cdfcCs t|Sr<)r[Z kolmogorovrxr-r-r.r\ szkstwobign_gen._sfcCs t|Sr<)rVZ _kolmogcir6r^r-r-r.r_szkstwobign_gen._ppfcCs t|Sr<)r[Zkolmogiryr-r-r.r`szkstwobign_gen._isfN) rarbrcrdrUrYrZr\r_r`r-r-r-r.rws$rw kstwobign)rhrjrDcCst|d dtSNrD@)r>exp _norm_pdf_CrXr-r-r. _norm_pdf#srcCs|d dtSr{)_norm_pdf_logCrr-r-r. _norm_logpdf'srcCs t|Sr<)r[ndtrrr-r-r. _norm_cdf+srcCs t|Sr<)r[log_ndtrrr-r-r. _norm_logcdf/srcCs t|Sr<r[ndtrir^r-r-r. _norm_ppf3srcCs t| Sr<rrr-r-r._norm_sf7srcCs t| Sr<rrr-r-r. _norm_logsf;srcCs t| Sr<rrr-r-r. _norm_isf?src@seZdZdZddZd!ddZddZd d Zd d Zd dZ ddZ ddZ ddZ ddZ ddZddZeeeddddZdd ZdS)"norm_genaA normal continuous random variable. The location (``loc``) keyword specifies the mean. The scale (``scale``) keyword specifies the standard deviation. %(before_notes)s Notes ----- The probability density function for `norm` is: .. math:: f(x) = \frac{\exp(-x^2/2)}{\sqrt{2\pi}} for a real number :math:`x`. %(after_notes)s %(example)s cCsgSr<r-rTr-r-r.rUZsznorm_gen._shape_infoNcCs ||Sr<)standard_normalr6size random_stater-r-r._rvs]sz norm_gen._rvscCst|Sr<rrxr-r-r.rY`sz norm_gen._pdfcCst|Sr<)rrxr-r-r._logpdfdsznorm_gen._logpdfcCst|Sr<rrxr-r-r.rZgsz norm_gen._cdfcCst|Sr<rrxr-r-r._logcdfjsznorm_gen._logcdfcCst|Sr<)rrxr-r-r.r\msz norm_gen._sfcCst|Sr<)rrxr-r-r._logsfpsznorm_gen._logsfcCst|Sr<rryr-r-r.r_ssz norm_gen._ppfcCst|Sr<rryr-r-r.r`vsz norm_gen._isfcCsdS)N)rerfrerer-rTr-r-r.rysznorm_gen._statscCsdtdtjdSNrmrDrr>logpirTr-r-r._entropy|sznorm_gen._entropya} For the normal distribution, method of moments and maximum likelihood estimation give identical fits, and explicit formulas for the estimates are available. This function uses these explicit formulas for the maximum likelihood estimation of the normal distribution parameters, so the `optimizer` and `method` arguments are ignored. ZnotescKs|dd}|dd}t||dk r8|dk r8tdt|}t|sXtd|dkrj|}n|}|dkrt||d}n|}||fS)Nflocfscale3All parameters fixed. There is nothing to optimize.$The data contains non-finite values.rD) r*r/ ValueErrorr>asarrayisfiniteallmeansqrt)r6datar,rrr%r&r-r-r.r5s    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 rDrrreN)r[Z factorial2rMr-r-r._munps znorm_gen._munp)NN)rarbrcrdrUrrYrrZrr\rr_r`rrr;rrr5rr-r-r-r.rCs"   rnorm)rjc@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 cCstdddtjfdgSNrhFrFFrRrTr-r-r.rUszalpha_gen._shape_infocCs$d|dt|t|d|SNrfrD)rrr6rXrhr-r-r.rYszalpha_gen._pdfcCs,dt|t|d|tt|S)Nrf)r>rrrrr-r-r.rszalpha_gen._logpdfcCst|d|t|SNrfrrr-r-r.rZszalpha_gen._cdfc Cs dt|t|t|Sr)r>rr[rrr6r^rhr-r-r.r_szalpha_gen._ppfcCstjgdtjgdSNrDr>rSnanr6rhr-r-r.rszalpha_gen._statsN) rarbrcrdr_open_support_mask _support_maskrUrYrrZr_rr-r-r-r.rsralphac@s@eZdZdZddZddZddZdd Zd d Zd d Z 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 cCsgSr<r-rTr-r-r.rUszanglit_gen._shape_infocCstd|Sr)r>cosrxr-r-r.rYszanglit_gen._pdfcCst|tjddS)Nr|r>sinrrxr-r-r.rZ szanglit_gen._cdfcCstt|tjdSNr)r>arcsinrrryr-r-r.r_ szanglit_gen._ppfcCs>dtjtjddddtjddtjtjddfS) Nrermrr`rDr>rrTr-r-r.rszanglit_gen._statscCsdtdSNrrDr>rrTr-r-r.rszanglit_gen._entropyN rarbrcrdrUrYrZr_rrr-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 cCsgSr<r-rTr-r-r.rU-szarcsine_gen._shape_infoc Cs@tjdd*dtjt|d|W5QRSQRXdS)Nignoredividerfr)r>errstaterrrxr-r-r.rY0szarcsine_gen._pdfcCsdtjtt|SNr|)r>rrrrxr-r-r.rZ5szarcsine_gen._cdfcCsttjd|dSrrryr-r-r.r_8szarcsine_gen._ppfcCsd}d}d}d}||||fS)Nrmg?rr-r6mumu2g1g2r-r-r.r;s zarcsine_gen._statscCsdS)Ngοr-rTr-r-r.rBszarcsine_gen._entropyNrr-r-r-r.rsrarcsinec@seZdZdZddZdS) FitDataErrorz=Raised when input data is inconsistent with fixed parameters.cCsdj|||df|_dS)NzInvalid values in `data`. Maximum likelihood estimation with {distr!r} requires that {lower!r} < (x - loc)/scale < {upper!r} for each x in `data`.)distrr2upper)formatr7)r6rr2rr-r-r.__init__Ns zFitDataError.__init__Nrarbrcrdrr-r-r-r.rIsrc@seZdZdZddZdS)rFzN Raised when a solver fails to converge while fitting a distribution. cCs d}||dd7}|f|_dS)Nz1Solver for the MLE equations failed to converge:  )replacer7)r6mesgemsgr-r-r.r]szFitSolverError.__init__Nrr-r-r-r.rFWsrFcCs*t||}||| t|}|Sr<r[psi)rhrirNs1psiabfuncr-r-r. _beta_mle_acsrcCsJ|\}}t||}||| t|||| t|g}|Sr<r)thetarNrs2rhrirrr-r-r. _beta_mle_abls rcseZdZdZddZdddZddZd d Zd d Zd dZ ddZ ddZ ddZ fddZ eeeddfddZZS)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}||gSNrhFrrrirRr6iaibr-r-r.rUszbeta_gen._shape_infoNcCs||||Sr<beta)r6rhrirrr-r-r.rsz beta_gen._rvscCst|||Sr<)_boostZ _beta_pdfr6rXrhrir-r-r.rYsz beta_gen._pdfcCs6t|d| t|d|}|t||8}|Sr)r[xlog1pyxlogybetaln)r6rXrhrilPxr-r-r.rs"zbeta_gen._logpdfcCst|||Sr<)rZ _beta_cdfrr-r-r.rZsz beta_gen._cdfcCst|||Sr<)rZ_beta_sfrr-r-r.r\sz beta_gen._sfc Cs@t.d}tjd|dt|||W5QRSQRXdS)Nz!overflow encountered in _beta_isfrmessage)warningscatch_warningsfilterwarningsrZ _beta_isf)r6rXrhrirr-r-r.r`s z beta_gen._isfc Cs@t.d}tjd|dt|||W5QRSQRXdS)Nz!overflow encountered in _beta_ppfrr)rrrrZ _beta_ppf)r6r^rhrirr-r-r.r_s z beta_gen._ppfcCs,t||t||t||t||fSr<)rZ _beta_meanZ_beta_varianceZ_beta_skewnessZ_beta_kurtosis_excessr6rhrir-r-r.rs     zbeta_gen._statscsBt|t|fdd}t|d\}}tj|||fdS)Ncs|\}}d||t||d||dt||}|d|dd|d|d|dd|||d}|||||d||d}|d9}||gS)NrDrr>r)rXrhriskkurrr-r.rs 8@$z beta_gen._fitstart..func)rfrfr7)rrr fsolver3 _fitstart)r6rrrhri __class__rr.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`. rcs8|dd}|dd}|dks(|dkrrrZravelanyrrr rrlenrsumrFr[log1pvarr)r6rr7r,rrr rxbarrirhrinfoierrrrfacr r-r.r5sd               z beta_gen.fit)NN)rarbrcrdrUrrYrrZr\r`r_rr r;rrr5 __classcell__r-r-r r.rzs   rrc@sHeZdZdZejZddZdddZddZ d d Z d d Z d dZ dS) 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. %(after_notes)s %(example)s cCs0tdddtjfd}tdddtjfd}||gSrrRrr-r-r.rUUszbetaprime_gen._shape_infoNcCs(tj|||d}tj|||d}||SNrr)gammarvs)r6rhrirru1u2r-r-r.rZszbetaprime_gen._rvscCst||||Sr<r>r}rrr-r-r.rY_szbetaprime_gen._pdfcCs,t|d|t|||t||Sr)r[rrrrr-r-r.rcszbetaprime_gen._logpdfcCst|||d|Sr)r[Zbetaincrr-r-r.rZfszbetaprime_gen._cdfcCs|dkr$t|dk||dtjS|dkrXt|dk||d|d|dtjS|dkrt|dk||d|d|d|d|dtjS|dkrt|dk||d|d|d|d|d|d|dtjStdS) Nrfrr|rD@r@r)r>whererSNotImplementedErrorr6rNrhrir-r-r.ris.    * zbetaprime_gen._munp)NN) rarbrcrdrrrrUrrYrrZrr-r-r-r.r<s 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 cCstdddtjfdgSNcFrrrRrTr-r-r.rUszbradford_gen._shape_infocCs|||dt|Srr[rr6rXr,r-r-r.rYszbradford_gen._pdfcCst||t|Sr<r-r.r-r-r.rZszbradford_gen._cdfcCst|t||Sr<r[expm1rr6r^r,r-r-r.r_szbradford_gen._ppfmvcCsxtd|}||||}|d|d|d|||}d}d}d|krtdd||d|||dd||||dd}|t|||dd|d||dd|}d |krl|d|d|d|d d d||||d |dd|||d|d d|d}|d|||dd|d}||||fS)Nrfr|rDs rrkrr)r>rr)r6r,momentsr6rrrrr-r-r.rs $F: B $zbradford_gen._statscCs$td|}|dt||SNrr|r)r6r,r6r-r-r.rszbradford_gen._entropyN)r2rr-r-r-r.r*s r*bradfordc@s`eZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ dS)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}||gSNr,FrrdrRr6icidr-r-r.rUszburr_gen._shape_infocCs8t|dk|||gddddd}|jdkr4|dS|S)NrcSs$|||||dd||SrJr-Zx_Zc_Zd_r-r-r.zburr_gen._pdf..cSs,|||| dd|| |dSNrfrr-rBr-r-r.rCsf2r-rndimr6rXr,r>outputr-r-r.rYs z burr_gen._pdfcCs8t|dk|||gddddd}|jdkr4|dS|S)NrcSs>t|t|t||d||dt||SrJ)r>rr[rrrBr-r-r.rCs&z"burr_gen._logpdf..cSs<t|t|t| d|t|d|| SrJr>rr[rrrBr-r-r.rCsrFr-rHrJr-r-r.rs zburr_gen._logpdfcCsd|| | SrJr-r6rXr,r>r-r-r.rZsz burr_gen._cdfcCst|| | Sr<r-rMr-r-r.r szburr_gen._logcdfcCst||||Sr<r>r}rrMr-r-r.r\sz burr_gen._sfcCstd|| |  SrJr>rrMr-r-r.rszburr_gen._logsfcCs|d|dd|SNrr-r6r^r,r>r-r-r.r_sz burr_gen._ppfc Cstdddd|}t||d||\}}}}t|dk|tj}||d} t|dk| tj} t|dk||||| fdd tjd } t|d k|||||| fd d tjd } t|d kr| | | | fS|| | | fS)NrrrfrDr|r$cSs*|d||d|dt|dS)NrrDr)r,e1e2e3mu2_if_cr-r-r.rC!rDz!burr_gen._stats.. fillvaluer%cSs8|d||d||dd|d|ddS)NrrrDrr-)r,rTrUrVe4rWr-r-r.rC&sr) r>arangereshaper[rr&rrrIitem) r6r,r>ncrTrUrVrZrrWrrrr-r-r.rs(   zburr_gen._statscs\ddt|t|t|}}}t||k||k@||k@|||ffddtjS)NcSs$d||}|td|||Srr[r)rNr,r>r^r-r-r.__munp.s zburr_gen._munp..__munpcs |||Sr<r-)r,r>rNZ_burr_gen__munpr-r.rC3rDz burr_gen._munp..)r>rrr)r6rNr,r>r-rar.r-s "  zburr_gen._munpN)rarbrcrdrUrYrrZrr\rr_rrr-r-r-r.r<s0  r<burrc@sXeZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ dS) burr12_gena}A Burr (Type XII) continuous random variable. %(before_notes)s See Also -------- fisk : a special case of either `burr` or `burr12` with ``d=1`` burr : Burr Type III distribution Notes ----- The probability density function for `burr12` is: .. math:: f(x; c, d) = c d \frac{x^{c-1}} {(1 + x^c)^{d + 1}} for :math:`x >= 0` and :math:`c, d > 0`. `burr12` takes ``c`` and ``d`` as shape parameters for :math:`c` and :math:`d`. This is the PDF corresponding to the twelfth CDF given in Burr's list; specifically, it is equation (20) in Burr's paper [1]_. %(after_notes)s The Burr type 12 distribution is also sometimes referred to as the Singh-Maddala distribution from NIST [2]_. References ---------- .. [1] Burr, I. W. "Cumulative frequency functions", Annals of Mathematical Statistics, 13(2), pp 215-232 (1942). .. [2] https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/b12pdf.htm .. [3] "Burr distribution", https://en.wikipedia.org/wiki/Burr_distribution %(example)s cCs0tdddtjfd}tdddtjfd}||gSr=rRr?r-r-r.rUgszburr12_gen._shape_infocCst||||Sr<r#rMr-r-r.rYlszburr12_gen._pdfcCs:t|t|t|d|t| d||SrJrLrMr-r-r.rpszburr12_gen._logpdfcCst|||| Sr<)r[r0rrMr-r-r.rZsszburr12_gen._cdfcCstd|||  SrJr-rMr-r-r.rvszburr12_gen._logcdfcCst||||Sr<rNrMr-r-r.r\yszburr12_gen._sfcCst| ||Sr<r[rrMr-r-r.r|szburr12_gen._logsfcCs"td|t| d|S)Nrr/rRr-r-r.r_szburr12_gen._ppfcCs$d||}|td|||Srr_)r6rNr,r>r^r-r-r.rs zburr12_gen._munpN) rarbrcrdrUrYrrZrr\rr_rr-r-r-r.rc:s,rcburr12c@sheZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ ddZdS)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``. %(after_notes)s %(example)s cCstdddtjfdgSr+rRrTr-r-r.rUszfisk_gen._shape_infocCst||dSr)rbrYr.r-r-r.rYsz fisk_gen._pdfcCst||dSr)rbrZr.r-r-r.rZsz fisk_gen._cdfcCst||dSr)rbr\r.r-r-r.r\sz fisk_gen._sfcCst||dSr)rbrr.r-r-r.rszfisk_gen._logpdfcCst||dSr)rbrr.r-r-r.rszfisk_gen._logcdfcCst||dSr)rbrr.r-r-r.rszfisk_gen._logsfcCst||dSr)rbr_r.r-r-r.r_sz fisk_gen._ppfcCst||dSr)rbrr6rNr,r-r-r.rszfisk_gen._munpcCs t|dSr)rbrr6r,r-r-r.rszfisk_gen._statscCsdt|Srrrir-r-r.rszfisk_gen._entropyN)rarbrcrdrUrYrZr\rrrr_rrrr-r-r-r.rgs&rgfiskc@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 cCsgSr<r-rTr-r-r.rUszcauchy_gen._shape_infocCsdtjd||Srrrxr-r-r.rYszcauchy_gen._pdfcCsddtjt|Srlr>rarctanrxr-r-r.rZszcauchy_gen._cdfcCsttj|tjdSrr>tanrryr-r-r.r_szcauchy_gen._ppfcCsddtjt|Srlrlrxr-r-r.r\szcauchy_gen._sfcCsttjdtj|Srrnryr-r-r.r`szcauchy_gen._isfcCstjtjtjtjfSr<r>rrTr-r-r.rszcauchy_gen._statscCstdtjSrrrTr-r-r.rszcauchy_gen._entropyNcCs(t|dddg\}}}|||dfS)N2KrDr>Z percentile)r6rr7p25p50p75r-r-r.r szcauchy_gen._fitstart)N) rarbrcrdrUrYrZr_r\r`rrr r-r-r-r.rksrkcauchyc@sZeZdZdZddZdddZddZd d Zd d Zd dZ 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 cCstdddtjfdgSNdfFrrrRrTr-r-r.rU0szchi_gen._shape_infoNcCsttj|||dSr)r>rchi2r r6r{rrr-r-r.r3sz chi_gen._rvscCst|||Sr<r#r6rXr{r-r-r.rY6sz chi_gen._pdfcCsJtddtd|td|}|t|d|d|dS)NrDrmrf)r>rr[gammalnr)r6rXr{lr-r-r.r<s*zchi_gen._logpdfcCstd|d|dSNrmrDr[gammaincr~r-r-r.rZ@sz chi_gen._cdfcCstd|d|dSrr[ gammainccr~r-r-r.r\Csz chi_gen._sfcCstdtd||SNrDrmr>rr[ gammaincinvr6r^r{r-r-r.r_Fsz chi_gen._ppfcCstdtd||Srr>rr[ gammainccinvrr-r-r.r`Isz chi_gen._isfcCstdtt|ddt|d}|||}d|d|dd|tt|d}d|d|d|d d |dd|d}|t|d}||||fS) NrDr|rmr$r?rfrr)r>rr}r[rrpowerr6r{rrrrr-r-r.rLs 0 .4zchi_gen._stats)NN) rarbrcrdrUrrYrrZr\r_r`rr-r-r-r.rys rychic@sZeZdZdZddZdddZddZd d Zd d Zd dZ ddZ ddZ ddZ dS)chi2_genaA chi-squared continuous random variable. For the noncentral chi-square distribution, see `ncx2`. %(before_notes)s See Also -------- ncx2 Notes ----- The probability density function for `chi2` is: .. math:: f(x, k) = \frac{1}{2^{k/2} \Gamma \left( k/2 \right)} x^{k/2-1} \exp \left( -x/2 \right) for :math:`x > 0` and :math:`k > 0` (degrees of freedom, denoted ``df`` in the implementation). `chi2` takes ``df`` as a shape parameter. The chi-squared distribution is a special case of the gamma distribution, with gamma parameters ``a = df/2``, ``loc = 0`` and ``scale = 2``. %(after_notes)s %(example)s cCstdddtjfdgSrzrRrTr-r-r.rUzszchi2_gen._shape_infoNcCs |||Sr<)Z chisquarer}r-r-r.r}sz chi2_gen._rvscCst|||Sr<r#r~r-r-r.rYsz chi2_gen._pdfcCs<t|dd||dt|dtd|dS)Nr|rrD)r[rrr>rr~r-r-r.rszchi2_gen._logpdfcCs t||Sr<)r[Zchdtrr~r-r-r.rZsz chi2_gen._cdfcCs t||Sr<)r[Zchdtrcr~r-r-r.r\sz chi2_gen._sfcCs t||Sr<)r[Zchdtrir6pr{r-r-r.r`sz chi2_gen._isfcCsdt|d|Srr[rrr-r-r.r_sz chi2_gen._ppfcCs2|}d|}dtd|}d|}||||fS)NrDr|(@rrr-r-r.rs zchi2_gen._stats)NN) rarbrcrdrUrrYrrZr\r`r_rr-r-r-r.rXs! rr|c@sXeZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ dS) cosine_gena\A cosine continuous random variable. %(before_notes)s Notes ----- The cosine distribution is an approximation to the normal distribution. The probability density function for `cosine` is: .. math:: f(x) = \frac{1}{2\pi} (1+\cos(x)) for :math:`-\pi \le x \le \pi`. %(after_notes)s %(example)s cCsgSr<r-rTr-r-r.rUszcosine_gen._shape_infocCsdtjdt|S)Nrmrr>rrrxr-r-r.rYszcosine_gen._pdfcCs(t|}t|dk|fddtj dS)NrecSst|tdtjSr)r>rrrr,r-r-r.rCrDz$cosine_gen._logpdf..rX)r>rrrSr.r-r-r.rs   zcosine_gen._logpdfcCs t|Sr<rVZ _cosine_cdfrxr-r-r.rZszcosine_gen._cdfcCs t| Sr<rrxr-r-r.r\szcosine_gen._sfcCs t|Sr<rVZ_cosine_invcdfr6rr-r-r.r_szcosine_gen._ppfcCs t| Sr<rrr-r-r.r`szcosine_gen._isfcCsBdtjtjddddtjdddtjtjdd fS) Nrer$r|rZ@rrDrrTr-r-r.rszcosine_gen._statscCstdtjdS)NrrfrrTr-r-r.rszcosine_gen._entropyN rarbrcrdrUrYrrZr\r_r`rrr-r-r-r.rsrcosinec@sReZdZdZddZdddZddZd d Zd d Zd dZ ddZ ddZ dS) dgamma_genaA double gamma continuous random variable. %(before_notes)s Notes ----- The probability density function for `dgamma` is: .. math:: f(x, a) = \frac{1}{2\Gamma(a)} |x|^{a-1} \exp(-|x|) for a real number :math:`x` and :math:`a > 0`. :math:`\Gamma` is the gamma function (`scipy.special.gamma`). `dgamma` takes ``a`` as a shape parameter for :math:`a`. %(after_notes)s %(example)s cCstdddtjfdgSrrRrTr-r-r.rUszdgamma_gen._shape_infoNcCs2|j|d}tj|||d}|t|dkddSNrrrmrre)uniformrr r>r&)r6rhrruZgmr-r-r.rs zdgamma_gen._rvscCs2t|}ddt|||dt| Sr)absr[rr>r}r6rXrhaxr-r-r.rYszdgamma_gen._pdfcCs0t|}t|d||tdt|Sr)rr[rr>rrrr-r-r.rszdgamma_gen._logpdfcCs.dt|t|}t|dkd|d|SNrmrr[rrr>r&r6rXrhrr-r-r.rZszdgamma_gen._cdfcCs.dt|t|}t|dkd|d|Srrrr-r-r.r\szdgamma_gen._sfcCs0t|dtd|d}t|dk|| SNrrDrm)r[rrr>r&)r6r^rhrr-r-r.r_szdgamma_gen._ppfcCs,||d}d|d|d|d|dfS)Nrfrer|r$r-)r6rhrr-r-r.r s zdgamma_gen._stats)NN rarbrcrdrUrrYrrZr\r_rr-r-r-r.rs rdgammac@sReZdZdZddZdddZddZd d Zd d Zd dZ ddZ ddZ dS) 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 cCstdddtjfdgSr+rRrTr-r-r.rU(szdweibull_gen._shape_infoNcCs2|j|d}tj|||d}|t|dkddSr)r weibull_minr r>r&)r6r,rrrwr-r-r.r+s zdweibull_gen._rvscCs0t|}|d||dt|| }|SNr|rf)rr>r})r6rXr,rPxr-r-r.rY0s$zdweibull_gen._pdfcCs4t|}t|tdt|d|||Sr)rr>rr[r)r6rXr,rr-r-r.r6szdweibull_gen._logpdfcCs.dtt|| }t|dkd||S)Nrmrr)r>r}rr&)r6rXr,ZCx1r-r-r.rZ:szdweibull_gen._cdfcCsFdt|dk|d|}tt| d|}t|dk|| S)Nr|rmrf)r>r&rr)r6r^r,rr-r-r.r_>szdweibull_gen._ppfcCs"d|dtdd||S)NrrDrfr[rrhr-r-r.rCszdweibull_gen._munpcCsdSN)rNrNr-rir-r-r.rIszdweibull_gen._stats)NN) rarbrcrdrUrrYrrZr_rrr-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 cCsgSr<r-rTr-r-r.rUkszexpon_gen._shape_infoNcCs ||Sr<)standard_exponentialrr-r-r.rnszexpon_gen._rvscCs t| Sr<r>r}rxr-r-r.rYqszexpon_gen._pdfcCs| Sr<r-rxr-r-r.ruszexpon_gen._logpdfcCst|  Sr<r[r0rxr-r-r.rZxszexpon_gen._cdfcCst|  Sr<r-ryr-r-r.r_{szexpon_gen._ppfcCs t| Sr<rrxr-r-r.r\~sz expon_gen._sfcCs| Sr<r-rxr-r-r.rszexpon_gen._logsfcCs t| Sr<rryr-r-r.r`szexpon_gen._isfcCsdS)N)rfrfr|@r-rTr-r-r.rszexpon_gen._statscCsdSrr-rTr-r-r.rszexpon_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||dk rL|dk rLtdt|}t|sltd| }|dkr|}n|}||krt d|tj d|dkr| |}n|}t |t |fS) NrToo many arguments.rrrrexponr)rr+r*r/rr>rrrminrrSrfloat) r6rr7r,rrZdata_minr%r&r-r-r.r5s(    z expon_gen.fit)NN)rarbrcrdrUrrYrrZr_r\rr`rrr;rrr5r-r-r-r.rPs  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 Wikpedia 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)NKFrrrRrTr-r-r.rUszexponnorm_gen._shape_infoNcCs |||}||}||Sr<)rr)r6rrrexpvalZgvalr-r-r.rs zexponnorm_gen._rvscCst|||Sr<r#)r6rXrr-r-r.rYszexponnorm_gen._pdfcCs2d|}|d||}|t||t|SNrfrmrr>r)r6rXrinvKZexpargr-r-r.rszexponnorm_gen._logpdfcCs:d|}|d||}|t||}t|t|Srrrr>r}r6rXrrrZlogprodr-r-r.rZszexponnorm_gen._cdfcCs<d|}|d||}|t||}t| t|Srrrr-r-r.r\szexponnorm_gen._sfcCsD||}d|}d|d|d}d|||d}||||fS)NrfrDrrrrr-)r6rZK2ZopK2ZskwZkrtr-r-r.rs zexponnorm_gen._stats)NN) rarbrcrdrUrrYrrZr\rr-r-r-r.rs) r exponnormc@s8eZdZdZddZddZddZdd Zd d Zd S) 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}||gSNrhFrrr,rRr6rr@r-r-r.rU:szexponweib_gen._shape_infocCst||||Sr<r#r6rXrhr,r-r-r.rY?szexponweib_gen._pdfcCsR|| }t| }t|t|t|d||t|d|}|Sr)r[r0r>rr)r6rXrhr,Znegxcexm1cZlogpr-r-r.rDs  "zexponweib_gen._logpdfcCst||  }||Sr<r)r6rXrhr,rr-r-r.rZKszexponweib_gen._cdfcCs$t|d|  td|Sr)r[rr>r)r6r^rhr,r-r-r.r_Oszexponweib_gen._ppfN rarbrcrdrUrYrrZr_r-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 cCstdddtjfdgSNriFrrrRrTr-r-r.rUrszexponpow_gen._shape_infocCst|||Sr<r#r6rXrir-r-r.rYuszexponpow_gen._pdfcCs8||}dt|t|d||t|}|SNrrf)r>rr[rr})r6rXrixbfr-r-r.rys,zexponpow_gen._logpdfcCstt||  Sr<rrr-r-r.rZ~szexponpow_gen._cdfcCstt|| Sr<r>r}r[r0rr-r-r.r\szexponpow_gen._sfcCstt| d|Srr[rr>rrr-r-r.r`szexponpow_gen._isfcCsttt|  d|Srpowr[rr6r^rir-r-r.r_szexponpow_gen._ppfN) rarbrcrdrUrYrrZr\r`r_r-r-r-r.rVsrexponpowc@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 cCstdddtjfdgSr+rRrTr-r-r.rUszfatiguelife_gen._shape_infoNcCsD||}d||}||}dd|d|td|}|S)NrmrfrDr)rr>r)r6r,rrzrXx2tr-r-r.rs   "zfatiguelife_gen._rvscCst|||Sr<r#r.r-r-r.rYszfatiguelife_gen._pdfcCsZt|d|ddd||dtd|dtdtjdt|S)NrrDr|rmrrr.r-r-r.rs6 zfatiguelife_gen._logpdfcCs$td|t|dt|Sr)rr>rr.r-r-r.rZszfatiguelife_gen._cdfcCs,|t|}d|t|dddSN?rDrr[rr>rr6r^r,tmpr-r-r.r_szfatiguelife_gen._ppfcCs$td|t|dt|Sr)rr>rr.r-r-r.r\szfatiguelife_gen._sfcCs.| t|}d|t|dddSrrrr-r-r.r`szfatiguelife_gen._isfcCst||}|dd}d|d}||d}d|d|dt|d}d |d |d |d}||||fS) Nr|rfrr%r rrr]gD@r>r)r6r,c2rdenrrrr-r-r.rs    zfatiguelife_gen._stats)NN)rarbrcrdrrrrUrrYrrZr_r\r`rr-r-r-r.rs r fatiguelifec@sBeZdZdZddZddZdddZd d Zd d Zd dZ 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-rir-r-r.rOszfoldcauchy_gen._argcheckcCstdddtjfdgSNr,FrrQrRrTr-r-r.rUszfoldcauchy_gen._shape_infoNcCsttj|||dS)Nr%rr)rrxr r6r,rrr-r-r.rs zfoldcauchy_gen._rvscCs2dtjdd||ddd||dSNrfrrDrr.r-r-r.rYszfoldcauchy_gen._pdfcCs&dtjt||t||Srrlr.r-r-r.rZszfoldcauchy_gen._cdfcCstjtjtjtjfSr<rrir-r-r.rszfoldcauchy_gen._stats)NN rarbrcrdrOrUrrYrZrr-r-r-r.rs r foldcauchyc@sReZdZdZddZdddZddZd d Zd d Zd dZ ddZ ddZ dS)f_genaEAn F continuous random variable. For the noncentral F distribution, see `ncf`. %(before_notes)s See Also -------- ncf Notes ----- 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` and parameters :math:`df_1, df_2 > 0` . `f` takes ``dfn`` and ``dfd`` as shape parameters. %(after_notes)s %(example)s cCs0tdddtjfd}tdddtjfd}||gS)NdfnFrrdfdrR)r6ZidfnZidfdr-r-r.rU(szf_gen._shape_infoNcCs||||Sr<)r)r6rrrrr-r-r.r-sz f_gen._rvscCst||||Sr<r#r6rXrrr-r-r.rY0sz f_gen._pdfcCs~d|}d|}|dt||dt|t|dd|||dt|||t|d|d}|SNrfrDr)r>rr[rr)r6rXrrrNmrr-r-r.r6s 60z f_gen._logpdfcCst|||Sr<)r[Zfdtrrr-r-r.rZ=sz f_gen._cdfcCst|||Sr<)r[Zfdtrcrr-r-r.r\@sz f_gen._sfcCst|||Sr<)r[Zfdtri)r6r^rrr-r-r.r_Csz 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)Nrfr|r%r @rDcSs||Sr<r-)v2v2_2r-r-r.rCLrDzf_gen._stats..rcSs$d||||||d|Srr-)v1rrv2_4r-r-r.rCQsrcSs&d|||t||||Srr)rrrv2_6r-r-r.rCWsrcSsd||||S)Nrr-)rrv2_8r-r-r.rC^rDr)rr>rSrr) r6rrrrrrrrrrrrr-r-r.rFs:$  z f_gen._stats)NNrr-r-r-r.r s rrc@sBeZdZdZddZddZdddZd d Zd d Zd dZ 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|dkSrr-rir-r-r.rOszfoldnorm_gen._argcheckcCstdddtjfdgSrrRrTr-r-r.rUszfoldnorm_gen._shape_infoNcCst|||Sr<rrrr-r-r.rszfoldnorm_gen._rvscCst||t||Sr<rr.r-r-r.rYszfoldnorm_gen._pdfcCst||t||dSrrr.r-r-r.rZszfoldnorm_gen._cdfcCs||}td|tdtj}d||t|td}|d||}d||||||}|t|d}||ddd||}|d|d d |d|d7}||dd }||||fS) Nr|rDrrrrrr$)r>r}rrr[erfr)r6r,rZexpfacrrrrr-r-r.rs $zfoldnorm_gen._stats)NNrr-r-r-r.rps rfoldnormcsteZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ e eddfddZZS)weibull_min_genaFWeibull minimum continuous random variable. The Weibull Minimum Extreme Value distribution, from extreme value theory (Fisher-Gnedenko theorem), is also often simply called the Weibull distribution. It arises as the limiting distribution of the rescaled minimum of iid random variables. %(before_notes)s See Also -------- weibull_max, numpy.random.Generator.weibull, exponweib Notes ----- The probability density function for `weibull_min` is: .. math:: f(x, c) = c x^{c-1} \exp(-x^c) for :math:`x > 0`, :math:`c > 0`. `weibull_min` takes ``c`` as a shape parameter for :math:`c`. (named :math:`k` in Wikipedia article and :math:`a` in ``numpy.random.weibull``). Special shape values are :math:`c=1` and :math:`c=2` where Weibull distribution reduces to the `expon` and `rayleigh` distributions respectively. %(after_notes)s References ---------- https://en.wikipedia.org/wiki/Weibull_distribution https://en.wikipedia.org/wiki/Fisher-Tippett-Gnedenko_theorem %(example)s cCstdddtjfdgSr+rRrTr-r-r.rUszweibull_min_gen._shape_infocCs$|t||dtt|| SrJrr>r}r.r-r-r.rYszweibull_min_gen._pdfcCs$t|t|d|t||SrJr>rr[rrr.r-r-r.rszweibull_min_gen._logpdfcCstt||  Sr<r[r0rr.r-r-r.rZszweibull_min_gen._cdfcCstt|| Sr<r>r}rr.r-r-r.r\szweibull_min_gen._sfcCs t|| Sr<rr.r-r-r.rszweibull_min_gen._logsfcCstt|  d|Srrr1r-r-r.r_szweibull_min_gen._ppfcCstd|d|Srrrhr-r-r.rszweibull_min_gen._munpcCst |t|tdSrJrr>rrir-r-r.rszweibull_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 s|ddr tj|f||St||||\}}}}|dd}ddt|d}|} | kr|dkr|dkr|stj|f||S|dkrd \} } } n,t|r|d nd} |d d} |d d} |dkr| dkrt fd dd|gddj } n|dk r|} |dkrh| dkrht |} t | tdd| tdd| d} n|dk rv|} |dkr| dkrt |}|| tdd| } n|dk r|} |dkr| | | fStj|| f| | d|SdS)NsuperfitFr(r0cSsjtdd|}tdd|}tdd|}d|dd|||}||dd}||S)NrrDrrr)r,Zgamma1Zgamma2Zgamma3numrr-r-r.skew s z!weibull_min_gen.fit..skewg@mmNNNrr%r&cs |Sr<r-rr3r r-r.rC' rDz%weibull_min_gen.fit..g{Gz?bisect)bracketr(rrDr%r&)r*r3r5_check_fit_input_parametersr1r2statsr rr#rootr>rrr[rr)r6rr7r,Zfcrrr(Zmax_cZs_minr,r%r&vrr rr.r5sJ         4     zweibull_min_gen.fit)rarbrcrdrUrYrrZr\rr_rrrrr5rr-r-r r.rs( rrcseZdZdZddZddZfddZdd 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@SNrer-r6r,rhrir-r-r.rOp sztruncweibull_min_gen._argcheckcCsFtdddtjfd}tdddtjfd}tdddtjfd}|||gS)Nr,FrrrhrQrirR)r6r@rrr-r-r.rUs sz truncweibull_min_gen._shape_infocstj|ddS)N)rrrrr3r r6rr r-r.r y sztruncweibull_min_gen._fitstartcCs||fSr<r-rr-r-r.ro} sz!truncweibull_min_gen._get_supportcCsLtt|| tt|| }|t||dtt|| |SrJr)r6rXr,rhridenumr-r-r.rY s$ztruncweibull_min_gen._pdfcCsRttt|| tt|| }t|t|d|t|||SrJ)r>rr}rr[r)r6rXr,rhrilogdenumr-r-r.r s*ztruncweibull_min_gen._logpdfcCsPtt|| tt|| }tt|| tt|| }||Sr<rr6rXr,rhrir rr-r-r.rZ s$$ztruncweibull_min_gen._cdfcCs\ttt|| tt|| }ttt|| tt|| }||Sr<r>rr}rr6rXr,rhriZlognumrr-r-r.r s**ztruncweibull_min_gen._logcdfcCsPtt|| tt|| }tt|| tt|| }||Sr<rrr-r-r.r\ s$$ztruncweibull_min_gen._sfcCs\ttt|| tt|| }ttt|| tt|| }||Sr<r r!r-r-r.r s**ztruncweibull_min_gen._logsfc CsBttd|tt|| |tt||  d|SrJrr>rr}r6r^r,rhrir-r-r.r` s6ztruncweibull_min_gen._isfc CsBttd|tt|| |tt||  d|SrJr"r#r-r-r.r_ s6ztruncweibull_min_gen._ppfcCsrt||dt||dt||t||dt||}tt|| tt|| }||Sr)r[rrrr>r})r6rNr,rhriZ gamma_funrr-r-r.r s 2$ztruncweibull_min_gen._munp)rarbrcrdrOrUr rorYrrZrr\rr`r_rrr-r-r r.rC s, rtruncweibull_minc@sXeZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ dS)weibull_max_gena0Weibull maximum continuous random variable. The Weibull Maximum Extreme Value distribution, from extreme value theory (Fisher-Gnedenko theorem), is the limiting distribution of rescaled maximum of iid random variables. This is the distribution of -X if X is from the `weibull_min` function. %(before_notes)s See Also -------- weibull_min Notes ----- The probability density function for `weibull_max` is: .. math:: f(x, c) = c (-x)^{c-1} \exp(-(-x)^c) for :math:`x < 0`, :math:`c > 0`. `weibull_max` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s References ---------- https://en.wikipedia.org/wiki/Weibull_distribution https://en.wikipedia.org/wiki/Fisher-Tippett-Gnedenko_theorem %(example)s cCstdddtjfdgSr+rRrTr-r-r.rU szweibull_max_gen._shape_infocCs(|t| |dtt| | SrJrr.r-r-r.rY szweibull_max_gen._pdfcCs(t|t|d| t| |SrJrr.r-r-r.r szweibull_max_gen._logpdfcCstt| | Sr<rr.r-r-r.rZ szweibull_max_gen._cdfcCst| | Sr<r r.r-r-r.r szweibull_max_gen._logcdfcCstt| |  Sr<rr.r-r-r.r\ szweibull_max_gen._sfcCstt| d| Sr)rr>rr1r-r-r.r_ szweibull_max_gen._ppfcCs4td|d|}t|dr(d}nd}||S)NrfrDrer)r[rint)r6rNr,valZsgnr-r-r.r s  zweibull_max_gen._munpcCst |t|tdSrJr rir-r-r.r szweibull_max_gen._entropyN) rarbrcrdrUrYrrZrr\r_rrr-r-r-r.r% s$r% weibull_max)rirjc@s@eZdZdZddZddZddZdd Zd d Zd d Z 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 :math:`x >= 0`, :math:`c > 0`. `genlogistic` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s %(example)s cCstdddtjfdgSr+rRrTr-r-r.rU szgenlogistic_gen._shape_infocCst|||Sr<r#r.r-r-r.rY szgenlogistic_gen._pdfcCsL|d |dkd}t|}t||||dtt| SNrr)r>rrr[rr})r6rXr,ZmultZabsxr-r-r.r s zgenlogistic_gen._logpdfcCsdt| | }|SrJr)r6rXr,ZCxr-r-r.rZ! szgenlogistic_gen._cdfcCstt|d|d }|SrPr>rr)r6r^r,valsr-r-r.r_% szgenlogistic_gen._ppfcCstt|}tjtjdtd|}dtd|dt}|t|d}tjdddtd|}||d }||||fS) NrrDrrrr.@rr|)rr[rr>rzetarrr6r,rrrrr-r-r.r) s zgenlogistic_gen._statsN) rarbrcrdrUrYrrZr_rr-r-r-r.r) sr) 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|Sr<r>rrir-r-r.rOZ szgenpareto_gen._argcheckcCstddtj tjfdgSNr,FrrRrTr-r-r.rU] szgenpareto_gen._shape_infocCsBt|}t|dk|fddtj}t|dk|j|j}||fS)NrcSsd|SNrQr-rr-r-r.rCc rDz,genpareto_gen._get_support..)r>rrrSr&rh)r6r,rirhr-r-r.ro` s  zgenpareto_gen._get_supportcCst|||Sr<r#r.r-r-r.rYh szgenpareto_gen._pdfcCs$t||k|dk@||fdd| S)NrcSst|d|| |SrrdrXr,r-r-r.rCn rDz'genpareto_gen._logpdf..rr.r-r-r.rl szgenpareto_gen._logpdfcCst| |  Sr<)r[Z inv_boxcox1pr.r-r-r.rZq szgenpareto_gen._cdfcCst| | Sr<)r[Z inv_boxcoxr.r-r-r.r\t szgenpareto_gen._sfcCs$t||k|dk@||fdd| S)NrcSst|| |Sr<r-r5r-r-r.rCy rDz&genpareto_gen._logsf..r6r.r-r-r.rw szgenpareto_gen._logsfcCst| |  Sr<)r[boxcox1pr1r-r-r.r_| szgenpareto_gen._ppfcCst||  Sr<)r[boxcoxr1r-r-r.r` szgenpareto_gen._isfr2cCsd|krd}nt|dk|fddtj}d|kr6d}nt|dk|fddtj}d|kr^d}nt|d k|fd dtj}d |krd}nt|d k|fd dtj}||||fS)NrrcSs dd|SrJr-xir-r-r.rC rDz&genpareto_gen._stats..rrmcSsdd|ddd|Srr-r9r-r-r.rC rDr3gUUUUUU?cSs*dd|tdd|dd|S)NrDrrrr9r-r-r.rC s r6rcSs@ddd|d|d|ddd|dd|dS)NrrrDrr-r9r-r-r.rC s "  rr>rSr)r6r,r9rrr3r6r-r-r.r s2    zgenpareto_gen._statscs0ddt|dk|ffddtdS)NcSspd}td|d}t|t||D]$\}}||d|d||}q&t||dk|d||tjS)NrerrrerfrQ)r>r[zipr[combr&rS)rNr,r'r6ZkiZcnkr-r-r.r` s z#genpareto_gen._munp..__munprcs |Sr<r-rZ_genpareto_gen__munprNr-r.rC rDz%genpareto_gen._munp..r)rr[rrhr-r>r.r s    zgenpareto_gen._munpcCsd|Srr-rir-r-r.r szgenpareto_gen._entropyN)r2)rarbrcrdrOrUrorYrrZr\rr_r`rrrr-r-r-r.r16 s#  r1 genparetoc@s8eZdZdZddZddZddZdd Zd d Zd S) genexpon_genaA 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, "The Exponential Distribution: Theory, Methods and Applications", Asit P. Basu. %(example)s cCsFtdddtjfd}tdddtjfd}tdddtjfd}|||gS)NrhFrrrir,rR)r6rrr@r-r-r.rU szgenexpon_gen._shape_infoc CsH||t| | t| |||t| | |Sr<r[r0r>r}r6rXrhrir,r-r-r.rY s(zgenexpon_gen._pdfcCsHt||t| | | |||t| | |Sr<r>rr[r0rBr-r-r.r szgenexpon_gen._logpdfcCs0t| |||t| | | Sr<rrBr-r-r.rZ szgenexpon_gen._cdfcCs.t| |||t| | |Sr<rrBr-r-r.r\ szgenexpon_gen._sfN) rarbrcrdrUrYrrZr\r-r-r-r.r@ s r@genexponcseZdZdZddZddZddZdd 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|Sr<r2rir-r-r.rO szgenextreme_gen._argcheckcCstddtj tjfdgSr3rRrTr-r-r.rU szgenextreme_gen._shape_infocCsLt|dkdt|ttj}t|dkdt|t tj }||fSNrrf)r>r&maximumrrSZminimum)r6r,_b_ar-r-r.ro s $zgenextreme_gen._get_supportcCs$t||k|dk@||fdd| S)NrcSst| ||Sr<r-r5r-r-r.rC rDz+genextreme_gen._loglogcdf..r6r.r-r-r. _loglogcdf szgenextreme_gen._loglogcdfcCst|||Sr<r#r.r-r-r.rY szgenextreme_gen._pdfcCst||k|dk@||fddd}t| }|||}t|}t||dk|tj k@dt|dk|tj kB|||fddtj d}t||dk|dk@d|S)NrcSs||Sr<r-r5r-r-r.rC' rDz(genextreme_gen._logpdf..rercSs| ||Sr<r-)pex2Zlpex2Zlex2r-r-r.rC/ rDrX)rr[rrJr>r}ZputmaskrS)r6rXr,ZcxZlogex2Zlogpex2rKlogpdfr-r-r.r& s"   zgenextreme_gen._logpdfcCst||| Sr<)r>r}rJr.r-r-r.r4 szgenextreme_gen._logcdfcCst|||Sr<r>r}rr.r-r-r.rZ7 szgenextreme_gen._cdfcCst||| Sr<)r[r0rr.r-r-r.r\: szgenextreme_gen._sfcCs6tt|  }t||k|dk@||fdd|S)NrcSst| | |Sr<rr5r-r-r.rC@ rDz%genextreme_gen._ppf..)r>rrr6r^r,rXr-r-r.r_= s zgenextreme_gen._ppfcCs8tt|   }t||k|dk@||fdd|S)NrcSst| | |Sr<rr5r-r-r.rCE rDz%genextreme_gen._isf..)r>rr[rrrNr-r-r.r`B s zgenextreme_gen._isfc sfdd}|d}|d}|d}|d}ttdktjdd ||dttdktjdd ttdd dtd d}d }tt|kt ttd} td ktj| } td ktj|d|} t dk|||ffddtjd} tt|dkdt dt tjd| } t dk||||fddtjd}tt|dkd|d}| | | |fS)Ncst|dSrJrrNrr-r.rCH rDz'genextreme_gen._stats..rrDrrgHz>r|rrf+=rQrgUUUUUUտcs(t|| |d|dSNrDrr=)r,rrg3Zg2gm12)g2mg12r-r.rCY srXg(\?r4rgпcSs(|d|d|||||dS)NrrDr-)rrrRg4rSr-r-r.rCa sgq= ףp?333333@r$) r>r&rrr[r0rrrrrr)r6r,grrrRrUZgam2kZepsZgamkrrZsk1rZku1rr-)r,rSr.rG s4 ,0,  2 zgenextreme_gen._statscs,t|}|dkrd}nd}tj||fdS)Nrrmrrrr3r )r6rrWrhr r-r.r g s zgenextreme_gen._fitstartcCsdtd|d}d||tjt||d|t||ddd}t||dk|tjS)NrrrfreZaxis)r>r[rr[r=rr&rS)r6rNr,r6r,r-r-r.rp s $zgenextreme_gen._munpcCstd|dSrJrrir-r-r.rw szgenextreme_gen._entropy)rarbrcrdrOrUrorJrYrrrZr\r_r`rr rrrr-r-r r.rE s & rE 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|Sr<)r[digammaryr-r.rC rDz_digammainv..grm g|=)Ztolg-@g뭁,?rfdy=T)xtolrrz"_digammainv: fsolve failed, y = %rr)r>r}r Znewtonr RuntimeError)r^Z_emrx0valuerrrr-r]r. _digammainv~ s    rfcseZdZdZddZdddZddZd 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 cCstdddtjfdgSrrRrTr-r-r.rU szgamma_gen._shape_infoNcCs |||Sr<standard_gamma)r6rhrrr-r-r.r szgamma_gen._rvscCst|||Sr<r#rr-r-r.rY szgamma_gen._pdfcCst|d||t|Sr)r[rrrr-r-r.r szgamma_gen._logpdfcCs t||Sr<rrr-r-r.rZ szgamma_gen._cdfcCs t||Sr<rrr-r-r.r\ sz gamma_gen._sfcCs t||Sr<rrr-r-r.r_ szgamma_gen._ppfcCs t||Sr<r[rrr-r-r.r` szgamma_gen._isfcCs||dt|d|fS)Nr|rrrr-r-r.r szgamma_gen._statscCs t|d||t|SrJr[rrrr-r-r.r szgamma_gen._entropycs&ddt|d}tj||fdS)Nr:0yE>rDrrXr6rrhr r-r.r  szgamma_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}|dks,|dkr@tj|f||S|ddt|dddg}|dd}t||dk r|dk rtd t |}t | std t ||krt d |tjd |d kr||}|}|dkrp|dk r|} npt|t|fdd} dtdddd} | d} | d} tj| | | d d} || }n$t|t|}t|} |}| ||fS)Nrr(r0rr r rrrrrrrcst|t|Sr<)r>rr[r\rhr3r-r.rC< rDzgamma_gen.fit..rrDr7r4g333333?gffffff?)Zdisp)r1r2r3r5r*rr/rr>rrrrrrSrrrr brentqrf)r6rr7r,rr(r rrrhrZaestZxarr&r,r ror.r5 s@       * z gamma_gen.fit)NN)rarbrcrdrUrrYrrZr\r_r`rrr rrr5rr-r-r r.rg s'  rgrcsHeZdZdZddZddZfddZeedd 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. cCs2tt||k}|s*td|ft|dkS)NzUThe shape parameter of the erlang distribution has been given a non-integer value %r.r)r>rfloorrwarnRuntimeWarning)r6rhZallintr-r-r.rOg szerlang_gen._argcheckcCstdddtjfdgS)NrhTrrQrRrTr-r-r.rUr szerlang_gen._shape_infocs.tddt|d}tt|j||fdS)Nr%rlrDr)r&rr3rgr rmr r-r.r u szerlang_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|f||Sr<)r3r5r6rr7r,r r-r.r5~ szerlang_gen.fit) rarbrcrdrOrUr rrr5rr-r-r r.rqS s   rqerlangc@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@Srr-)r6rhr,r-r-r.rO szgengamma_gen._argcheckcCs4tdddtjfd}tddtj tjfd}||gSrrRrr-r-r.rU szgengamma_gen._shape_infocCst||||Sr<r#rr-r-r.rY szgengamma_gen._pdfcs,t|dk|dkB||ffddtj dS)Nrcs4tt|t|d|||tSrJ)r>rrr[rrr5rnr-r.rC s z&gengamma_gen._logpdf..rXrr>rSrr-rnr.r s zgengamma_gen._logpdfcCs2||}t||}t||}t|dk||Srr[rrr>r&r6rXrhr,Zxcval1val2r-r-r.rZ s  zgengamma_gen._cdfNcCs|j||d}|d|S)Nrrfrh)r6rhr,rrrr-r-r.r szgengamma_gen._rvscCs2||}t||}t||}t|dk||Srryrzr-r-r.r\ s  zgengamma_gen._sfcCs2t||}t||}t|dk||d|SrFr[rrr>r&r6r^rhr,r{r|r-r-r.r_ s  zgengamma_gen._ppfcCs2t||}t||}t|dk||d|SrFr~rr-r-r.r` s  zgengamma_gen._isfcCst||d|Sr)r[Zpoch)r6rNrhr,r-r-r.r szgengamma_gen._munpcCs:t|}|d|d||t|tt|Sr)r[rrr>rr)r6rhr,r'r-r-r.r s zgengamma_gen._entropy)NN)rarbrcrdrOrUrYrrZrr\r_r`rrr-r-r-r.rw s rwgengammac@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 cCstdddtjfdgSr+rRrTr-r-r.rU szgenhalflogistic_gen._shape_infocCs|jd|fSrrnrir-r-r.ro sz genhalflogistic_gen._get_supportcCsBd|}td||}||d}||}d|d|dSrr>r)r6rXr,limitrZtmp0tmp2r-r-r.rY s  zgenhalflogistic_gen._pdfcCs2d|}td||}||}d|d|SrEr)r6rXr,rrrr-r-r.rZ szgenhalflogistic_gen._cdfcCs d|dd|d||SrEr-r1r-r-r.r_ szgenhalflogistic_gen._ppfcCsdd|dtdSNrDrrrir-r-r.r szgenhalflogistic_gen._entropyN) rarbrcrdrUrorYrZr_rr-r-r-r.r s rgenhalflogisticcsZeZdZdZddZddZfddZdd Zd d Zd d Z dddZ ddZ Z S)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-0.5} 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.kn`) `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|dkBSr)r> logical_andr)r6rrhrir-r-r.rOr szgenhyperbolic_gen._argcheckcCsNtddtj tjfd}tdddtjfd}tddtj tjfd}|||gS)NrFrrhrrQrirR)r6iprrr-r-r.rUv szgenhyperbolic_gen._shape_infocstj|ddS)N)rrrmrrrr r-r.r | szgenhyperbolic_gen._fitstartcCstjdd}|||||S)NcSst||||Sr<)rZgenhyperbolic_logpdfrXrrhrir-r-r._logpdf_single sz1genhyperbolic_gen._logpdf.._logpdf_singler> vectorize)r6rXrrhrirr-r-r.r s zgenhyperbolic_gen._logpdfcCstjdd}|||||S)NcSst||||Sr<)rZgenhyperbolic_pdfrr-r-r. _pdf_single sz+genhyperbolic_gen._pdf.._pdf_singler)r6rXrrhrirr-r-r.rY s zgenhyperbolic_gen._pdfcCstjdd}|||||S)NcSs^t|||gtjtj}ttd|}t |tj |d}t |rZd}t |t|S)NZ_genhyperbolic_pdfrzdInfinite values encountered in scipy.special.kve. Values replaced by NaN to avoid incorrect results.)r>arrayrctypesdata_asc_void_pr from_cythonrr quadrSisnanrrsrt)rXrrhri user_datallct1rGr-r-r. _cdf_single s   z+genhyperbolic_gen._cdf.._cdf_singler)r6rXrrhrirr-r-r.rZ s zgenhyperbolic_gen._cdfNc Csht|dt|d}t|d}t|d}tj|||||d} tj||d} || t| | S)NrDrmr)rrir&rrr)r> float_power geninvgaussr rr) r6rrhrirrrt2t3ZgigZnormstr-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)NrDrmrrerrrS)rcsg|] }|qSr-r-).0rib0r-r. sz,genhyperbolic_gen._stats..rrrrr`r4) r>broadcast_arraysrZlinspacer\shaperIr[kv)r6rrhrirrZintegersZb1b2Zb3Zb4Zr1Zr2Zr3Zr4rrZm3er3Zm4er6r-rr.r sT "  zgenhyperbolic_gen._stats)NN) rarbrcrdrOrUr rrYrZrrrr-r-r r.r sY    r genhyperbolicc@s@eZdZdZddZddZddZdd Zd d Zd d Z 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 cCstdddtjfdgSr+rRrTr-r-r.rUszgompertz_gen._shape_infocCst|||Sr<r#r.r-r-r.rYszgompertz_gen._pdfcCst|||t|Sr<rCr.r-r-r.rszgompertz_gen._logpdfcCst| t| Sr<rr.r-r-r.rZ szgompertz_gen._cdfcCstd|t| Sr4r-r1r-r-r.r_ szgompertz_gen._ppfcCs$dt|t|td|SrE)r>rr}r[Zexpnrir-r-r.rszgompertz_gen._entropyN) rarbrcrdrUrYrrZr_rr-r-r-r.r srgompertzcCs8t|}t|}|}t||}tj||dS)N)weights)r>rmaxr}Zaverage)rX logweightsZmaxlogwrr-r-r._average_with_log_weightss   rc@steZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ eeeddZdS) gumbel_r_genaA right-skewed Gumbel continuous random variable. %(before_notes)s See Also -------- gumbel_l, gompertz, genextreme Notes ----- The probability density function for `gumbel_r` is: .. math:: f(x) = \exp(-(x + e^{-x})) The Gumbel distribution is sometimes referred to as a type I Fisher-Tippett distribution. It is also related to the extreme value distribution, log-Weibull and Gompertz distributions. %(after_notes)s %(example)s cCsgSr<r-rTr-r-r.rU9szgumbel_r_gen._shape_infocCst||Sr<r#rxr-r-r.rY<szgumbel_r_gen._pdfcCs| t| Sr<rrxr-r-r.r@szgumbel_r_gen._logpdfcCstt|  Sr<rrxr-r-r.rZCszgumbel_r_gen._cdfcCst|  Sr<rrxr-r-r.rFszgumbel_r_gen._logcdfcCstt|  Sr<rryr-r-r.r_Iszgumbel_r_gen._ppfcCstt|   Sr<rArxr-r-r.r\Lszgumbel_r_gen._sfcCstt|   Sr<r>rrrr-r-r.r`Oszgumbel_r_gen._isfcCs0ttjtjddtdtjdtdfS)Nrr4rrrVrr>rrrrTr-r-r.rRszgumbel_r_gen._statscCstdSrrZrTr-r-r.rUszgumbel_r_gen._entropyc st|||\}}fdd}|dk r6|}||n|dk rR|fddn fdd|dd}|d|d} } fd d } | | | s| d ks| tjkr| d} | d9} qtj| | fd d d } | j}|dk r|n|||fS)Ncs$| t |ttSr<)r[ logsumexpr>rr)r&rr-r.get_loc_from_scalefsz,gumbel_r_gen.fit..get_loc_from_scalecs:t|}t|}||Sr<)r>r}rr)r&Zterm1Zterm2rr%r-r.ryszgumbel_r_gen.fit..funccs& |}t|d}||S)N)r)rr)r&ZsdataZwavgrr-r.rs  r&rrDcst|t|kSr<r=r@)rr-r.rCs  z0gumbel_r_gen.fit..interval_contains_rootrrP)rZrtolrb)rr1r>rSr r#r) r6rr7r,rrrr& brack_startrArBrCresr-)rrr%r.r5Ys:        zgumbel_r_gen.fitN)rarbrcrdrUrYrrZrr_r\r`rrr;r rr5r-r-r-r.rsrgumbel_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 cCsgSr<r-rTr-r-r.rUszgumbel_l_gen._shape_infocCst||Sr<r#rxr-r-r.rYszgumbel_l_gen._pdfcCs|t|Sr<rrxr-r-r.rszgumbel_l_gen._logpdfcCstt|  Sr<rArxr-r-r.rZszgumbel_l_gen._cdfcCstt|  Sr<r>rr[rryr-r-r.r_szgumbel_l_gen._ppfcCs t| Sr<rrxr-r-r.rszgumbel_l_gen._logsfcCstt| Sr<rrxr-r-r.r\szgumbel_l_gen._sfcCstt| Sr<rrxr-r-r.r`szgumbel_l_gen._isfcCs2t tjtjddtdtjdtdfS)NrrrrVrrTr-r-r.rszgumbel_l_gen._statscCstdSrrZrTr-r-r.rszgumbel_l_gen._entropycOsD|ddk r|d |d<tjt| f||\}}| |fS)Nr)r1rr5r>r)r6rr7r,Zloc_rZscale_rr-r-r.r5s zgumbel_l_gen.fitN)rarbrcrdrUrYrrZr_rr\r`rrr;r rr5r-r-r-r.rsrgumbel_lc@sHeZdZdZddZddZddZdd Zd d Zd d Z ddZ dS)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 cCsgSr<r-rTr-r-r.rUszhalfcauchy_gen._shape_infocCsdtjd||Srrrxr-r-r.rYszhalfcauchy_gen._pdfcCstdtjt||Srr>rrr[rrxr-r-r.r szhalfcauchy_gen._logpdfcCsdtjt|Srrlrxr-r-r.rZszhalfcauchy_gen._cdfcCsttjd|Srrnryr-r-r.r_szhalfcauchy_gen._ppfcCstjtjtjtjfSr<rrTr-r-r.rszhalfcauchy_gen._statscCstdtjSrrrTr-r-r.rszhalfcauchy_gen._entropyN) rarbrcrdrUrYrrZr_rrr-r-r-r.rsr halfcauchyc@sHeZdZdZddZddZddZdd Zd d Zd d Z ddZ dS)halflogistic_genaGA 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 %(example)s cCsgSr<r-rTr-r-r.rU3szhalflogistic_gen._shape_infocCst||Sr<r#rxr-r-r.rY6szhalflogistic_gen._pdfcCs$td|dtt| Sr{)r>rr[rr}rxr-r-r.r;szhalflogistic_gen._logpdfcCst|dSr)r>tanhrxr-r-r.rZ>szhalflogistic_gen._cdfcCsdt|Sr)r>Zarctanhryr-r-r.r_Aszhalflogistic_gen._ppfcCs|dkrdtdS|dkr.tjtjdS|dkr>dtS|dkrXdtjddSddtd d|t|dt|dS) NrrDr$rr5rr-r|)r>rrrrr[rr.rMr-r-r.rDszhalflogistic_gen._munpcCsdtdSrrrTr-r-r.rOszhalflogistic_gen._entropyN) rarbrcrdrUrYrrZr_rrr-r-r-r.rs r halflogisticc@sReZdZdZddZdddZddZd d Zd d Zd dZ ddZ ddZ dS) 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 cCsgSr<r-rTr-r-r.rUlszhalfnorm_gen._shape_infoNcCst|j|dSNrrrr-r-r.roszhalfnorm_gen._rvscCs$tdtjt| |dSrr>rrr}rxr-r-r.rYrszhalfnorm_gen._pdfcCs dtdtj||dSNrmr|rrxr-r-r.rvszhalfnorm_gen._logpdfcCst|ddSNrDrfrrxr-r-r.rZyszhalfnorm_gen._cdfcCstd|dSr:rryr-r-r.r_|szhalfnorm_gen._ppfcCsXtdtjddtjtddtjtjdddtjdtjddfS)Nr|rrDrrrrr>rrrTr-r-r.rs   zhalfnorm_gen._statscCsdttjddSrrrTr-r-r.rszhalfnorm_gen._entropy)NN rarbrcrdrUrrYrrZr_rrr-r-r-r.rVs rhalfnormc@s@eZdZdZddZddZddZdd Zd d Zd d Z 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 cCsgSr<r-rTr-r-r.rUszhypsecant_gen._shape_infocCsdtjt|Sr)r>rcoshrxr-r-r.rYszhypsecant_gen._pdfcCsdtjtt|Sr)r>rrmr}rxr-r-r.rZszhypsecant_gen._cdfcCstttj|dSr)r>rrorryr-r-r.r_szhypsecant_gen._ppfcCsdtjtjdddfS)NrrrDrrTr-r-r.rszhypsecant_gen._statscCstdtjSrrrTr-r-r.rszhypsecant_gen._entropyNrr-r-r-r.rsr 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)Nrrer-)r6rhrir,rr-r-r.rOszgausshyper_gen._argcheckcCs`tdddtjfd}tdddtjfd}tddtj tjfd}tdddtjfd}||||gS) NrhFrrrir,rrerR)r6rrr@Zizr-r-r.rUs zgausshyper_gen._shape_infocCslt|t|t||t||||| }d|||dd||dd|||Sr)r[rhyp2f1)r6rXrhrir,rZCinvr-r-r.rYs8zgausshyper_gen._pdfc Cs\t|||t||}t||||||| }t||||| }|||Sr<)r[rr) r6rNrhrir,rrr rr-r-r.rszgausshyper_gen._munpN)rarbrcrdrOrUrYrr-r-r-r.rs  r 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 cCstdddtjfdgSr+rRrTr-r-r.rUszinvgamma_gen._shape_infocCst|||Sr<r#rr-r-r.rYszinvgamma_gen._pdfcCs&|d t|t|d|Srr>rr[rrr-r-r.rszinvgamma_gen._logpdfcCst|d|Srrrr-r-r.rZszinvgamma_gen._cdfcCsdt||Srrjrr-r-r.r_ szinvgamma_gen._ppfcCst|d|Srrrr-r-r.r\#szinvgamma_gen._sfcCsdt||Srrrr-r-r.r`&szinvgamma_gen._isfmvskcCst|dk|fddtj}t|dk|fddtj}d\}}d|kr^t|dk|fd dtj}d |krt|d k|fd dtj}||||fS) NrcSs d|dSrr-rr-r-r.rC*rDz%invgamma_gen._stats..rDcSsd|dd|dS)NrfrDr|r-rr-r-r.rC+rD)NNr3rcSsdt|d|dS)Nr%r|r$rrr-r-r.rC2rDr6rcSs dd|d|d|dS)Nrrg&@r$r%r-rr-r-r.rC6rDr;)r6rhr9m1m2rrr-r-r.r)s(zinvgamma_gen._statscCs ||dt|t|Srrkrr-r-r.r9szinvgamma_gen._entropyN)r)rarbrcrdrrrrUrYrrZr_r\r`rrr-r-r-r.rs rinvgammacseZdZdZejZddZdddZddZ d d Z d d Z d dZ ddZ ddZfddZfddZddZeefddZZS) invgauss_genaAn inverse Gaussian continuous random variable. %(before_notes)s Notes ----- The probability density function for `invgauss` is: .. math:: f(x, \mu) = \frac{1}{\sqrt{2 \pi x^3}} \exp(-\frac{(x-\mu)^2}{2 x \mu^2}) for :math:`x >= 0` and :math:`\mu > 0`. `invgauss` takes ``mu`` as a shape parameter for :math:`\mu`. %(after_notes)s %(example)s cCstdddtjfdgSNrFrrrRrTr-r-r.rUYszinvgauss_gen._shape_infoNcCs|j|d|dSNrfrwaldr6rrrr-r-r.r\szinvgauss_gen._rvscCs>dtdtj|dtdd||||dS)NrfrDr$rQrr6rXrr-r-r.rY_szinvgauss_gen._pdfcCs:dtdtjdt||||dd|S)NrrDrrrr-r-r.rdszinvgauss_gen._logpdfcCsXdt|}t|||d}d|t| ||d}|tt||Sr)r>rrrr}r6rXrrrhrir-r-r.rkszinvgauss_gen._logcdfcCsZdt|}t|||d}d|t| |||}|tt|| Sr)r>rrrrr}rr-r-r.rqszinvgauss_gen._logsfcCst|||Sr<rNrr-r-r.r\wszinvgauss_gen._sfcCst|||Sr<rMrr-r-r.rZzszinvgauss_gen._cdfc stjddddnt||\}}t||d}|dk}td||||d||<t|}t||||||<W5QRX|SNr)roverinvalidrrm) r>rrr _invgauss_ppf _invgauss_isfrr3r_)r6rXrppfi_wti_nanr r-r.r_}s $zinvgauss_gen._ppfc stjddddnt||\}}t||d}|dk}td||||d||<t|}t||||||<W5QRX|Sr) r>rrrrrrr3r`)r6rXrisfrrr r-r.r`s $zinvgauss_gen._isfcCs||ddt|d|fS)Nr$rr)r6rr-r-r.rszinvgauss_gen._statsc s|dd}t|tks$|dkr8tj|f||St||||\}}}}|dks^|dk rrtj|f||St||dkrt ddtj dn@||}t |}|dkrt |t |d|d}||}|||fS)Nr(r0rrinvgaussrre)r1r4wald_genr2r3r5rr>rrrSrrr) r6rr7r,r(Zfshape_srrZfshape_nr r-r.r5s"   zinvgauss_gen.fit)NN)rarbrcrdrrrrUrrYrrrr\rZr_r`rr r5rr-r-r r.r@s  rrc@sbeZdZdZddZddZddZdd Zd d Zd d Z dddZ ddZ ddZ ddZ dS)geninvgauss_genaWA 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@Srr-r6rrir-r-r.rOszgeninvgauss_gen._argcheckcCs4tddtj tjfd}tdddtjfd}||gS)NrFrrirrR)r6rrr-r-r.rUszgeninvgauss_gen._shape_infocCs<tjdd}||||}t|r8d}t|t|S)NcSst|||Sr<)rZgeninvgauss_logpdfrXrrir-r-r. logpdf_singlesz.geninvgauss_gen._logpdf..logpdf_singlezjInfinite values encountered in scipy.special.kve(p, b). Values replaced by NaN to avoid incorrect results.)r>rrrrrsrt)r6rXrrirrrGr-r-r.rs   zgeninvgauss_gen._logpdfcCst||||Sr<r#r6rXrrir-r-r.rYszgeninvgauss_gen._pdfcs.|j|\}tjfdd}||f|S)NcsB|\}}t||gtjtj}ttd|}t ||dS)NZ_geninvgauss_pdfr) r>rrrrrr rrr r)rXr7rrirrrIr-r.rs z)geninvgauss_gen._cdf.._cdf_single)ror>r)r6rXr7rHrr-rr.rZszgeninvgauss_gen._cdfcCs t|dk|||fddtj S)NrcSs&|dt|||d|dSrrrr-r-r.rC rDz.geninvgauss_gen._logquasipdf..rxrr-r-r. _logquasipdfszgeninvgauss_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|ntdVqdSr<rslicerjZbcitr-r. :sz'geninvgauss_gen._rvs..rr-)r>isscalar _rvs_scalarrr]rrrr&prodemptynditerfinishedtuplerangerr\iternext) r6rrirroutshp numsamplesidxr-rr.r s2       zgeninvgauss_gen._rvsc3 sd}|s d}dkr d}}d}dks>dkrDd}n*tddtddkrjd}nd}tt|} t| } t| } d} |rT|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} | dkrD|| dkrDd|| }$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)NFrrTrmrDrrirTcs|Sr<rrriZlmrr6r-r.rC}rDz-geninvgauss_gen._rvs_scalar..cs|Sr<r r)rirr6r-r.rCrDzvmin must be smaller than vmax.zumax must be positive.riPz|Not a single random variate could be generated in {} attempts. Sampling does not appear to work for the provided parameters.)rr)_moderr>rr atleast_1drzerosZarccosrrrr}rrrrrrcrZ logical_notr\)3r6rrirrZ invert_resrZ ratio_unifZ mode_shiftsize1dNrX simulatedZa2Za1p1Zq1phirZroot1root2d1Zd2ZvminZvmaxZumaxZlogqpdfr,Zxplusir6rrr accept num_acceptrGrdxsZk1A1Zk2A2Zk3ZA3AhZcond1Zcond2Zcond3rr-r r.rDs     "$&                *$0    zgeninvgauss_gen._rvs_scalarcCsX|dkr.|t|dd|dd|Std|d|dd||SdSrrrr-r-r.r s&zgeninvgauss_gen._modec Cst|||}t||}t|t|B}|rxd}t|ttj|tj tj d}||||||<n||}|S)NzInfinite values encountered in the moment calculation involving scipy.special.kve. Values replaced by NaN to avoid incorrect results.dtype) r[Zkver>rErrrsrt full_likerdouble) r6rNrrir ZdenomZinf_valsrGrr-r-r.rs  zgeninvgauss_gen._munp)NN)rarbrcrdrOrUrrYrZrrrr rr-r-r-r.rs$ 7rrcs`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@Sr)r>absoluterr-r-r.rO+sznorminvgauss_gen._argcheckcCs4tdddtjfd}tddtj tjfd}||gSrrRrr-r-r.rU.sznorminvgauss_gen._shape_infocstj|ddS)N)rrmrrrr r-r.r 3sznorminvgauss_gen._fitstartcCsbt|d|d}|tjt|}td|}|t||t|||||Sr)r>rrr}hypotr[Zk1e)r6rXrhrirfac1sqr-r-r.rY7s znorminvgauss_gen._pdfc Csvt|r(tj|j|tj||fddSg}t|||D].\}}}|tj|j|tj||fddq8t|SdS)Nrr) r>rr rrYrSr<appendr)r6rXrhriresultrda0rr-r-r.r\=s znorminvgauss_gen._sfc s^fdd}t|r"||||Sg}t|||D]\}}}|||||q2t|SdS)Nc sfdd}||}|||||}|dkr2|S|dkrpd}|}||}|||||dkrd|}||}qJn4d}|}||}|||||dkrd|}||}qtj||||||fjd} | S)Ncs||||Sr<r\)rXrhrir^rTr-r.eqKsz6norminvgauss_gen._isf.._isf_scalar..eqrrrD)r7rb)rr rprb) r^rhrir+xmemdeltaleftrightr(rTr-r. _isf_scalarIs,    z*norminvgauss_gen._isf.._isf_scalar)r>rr<r'r) r6r^rhrir1r(Zq0r)rr-rTr.r`Hs !  znorminvgauss_gen._isfNcCsJt|d|d}tjd|||d}||t|tj||dS)NrDr)rrrr)r>rrr r)r6rhrirrrZigr-r-r.rrs znorminvgauss_gen._rvscCspt|d|d}||}|d|d}d||t|}ddd|d|d|}||||fS)NrDrr$rrr)r6rhrirrZvarianceskewnessZkurtosisr-r-r.rzs  znorminvgauss_gen._stats)NN)rarbrcrdrrrrOrUr rYr\r`rrrr-r-r r.r"s5  * r" norminvgausscsheZdZdZejZddZddZddZ dd Z d d Z d d Z ddZ ddZdfdd ZZS)invweibull_genuAn inverted Weibull continuous random variable. This distribution is also known as the Fréchet distribution or the type II extreme value distribution. %(before_notes)s Notes ----- The probability density function for `invweibull` is: .. math:: f(x, c) = c x^{-c-1} \exp(-x^{-c}) for :math:`x > 0`, :math:`c > 0`. `invweibull` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s References ---------- F.R.S. de Gusmao, E.M.M Ortega and G.M. Cordeiro, "The generalized inverse Weibull distribution", Stat. Papers, vol. 52, pp. 591-619, 2011. %(example)s cCstdddtjfdgSr+rRrTr-r-r.rUszinvweibull_gen._shape_infocCs8t|| d}t|| }t| }|||Srr>rr})r6rXr,xc1Zxc2r-r-r.rYs zinvweibull_gen._pdfcCst|| }t| Sr<r5)r6rXr,r6r-r-r.rZszinvweibull_gen._cdfcCst||   Sr<)r>r0r.r-r-r.r\szinvweibull_gen._sfcCstt| d|Sr4)r>rrr1r-r-r.r_szinvweibull_gen._ppfcCst|  d|S)NrerO)r6rr,r-r-r.r`szinvweibull_gen._isfcCstd||SrJrrhr-r-r.rszinvweibull_gen._munpcCsdtt|t|SrJr rir-r-r.rszinvweibull_gen._entropyNcs$|dkr dn|}tt|j||dS)N)r|r)r3r4r )r6rr7r r-r.r szinvweibull_gen._fitstart)N)rarbrcrdrrrrUrYrZr\r_r`rrr rr-r-r r.r4sr4 invweibullc@s>eZdZdZejZddZddZddZ dd Z d d Z d S) 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@Srr-rr-r-r.rOszjohnsonsb_gen._argcheckcCs4tddtj tjfd}tdddtjfd}||gSNrhFrrirrRrr-r-r.rUszjohnsonsb_gen._shape_infocCs6t||t|d|}|d|d||SrE)rr>r)r6rXrhritrmr-r-r.rYszjohnsonsb_gen._pdfcCst||t|d|Srrr>rrr-r-r.rZszjohnsonsb_gen._cdfcCs"ddtd|t||S)NrfrrQr>r}rr6r^rhrir-r-r.r_szjohnsonsb_gen._ppfN) rarbrcrdrrrrOrUrYrZr_r-r-r-r.r8sr8 johnsonsbc@s8eZdZdZddZddZddZdd Zd d Zd S) 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. %(after_notes)s %(example)s cCs|dk||k@Srr-rr-r-r.rOszjohnsonsu_gen._argcheckcCs4tddtj tjfd}tdddtjfd}||gSr9rRrr-r-r.rU szjohnsonsu_gen._shape_infoc CsF||}t||t|t|d}|dt|d|Sr)rr>rr)r6rXrhrirr:r-r-r.rY%s$zjohnsonsu_gen._pdfc Cs(t||t|t||dSrJ)rr>rrrr-r-r.rZ,szjohnsonsu_gen._cdfcCstt|||Sr<)r>sinhrr=r-r-r.r_/szjohnsonsu_gen._ppfN) rarbrcrdrOrUrYrZr_r-r-r-r.r?s 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 cCsgSr<r-rTr-r-r.rUJszlaplace_gen._shape_infoNcCs|jdd|dS)Nrrr)laplacerr-r-r.rMszlaplace_gen._rvscCsdtt| SNrm)r>r}rrxr-r-r.rYPszlaplace_gen._pdfc CsPtjdd:t|dkddt| dt|W5QRSQRXdS)Nr)rrrfrm)r>rr&r}rxr-r-r.rZTszlaplace_gen._cdfcCs || Sr<rZrxr-r-r.r\Xszlaplace_gen._sfcCs,t|dktdd| td|Srr>r&rryr-r-r.r_\szlaplace_gen._ppfcCs || Sr<r_ryr-r-r.r`_szlaplace_gen._isfcCsdS)N)rrDrrr-rTr-r-r.rcszlaplace_gen._statscCstddSrrrTr-r-r.rfszlaplace_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||||\}}}|dkr&t|}|dkrJtt||t|}||fSr<)rr>medianrrr)r6rr7r,rrr-r-r.r5is  zlaplace_gen.fit)NN)rarbrcrdrUrrYrZr\r_r`rrr;rrr5r-r-r-r.rB6s  rBrCc@sXeZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ dS)laplace_asymmetric_genu An 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 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 cCstdddtjfdgSNkappaFrrrRrTr-r-r.rUsz"laplace_asymmetric_gen._shape_infocCst|||Sr<r#r6rXrKr-r-r.rYszlaplace_asymmetric_gen._pdfcCs6d|}|t|dk| |}|t||8}|Sr*rF)r6rXrKkapinvrr-r-r.rszlaplace_asymmetric_gen._logpdfcCsLd|}||}t|dkdt| |||t||||Sr*r>r&r}r6rXrKrM kappkapinvr-r-r.rZs  zlaplace_asymmetric_gen._cdfc CsLd|}||}t|dkt| |||dt||||Sr*rNrOr-r-r.r\s  zlaplace_asymmetric_gen._sfcCsPd|}||}t|||ktd||| |t||||SrJrFr6r^rKrMrPr-r-r.r_s zlaplace_asymmetric_gen._ppfcCsPd|}||}t|||kt||| |td||||SrJrFrQr-r-r.r`s zlaplace_asymmetric_gen._isfcCsd|}||}||||}ddt|dtdt|dd}ddt|dtdt|dd}||||fS) Nrr|rrrrrrDr)r6rKrMmnrrrr-r-r.rs ,,zlaplace_asymmetric_gen._statscCsdt|d|SrJrr6rKr-r-r.rszlaplace_asymmetric_gen._entropyNrr-r-r-r.rIs&rIlaplace_asymmetriccCsHt|}|dd}|dd}|jr8t|jdnd}g}g}|jr|jdd} t| D]T\} } dt| } | d| d| g} t || }| | | ||dk rd||| <qddd d d ddh|}t | |}|rt d |d t||krt dd||h|kr tdt|s8td|f|||fS)Nrr,r rZfix_r%r&r'r(zUnknown keyword arguments: .zToo many positional arguments.rr)r>rr1shapesrsplitr enumeratestrrr'set differencer+rcrrr)distrr7r,rrZ num_shapesZ fshape_keysZfshapesrXrr3keynamesr'Z known_keysZ unknown_keysr-r-r.rs@        rc@sNeZdZdZejZddZddZddZ dd Z d d Z d d Z ddZ dS)levy_genahA 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() cCsgSr<r-rTr-r-r.rUXszlevy_gen._shape_infocCs.dtdtj||tdd|SNrrDrerrxr-r-r.rY[sz levy_gen._pdfcCsttd|SrD)r[erfcr>rrxr-r-r.rZ_sz levy_gen._cdfcCsttd|SrD)r[rr>rrxr-r-r.r\csz levy_gen._sfcCst|d }d||Srrr6r^r'r-r-r.r_fsz levy_gen._ppfcCsddt|dSr)r[Zerfinvrr-r-r.r`ksz levy_gen._isfcCstjtjtjtjfSr<rrTr-r-r.rnszlevy_gen._statsNrarbrcrdrrrrUrYrZr\r_r`rr-r-r-r.ra sHralevyc@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() cCsgSr<r-rTr-r-r.rUszlevy_l_gen._shape_infocCs6t|}dtdtj||tdd|Srb)rr>rrr}r6rXrr-r-r.rYszlevy_l_gen._pdfcCs"t|}dtdt|dSr)rrr>rrhr-r-r.rZszlevy_l_gen._cdfcCst|}dtdt|Sr)rrr>rrhr-r-r.r\szlevy_l_gen._sfcCst|dd}d||S)NrfrDrQrrdr-r-r.r_szlevy_l_gen._ppfcCsdt|ddS)NrerDrrr-r-r.r`szlevy_l_gen._isfcCstjtjtjtjfSr<rrTr-r-r.rszlevy_l_gen._statsNrer-r-r-r.rgusGrglevy_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 cCsgSr<r-rTr-r-r.rUszlogistic_gen._shape_infoNcCs |j|dSr)logisticrr-r-r.rszlogistic_gen._rvscCst||Sr<r#rxr-r-r.rYszlogistic_gen._pdfcCs$t| }|dtt|Sr)r>rr[rr})r6rXr^r-r-r.rs zlogistic_gen._logpdfcCs t|Sr<r[expitrxr-r-r.rZszlogistic_gen._cdfcCs t|Sr<r[Z log_expitrxr-r-r.rszlogistic_gen._logcdfcCs t|Sr<r[Zlogitryr-r-r.r_ szlogistic_gen._ppfcCs t| Sr<rlrxr-r-r.r\ szlogistic_gen._sfcCs t| Sr<rnrxr-r-r.rszlogistic_gen._logsfcCs t| Sr<roryr-r-r.r`szlogistic_gen._isfcCsdtjtjdddfS)Nrr$g333333?rrTr-r-r.rszlogistic_gen._statscCsdSrr-rTr-r-r.rszlogistic_gen._entropyc s0|ddr tjf||St|||\}}t|\}}|d||d|}}|ffdd |ffdd fd d }|dk r|dkrt|f} | j d }|}nH|dk r|dkrt|f} | j d }|}nt|||f} | j \}}| j r||fStjf||S) Nr Fr%r&cs$||}tt|dSr)r>rr[rm)r%r&r,rrNr-r.dl_dloc/s z!logistic_gen.fit..dl_dloccs(||}t|t|dSr)r>rr)r&r%r,rpr-r. dl_dscale3s z#logistic_gen.fit..dl_dscalecs|\}}||||fSr<r-)paramsr%r&)rqrrr-r.r7szlogistic_gen.fit..funcr) r*r3r5rrr r1r rrXsuccess) r6rr7r,rrr%r&rrr )rrqrrrNr.r5s2     zlogistic_gen.fit)NN)rarbrcrdrUrrYrrZrr_r\rr`rrr;r rr5rr-r-r r.rjs  rjrkc@sZeZdZdZddZdddZddZd d Zd d Zd dZ 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 cCstdddtjfdgSr+rRrTr-r-r.rUgszloggamma_gen._shape_infoNcCs.t|j|d|dt|j|d|S)Nrr)r>rrrrr-r-r.rjs zloggamma_gen._rvscCs"t||t|t|Sr<r>r}r[rr.r-r-r.rYxszloggamma_gen._pdfcCs||t|t|Sr<rvr.r-r-r.r|szloggamma_gen._logpdfcCst|t|Sr<)r[rr>r}r.r-r-r.rZszloggamma_gen._cdfcCstt||Sr<)r>rr[rr1r-r-r.r_szloggamma_gen._ppfcCst|t|Sr<)r[rr>r}r.r-r-r.r\szloggamma_gen._sfcCstt||Sr<)r>rr[rr1r-r-r.r`szloggamma_gen._isfcCsNt|}td|}td|t|d}td|||}||||fS)NrrDrr)r[r\Z polygammar>r)r6r,rrr2Zexcess_kurtosisr-r-r.rs   zloggamma_gen._stats)NN) rarbrcrdrUrrYrrZr_r\r`rr-r-r-r.ruNs ruloggammac@s@eZdZdZddZddZddZdd Zd d Zd d Z dS)loglaplace_gena|A 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 References ---------- T.J. Kozubowski and K. Podgorski, "A log-Laplace growth rate model", The Mathematical Scientist, vol. 28, pp. 49-60, 2003. %(example)s cCstdddtjfdgSr+rRrTr-r-r.rUszloglaplace_gen._shape_infocCs,|d}t|dk|| }|||dSNr|rr>r&)r6rXr,Zcd2r-r-r.rYszloglaplace_gen._pdfcCs(t|dkd||dd|| SNrrmrzr.r-r-r.rZszloglaplace_gen._cdfcCs.t|dkd|d|dd|d|S)Nrmr|rfrDrQrzr1r-r-r.r_szloglaplace_gen._ppfcCs|d|d|dSrr-rhr-r-r.rszloglaplace_gen._munpcCstd|dSrrrir-r-r.rszloglaplace_gen._entropyN) rarbrcrdrUrYrZr_rrr-r-r-r.rxsrx loglaplacecCst|dk||fddtj S)NrcSs:t|d d|dt||tdtjSr)r>rrrrXr3r-r-r.rCrDz!_lognorm_logpdf..rxr}r-r-r._lognorm_logpdfsr~cseZdZdZejZddZdddZddZ d d Z d d Z d dZ ddZ ddZddZddZddZeeeddfddZZS) lognorm_genabA 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 cCstdddtjfdgS)Nr3FrrrRrTr-r-r.rUszlognorm_gen._shape_infoNcCst|||Sr<r>r}r)r6r3rrr-r-r.rszlognorm_gen._rvscCst|||Sr<r#r6rXr3r-r-r.rYszlognorm_gen._pdfcCs t||Sr<r~rr-r-r.rszlognorm_gen._logpdfcCstt||Sr<r;rr-r-r.rZszlognorm_gen._cdfcCstt||Sr<rrr-r-r.rszlognorm_gen._logcdfcCst|t|Sr<r<)r6r^r3r-r-r.r_szlognorm_gen._ppfcCstt||Sr<)rr>rrr-r-r.r\szlognorm_gen._sfcCstt||Sr<)rr>rrr-r-r.r szlognorm_gen._logsfcCs\t||}t|}||d}t|dd|}tdddddg|}||||fSNrrDrrr)r>r}rpolyval)r6r3rrrrrr-r-r.rs   zlognorm_gen._statscCs&ddtdtjdt|S)NrmrrDr)r6r3r-r-r.rszlognorm_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. rc sp|dd}|dkr(tj|f||S|ddpJ|ddpJ|dd}|dd}t|dkrltddD]}||dqp|rtd ||dk r|dk rtd t|}t | std t |}|d kr||}t |d krt d |tjdt|}|dkrBt|} |dkr8|} nt |} n$t |} t|t| d} | || fS)Nrr fsfix_srrzToo many input arguments.) r rrrrr%r&r'r(r)rrrlognormrrD)r1r3r5rr+r*rr>rrrrrrrSrr}rstdr) r6rr7r,rr rrjZlndatar&rr r-r.r5s@           zlognorm_gen.fit)NN)rarbrcrdrrrrUrrYrrZrr_r\rrrr;rr5rr-r-r r.rs   rrc@sXeZdZdZejZddZdddZddZ d d Z d d Z d dZ ddZ ddZdS) gibrat_genaBA Gibrat continuous random variable. %(before_notes)s Notes ----- The probability density function for `gibrat` is: .. math:: f(x) = \frac{1}{x \sqrt{2\pi}} \exp(-\frac{1}{2} (\log(x))^2) `gibrat` is a special case of `lognorm` with ``s=1``. %(after_notes)s %(example)s cCsgSr<r-rTr-r-r.rUxszgibrat_gen._shape_infoNcCst||Sr<rrr-r-r.r{szgibrat_gen._rvscCst||Sr<r#rxr-r-r.rY~szgibrat_gen._pdfcCs t|dSrrrxr-r-r.rszgibrat_gen._logpdfcCstt|Sr<r;rxr-r-r.rZszgibrat_gen._cdfcCstt|Sr<r<ryr-r-r.r_szgibrat_gen._ppfcCsTtj}t|}||d}t|dd|}tdddddg|}||||fSr)r>err)r6rrrrrr-r-r.rs   zgibrat_gen._statscCsdtdtjdSrrrTr-r-r.rszgibrat_gen._entropy)NN)rarbrcrdrrrrUrrYrrZr_rrr-r-r-r.rbs rzp`gilbrat` is a misspelling of the correct name for the `gibrat` distribution, and will be removed in SciPy 1.11.c@seZdZdZddZdS) gilbrat_genz .. deprecated:: 1.9.0 `gilbrat` is deprecated, use `gibrat` instead! `gilbrat` is a misspelling of the correct name for the `gibrat` distribution, and will be removed in SciPy 1.11. cOs$dt}tj|tdd|j||S)Nz/`gilbrat` is deprecated, use `gibrat` instead! rD stacklevel)deprmsgrrsDeprecationWarningfreeze)r6r7r,rGr-r-r.__call__szgilbrat_gen.__call__N)rarbrcrdrr-r-r-r.rsrgibratgilbratrsZentropyexpectr5intervalrlogcdfrLlogsfrrHZmomentpdfrr sfrrrzgilbrat.zgibrat.c@sReZdZdZddZdddZddZd d Zd d Zd dZ 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 cCsgSr<r-rTr-r-r.rUszmaxwell_gen._shape_infoNcCstjd||dS)Nr$rrr rr-r-r.rszmaxwell_gen._rvscCs t||t| |dSr)r!r>r}rxr-r-r.rYszmaxwell_gen._pdfc CsBtjdd,tdt|d||W5QRSQRXdS)NrrrDrm)r>rr"rrxr-r-r.rszmaxwell_gen._logpdfcCstd||dS)Nrr|rrxr-r-r.rZszmaxwell_gen._cdfcCstdtd|SrQrryr-r-r.r_szmaxwell_gen._ppfcCsrdtjd}dtdtjddtjtdddtj|ddtjtjd tjd |dfS) NrrrDr| r_rrir>rrr6r'r-r-r.rs  $zmaxwell_gen._statscCstdtdtjdSr)rr>rrrTr-r-r.rszmaxwell_gen._entropy)NNrr-r-r-r.rs rmaxwellc@s@eZdZdZddZddZddZdd Zd d Zd d Z dS) mielke_genaA Mielke Beta-Kappa / Dagum continuous random variable. %(before_notes)s Notes ----- The probability density function for `mielke` is: .. math:: f(x, k, s) = \frac{k x^{k-1}}{(1+x^s)^{1+k/s}} for :math:`x > 0` and :math:`k, s > 0`. The distribution is sometimes called Dagum distribution ([2]_). It was already defined in [3]_, called a Burr Type III distribution (`burr` with parameters ``c=s`` and ``d=k/s``). `mielke` takes ``k`` and ``s`` as shape parameters. %(after_notes)s References ---------- .. [1] Mielke, P.W., 1973 "Another Family of Distributions for Describing and Analyzing Precipitation Data." J. Appl. Meteor., 12, 275-280 .. [2] Dagum, C., 1977 "A new model for personal income distribution." Economie Appliquee, 33, 327-367. .. [3] Burr, I. W. "Cumulative frequency functions", Annals of Mathematical Statistics, 13(2), pp 215-232 (1942). %(example)s cCs0tdddtjfd}tdddtjfd}||gS)Nr6Frrr3rR)r6iki_sr-r-r.rU szmielke_gen._shape_infocCs,|||dd||d|d|Srr-r6rXr6r3r-r-r.rY%szmielke_gen._pdfc CsZtjddDt|t||dt||d||W5QRSQRXdS)Nrrr)r>rrrrr-r-r.r(szmielke_gen._logpdfcCs ||d|||d|Srr-rr-r-r.rZ-szmielke_gen._cdfcCs(t||d|}t|d|d|Srr )r6r^r6r3Zqskr-r-r.r_0szmielke_gen._ppfcCs"dd}t||k|||f|tjS)NcSs2t|||td||t||SrJr)rNr6r3r-r-r. nth_moment5sz$mielke_gen._munp..nth_momentrx)r6rNr6r3rr-r-r.r4szmielke_gen._munpN) rarbrcrdrUrYrrZr_rr-r-r-r.rs!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)r>rrfull)r6rr6rr-r-r.rOszkappa4_gen._argcheckcCs8tddtj tjfd}tddtj tjfd}||gS)NrFrr6rR)r6Zihrr-r-r.rUszkappa4_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)r>rrr6r-r-r.r sz#kappa4_gen._get_support..f0cSs t|Sr<rrr-r-r.rsz#kappa4_gen._get_support..f1cSs$tt|}tj |dd<|Sr<r>rrrSrr6rhr-r-r.f3sz#kappa4_gen._get_support..f3cSsd|Srr-rr-r-r.f5sz#kappa4_gen._get_support..f5defaultcSsd|Srr-rr-r-r.r scSs"tt|}tj|dd<|Sr<rrr-r-r.rsr>rr r) r6rr6condlistr rrrrIrHr-r-r.ros0zkappa4_gen._get_supportcCst||||Sr<r#r6rXrr6r-r-r.rYszkappa4_gen._pdfc Cst|dk|dkt|dk|dkt|dk|dkt|dk|dkg}dd}dd}dd}dd }t|||||g|||gtjd S) NrcSsDtd|d| |td|d| d||d|S)zpdf = (1.0 - k*x)**(1.0/k - 1.0)*( 1.0 - h*(1.0 - k*x)**(1.0/k))**(1.0/h-1.0) logpdf = ... rfrdrXrr6r-r-r.r s(zkappa4_gen._logpdf..f0cSs.td|d| |d||d|S)z~pdf = (1.0 - k*x)**(1.0/k - 1.0)*np.exp(-( 1.0 - k*x)**(1.0/k)) logpdf = ... rfrdrr-r-r.rszkappa4_gen._logpdf..f1cSs(| td|d| t| S)z]pdf = np.exp(-x)*(1.0 - h*np.exp(-x))**(1.0/h - 1.0) logpdf = ... rf)r[rr>r}rr-r-r.rGszkappa4_gen._logpdf..f2cSs| t| S)zDpdf = np.exp(-x-np.exp(-x)) logpdf = ... rrr-r-r.rszkappa4_gen._logpdf..f3rr r6rXrr6rr rrGrr-r-r.rs zkappa4_gen._logpdfcCst||||Sr<rMrr-r-r.rZszkappa4_gen._cdfc Cst|dk|dkt|dk|dkt|dk|dkt|dk|dkg}dd}dd}dd}dd }t|||||g|||gtjd S) NrcSs(d|t| d||d|S)zVcdf = (1.0 - h*(1.0 - k*x)**(1.0/k))**(1.0/h) logcdf = ... rfr-rr-r-r.r szkappa4_gen._logcdf..f0cSsd||d| S)zLcdf = np.exp(-(1.0 - k*x)**(1.0/k)) logcdf = ... rfr-rr-r-r.rszkappa4_gen._logcdf..f1cSs d|t| t| S)zLcdf = (1.0 - h*np.exp(-x))**(1.0/h) logcdf = ... rf)r[rr>r}rr-r-r.rG szkappa4_gen._logcdf..f2cSst|  S)zBcdf = np.exp(-np.exp(-x)) logcdf = ... rrr-r-r.rszkappa4_gen._logcdf..f3rrrr-r-r.rs zkappa4_gen._logcdfc Cst|dk|dkt|dk|dkt|dk|dkt|dk|dkg}dd}dd}dd}dd }t|||||g|||gtjd S) NrcSs d|dd||||Srr-r^rr6r-r-r.r !szkappa4_gen._ppf..f0cSsd|dt| |Srrrr-r-r.r$szkappa4_gen._ppf..f1cSst||  t|S)z,ppf = -np.log((1.0 - (q**h))/h) rrr-r-r.rG'szkappa4_gen._ppf..f2cSstt|  Sr<rrr-r-r.r,szkappa4_gen._ppf..f3rr) r6r^rr6rr rrGrr-r-r.r_s zkappa4_gen._ppfcCsDt|dk|dk|dkg}dd}dd}t|||g||gddS)NrcSsd||tSr4Zastyper&rr-r-r.r :sz&kappa4_gen._get_stats_info..f0cSsd|tSr4rrr-r-r.r=sz&kappa4_gen._get_stats_info..f1rSr)r>rr )r6rr6rr rr-r-r._get_stats_info4s zkappa4_gen._get_stats_infocs0|||fddtddD}|ddS)Ncs$g|]}t|krdntjqSr<r>rr)rr}maxrr-r.rDsz%kappa4_gen._stats..rrS)rr)r6rr6outputsr-rr.rBs zkappa4_gen._statscGs@||d|d}||kr"tjStj|jdd|f|ddSNrrr)rr>rr r _mom_integ1)r6rr7rr-r-r._mom1_scGszkappa4_gen._mom1_scN)rarbrcrdrOrUrorYrrZrr_rrrr-r-r-r.r?sX)&#rkappa4c@s@eZdZdZddZddZddZdd Zd d Zd d Z dS) kappa3_gena*Kappa 3 parameter distribution. %(before_notes)s Notes ----- The probability density function for `kappa3` is: .. math:: f(x, a) = a (a + x^a)^{-(a + 1)/a} for :math:`x > 0` and :math:`a > 0`. `kappa3` takes ``a`` as a shape parameter for :math:`a`. References ---------- P.W. Mielke and E.S. Johnson, "Three-Parameter Kappa Distribution Maximum Likelihood and Likelihood Ratio Tests", Methods in Weather Research, 701-707, (September, 1973), :doi:`10.1175/1520-0493(1973)101<0701:TKDMLE>2.3.CO;2` B. Kumphon, "Maximum Entropy and Maximum Likelihood Estimation for the Three-Parameter Kappa Distribution", Open Journal of Statistics, vol 2, 415-419 (2012), :doi:`10.4236/ojs.2012.24050` %(after_notes)s %(example)s cCstdddtjfdgSrrRrTr-r-r.rUrszkappa3_gen._shape_infocCs||||d|dSrPr-rr-r-r.rYuszkappa3_gen._pdfcCs||||d|Sr4r-rr-r-r.rZyszkappa3_gen._cdfcCs||| dd|Srr-rr-r-r.r_|szkappa3_gen._ppfcs$fddtddD}|ddS)Ncs$g|]}t|krdntjqSr<r)rrrnr-r.rsz%kappa3_gen._stats..rrS)r)r6rhrr-rnr.rszkappa3_gen._statscGs6t||dkrtjStj|jdd|f|ddSr)r>rrr rr)r6rr7r-r-r.rszkappa3_gen._mom1_scN) rarbrcrdrUrYrZr_rrr-r-r-r.rQs rkappa3c@sReZdZdZddZdddZddZd d Zd d Zd dZ ddZ ddZ dS) moyal_genaA Moyal continuous random variable. %(before_notes)s Notes ----- The probability density function for `moyal` is: .. math:: f(x) = \exp(-(x + \exp(-x))/2) / \sqrt{2\pi} for a real number :math:`x`. %(after_notes)s This distribution has utility in high-energy physics and radiation detection. It describes the energy loss of a charged relativistic particle due to ionization of the medium [1]_. It also provides an approximation for the Landau distribution. For an in depth description see [2]_. For additional description, see [3]_. References ---------- .. [1] J.E. Moyal, "XXX. Theory of ionization fluctuations", The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, vol 46, 263-280, (1955). :doi:`10.1080/14786440308521076` (gated) .. [2] G. Cordeiro et al., "The beta Moyal: a useful skew distribution", International Journal of Research and Reviews in Applied Sciences, vol 10, 171-192, (2012). http://www.arpapress.com/Volumes/Vol10Issue2/IJRRAS_10_2_02.pdf .. [3] C. Walck, "Handbook on Statistical Distributions for Experimentalists; International Report SUF-PFY/96-01", Chapter 26, University of Stockholm: Stockholm, Sweden, (2007). http://www.stat.rice.edu/~dobelman/textfiles/DistributionsHandbook.pdf .. versionadded:: 1.1.0 %(example)s cCsgSr<r-rTr-r-r.rUszmoyal_gen._shape_infoNcCstjdd||d}t| S)NrmrD)rhr&rr)rr r>r)r6rrr!r-r-r.rs zmoyal_gen._rvscCs*td|t| tdtjSNrrD)r>r}rrrxr-r-r.rYszmoyal_gen._pdfcCsttd|tdSr)r[rcr>r}rrxr-r-r.rZszmoyal_gen._cdfcCsttd|tdSr)r[rr>r}rrxr-r-r.r\sz moyal_gen._sfcCstdt|d Sr)r>rr[Zerfcinvrxr-r-r.r_szmoyal_gen._ppfcCsPtdtj}tjdd}dtdtdtjd}d}||||fS)NrDrr%)r>r euler_gammarrr[r.rr-r-r.rs "zmoyal_gen._statscCs$|dkrtdtjS|dkrBtjddtdtjdS|dkrdtjdtdtj}tdtjd}dtd}|||S|dkrd tdtdtj}dtjdtdtjd}tdtjd }d tjd d }||||S||SdS) NrfrDr|r$rrr8r%8rr)r>rrrr[r.r)r6rNZtmp1rZtmp3Ztmp4r-r-r.rs "  "zmoyal_gen._munp)NN) rarbrcrdrUrrYrZr\r_rrr-r-r-r.rs* rmoyalc@sdeZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ dddZ dddZ dS) 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 cCstdddtjfdgS)NnuFrrrRrTr-r-r.rU sznakagami_gen._shape_infocCst|||Sr<r#r6rXrr-r-r.rYsznakagami_gen._pdfcCs@tdt||t|td|d|||dSr)r>rr[rrrr-r-r.rs  znakagami_gen._logpdfcCst||||Sr<rrr-r-r.rZsznakagami_gen._cdfcCstd|t||Srr)r6r^rr-r-r.r_sznakagami_gen._ppfcCst||||Sr<rrr-r-r.r\sznakagami_gen._sfcCstd|t||SrJr)r6rrr-r-r.r` sznakagami_gen._isfcCst|dt|t|}d||}|dd||d|t|d}d|d|d|d |d d |d}|||d}||||fS) Nrmrfrrr|rrrD)r[rr>rr)r6rrrrrr-r-r.r#s " (0znakagami_gen._statsNcCst|j||d|Sr)r>rri)r6rrrr-r-r.r+sznakagami_gen._rvscCsH|dkrd|j}t|}tt||dt|}|||fS)N)rfrD)Znumargsr>rrrr)r6rr7r%r&r-r-r.r /s    znakagami_gen._fitstart)NN)N)rarbrcrdrUrYrrZr_r\r`rrr r-r-r-r.rs rnakagamicCsz|dd}t|t|}}t|d||d||d}t|||d}t|dk||fddtj dS) Nr|rfrmrDrcSs|t|Sr<r)r}r,r-r-r.rCIrDz_ncx2_log_pdf..)rrY)r>rr[riverrS)rXr{r^df2rnsrZcorrr-r-r. _ncx2_log_pdf=s $rc@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@Srr2r6r{r^r-r-r.rOiszncx2_gen._argcheckcCs0tdddtjfd}tdddtjfd}||gS)Nr{Frrr^rQrRr6idfincr-r-r.rUlszncx2_gen._shape_infoNcCs||||Sr<)Znoncentral_chisquare)r6r{r^rrr-r-r.rqsz ncx2_gen._rvscCs0tj|td|dk@}t||||ftdddS)NrrcSs t||Sr<)r|rrXr{_r-r-r.rCwrDz"ncx2_gen._logpdf..rrG)r> ones_likeboolrrr6rXr{r^condr-r-r.rtszncx2_gen._logpdfc Csdtj|td|dk@}t<d}tjd|dt||||ftjdddW5QRSQRXdS) Nrrz!overflow encountered in _ncx2_pdfrrcSs t||Sr<)r|rYrr-r-r.rCrDzncx2_gen._pdf..r) r>rrrrrrrZ _ncx2_pdfr6rXr{r^rrr-r-r.rYys z ncx2_gen._pdfcCs2tj|td|dk@}t||||ftjdddS)NrrcSs t||Sr<)r|rZrr-r-r.rCrDzncx2_gen._cdf..r)r>rrrrZ _ncx2_cdfrr-r-r.rZsz ncx2_gen._cdfc Csdtj|td|dk@}t<d}tjd|dt||||ftjdddW5QRSQRXdS) Nrrz!overflow encountered in _ncx2_ppfrrcSs t||Sr<)r|r_rr-r-r.rCrDzncx2_gen._ppf..r) r>rrrrrrrZ _ncx2_ppf)r6r^r{r^rrr-r-r.r_s z ncx2_gen._ppfcCs2tj|td|dk@}t||||ftjdddS)NrrcSs t||Sr<)r|r\rr-r-r.rCrDzncx2_gen._sf..r)r>rrrrZ_ncx2_sfrr-r-r.r\sz ncx2_gen._sfc Csdtj|td|dk@}t<d}tjd|dt||||ftjdddW5QRSQRXdS) Nrrz!overflow encountered in _ncx2_isfrrcSs t||Sr<)r|r`rr-r-r.rCrDzncx2_gen._isf..r) r>rrrrrrrZ _ncx2_isfrr-r-r.r`s z ncx2_gen._isfcCs,t||t||t||t||fSr<)rZ _ncx2_meanZ_ncx2_varianceZ_ncx2_skewnessZ_ncx2_kurtosis_excessrr-r-r.rs     zncx2_gen._stats)NN)rarbrcrdrOrUrrrYrZr_r\r`rr-r-r-r.rMs rncx2c@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@Srr-)r6df1rr^r-r-r.rOszncf_gen._argcheckcCsFtdddtjfd}tdddtjfd}tdddtjfd}|||gS)NrFrrrr^rQrR)r6Zidf1Zidf2rr-r-r.rUszncf_gen._shape_infoNcCs|||||Sr<)Z noncentral_f)r6rrr^rrr-r-r.rsz ncf_gen._rvscCst||||Sr<)rZ_ncf_pdfr6rXrrr^r-r-r.rYsz ncf_gen._pdfcCst||||Sr<)rZ_ncf_cdfrr-r-r.rZsz ncf_gen._cdfcCst||||Sr<)rZ_ncf_ppf)r6r^rrr^r-r-r.r_sz ncf_gen._ppfcCst||||Sr<)rZ_ncf_sfrr-r-r.r\sz ncf_gen._sfcCst||||Sr<)rZ_ncf_isfrr-r-r.r`sz ncf_gen._isfcCs|d||}t|d|td||t|d}|t| d|9}|t|d|d|d|9}|S)Nrfrmr|)r[rr>r}hyp1f1)r6rNrrr^r'Ztermr-r-r.rs 2"z ncf_gen._munpr2c Cs\t|||}t|||}d|kr2t|||nd}d|krLt|||nd}||||fS)Nr3r6)rZ _ncf_meanZ _ncf_varianceZ _ncf_skewnessZ_ncf_kurtosis_excess) r6rrr^r9rrrrr-r-r.rszncf_gen._stats)NN)r2)rarbrcrdrOrUrrYrZr_r\r`rrr-r-r-r.rs(  rncfc@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 cCstdddtjfdgSrzrRrTr-r-r.rU#szt_gen._shape_infoNcCs|j||dSr)Z standard_tr}r-r-r.r&sz t_gen._rvscCs"t|tjk||fdddddS)NcSs t|Sr<)rrYrXr{r-r-r.rC,rDzt_gen._pdf..cSsRtt|ddt|dt|tjd|d||ddSr)r>r}r[rrrrr-r-r.rC-s$*rrxr~r-r-r.rY)s z t_gen._pdfcCs"t|tjk||fdddddS)NcSs t|Sr<)rrrr-r-r.rC6rDzt_gen._logpdf..cSsVt|ddt|ddt|tj|ddtd|d|Sr)r[rr>rrrr-r-r.rC7s  rrxr~r-r-r.r3s z t_gen._logpdfcCs t||Sr<r[Zstdtrr~r-r-r.rZ>sz t_gen._cdfcCst|| Sr<rr~r-r-r.r\Asz t_gen._sfcCs t||Sr<r[Zstdtritrr-r-r.r_Dsz t_gen._ppfcCst|| Sr<rrr-r-r.r`Gsz 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) NrrerDcSsttj|jSr<r> broadcast_torSrr{r-r-r.rCSrDzt_gen._stats..cSs ||dSrr-rr-r-r.rCTrDcSstd|jSrJr>rrrr-r-r.rCUrDrrcSsttj|jSr<rrr-r-r.rC]rDcSs d|dS)Nrr%r-rr-r-r.rC^rDcSstd|jSrrrr-r-r.rC_rD)r>Zisposinfr&rSrr r) r6r{Z infinite_dfrr choicelistrrrr-r-r.rJs, z t_gen._statscCsZ|tjkrtS|d}|dd}|t|t|tt|t|dS)NrDrrm) r>rSrrr[r\rrr)r6r{ZhalfZhalf1r-r-r.rds  zt_gen._entropy)NN)rarbrcrdrUrrYrrZr\r_r`rrr-r-r-r.rs   rrc@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@Srr-rr-r-r.rOsznct_gen._argcheckcCs4tdddtjfd}tddtj tjfd}||gS)Nr{Frrr^rRrr-r-r.rUsznct_gen._shape_infoNcCs8tj|||d}tj|||d}|t|t|S)Nrr)rr r|r>r)r6r{r^rrrNrr-r-r.rsz 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)Nrfr|rrDrrmr) r>rr[rr}rrrrclip) r6rXr{r^rNrrr%trm1rZvalFtrm2r-r-r.rYs( *   &  z nct_gen._pdfcCstt|||ddSNrr)r>rrZ_nct_cdfr6rXr{r^r-r-r.rZsz nct_gen._cdfcCst|||Sr<)rZ_nct_ppf)r6r^r{r^r-r-r.r_sz nct_gen._ppfcCstt|||ddSr)r>rrZ_nct_sfrr-r-r.r\sz nct_gen._sfcCst|||Sr<)rZ_nct_isfrr-r-r.r`sz nct_gen._isfr2cCsXt||}t||}d|kr,t||nd}d|krHt||dnd}||||fS)Nr3r6r)rZ _nct_meanZ _nct_varianceZ _nct_skewnessZ_nct_kurtosis_excess)r6r{r^r9rrrrr-r-r.rs   znct_gen._stats)NN)r2) rarbrcrdrOrUrrYrZr_r\r`rr-r-r-r.rps rnctcsfeZdZdZddZddZddZdd Zd d Zdd dZ ddZ e e e fddZZS) 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 cCstdddtjfdgSrrRrTr-r-r.rUszpareto_gen._shape_infocCs||| dSrJr-rr-r-r.rYszpareto_gen._pdfcCsd|| SrJr-rr-r-r.rZszpareto_gen._cdfcCstd|d|S)NrrQr rr-r-r.r_szpareto_gen._ppfcCs || Sr<r-rr-r-r.r\szpareto_gen._sfr2c Csd\}}}}d|krT|dk}t||}tjt|tjd}t||||dd|kr|dk}t||}tjt|tjd}t||||d|ddd |kr|d k}t||}tjt|tjd}d|dt|d|d t|} t||| d |kr|d k}t||}tjt|tjd}dtddddg|tddddg|} t||| ||||fS)N)NNNNrrrrfrrDr|r3rr$r6rrrrgrre) r>extractrrrSZplacerrr) r6rir9rrrrmaskZbtr,r-r-r.rs4   "  ,  zpareto_gen._statscCsdd|t|Srrrir-r-r.rszpareto_gen._entropycst|||}|\}}|dk rHt||p4dkrHtddtjdjdfdd||krvdkrnnfddfd d fd d fd d}|dd}|d|d} } || | s| dks| tjkr| d} | d9} qt| | gd} | jrx| j } t| } p>| | }| | tksnt| } t | d} || | fSt j f|Sn|dkrt|} n|} |pt| } pʈ| | }|| | fS)Nrparetorrcstt||Sr<)r>rr)r&location)rndatar-r. get_shapesz!pareto_gen.fit..get_shapecs ||Sr<r-)rr&)rr-r. dL_dScalesz!pareto_gen.fit..dL_dScalecs|dtd|SrJr>r)rrrr-r. dL_dLocation!sz$pareto_gen.fit..dL_dLocationcs0t|}p||}||||Sr<)r>r)r&rr)rrrfshaperr-r. fun_to_solve&sz$pareto_gen.fit..fun_to_solvecst|t|kSr<r=r@rr-r.rC-s  z.pareto_gen.fit..interval_contains_rootr&rDr) rr>rrrSrr1r# convergedr nextafterr3r5)r6rr7r, parametersrrrCrrArBrr&r%rr )rrrrrrrr.r5sH           zpareto_gen.fit)r2)rarbrcrdrUrYrZr_r\rrr;r rr5rr-r-r r.rs rrc@sXeZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ dS) lomax_genaA Lomax (Pareto of the second kind) continuous random variable. %(before_notes)s Notes ----- The probability density function for `lomax` is: .. math:: f(x, c) = \frac{c}{(1+x)^{c+1}} for :math:`x \ge 0`, :math:`c > 0`. `lomax` takes ``c`` as a shape parameter for :math:`c`. `lomax` is a special case of `pareto` with ``loc=-1.0``. %(after_notes)s %(example)s cCstdddtjfdgSr+rRrTr-r-r.rUvszlomax_gen._shape_infocCs|dd||dSrr-r.r-r-r.rYyszlomax_gen._pdfcCst||dt|SrJrr.r-r-r.r}szlomax_gen._logpdfcCst| t| Sr<r/r.r-r-r.rZszlomax_gen._cdfcCst| t|Sr<)r>r}r[rr.r-r-r.r\sz lomax_gen._sfcCs| t|Sr<r-r.r-r-r.rszlomax_gen._logsfcCstt|  |Sr<r/r1r-r-r.r_szlomax_gen._ppfcCs$tj|ddd\}}}}||||fS)NrQr)r%r9)rrr/r-r-r.rszlomax_gen._statscCsdd|t|Srrrir-r-r.rszlomax_gen._entropyN) rarbrcrdrUrYrrZr\rr_rrr-r-r-r.r^srlomaxcszeZdZdZddZddZddZdd Zd d Zd d Z ddZ dddZ ddZ e eeddfddZZS) 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)Nrerfg>r|rD)r>rcopyr#) r6rXr r%r&Znorm2pearson_transitionansrinvmaskrrr.transxr-r-r. _preprocesss  zpearson3_gen._preprocesscCs t|Sr<r2)r6r r-r-r.rOszpearson3_gen._argcheckcCstddtj tjfdgS)Nr FrrRrTr-r-r.rUszpearson3_gen._shape_infocCs$d}d}|}d|d}||||fS)NrerfrrDr-)r6r rrr3r6r-r-r.rs  zpearson3_gen._statscCs@t|||}|jdkr.t|r*dS|Sd|t|<|S)Nrre)r>r}rrIr)r6rXr r r-r-r.rYs  zpearson3_gen._pdfc CsT|||\}}}}}}}} tt||||<tt|t||||<|Sr<)rr>rrrrrL) r6rXr r r rr rrrr-r-r.rs  zpearson3_gen._logpdfc Cs|||\}}}}}}}}t||||<t||j}t||dk} ||dk} t|| || || <t||dk} ||dk} t|| || || <|Sr) rrr>rrrrrsr) r6rXr r r rr rrZ invmask1aZ invmask1bZ invmask2aZ invmask2br-r-r.rZs   zpearson3_gen._cdfNc Csvt||}|dg|\}}}}}}} } |} |j| } || ||<|| | || ||<|dkrr|d}|S)Nrr-)r>rrrrrri) r6r rrr rrr rrr.ZnsmallZnbigr-r-r.r&s   zpearson3_gen._rvsc Csh|||\}}}}}}}} t||||<||}d||dk||dk<t|||| ||<|Sr*)rrr[r) r6r^r r rrr rrr.r-r-r.r_4s zpearson3_gen._ppfze Note that method of moments (`method='MM'`) is not available for this distribution. rcs:|dddkrtdntt||j|f||SdS)Nr(ZMMzhFit `method='MM'` is not available for the Pearson3 distribution. Please try the default `method='MLE'`.)r1r'r3r4r5rur r-r.r5=s zpearson3_gen.fit)NN)rarbrcrdrrOrUrrYrrZrr_r;rrr5rr-r-r r.r s-    r pearson3cs|eZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ fddZ e eeddfddZZS) powerlaw_genaA power-function continuous random variable. %(before_notes)s See Also -------- pareto Notes ----- The probability density function for `powerlaw` is: .. math:: f(x, a) = a x^{a-1} for :math:`0 \le x \le 1`, :math:`a > 0`. `powerlaw` takes ``a`` as a shape parameter for :math:`a`. %(after_notes)s For example, the support of `powerlaw` can be adjusted from the default interval ``[0, 1]`` to the interval ``[c, c+d]`` by setting ``loc=c`` and ``scale=d``. For a power-law distribution with infinite support, see `pareto`. `powerlaw` is a special case of `beta` with ``b=1``. %(example)s cCstdddtjfdgSrrRrTr-r-r.rUnszpowerlaw_gen._shape_infocCs|||dSrr-rr-r-r.rYqszpowerlaw_gen._pdfcCst|t|d|SrJ)r>rr[rrr-r-r.ruszpowerlaw_gen._logpdfcCs ||dSrr-rr-r-r.rZxszpowerlaw_gen._cdfcCs|t|Sr<rrr-r-r.r{szpowerlaw_gen._logcdfcCst|d|Srr rr-r-r.r_~szpowerlaw_gen._ppfc Csr||d||d|ddd|d|dt|d|dtddd dg|||d|d fS) Nrfr|rDgr$rrrerr)r>rrrr-r-r.rs  $*zpowerlaw_gen._statscCsdd|t|Srrrr-r-r.rszpowerlaw_gen._entropycs"tt||||dk|dkB@Sr)r3rrrr r-r.rszpowerlaw_gen._support_maska: Notes specifically for ``powerlaw.fit``: If the location is a free parameter and the value returned for the shape parameter is less than one, the true maximum likelihood approaches infinity. This causes numerical difficulties, and the resulting estimates are approximate. rcs|ddr tjf||SttdkrFtjf||St|||\}}|fg}||id}|dk r |kst ddd|dk r ||kst ddd|dk r|dkrt d| krd}t |dd d d |dk r&|dk r&||||fS|dk rt tj } pT| |} || | |f} t |tj} p| |} || | |f}| |kr| | |fS| | |fS|dk r|}p||}|||fSfd d }ddddfddfddfdd}dk rhdkrh|Sdk rdkr|S|}||} |}||}| |kr|ddkr|S| |kr|ddkr|Stjf||SdS)Nr FrpowerlawrzKNegative or zero `fscale` is outside the range allowed by the distribution.z0`fscale` must be greater than the range of data.cSs0t|}| tt|||t|Sr<)rr>rr)rr%r&rr-r-r.rsz#powerlaw_gen.fit..get_shapecSs ||Sr<)rrr-r-r. get_scalesz#powerlaw_gen.fit..get_scalecsrttj }t|t|jjkrDt|t|jj}t|tj}pf||}|||fSr<) r>rrrSrZfinforZtinyr?r%r&r)rrrrr-r.fit_loc_scale_w_shape_lt_1s z4powerlaw_gen.fit..fit_loc_scale_w_shape_lt_1cSs|jd ||Sr)r)rrr&r-r-r.rsz#powerlaw_gen.fit..dL_dScalecSs|dtd||SrJr)rrr%r-r-r.rsz&powerlaw_gen.fit..dL_dLocationcs2t|tj }p$||}||Sr<r>rrSr)rrrrrr-r.dL_dLocation_starsz+powerlaw_gen.fit..dL_dLocation_starcs>t|tj }p$||}||||Sr<rr)rrrrrrr-r.r s   z&powerlaw_gen.fit..fun_to_solvec sttj }|}|dkrB|}|d9}q fdd}|d}d}|||s|tj kr|}|d9}qZtj||fd}t|jtj }t|tj}p̈||}|||fS)NrrDcst|t|kSr<r=r@rr-r.rC4s  zTpowerlaw_gen.fit..fit_loc_scale_w_shape_gt_1..interval_contains_rootrrfr)r>rrrSr r#r) rBr.rCrArrr%r&r)rrrrrrr-r.fit_loc_scale_w_shape_gt_1(s$         z4powerlaw_gen.fit..fit_loc_scale_w_shape_gt_1)r*r3r5rr>uniquerr Z _reduce_funcrrrrptprrSZnnlf)r6rr7r,rrZpenalized_nllf_argsZpenalized_nllfrGZloc_lt1Z shape_lt1Zll_lt1Zloc_gt1Z shape_gt1Zll_gt1r&rrrZ fit_shape_lt1Z fit_shape_gt1r )rrrrrrrrr.r5st(             $  zpowerlaw_gen.fit)rarbrcrdrUrYrrZrr_rrrr;rrr5rr-r-r r.rMs   rrc@s6eZdZdZejZddZddZddZ dd Z d S) 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)Nr,Frrr3rR)r6r@rr-r-r.rU}szpowerlognorm_gen._shape_infocCs@|||tt||ttt| ||ddSr)rr>rrrr6rXr,r3r-r-r.rYs zpowerlognorm_gen._pdfcCs"dttt| ||dSr)rrr>rrr-r-r.rZszpowerlognorm_gen._cdfcCs"t| ttd|d|Sr)r>r}rr)r6r^r,r3r-r-r.r_szpowerlognorm_gen._ppfN) rarbrcrdrrrrUrYrZr_r-r-r-r.rcs r powerlognormc@s8eZdZdZddZddZddZdd Zd d Zd S) powernorm_genaA 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, and :math:`\Phi` is the normal cdf, and :math:`x >= 0`, :math:`c > 0`. `powernorm` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s %(example)s cCstdddtjfdgSr+rRrTr-r-r.rUszpowernorm_gen._shape_infocCs|t|t| |dSrrrr.r-r-r.rYszpowernorm_gen._pdfcCs$t|t||dt| SrJ)r>rrrr.r-r-r.rszpowernorm_gen._logpdfcCsdt| |dSrrr.r-r-r.rZszpowernorm_gen._cdfcCsttd|d| Sr)rrr1r-r-r.r_szpowernorm_gen._ppfNrr-r-r-r.rs r powernormc@sJeZdZdZddZddZddZdd Zd d Zdd dZ ddZ d S) rdist_gena/An R-distributed (symmetric beta) continuous random variable. %(before_notes)s Notes ----- The probability density function for `rdist` is: .. math:: f(x, c) = \frac{(1-x^2)^{c/2-1}}{B(1/2, c/2)} for :math:`-1 \le x \le 1`, :math:`c > 0`. `rdist` is also called the symmetric beta distribution: if B has a `beta` distribution with parameters (c/2, c/2), then X = 2*B - 1 follows a R-distribution with parameter c. `rdist` takes ``c`` as a shape parameter for :math:`c`. This distribution includes the following distribution kernels as special cases:: c = 2: uniform c = 3: `semicircular` c = 4: Epanechnikov (parabolic) c = 6: quartic (biweight) c = 8: triweight %(after_notes)s %(example)s cCstdddtjfdgSr+rRrTr-r-r.rUszrdist_gen._shape_infocCst|||Sr<r#r.r-r-r.rYszrdist_gen._pdfcCs*td t|dd|d|dSr)r>rrrr.r-r-r.rszrdist_gen._logpdfcCst|dd|d|dSr)rrZr.r-r-r.rZszrdist_gen._cdfcCsdt||d|ddSr)rr_r1r-r-r.r_szrdist_gen._ppfNcCsd||d|d|dSrrrr-r-r.rszrdist_gen._rvscCs8d|dt|dd|d}|td|dS)NrrDrfr|rmr_)r6rNr, numeratorr-r-r.rs$zrdist_gen._munp)NN) rarbrcrdrUrYrrZr_rrr-r-r-r.r s! r rQrdistcseZdZdZejZddZdddZddZ 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 cCsgSr<r-rTr-r-r.rUszrayleigh_gen._shape_infoNcCstjd||dS)NrDrrrr-r-r.rszrayleigh_gen._rvscCst||Sr<r#r6r}r-r-r.rYszrayleigh_gen._pdfcCst|d||SrDrr$r-r-r.rszrayleigh_gen._logpdfcCstd|d Srrr$r-r-r.rZszrayleigh_gen._cdfcCstdt| SNr)r>rr[rryr-r-r.r_"szrayleigh_gen._ppfcCst||Sr<rNr$r-r-r.r\%szrayleigh_gen._sfcCs d||S)Nrr-r$r-r-r.r(szrayleigh_gen._logsfcCstdt|Sr%)r>rrryr-r-r.r`+szrayleigh_gen._isfcCsZdtj}ttjd|ddtjdttj|ddtj|d|dfS)NrrDrrrrrrr-r-r.r.s   zrayleigh_gen._statscCstdddtdS)Nr|rrmrDr rTr-r-r.r5szrayleigh_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$|ddr tjf||St|||\}}fdd}fdd}|ffdd }|dk rt|d krtd d tjd n |||fS|d } | dkr| d } |dkr|n|} t t tj } t | | } t j| | | fd} | jst| j| j}|p||}||fS)Nr Fcs"t|ddtdSr)r>rr)r%rr-r. scale_mleGsz#rayleigh_gen.fit..scale_mlecs@|}|}|d}d|}||dt|Sr)rr)r%r,rrZs3rr-r.loc_mleLs   z!rayleigh_gen.fit..loc_mlecs$|}||dd|Sr)r)r%r&r,rr-r.loc_mle_scale_fixedUsz-rayleigh_gen.fit..loc_mle_scale_fixedrrayleighrrr%r)r*r3r5rr>rrrSr1r rrrHr r#rrFflagr)r6rr7r,rrr&r'r(Zloc0r9rBrArr%r&r rr.r58s2        zrayleigh_gen.fit)NN)rarbrcrdrrrrUrrYrrZr_r\rr`rrr;rr5rr-r-r r.r#s   r#r)cseZdZdZddZddZfddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ dZeeedfddZZS)reciprocal_genaA 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 >>> 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@Srr-rr-r-r.rOszreciprocal_gen._argcheckcCs0tdddtjfd}tdddtjfd}||gSrrRrr-r-r.rUszreciprocal_gen._shape_infocs tj|t|t|fdSNrr3r r>rrrr r-r.r szreciprocal_gen._fitstartcCs||fSr<r-rr-r-r.roszreciprocal_gen._get_supportcCsd|t|d|Srrrr-r-r.rYszreciprocal_gen._pdfcCs$t| tt|d|Srrrr-r-r.rszreciprocal_gen._logpdfcCs&t|t|t|d|Srrrr-r-r.rZszreciprocal_gen._cdfcCs|t|d||Srr r=r-r-r.r_szreciprocal_gen._ppfcCs6dt|d||t|d|t|d|Srr+r(r-r-r.rszreciprocal_gen._munpcCs*dt||tt|d|Srlrrr-r-r.rszreciprocal_gen._entropyz `loguniform`/`reciprocal` is over-parameterized. `fit` automatically fixes `scale` to 1 unless `fscale` is provided by the user. rcs(|dd}tj|f|d|i|S)Nrr)r*r3r5)r6rr7r,rr r-r.r5s zreciprocal_gen.fit)rarbrcrdrOrUr rorYrrZr_rrZfit_noterrr5rr-r-r r.r+vs2  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|dkSrr-r6rir-r-r.rOszrice_gen._argcheckcCstdddtjfdgS)NriFrrQrRrTr-r-r.rUszrice_gen._shape_infoNcCs4|td|jd|d}t||jddS)NrD)rDrrrY)r>rrr)r6rirrrr-r-r.rsz rice_gen._rvscCstt|dt|Sr)r[Zchndtrr>squarerr-r-r.rZsz rice_gen._cdfc Cstt|dt|Sr)r>rr[Zchndtrixr2rr-r-r.r_sz rice_gen._ppfcCs.|t|| ||dt||Sr)r>r}r[i0err-r-r.rYsz rice_gen._pdfcCsH|d}d|}||d}d|t| t|t|d|Sry)r>r}r[rr)r6rNriZnd2Zn1rr-r-r.rs   zrice_gen._munp)NN) rarbrcrdrOrUrrZr_rYrr-r-r-r.r0s  r0ricec@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 cCstdddtjfdgSrrRrTr-r-r.rU2szrecipinvgauss_gen._shape_infocCst|||Sr<r#rr-r-r.rY5szrecipinvgauss_gen._pdfcCs t|dk||fddtj dS)NrcSs:d||d d||ddtdtj|S)Nrr|rDrmr)rXrr-r-r.rC<s z+recipinvgauss_gen._logpdf..rXrxrr-r-r.r:szrecipinvgauss_gen._logpdfcCsPd||}d||}dt|}t| |td|t| |SNrfr|r>rrr}r6rXrrrZisqxr-r-r.rZ@s  zrecipinvgauss_gen._cdfcCsNd||}d||}dt|}t||td|t| |Sr6r7r8r-r-r.r\Fs  zrecipinvgauss_gen._sfNcCsd|j|d|dSrrrr-r-r.rLszrecipinvgauss_gen._rvs)NN) rarbrcrdrUrYrrZr\rr-r-r-r.r5sr5 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 cCsgSr<r-rTr-r-r.rUrszsemicircular_gen._shape_infocCsdtjtd||Sryrrxr-r-r.rYuszsemicircular_gen._pdfcCs$tdtjdt| |Srrrxr-r-r.rxszsemicircular_gen._logpdfcCs.ddtj|td||t|S)Nrmrfr)r>rrrrxr-r-r.rZ{szsemicircular_gen._cdfcCs t|dSNr)r"r_ryr-r-r.r_~szsemicircular_gen._ppfNcCs2t|j|d}ttj|j|d}||Sr)r>rrrr)r6rrr}rhr-r-r.rszsemicircular_gen._rvscCsdS)N)rrrrQr-rTr-r-r.rszsemicircular_gen._statscCsdS)NgzCϑ?r-rTr-r-r.rszsemicircular_gen._entropy)NN) rarbrcrdrUrYrrZr_rrrr-r-r-r.r:Ss 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|dkSrJ)r>rrr-r-r.rOszskewcauchy_gen._argcheckcCstddddgS)NrhF)rQrfrrrTr-r-r.rUszskewcauchy_gen._shape_infocCs,dtj|d|t|dddSr)r>rr?rr-r-r.rYszskewcauchy_gen._pdfc Csbt|dkd|dd|tjt|d|d|dd|tjt|d|SNrrrD)r>r&rrmrr-r-r.rZs **zskewcauchy_gen._cdfc Csn||d|k}t|ttjd||d|dd|ttjd||d|dd|Sr?)rZr>r&ror)r6rXrhrr-r-r.r_s **zskewcauchy_gen._ppfrcCstjtjtjtjfSr<rp)r6rhr9r-r-r.rszskewcauchy_gen._statscCs*t|dddg\}}}d|||dfS)NrqrrrsrerDrt)r6rrurvrwr-r-r.r szskewcauchy_gen._fitstartN)r) rarbrcrdrOrUrYrZr_rr r-r-r-r.r=s! r= skewcauchycseZdZdZddZddZddZfdd Zd d Zd d Z ddZ dddZ dddZ e ddZddZeeddfddZZS) skew_norm_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|Sr<r2rr-r-r.rOszskew_norm_gen._argcheckcCstddtj tjfdgS)NrhFrrRrTr-r-r.rUszskew_norm_gen._shape_infocCs t|dk||fdddddS)NrcSst|Sr<rrXrhr-r-r.rCrDz$skew_norm_gen._pdf..cSsdt|t||SrrrBr-r-r.rCrDrFr6rr-r-r.rYs zskew_norm_gen._pdfcsVt|dd|}t||j}|dk|dk@}t||||||<t|ddS)Nrrgư>)rZ _skewnorm_cdfr>rrr3rZr)r6rXrhrsZ i_small_cdfr r-r.rZs zskew_norm_gen._cdfcCst|dd|Sr)rZ _skewnorm_ppfrr-r-r.r_ szskew_norm_gen._ppfcCs|| | Sr<rErr-r-r.r\ szskew_norm_gen._sfcCst|dd|Sr)rZ _skewnorm_isfrr-r-r.r` szskew_norm_gen._isfNcCs`|j|d}|j|d}|td|d}|||td|d}t|dk|| S)NrrrDr)normalr>rr&)r6rhrrZu0rr>r!r-r-r.r s   zskew_norm_gen._rvsrcCsddddg}tdtj|td|d}d|krB||d<d|krZd|d|d<d|krdtjd|td|dd|d<d |krdtjd|dd|dd|d<|S) NrDrrrrr3rrr6r)r6rhr9rKconstr-r-r.r s &,*zskew_norm_gen._statscCstdgtddgtdddgtdddd gtd d d d dgtddddddgtdddddddgtddddddd d!gtd"d#d$d%d&d'd(d)dg td*d+d,d-d.d/d0d1d2d3g d4 }|S)5Nrrreriii?iiiinii(iSi6QiiiOiiBi/iioiiiԷiiYei{Hxiii!i! iׅi쇀 iiViX'i liis_'il     z#skew_norm_gen._skewnorm_odd_momentscCsn|d@rH|dkrtd|td|d}||j||dtSt|ddd|dtSdS)NrrIzKskewnorm noncentral moments not implemented for odd orders greater than 19.rD)r'r>rrJr!r[rr )r6orderrhr.r-r-r.rB s zskew_norm_gen._munpa 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. Note that the maximum possible skewness magnitude of a `scipy.stats.skewnorm` distribution is approximately 0.9952717; if the magnitude of the data's sample skewness exceeds this, the returned shape parameter ``a`` will be infinite. rc s*t||||\}}}}|dd}ddt|d}t|krt|dkrt|dkrt|sttj|f||S|dkrd\} } } n,t|r|dnd} | d d} | d d} |dkr6| dkr6t | |t fd d d dgdj } t jdd.t t | dd| dt } W5QRXn(|dk rD|n| } | t d| d} |dkr| dkrt |} t | dd| dt j} n|dk r|} |dkr| dkrt |}|| | t dt j} n|dk r|} |dkr | | | fStj|| f| | d|SdS)Nr(r0cSs@dtjd|tdtjddd|dtjdS)NrrDrrrrr>r-r-r.skew_dk s"z!skew_norm_gen.fit..skew_drrrrr%r&cs |Sr<r-rLr3rMr-r.rC rDz#skew_norm_gen.fit..rerrrrDr)rr1r2rr rr3r5rr*r>rr#rrrrr?rrr)r6rr7r,r rrr(Zs_maxrhr%r&r>rrr rNr.r5V sF      4 "     zskew_norm_gen.fit)NN)r)rarbrcrdrOrUrYrZr_r\r`rrrrJrrrr5rr-r-r r.rAs     rAskewnormc@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-r6r,r>r-r-r.rO sztrapezoid_gen._argcheckcCs$tdddd}tdddd}||gS)Nr,FrrfTTr>r>r?r-r-r.rU sztrapezoid_gen._shape_infocCsPd||d}t||k||k||k@||kgddddddg||||fS)NrDrcSs |||Sr<r-rXr,r>rr-r-r.rC rDz$trapezoid_gen._pdf..cSs|Sr<r-rTr-r-r.rC rDcSs|d|d|SrJr-rTr-r-r.rC rDr )r6rXr,r>rr-r-r.rY s ztrapezoid_gen._pdfcCs>t||k||k||k@||kgddddddg|||fS)NcSs|d|||dSrr-rXr,r>r-r-r.rC rDz$trapezoid_gen._cdf..cSs|d||||dSrr-rVr-r-r.rC rDcSs$dd|d||dd|Srr-rVr-r-r.rC s  rUrMr-r-r.rZ sztrapezoid_gen._cdfcCs||||||||}}||k||k||kg}t||d||d|d||d|dtd|||dd|g}t||Sr{)rZr>rselect)r6r^r,r>ZqcZqdrrr-r-r.r_ s$ztrapezoid_gen._ppfcsz|d}t|dkd|k|dk@|dkgddfddfddg|g}dd||||dd }|S) NrrerfcSsdSrr-rLr-r-r.rC rDz%trapezoid_gen._munp..cs tdt||dSr)r>r0rrLrOr-r.rC rDcsdSrr-rLrOr-r.rC rDr|rDrU)r6rNr,r>Zab_termZdc_termr'r-rOr.r s   (ztrapezoid_gen._munpcCs2dd||d||tdd||SrlrrQr-r-r.r!sztrapezoid_gen._entropyN) rarbrcrdrOrUrYrZr_rrr-r-r-r.rP s!  rP trapezoidtrapzz!trapz is an alias for `trapezoid`c@sReZdZdZdddZddZddZd d Zd d Zd dZ ddZ ddZ dS) triang_gena5A triangular continuous random variable. %(before_notes)s Notes ----- The triangular distribution can be represented with an up-sloping line from ``loc`` to ``(loc + c*scale)`` and then downsloping for ``(loc + c*scale)`` to ``(loc + scale)``. `triang` takes ``c`` as a shape parameter for :math:`0 \le c \le 1`. %(after_notes)s The standard form is in the range [0, 1] with c the mode. The location parameter shifts the start to `loc`. The scale parameter changes the width from 1 to `scale`. %(example)s NcCs|d|d|Sr) triangularrr-r-r.r)!sztriang_gen._rvscCs|dk|dk@Srr-rir-r-r.rO,!sztriang_gen._argcheckcCstddddgS)Nr,FrRrSr>rTr-r-r.rU/!sztriang_gen._shape_infocCsLt|dk||k||k|dk@|dkgddddddddg||f}|S)NrrcSs dd|Srr-r5r-r-r.rC.cSs d||Srr-r5r-r-r.rC=!rDcSsdd|d|Srr-r5r-r-r.rC>!rDcSsd|Srr-r5r-r-r.rC?!rDrUr6rXr,r}r-r-r.rY2!s ztriang_gen._pdfcCsLt|dk||k||k|dk@|dkgddddddddg||f}|S)NrrcSsd|||Srr-r5r-r-r.rCH!rDz!triang_gen._cdf..cSs |||Sr<r-r5r-r-r.rCI!rDcSs||d|||dSrr-r5r-r-r.rCJ!rDcSs||Sr<r-r5r-r-r.rCK!rDrUr\r-r-r.rZC!s ztriang_gen._cdfc Cs2t||kt||dtd|d|SrJ)r>r&rr1r-r-r.r_O!sztriang_gen._ppfc Csb|ddd|||dtdd|d|d|ddtd|||ddfS) Nrfr$rDrrSrg333333)r>rrrir-r-r.rR!s  @ztriang_gen._statscCsdtdSrrrir-r-r.rX!sztriang_gen._entropy)NN) rarbrcrdrrOrUrYrZr_rrr-r-r-r.rZ!s  rZtriangc@sPeZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ dS)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 cCstdddtjfdgSrrRrTr-r-r.rUu!sztruncexpon_gen._shape_infocCs |j|fSr<rnr1r-r-r.rox!sztruncexpon_gen._get_supportcCst| t|  Sr<rrr-r-r.rY{!sztruncexpon_gen._pdfcCs| tt|  Sr<rCrr-r-r.r!sztruncexpon_gen._logpdfcCst| t| Sr<rrr-r-r.rZ!sztruncexpon_gen._cdfcCst|t|  Sr<)r[rr0rr-r-r.r_!sztruncexpon_gen._ppfcCs|dkr.d|dt| t|  S|dkrpddd||d|dt| t|  S|||SdSr)r>r}r[r0r)r6rNrir-r-r.r!s &:ztruncexpon_gen._munpcCs0t|}t|dd||dd|Sr)r>r}r)r6riZeBr-r-r.r!s ztruncexpon_gen._entropyN) rarbrcrdrUrorYrrZr_rrr-r-r-r.r__!s r_ truncexponcCstj||gddS)NrrY)r[rZlog_pZlog_qr-r-r._log_sum!srbcCstj||tjdgddS)Ny?rrY)r[rr>rrar-r-r. _log_diff!srccst|t|}}t||\}}|dk}|dk}||B}ddfdd}dd}tj|tjtjd}||jr||||||<||jr|||||||<||jr|||||||<t|S) z3Log of Gaussian probability mass within an intervalrcSstt|t|Sr<)rcr[rrhrir-r-r.mass_case_left!sz'_log_gauss_mass..mass_case_leftcs| | Sr<r-rdrer-r.mass_case_right!sz(_log_gauss_mass..mass_case_rightcSstt| t| Sr<)r[rrrdr-r-r.mass_case_central!s z*_log_gauss_mass..mass_case_central)rr)r>r rr rZ complex128rreal)rhri case_left case_rightZ case_centralrgrhrr-rfr._log_gauss_mass!s      rlcseZdZdZddZddZfddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ ddZddZddZdddZZS) truncnorm_gena^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 clipped at ``a``, ``b`` standard deviations to the left, right (respectively) from ``loc``. If ``myclip_a`` and ``myclip_b`` are clip values in the sample space (as opposed to the number of standard deviations) then they can be converted to the required form according to:: a, b = (myclip_a - loc) / scale, (myclip_b - loc) / scale %(example)s cCs||kSr<r-rr-r-r.rO!sztruncnorm_gen._argcheckcCs8tddtj tjfd}tddtj tjfd}||gS)NrhFrQri)FTrRrr-r-r.rU!sztruncnorm_gen._shape_infocs tj|t|t|fdSr,r-rr r-r.r !sztruncnorm_gen._fitstartcCs||fSr<r-rr-r-r.ro!sztruncnorm_gen._get_supportcCst||||Sr<r#rr-r-r.rY!sztruncnorm_gen._pdfcCst|t||Sr<)rrlrr-r-r.r!sztruncnorm_gen._logpdfcCst||||Sr<rMrr-r-r.rZ!sztruncnorm_gen._cdfc Csjt|||\}}}t||t||}|dk}t|rftt||||||| ||<|SNg)r>rrlrrr}r)r6rXrhrirrr-r-r.r!s  ,ztruncnorm_gen._logcdfcCst||||Sr<rNrr-r-r.r\"sztruncnorm_gen._sfc Csjt|||\}}}t||t||}|dk}t|rftt||||||| ||<|Srn)r>rrlrrr}r)r6rXrhrirrr-r-r.r"s  ,ztruncnorm_gen._logsfc Cst|||\}}}|dk}|}dd}dd}t|}||} ||} | jrj|| ||||||<| jr|| ||||||<|S)NrcSs*tt|t|t||}t|Sr<)rbr[rr>rrl ndtri_expr^rhriZ log_Phi_xr-r-r.ppf_left"s z$truncnorm_gen._ppf..ppf_leftcSs0tt| t| t||}t| Sr<)rbr[rr>rrlrorpr-r-r. ppf_right"s z%truncnorm_gen._ppf..ppf_rightr>r empty_liker) r6r^rhrirjrkrqrrrq_leftq_rightr-r-r.r_"s ztruncnorm_gen._ppfc Cst|||\}}}|dk}|}dd}dd}t|}||} ||} | jrj|| ||||||<| jr|| ||||||<|S)NrcSs0tt|t|t||}tt|Sr<)rcr[rr>rrlrorirpr-r-r.isf_left1"s z$truncnorm_gen._isf..isf_leftcSs6tt| t| t||}tt| Sr<)rcr[rr>rrlrorirpr-r-r. isf_right6"s z%truncnorm_gen._isf..isf_rightrs) r6r^rhrirjrkrwrxrrurvr-r-r.r`*"s ztruncnorm_gen._isfcsDfdd}t|dk||k@||k@|||ftj|tjgdtjS)Nc s||g||\}}|| g}ddg}td|dD]Ht||||ggfdddd}t|d|d}||q6|dS)z Returns n-th moment. Defined only if n >= 0. Function cannot broadcast due to the loop over n rrcs||dSrJr-rXr^r6r-r.rCV"rDz:truncnorm_gen._munp..n_th_moment..rXrre)rYrrr>rr') rNrhripApBprobsr9r,mkrTrzr. n_th_momentH"s   z(truncnorm_gen._munp..n_th_momentrZotypes)rr>rfloat64r)r6rNrhrirr-rTr.rG"s   ztruncnorm_gen._munpr2cCsB|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||Sr<r-ryr-r-r.rCg"rDzGtruncnorm_gen._stats.._truncnorm_stats_scalar..rrXrcSs||Sr<r-ryr-r-r.rCj"rDcSs ||dSrr-ryr-r-r.rCo"rDrDcSs ||dSr;r-ryr-r-r.rCr"rDrr`rrT)rr>rr)rhrir{r|r9rrr}r,rrZm3Zm4Zmu3rZmu4rr-r-r._truncnorm_stats_scalarb"s0 (z5truncnorm_gen._stats.._truncnorm_stats_scalar)r9)Zexcluded)rr>rr)r6rhrir9r{r|rZ_truncnorm_statsr-r-r.r_"s ztruncnorm_gen._stats)r2)rarbrcrdrOrUr rorYrrZrr\rr_r`rrrr-r-r r.rm!s rm truncnorm)rjrvc@seZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ ddZddZddZddZdS)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)NriFrerr,rfrR)r6rr@r-r-r.rU"sztruncpareto_gen._shape_infocCs|dk|dk@Srr-r6rir,r-r-r.rO"sztruncpareto_gen._argcheckcCs |j|fSr<rnrr-r-r.ro"sztruncpareto_gen._get_supportcCs |||d d|| SrJr-r6rXrir,r-r-r.rY"sztruncpareto_gen._pdfcCs.t|t||  |dt|SrJrrr-r-r.r"sztruncpareto_gen._logpdfcCsd|| d|| SrJr-rr-r-r.rZ"sztruncpareto_gen._cdfcCs$t||  t||  Sr<rOrr-r-r.r"sztruncpareto_gen._logcdfcCs tdd|| |d|SNrrer r6r^rir,r-r-r.r_"sztruncpareto_gen._ppfcCs"|| || d|| SrJr-rr-r-r.r\"sztruncpareto_gen._sfcCs,t|| || t||  Sr<rrr-r-r.r"sztruncpareto_gen._logsfcCs&t|| d|| |d|Srr rr-r-r.r`"sztruncpareto_gen._isfcCs@t|d|| |dt|||dd| SrJrrr-r-r.r"s$ztruncpareto_gen._entropycCsP||kr$|t|d|| S|||||||||dSdSrJr)r6rNrir,r-r-r.r"sztruncpareto_gen._munpcCs,t|\}}}t|||}||||fSr<)rr5r)r6rrir%r&r,r-r-r.r "sztruncpareto_gen._fitstartN)rarbrcrdrUrOrorYrrZrr_r\rr`rrr r-r-r-r.r"s)r truncparetoc@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|Sr<r2r6lamr-r-r.rO"sztukeylambda_gen._argcheckcCstddtj tjfdgS)NrFrrRrTr-r-r.rU"sztukeylambda_gen._shape_infocCsjtt||}||dtd||d}dt|}t|dkt|dt|kB|dS)Nrfrrre)r>rr[tklmbdar&r)r6rXrZFxrr-r-r.rY#s"ztukeylambda_gen._pdfcCs t||Sr<)r[r)r6rXrr-r-r.rZ#sztukeylambda_gen._cdfcCst||t| |Sr<)r[r8r7)r6r^rr-r-r.r_ #sztukeylambda_gen._ppfcCsdt|dt|fSr)_tlvar_tlkurtrr-r-r.r #sztukeylambda_gen._statscsfdd}t|dddS)Ncs&tt|dtd|dSrJr+)rrr-r.integ#sz'tukeylambda_gen._entropy..integrr)r r)r6rrr-rr.r#s ztukeylambda_gen._entropyN) rarbrcrdrOrUrYrZr_rrr-r-r-r.r"sr tukeylambdac@seZdZddZdS)FitUniformFixedScaleDataErrorcCsd||ff|_dS)NzInvalid values in `data`. Maximum likelihood estimation with the uniform distribution and fixed scale requires that data.ptp() <= fscale, but data.ptp() = %r and fscale = %r.r)r6rrr-r-r.r#sz&FitUniformFixedScaleDataError.__init__N)rarbrcrr-r-r-r.r#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 cCsgSr<r-rTr-r-r.rU/#szuniform_gen._shape_infoNcCs|dd|S)Nrerf)rrr-r-r.r2#szuniform_gen._rvscCs d||kSrr-rxr-r-r.rY5#szuniform_gen._pdfcCs|Sr<r-rxr-r-r.rZ8#szuniform_gen._cdfcCs|Sr<r-ryr-r-r.r_;#szuniform_gen._ppfcCsdS)N)rmgUUUUUU?rg333333r-rTr-r-r.r>#szuniform_gen._statscCsdSrr-rTr-r-r.rA#szuniform_gen._entropyc Ost|dkrtd|dd}|dd}t||dk rL|dk rLtdt|}t|sltd|dkr|dkr| }| }q|}| |}| |krt d|||d n4| }||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)rrrm)rr+r*r/rr>rrrrrrrrr) r6rr7r,rrr%r&rr-r-r.r5D#s0K        zuniform_gen.fit)NN) rarbrcrdrUrrYrZr_rrr;r5r-r-r-r.r##s  rrcs|eZdZdZddZdddZeefddZd d Z d d Z d dZ ddZ ddZ eedddfdd ZZS) vonmises_genaA Von Mises continuous random variable. %(before_notes)s 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 > 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. `vonmises` and `vonmises_line` take ``kappa`` as a shape parameter. %(after_notes)s %(example)s cCstdddtjfdgSrJrRrTr-r-r.rU#szvonmises_gen._shape_infoNcCs|jd||dS)Nrer)vonmises)r6rKrrr-r-r.r#szvonmises_gen._rvscs,tj||}t|tjdtjtjSr)r3r r>modr)r6r7r,r r r-r.r $szvonmises_gen.rvscCs(t|t|dtjt|Sr)r>r}r[cosm1rr3rLr-r-r.rY$szvonmises_gen._pdfcCs.|t|tdtjtt|Sr)r[rr>rrr3rLr-r-r.r$szvonmises_gen._logpdfcCs t||Sr<)rZ von_mises_cdfrLr-r-r.rZ$szvonmises_gen._cdfcCsdSrr-rSr-r-r. _stats_skip$szvonmises_gen._stats_skipcCs8| t|t|tdtjt||Sr)r[Zi1er3r>rrrSr-r-r.r$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  sLtj tj} } |dkr || }|dkr0|| }tj|||||||f|Sr<)r>rr3r) r6rr7r%r&ZlbZubZ conditionalr,rIrHr r-r.r$$s zvonmises_gen.expect)NN)Nr-rrNNF)rarbrcrdrUrr rr rYrrZrrrrrr-r-r r.r#s  rr vonmises_linec@speZdZdZejZddZdddZddZ d d Z d d Z d dZ ddZ ddZddZddZddZdS)raXA 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 cCsgSr<r-rTr-r-r.rUP$szwald_gen._shape_infoNcCs|jdd|dSrrrr-r-r.rS$sz wald_gen._rvscCs t|dSr)rrYrxr-r-r.rYV$sz wald_gen._pdfcCs t|dSr)rrZrxr-r-r.rZZ$sz wald_gen._cdfcCs t|dSr)rr\rxr-r-r.r\]$sz wald_gen._sfcCs t|dSr)rr_rxr-r-r.r_`$sz wald_gen._ppfcCs t|dSr)rr`rxr-r-r.r`c$sz wald_gen._isfcCs t|dSr)rrrxr-r-r.rf$szwald_gen._logpdfcCs t|dSr)rrrxr-r-r.ri$szwald_gen._logcdfcCs t|dSr)rrrxr-r-r.rl$szwald_gen._logsfcCsdS)N)rfrfr$r-r-rTr-r-r.ro$szwald_gen._stats)NN)rarbrcrdrrrrUrrYrZr\r_r`rrrrr-r-r-r.r9$s rrc@sHeZdZdZddZddZddZdd Zd d Zd d Z ddZ dS)wrapcauchy_genaA wrapped Cauchy continuous random variable. %(before_notes)s Notes ----- The probability density function for `wrapcauchy` is: .. math:: f(x, c) = \frac{1-c^2}{2\pi (1+c^2 - 2c \cos(x))} for :math:`0 \le x \le 2\pi`, :math:`0 < c < 1`. `wrapcauchy` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s %(example)s cCs|dk|dk@Srr-rir-r-r.rO$szwrapcauchy_gen._argcheckcCstddddgS)Nr,F)rrrr>rTr-r-r.rU$szwrapcauchy_gen._shape_infocCs4d||dtjd||d|t|Srrr.r-r-r.rY$szwrapcauchy_gen._pdfcCs:dd}dd}d|d|}t|tjk||f||dS)NcSs"dtjt|t|dSrr>rrmrorXcrr-r-r.r$szwrapcauchy_gen._cdf..f1c Ss0ddtjt|tdtj|dSrrrr-r-r.rG$szwrapcauchy_gen._cdf..f2rr)rr>r)r6rXr,rrGrr-r-r.rZ$szwrapcauchy_gen._cdfc Csld|d|}dt|ttj|}dtjdt|ttjd|}t|dk||S)NrfrDrrm)r>rmrorr&)r6r^r,r'ZrcqZrcmqr-r-r.r_$s,zwrapcauchy_gen._ppfcCstdtjd||Srrrir-r-r.r$szwrapcauchy_gen._entropycCs dt|t|dtjfSr)r>rrrrr-r-r.r $szwrapcauchy_gen._fitstartN) rarbrcrdrOrUrYrZr_rr r-r-r-r.rv$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 cCstdddtjfdgSNrFrrrRrTr-r-r.rU$szgennorm_gen._shape_infocCst|||Sr<r#r6rXrr-r-r.rY$szgennorm_gen._pdfcCs(td|td|t||Srl)r>rr[rrrr-r-r.r$szgennorm_gen._logpdfcCs2dt|}d||td|t||Srl)r>r?r[rrr6rXrr,r-r-r.rZ$szgennorm_gen._cdfcCs:t|d}|td|d|d||d|S)Nrmrfr|)r>r?r[rrr-r-r.r_$szgennorm_gen._ppfcCs|| |Sr<rErr-r-r.r\$szgennorm_gen._sfcCs||| Sr<rGrr-r-r.r`$szgennorm_gen._isfcCsNtd|d|d|g\}}}dt||dt||d|dfS)Nrfr$rrer|)r[rr>r})r6rZc1c3Zc5r-r-r.r$s"zgennorm_gen._statscCs$d|td|td|Srrr6rr-r-r.r$szgennorm_gen._entropyNcCsL|jd||d}|d|}t|}|j|jddk}|| ||<|S)Nrrrm)rr>rrandomr)r6rrrrr^rr-r-r.r%s   zgennorm_gen._rvs)NN)rarbrcrdrUrYrrZr_r\r`rrrr-r-r-r.r$s*rgennormc@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 cCstdddtjfdgSrrRrTr-r-r.rU4%szhalfgennorm_gen._shape_infocCst|||Sr<r#rr-r-r.rY7%szhalfgennorm_gen._pdfcCs t|td|||Srrrr-r-r.r=%szhalfgennorm_gen._logpdfcCstd|||Srrrr-r-r.rZ@%szhalfgennorm_gen._cdfcCstd||d|Srrrr-r-r.r_C%szhalfgennorm_gen._ppfcCstd|||Srrrr-r-r.r\F%szhalfgennorm_gen._sfcCstd||d|Srrjrr-r-r.r`I%szhalfgennorm_gen._isfcCs d|t|td|Srrrr-r-r.rL%szhalfgennorm_gen._entropyN) rarbrcrdrUrYrrZr_r\r`rr-r-r-r.r%s#r 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. References ---------- .. [1] "Crystal Ball Function", https://en.wikipedia.org/wiki/Crystal_Ball_function %(after_notes)s .. versionadded:: 0.19.0 %(example)s cCs|dk|dk@S)z@ Shape parameter bounds are m > 1 and beta > 0. rrr-)r6rrr-r-r.rOw%szcrystalball_gen._argcheckcCs0tdddtjfd}tdddtjfd}||gS)NrFrrrrrR)r6ZibetaZimr-r-r.rU}%szcrystalball_gen._shape_infocstj|ddS)N)rrrrrr 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 -- rfrrDr|cSst|d dSrrrXrrr-r-r.rhs%sz!crystalball_gen._pdf..rhscSs6|||t|d d||||| Sr{rrr-r-r.lhs%sz!crystalball_gen._pdf..lhsrr>r}r~rrr6rXrrrrrr-r-r.rY%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. rfrrDr|cSs|d dSrr-rr-r-r.r%sz$crystalball_gen._logpdf..rhscSs8|t|||dd|t||||Srrrr-r-r.r%sz$crystalball_gen._logpdf..lhsr)r>r}r~rrrrr-r-r.r%s $ zcrystalball_gen._logpdfcCsdd|||dt|d dtt|}dd}dd}|t|| k|||f||d S) z8 Return CDF of the crystalball function rfrrDr|cSs:||t|d d|dtt|t| SNrDr|rr>r}r~rrr-r-r.r%s"z!crystalball_gen._cdf..rhscSsB|||t|d d||||| d|dSrrrr-r-r.r%s z!crystalball_gen._cdf..lhsrrrr-r-r.rZ%s $ zcrystalball_gen._cdfcCsd|||dt|d dtt|}|||t|d d|d}dd}dd}t||k|||f||d S) NrfrrDr|cSsvt|d d}||||d}d|tt|}||||d||| |||dd|SrrrrrZeb2Crr-r-r.ppf_less%s  ,z&crystalball_gen._ppf..ppf_lesscSs^t|d d}||||d}d|tt|}tt| dt|||Sr)r>r}r~rrrr-r-r. ppf_greater%sz)crystalball_gen._ppf..ppf_greaterrr)r6rrrrZpbetarrr-r-r.r_%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. rfrrDr|cSs|||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 rDr|rrfre) r>r}r[rrrrrZbinom)rNrrrBrrr6r-r-r.r%s   & ,z*crystalball_gen._munp..n_th_momentr)r>r}r~rrrrrS)r6rNrrrrr-r-r.r%s$ zcrystalball_gen._munp) rarbrcrdrOrUr rYrrZr_rrr-r-r r.rS%s# r crystalballzA Crystalball Function)rjlongnamecCstd|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. rrDr)rr-r-r. _argus_phi%s rc@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. %(after_notes)s References ---------- .. [1] "ARGUS distribution", https://en.wikipedia.org/wiki/ARGUS_distribution .. versionadded:: 0.19.0 %(example)s cCstdddtjfdgS)NrFrrrRrTr-r-r.rU!&szargus_gen._shape_infoc Cstjddld||}dt|ttt|}|t|dt| ||d|dW5QRSQRXdS)NrrrfrrmrD)r>rrrrr)r6rXrr^rr-r-r.r$&s  zargus_gen._logpdfcCst|||Sr<r#r6rXrr-r-r.rY+&szargus_gen._pdfcCsd|||Srr*rr-r-r.rZ.&szargus_gen._cdfcCs"t|td|dt|Sr)rr>rrr-r-r.r\1&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)rrrrrc3s(|] }|sj|ntdVqdSr<rrrr-r.rA&sz!argus_gen._rvs..rr-)r>rrrrrr&rrrrrrrr\r) r6rrrrrrrr}r-rr.r4&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) NrrmrDrgUUUUUU?rg?r) rr>r r&rrrrrrr}rir\)r6rrrrrrXrr|r>r6rrrrrr ZechirWr-r-r.rL&sR4                zargus_gen._rvs_scalarc Cstj|td}t|}ttjd|td|dd|}t|}|dk}||}dd|d|t |||||<||}dd d d d d d d d g }t ||||<|||dddfS)NrrrrDrg?rg_1g־rgWBagp|RH?gE'卡?g?) r>rrrrrr[rrtrr)r6rrrrrr,Zcoefr-r-r.r&s, ( zargus_gen._stats)NN)NN) rarbrcrdrUrrYrZr\rrrr-r-r-r.r%s$  erarguszAn Argus Function)rjrrhrics`eZdZdZejZddfdd ZddZdd Zd d Z d d Z ddZ fddZ Z S) rv_histograma 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)densitycs^||_||_t|dkr tdt|d|_t|d|_t|jdt|jkr`td|jdd|jdd|_t |j|jd }|dkr|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||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. rDz)Expected length 2 for parameter histogramrrzbNumber of elements in histogram content and histogram boundaries do not match, expected n and n+1.NrezjBin widths are not constant. Assuming `density=True`.Specify `density` explicitly to silence this warning.rTrerhri) _histogram_densityrrr>r_hpdf_hbins _hbin_widthsZallcloserrsrtrrZcumsum_hcdfZhstackrhrir3r)r6 histogramrr7kwargsZ bins_varyrr r-r.r"'s.  zrv_histogram.__init__cCs|jtj|j|ddS)z& PDF of the histogram r0)Zside)rr>Z searchsortedrrxr-r-r.rYR'szrv_histogram._pdfcCst||j|jS)z3 CDF calculated from the histogram )r>interprrrxr-r-r.rZX'szrv_histogram._cdfcCst||j|jS)zC Percentile function calculated from the histogram )r>rrrrxr-r-r.r_^'szrv_histogram._ppfcCsL|jdd|d|jdd|d|d}t|jdd|S)z$Compute the n-th non-central moment.rNre)rr>rr)r6rNZ integralsr-r-r.rd's4zrv_histogram._munpcCsJt|jdddk|jddftjd}t|jdd||j S)zCompute entropy of distributionrrere)rrr>rrr)r6rr-r-r.ri's zrv_histogram._entropycs"t}|j|d<|j|d<|S)zF Set the histogram as additional constructor argument rr)r3_updated_ctor_paramrr)r6dctr r-r.rq's   z rv_histogram._updated_ctor_param)rarbrcrdrrrrYrZr_rrrrr-r-r r.r&sZ0rcsHeZdZdZddZddZfddZdd 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. 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