o EOfN@s&ddlZddlZddlmmZddlm Z dZ e e de Z e e de ZedejZedejZdZedZejdZejdZejd Zgd Zd d Zd dZd&ddZd&ddZddZd&ddZd&ddZ d&ddZ!ddZ"ddZ#d&d d!Z$d"d#Z%d&d$d%Z&dS)'N) _derivativei<)gSˆBgAAz?g}<ٰj_g#+K?g88CgJ?gllfgUUUUUU?cCs6d|}t|d|td|tt||S)N?r)nplog_LOG_2PIZpolyval_STIRLING_COEFFS)nZrnr4lib/python3.10/site-packages/scipy/stats/_ksstats.py_log_nfactorial_div_n_pow_n]s.rcCst|ddS)z%clips a probability to range 0<=p<=1.r )r Zclip)prrr _clip_probgsrTcCst|||}t|S)z>Selects either the CDF or SF, and then clips to range 0<=p<=1.)r wherer)cdfprobZsfprobcdfrrrr_select_and_clip_problsrcCs^|dkr tdd|S||}|dkrtdd|Stt|}||}d|d}t||g}td|d}d||} t|} d} |D]} | | | d<| | } | | d| 9<qGtd|dd|d||} d| | | d<td|D]}| d||d||dd|f<q|| |dddf<tj | dd |dddf<t t |d}|}d}d}|dkr|drt ||}||7}t ||}|d9}t ||d|dftkr|t}|t7}|d}|dks||d|df}td|dD]}|||}t |tkr|t9}|t8}q|dkr't||}t|d||S) zComputes the Kolmogorov CDF: Pr(D_n <= d) using the MTW approach to the Durbin matrix algorithm. Durbin (1968); Marsaglia, Tsang, Wang (2003). [1], [3]. r r?rrrN)Zaxis)rintr ceilzerosarangeemptymaxrangeZflipZeyeshapematmulabs_EP128_E128_EM128ldexp)rdrZndkhmHZintmvwZfacjttiZHpwrnnexpntZHexpntrrrr _kolmogn_DMTWrs`      "&      r5c Cs|dkr| |d||d}}nLt|dd\}}|dkrN||dkr8|||d|||d}}n'|d||d||d|d}}n|d|d|||d}}t|ddt||fS)z0Compute the endpoints of the interval for row i.rrr)divmodr min) r2rllceilfroundfj1j2Zip1div2Zip1mod2rrr_pomeranz_compute_j1j2s $,"r=c Cs||}tt|}d||}t|d|}|dkrdnd}|dkr&dnd}d|d} t| } t| } t| } d| d<d| d<d| d<d} ||d||dd||}}}td| D]&}| |d||| |<| |d||| |<| |d||| |<qdt| g}t| g}d|d<d\}}td||||\}}tdd|dD]}|}||}}||}}|dt|||||\}}|dks|d|dkr| }n|dr| n| }||d}|dkr:t |||||||d|}||}||d}|||||d|<dt |kr*t kr4nn|t 9}| t 8} |||}q|||}td|dD]}t|t krZ|t 9}| t 7} ||9}qH| dkrkt|| }t|d||}|S) z[Computes Pr(D_n <= d) using the Pomeranz recursion algorithm. Pomeranz (1974) [2] r rrrr)rrrN)rr Zfloorr7rr!rr=fillZconvolver r'r%r&r$r(r) rxrtr8fgr9r:ZnpwrsZgpowerZ twogpowerZ onem2gpowerr4Zg_over_nZ two_g_over_nZone_minus_two_g_over_nr,ZV0ZV1ZV0sZV1sr;r<r2Zk1ZpwrsZln2convZ conv_startZconv_lenZansrrr_kolmogn_Pomeranzsl     (       ( "     rDc% Cs.|dkr tdd|dS|dkrtdd|dSt||}|d|d|d|df\}}}}t d|}|tkrAtdd|dSt|} | } td} d|d|} d|d |td} td d|d }td d |d }td|d|d }td|d|d}d|d|d}td}t t d |tj }t |ddD]G}d|d }|d|d|d}}}t | d|}td| | || | ||||||||||g}||9}||7}q|| 9}|t9}|t|d|d|dd|dg}tt d|} t|dd}|d}t|}tj |}| |} t|| }!|!ttd|9}!|d|!7<t|||||| }"|"ttd|9}"|d|"7<t |dtt|d}#||#}|s|d9}|dd 7<t|}$|$S)aPComputes the Pelz-Good approximation to Prob(Dn <= x) with 0<=x<=1. Start with Li-Chien, Korolyuk approximation: Prob(Dn <= x) ~ K0(z) + K1(z)/sqrt(n) + K2(z)/n + K3(z)/n**1.5 where z = x*sqrt(n). Transform each K_(z) using Jacobi theta functions into a form suitable for small z. Pelz-Good (1976). [6] rr rrrrrr@i`iZrrHiP i@)rr sqrt _PI_SQUARED_MIN_LOGexp_PI_FOUR_PI_SIXrrrpir!ZpowerZarray_SQRT2PIr_SQRT3sumlen)%rr?rzZzsquaredZzthreeZzfourZzsixZqlogqZk1aZk1bZk2aZk2bZk2cZk3dZk3cZk3bZk3aZK0to3Zmaxkr*r,ZmsquaredZmfourZmsixZqpowerZcoeffsksZksquaredZsqrt3zZkspiZqpwersZk2extraZk3extraZ powers_of_nZKsumrrr_kolmogn_PelzGood#sl $     * rbcCsPt|r|St||ks|dkrtjS|dkrtdd|dS|dkr*tdd|dS||}|dkrr|dkr=tdd|dS|dkrWttd|dd|d|d}ntt||t d|d}t|d||dS||dkrdd||}td|||dS|dkrdt j ||}td|||dS||}|dkr|d krt ||d d}t|d||dS|d krt||d d}t|d||dSdt j ||}td|||dS|s|d krdS|d krdt j ||}t|S|dkrd}n|dkr||ddkrt ||d d}nt||d d}t|d||dS)zComputes the CDF(or SF) for the two-sided Kolmogorov-Smirnov statistic. x must be of type float, n of type integer. Simard & L'Ecuyer (2011) [7]. rr rrErrrg0q&?Trg w@g@g2@ig?gffffff?)r isnanrnanrprodrrWrr scipyspecialZsmirnovr5rDrrb)rr?rr@ZprobZ nxsquaredrrrr_kolmognvsX ,$  ricsDtrStksdkrtjS|dks|dkrdS|}|dkr`|dkr,dSdkrDttddd|d}nttdtd|d}|ddS|dkrrdd|dS|dkrdt j j |S|d}t ||d}t |d|}fd d }t|||d d S) zvComputes the PDF for the two-sided Kolmogorov-Smirnov statistic. x must be of type float, n of type integer. rr rrrcrrg@cs t|SN)kolmogn)_xrrr_kks z_kolmogn_p.._kkrG)ZdxZorder)r rdrrerfrrWrr rgZstatsZksoneZpdfr7r)rr?r@Zprddeltarnrrmr _kolmogn_ps. ((  rpcstrStksdkrtjSdkrdS|dkr"dStttjd}|dkrB|ddSt t|d }|ddkrY|St t }t |dd}fdd}tjj|d|dd S) zeComputes the PPF/ISF of kolmogn. n of type integer, n>= 1 p is the CDF, q the SF, p+q=1 rr rrrScst|Srj)ri)r?rrrr_fsz_kolmogni.._fg+=)Zxtol)r rdrrerWr rgrhZloggammaZexpm1scuZ _kolmogcirTr7optimizeZbrentq)rrr`ror?Zx1rrrrqr _kolmognis$ $ ruc Cstj|||dgdtjtjtjgd}|D](\}}}}t|r$||d<qt||kr1td|tt|||d|d<q|jd}|S)aComputes the CDF for the two-sided Kolmogorov-Smirnov distribution. The two-sided Kolmogorov-Smirnov distribution has as its CDF Pr(D_n <= x), for a sample of size n drawn from a distribution with CDF F(t), where :math:`D_n &= sup_t |F_n(t) - F(t)|`, and :math:`F_n(t)` is the Empirical Cumulative Distribution Function of the sample. Parameters ---------- n : integer, array_like the number of samples x : float, array_like The K-S statistic, float between 0 and 1 cdf : bool, optional whether to compute the CDF(default=true) or the SF. Returns ------- cdf : ndarray CDF (or SF it cdf is False) at the specified locations. The return value has shape the result of numpy broadcasting n and x. N)Z op_dtypes.n is not integral: rEr) r nditerZfloat64Zbool_rdr ValueErrorrioperands) rr?rit_nrl_cdfr_resultrrrrks   rkcCsnt||dg}|D]%\}}}t|r||d<q t||kr&td|tt|||d<q |jd}|S)aComputes the PDF for the two-sided Kolmogorov-Smirnov distribution. Parameters ---------- n : integer, array_like the number of samples x : float, array_like The K-S statistic, float between 0 and 1 Returns ------- pdf : ndarray The PDF at the specified locations The return value has shape the result of numpy broadcasting n and x. N.rvr)r rwrdrrxrpry)rr?rzr{rlr_r}rrrkolmognps   r~c Cst|||dg}|D]7\}}}}t|r||d<q t||kr(td||r0|d|fnd||f\}} tt||| |d<q |jd} | S)aComputes the PPF(or ISF) for the two-sided Kolmogorov-Smirnov distribution. Parameters ---------- n : integer, array_like the number of samples q : float, array_like Probabilities, float between 0 and 1 cdf : bool, optional whether to compute the PPF(default=true) or the ISF. Returns ------- ppf : ndarray PPF (or ISF if cdf is False) at the specified locations The return value has shape the result of numpy broadcasting n and x. N.rvrr)r rwrdrrxrury) rr`rrzr{Z_qr|r_Z_pcdfZ_psfr}rrrkolmogni;s    r)T)'Znumpyr Z scipy.specialrgZscipy.special._ufuncsrhZ_ufuncsrsZscipy._lib._finite_differencesrr&r(Z longdoubler%r'rTrZr[r r rVr\rUrXrYr rrrr5r=rDrbrirprurkr~rrrrrs8C       K  T S<* %