597 lines
20 KiB
Python
597 lines
20 KiB
Python
# Compute the two-sided one-sample Kolmogorov-Smirnov Prob(Dn <= d) where:
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# D_n = sup_x{|F_n(x) - F(x)|},
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# F_n(x) is the empirical CDF for a sample of size n {x_i: i=1,...,n},
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# F(x) is the CDF of a probability distribution.
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#
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# Exact methods:
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# Prob(D_n >= d) can be computed via a matrix algorithm of Durbin[1]
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# or a recursion algorithm due to Pomeranz[2].
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# Marsaglia, Tsang & Wang[3] gave a computation-efficient way to perform
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# the Durbin algorithm.
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# D_n >= d <==> D_n+ >= d or D_n- >= d (the one-sided K-S statistics), hence
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# Prob(D_n >= d) = 2*Prob(D_n+ >= d) - Prob(D_n+ >= d and D_n- >= d).
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# For d > 0.5, the latter intersection probability is 0.
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#
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# Approximate methods:
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# For d close to 0.5, ignoring that intersection term may still give a
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# reasonable approximation.
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# Li-Chien[4] and Korolyuk[5] gave an asymptotic formula extending
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# Kolmogorov's initial asymptotic, suitable for large d. (See
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# scipy.special.kolmogorov for that asymptotic)
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# Pelz-Good[6] used the functional equation for Jacobi theta functions to
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# transform the Li-Chien/Korolyuk formula produce a computational formula
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# suitable for small d.
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#
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# Simard and L'Ecuyer[7] provided an algorithm to decide when to use each of
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# the above approaches and it is that which is used here.
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#
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# Other approaches:
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# Carvalho[8] optimizes Durbin's matrix algorithm for large values of d.
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# Moscovich and Nadler[9] use FFTs to compute the convolutions.
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# References:
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# [1] Durbin J (1968).
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# "The Probability that the Sample Distribution Function Lies Between Two
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# Parallel Straight Lines."
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# Annals of Mathematical Statistics, 39, 398-411.
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# [2] Pomeranz J (1974).
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# "Exact Cumulative Distribution of the Kolmogorov-Smirnov Statistic for
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# Small Samples (Algorithm 487)."
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# Communications of the ACM, 17(12), 703-704.
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# [3] Marsaglia G, Tsang WW, Wang J (2003).
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# "Evaluating Kolmogorov's Distribution."
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# Journal of Statistical Software, 8(18), 1-4.
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# [4] LI-CHIEN, C. (1956).
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# "On the exact distribution of the statistics of A. N. Kolmogorov and
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# their asymptotic expansion."
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# Acta Matematica Sinica, 6, 55-81.
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# [5] KOROLYUK, V. S. (1960).
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# "Asymptotic analysis of the distribution of the maximum deviation in
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# the Bernoulli scheme."
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# Theor. Probability Appl., 4, 339-366.
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# [6] Pelz W, Good IJ (1976).
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# "Approximating the Lower Tail-areas of the Kolmogorov-Smirnov One-sample
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# Statistic."
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# Journal of the Royal Statistical Society, Series B, 38(2), 152-156.
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# [7] Simard, R., L'Ecuyer, P. (2011)
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# "Computing the Two-Sided Kolmogorov-Smirnov Distribution",
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# Journal of Statistical Software, Vol 39, 11, 1-18.
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# [8] Carvalho, Luis (2015)
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# "An Improved Evaluation of Kolmogorov's Distribution"
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# Journal of Statistical Software, Code Snippets; Vol 65(3), 1-8.
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# [9] Amit Moscovich, Boaz Nadler (2017)
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# "Fast calculation of boundary crossing probabilities for Poisson
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# processes",
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# Statistics & Probability Letters, Vol 123, 177-182.
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import numpy as np
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import scipy.special
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import scipy.special._ufuncs as scu
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import scipy.misc
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_E128 = 128
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_EP128 = np.ldexp(np.longdouble(1), _E128)
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_EM128 = np.ldexp(np.longdouble(1), -_E128)
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_SQRT2PI = np.sqrt(2 * np.pi)
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_LOG_2PI = np.log(2 * np.pi)
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_MIN_LOG = -708
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_SQRT3 = np.sqrt(3)
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_PI_SQUARED = np.pi ** 2
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_PI_FOUR = np.pi ** 4
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_PI_SIX = np.pi ** 6
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# [Lifted from _loggamma.pxd.] If B_m are the Bernoulli numbers,
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# then Stirling coeffs are B_{2j}/(2j)/(2j-1) for j=8,...1.
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_STIRLING_COEFFS = [-2.955065359477124183e-2, 6.4102564102564102564e-3,
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-1.9175269175269175269e-3, 8.4175084175084175084e-4,
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-5.952380952380952381e-4, 7.9365079365079365079e-4,
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-2.7777777777777777778e-3, 8.3333333333333333333e-2]
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def _log_nfactorial_div_n_pow_n(n):
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# Computes n! / n**n
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# = (n-1)! / n**(n-1)
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# Uses Stirling's approximation, but removes n*log(n) up-front to
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# avoid subtractive cancellation.
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# = log(n)/2 - n + log(sqrt(2pi)) + sum B_{2j}/(2j)/(2j-1)/n**(2j-1)
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rn = 1.0/n
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return np.log(n)/2 - n + _LOG_2PI/2 + rn * np.polyval(_STIRLING_COEFFS, rn/n)
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def _clip_prob(p):
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"""clips a probability to range 0<=p<=1."""
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return np.clip(p, 0.0, 1.0)
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def _select_and_clip_prob(cdfprob, sfprob, cdf=True):
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"""Selects either the CDF or SF, and then clips to range 0<=p<=1."""
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p = np.where(cdf, cdfprob, sfprob)
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return _clip_prob(p)
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def _kolmogn_DMTW(n, d, cdf=True):
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r"""Computes the Kolmogorov CDF: Pr(D_n <= d) using the MTW approach to
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the Durbin matrix algorithm.
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Durbin (1968); Marsaglia, Tsang, Wang (2003). [1], [3].
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"""
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# Write d = (k-h)/n, where k is positive integer and 0 <= h < 1
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# Generate initial matrix H of size m*m where m=(2k-1)
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# Compute k-th row of (n!/n^n) * H^n, scaling intermediate results.
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# Requires memory O(m^2) and computation O(m^2 log(n)).
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# Most suitable for small m.
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if d >= 1.0:
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return _select_and_clip_prob(1.0, 0.0, cdf)
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nd = n * d
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if nd <= 0.5:
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return _select_and_clip_prob(0.0, 1.0, cdf)
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k = int(np.ceil(nd))
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h = k - nd
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m = 2 * k - 1
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H = np.zeros([m, m])
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# Initialize: v is first column (and last row) of H
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# v[j] = (1-h^(j+1)/(j+1)! (except for v[-1])
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# w[j] = 1/(j)!
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# q = k-th row of H (actually i!/n^i*H^i)
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intm = np.arange(1, m + 1)
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v = 1.0 - h ** intm
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w = np.empty(m)
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fac = 1.0
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for j in intm:
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w[j - 1] = fac
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fac /= j # This might underflow. Isn't a problem.
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v[j - 1] *= fac
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tt = max(2 * h - 1.0, 0)**m - 2*h**m
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v[-1] = (1.0 + tt) * fac
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for i in range(1, m):
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H[i - 1:, i] = w[:m - i + 1]
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H[:, 0] = v
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H[-1, :] = np.flip(v, axis=0)
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Hpwr = np.eye(np.shape(H)[0]) # Holds intermediate powers of H
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nn = n
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expnt = 0 # Scaling of Hpwr
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Hexpnt = 0 # Scaling of H
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while nn > 0:
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if nn % 2:
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Hpwr = np.matmul(Hpwr, H)
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expnt += Hexpnt
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H = np.matmul(H, H)
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Hexpnt *= 2
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# Scale as needed.
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if np.abs(H[k - 1, k - 1]) > _EP128:
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H /= _EP128
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Hexpnt += _E128
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nn = nn // 2
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p = Hpwr[k - 1, k - 1]
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# Multiply by n!/n^n
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for i in range(1, n + 1):
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p = i * p / n
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if np.abs(p) < _EM128:
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p *= _EP128
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expnt -= _E128
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# unscale
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if expnt != 0:
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p = np.ldexp(p, expnt)
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return _select_and_clip_prob(p, 1.0-p, cdf)
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def _pomeranz_compute_j1j2(i, n, ll, ceilf, roundf):
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"""Compute the endpoints of the interval for row i."""
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if i == 0:
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j1, j2 = -ll - ceilf - 1, ll + ceilf - 1
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else:
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# i + 1 = 2*ip1div2 + ip1mod2
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ip1div2, ip1mod2 = divmod(i + 1, 2)
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if ip1mod2 == 0: # i is odd
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if ip1div2 == n + 1:
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j1, j2 = n - ll - ceilf - 1, n + ll + ceilf - 1
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else:
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j1, j2 = ip1div2 - 1 - ll - roundf - 1, ip1div2 + ll - 1 + ceilf - 1
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else:
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j1, j2 = ip1div2 - 1 - ll - 1, ip1div2 + ll + roundf - 1
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return max(j1 + 2, 0), min(j2, n)
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def _kolmogn_Pomeranz(n, x, cdf=True):
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r"""Computes Pr(D_n <= d) using the Pomeranz recursion algorithm.
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Pomeranz (1974) [2]
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"""
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# V is n*(2n+2) matrix.
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# Each row is convolution of the previous row and probabilities from a
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# Poisson distribution.
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# Desired CDF probability is n! V[n-1, 2n+1] (final entry in final row).
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# Only two rows are needed at any given stage:
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# - Call them V0 and V1.
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# - Swap each iteration
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# Only a few (contiguous) entries in each row can be non-zero.
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# - Keep track of start and end (j1 and j2 below)
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# - V0s and V1s track the start in the two rows
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# Scale intermediate results as needed.
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# Only a few different Poisson distributions can occur
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t = n * x
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ll = int(np.floor(t))
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f = 1.0 * (t - ll) # fractional part of t
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g = min(f, 1.0 - f)
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ceilf = (1 if f > 0 else 0)
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roundf = (1 if f > 0.5 else 0)
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npwrs = 2 * (ll + 1) # Maximum number of powers needed in convolutions
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gpower = np.empty(npwrs) # gpower = (g/n)^m/m!
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twogpower = np.empty(npwrs) # twogpower = (2g/n)^m/m!
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onem2gpower = np.empty(npwrs) # onem2gpower = ((1-2g)/n)^m/m!
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# gpower etc are *almost* Poisson probs, just missing normalizing factor.
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gpower[0] = 1.0
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twogpower[0] = 1.0
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onem2gpower[0] = 1.0
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expnt = 0
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g_over_n, two_g_over_n, one_minus_two_g_over_n = g/n, 2*g/n, (1 - 2*g)/n
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for m in range(1, npwrs):
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gpower[m] = gpower[m - 1] * g_over_n / m
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twogpower[m] = twogpower[m - 1] * two_g_over_n / m
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onem2gpower[m] = onem2gpower[m - 1] * one_minus_two_g_over_n / m
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V0 = np.zeros([npwrs])
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V1 = np.zeros([npwrs])
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V1[0] = 1 # first row
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V0s, V1s = 0, 0 # start indices of the two rows
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j1, j2 = _pomeranz_compute_j1j2(0, n, ll, ceilf, roundf)
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for i in range(1, 2 * n + 2):
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# Preserve j1, V1, V1s, V0s from last iteration
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k1 = j1
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V0, V1 = V1, V0
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V0s, V1s = V1s, V0s
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V1.fill(0.0)
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j1, j2 = _pomeranz_compute_j1j2(i, n, ll, ceilf, roundf)
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if i == 1 or i == 2 * n + 1:
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pwrs = gpower
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else:
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pwrs = (twogpower if i % 2 else onem2gpower)
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ln2 = j2 - k1 + 1
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if ln2 > 0:
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conv = np.convolve(V0[k1 - V0s:k1 - V0s + ln2], pwrs[:ln2])
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conv_start = j1 - k1 # First index to use from conv
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conv_len = j2 - j1 + 1 # Number of entries to use from conv
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V1[:conv_len] = conv[conv_start:conv_start + conv_len]
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# Scale to avoid underflow.
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if 0 < np.max(V1) < _EM128:
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V1 *= _EP128
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expnt -= _E128
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V1s = V0s + j1 - k1
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# multiply by n!
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ans = V1[n - V1s]
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for m in range(1, n + 1):
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if np.abs(ans) > _EP128:
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ans *= _EM128
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expnt += _E128
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ans *= m
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# Undo any intermediate scaling
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if expnt != 0:
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ans = np.ldexp(ans, expnt)
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ans = _select_and_clip_prob(ans, 1.0 - ans, cdf)
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return ans
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def _kolmogn_PelzGood(n, x, cdf=True):
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"""Computes the Pelz-Good approximation to Prob(Dn <= x) with 0<=x<=1.
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Start with Li-Chien, Korolyuk approximation:
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Prob(Dn <= x) ~ K0(z) + K1(z)/sqrt(n) + K2(z)/n + K3(z)/n**1.5
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where z = x*sqrt(n).
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Transform each K_(z) using Jacobi theta functions into a form suitable
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for small z.
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Pelz-Good (1976). [6]
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"""
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if x <= 0.0:
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return _select_and_clip_prob(0.0, 1.0, cdf=cdf)
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if x >= 1.0:
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return _select_and_clip_prob(1.0, 0.0, cdf=cdf)
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z = np.sqrt(n) * x
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zsquared, zthree, zfour, zsix = z**2, z**3, z**4, z**6
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qlog = -_PI_SQUARED / 8 / zsquared
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if qlog < _MIN_LOG: # z ~ 0.041743441416853426
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return _select_and_clip_prob(0.0, 1.0, cdf=cdf)
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q = np.exp(qlog)
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# Coefficients of terms in the sums for K1, K2 and K3
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k1a = -zsquared
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k1b = _PI_SQUARED / 4
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k2a = 6 * zsix + 2 * zfour
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k2b = (2 * zfour - 5 * zsquared) * _PI_SQUARED / 4
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k2c = _PI_FOUR * (1 - 2 * zsquared) / 16
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k3d = _PI_SIX * (5 - 30 * zsquared) / 64
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k3c = _PI_FOUR * (-60 * zsquared + 212 * zfour) / 16
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k3b = _PI_SQUARED * (135 * zfour - 96 * zsix) / 4
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k3a = -30 * zsix - 90 * z**8
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K0to3 = np.zeros(4)
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# Use a Horner scheme to evaluate sum c_i q^(i^2)
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# Reduces to a sum over odd integers.
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maxk = int(np.ceil(16 * z / np.pi))
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for k in range(maxk, 0, -1):
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m = 2 * k - 1
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msquared, mfour, msix = m**2, m**4, m**6
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qpower = np.power(q, 8 * k)
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coeffs = np.array([1.0,
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k1a + k1b*msquared,
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k2a + k2b*msquared + k2c*mfour,
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k3a + k3b*msquared + k3c*mfour + k3d*msix])
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K0to3 *= qpower
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K0to3 += coeffs
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K0to3 *= q
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K0to3 *= _SQRT2PI
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# z**10 > 0 as z > 0.04
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K0to3 /= np.array([z, 6 * zfour, 72 * z**7, 6480 * z**10])
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# Now do the other sum over the other terms, all integers k
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# K_2: (pi^2 k^2) q^(k^2),
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# K_3: (3pi^2 k^2 z^2 - pi^4 k^4)*q^(k^2)
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# Don't expect much subtractive cancellation so use direct calculation
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q = np.exp(-_PI_SQUARED / 2 / zsquared)
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ks = np.arange(maxk, 0, -1)
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ksquared = ks ** 2
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sqrt3z = _SQRT3 * z
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kspi = np.pi * ks
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qpwers = q ** ksquared
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k2extra = np.sum(ksquared * qpwers)
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k2extra *= _PI_SQUARED * _SQRT2PI/(-36 * zthree)
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K0to3[2] += k2extra
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k3extra = np.sum((sqrt3z + kspi) * (sqrt3z - kspi) * ksquared * qpwers)
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k3extra *= _PI_SQUARED * _SQRT2PI/(216 * zsix)
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K0to3[3] += k3extra
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powers_of_n = np.power(n * 1.0, np.arange(len(K0to3)) / 2.0)
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K0to3 /= powers_of_n
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if not cdf:
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K0to3 *= -1
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K0to3[0] += 1
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Ksum = sum(K0to3)
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return Ksum
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def _kolmogn(n, x, cdf=True):
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"""Computes the CDF(or SF) for the two-sided Kolmogorov-Smirnov statistic.
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x must be of type float, n of type integer.
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Simard & L'Ecuyer (2011) [7].
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"""
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if np.isnan(n):
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return n # Keep the same type of nan
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if int(n) != n or n <= 0:
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return np.nan
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if x >= 1.0:
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return _select_and_clip_prob(1.0, 0.0, cdf=cdf)
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if x <= 0.0:
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return _select_and_clip_prob(0.0, 1.0, cdf=cdf)
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t = n * x
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if t <= 1.0: # Ruben-Gambino: 1/2n <= x <= 1/n
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if t <= 0.5:
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return _select_and_clip_prob(0.0, 1.0, cdf=cdf)
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if n <= 140:
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prob = np.prod(np.arange(1, n+1) * (1.0/n) * (2*t - 1))
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else:
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prob = np.exp(_log_nfactorial_div_n_pow_n(n) + n * np.log(2*t-1))
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return _select_and_clip_prob(prob, 1.0 - prob, cdf=cdf)
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if t >= n - 1: # Ruben-Gambino
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prob = 2 * (1.0 - x)**n
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return _select_and_clip_prob(1 - prob, prob, cdf=cdf)
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if x >= 0.5: # Exact: 2 * smirnov
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prob = 2 * scipy.special.smirnov(n, x)
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return _select_and_clip_prob(1.0 - prob, prob, cdf=cdf)
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nxsquared = t * x
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if n <= 140:
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if nxsquared <= 0.754693:
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prob = _kolmogn_DMTW(n, x, cdf=True)
|
|
return _select_and_clip_prob(prob, 1.0 - prob, cdf=cdf)
|
|
if nxsquared <= 4:
|
|
prob = _kolmogn_Pomeranz(n, x, cdf=True)
|
|
return _select_and_clip_prob(prob, 1.0 - prob, cdf=cdf)
|
|
# Now use Miller approximation of 2*smirnov
|
|
prob = 2 * scipy.special.smirnov(n, x)
|
|
return _select_and_clip_prob(1.0 - prob, prob, cdf=cdf)
|
|
|
|
# Split CDF and SF as they have different cutoffs on nxsquared.
|
|
if not cdf:
|
|
if nxsquared >= 370.0:
|
|
return 0.0
|
|
if nxsquared >= 2.2:
|
|
prob = 2 * scipy.special.smirnov(n, x)
|
|
return _clip_prob(prob)
|
|
# Fall through and compute the SF as 1.0-CDF
|
|
if nxsquared >= 18.0:
|
|
cdfprob = 1.0
|
|
elif n <= 100000 and n * x**1.5 <= 1.4:
|
|
cdfprob = _kolmogn_DMTW(n, x, cdf=True)
|
|
else:
|
|
cdfprob = _kolmogn_PelzGood(n, x, cdf=True)
|
|
return _select_and_clip_prob(cdfprob, 1.0 - cdfprob, cdf=cdf)
|
|
|
|
|
|
def _kolmogn_p(n, x):
|
|
"""Computes the PDF for the two-sided Kolmogorov-Smirnov statistic.
|
|
|
|
x must be of type float, n of type integer.
|
|
"""
|
|
if np.isnan(n):
|
|
return n # Keep the same type of nan
|
|
if int(n) != n or n <= 0:
|
|
return np.nan
|
|
if x >= 1.0 or x <= 0:
|
|
return 0
|
|
t = n * x
|
|
if t <= 1.0:
|
|
# Ruben-Gambino: n!/n^n * (2t-1)^n -> 2 n!/n^n * n^2 * (2t-1)^(n-1)
|
|
if t <= 0.5:
|
|
return 0.0
|
|
if n <= 140:
|
|
prd = np.prod(np.arange(1, n) * (1.0 / n) * (2 * t - 1))
|
|
else:
|
|
prd = np.exp(_log_nfactorial_div_n_pow_n(n) + (n-1) * np.log(2 * t - 1))
|
|
return prd * 2 * n**2
|
|
if t >= n - 1:
|
|
# Ruben-Gambino : 1-2(1-x)**n -> 2n*(1-x)**(n-1)
|
|
return 2 * (1.0 - x) ** (n-1) * n
|
|
if x >= 0.5:
|
|
return 2 * scipy.stats.ksone.pdf(x, n)
|
|
|
|
# Just take a small delta.
|
|
# Ideally x +/- delta would stay within [i/n, (i+1)/n] for some integer a.
|
|
# as the CDF is a piecewise degree n polynomial.
|
|
# It has knots at 1/n, 2/n, ... (n-1)/n
|
|
# and is not a C-infinity function at the knots
|
|
delta = x / 2.0**16
|
|
delta = min(delta, x - 1.0/n)
|
|
delta = min(delta, 0.5 - x)
|
|
|
|
def _kk(_x):
|
|
return kolmogn(n, _x)
|
|
|
|
return scipy.misc.derivative(_kk, x, dx=delta, order=5)
|
|
|
|
|
|
def _kolmogni(n, p, q):
|
|
"""Computes the PPF/ISF of kolmogn.
|
|
|
|
n of type integer, n>= 1
|
|
p is the CDF, q the SF, p+q=1
|
|
"""
|
|
if np.isnan(n):
|
|
return n # Keep the same type of nan
|
|
if int(n) != n or n <= 0:
|
|
return np.nan
|
|
if p <= 0:
|
|
return 1.0/n
|
|
if q <= 0:
|
|
return 1.0
|
|
delta = np.exp((np.log(p) - scipy.special.loggamma(n+1))/n)
|
|
if delta <= 1.0/n:
|
|
return (delta + 1.0 / n) / 2
|
|
x = -np.expm1(np.log(q/2.0)/n)
|
|
if x >= 1 - 1.0/n:
|
|
return x
|
|
x1 = scu._kolmogci(p)/np.sqrt(n)
|
|
x1 = min(x1, 1.0 - 1.0/n)
|
|
_f = lambda x: _kolmogn(n, x) - p
|
|
return scipy.optimize.brentq(_f, 1.0/n, x1, xtol=1e-14)
|
|
|
|
|
|
def kolmogn(n, x, cdf=True):
|
|
"""Computes 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
|
|
D_n &= sup_t |F_n(t) - F(t)|, and
|
|
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.
|
|
"""
|
|
it = np.nditer([n, x, cdf, None],
|
|
op_dtypes=[None, np.float64, np.bool_, np.float64])
|
|
for _n, _x, _cdf, z in it:
|
|
if np.isnan(_n):
|
|
z[...] = _n
|
|
continue
|
|
if int(_n) != _n:
|
|
raise ValueError(f'n is not integral: {_n}')
|
|
z[...] = _kolmogn(int(_n), _x, cdf=_cdf)
|
|
result = it.operands[-1]
|
|
return result
|
|
|
|
|
|
def kolmognp(n, x):
|
|
"""Computes 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.
|
|
"""
|
|
it = np.nditer([n, x, None])
|
|
for _n, _x, z in it:
|
|
if np.isnan(_n):
|
|
z[...] = _n
|
|
continue
|
|
if int(_n) != _n:
|
|
raise ValueError(f'n is not integral: {_n}')
|
|
z[...] = _kolmogn_p(int(_n), _x)
|
|
result = it.operands[-1]
|
|
return result
|
|
|
|
|
|
def kolmogni(n, q, cdf=True):
|
|
"""Computes 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.
|
|
"""
|
|
it = np.nditer([n, q, cdf, None])
|
|
for _n, _q, _cdf, z in it:
|
|
if np.isnan(_n):
|
|
z[...] = _n
|
|
continue
|
|
if int(_n) != _n:
|
|
raise ValueError(f'n is not integral: {_n}')
|
|
_pcdf, _psf = (_q, 1-_q) if _cdf else (1-_q, _q)
|
|
z[...] = _kolmogni(int(_n), _pcdf, _psf)
|
|
result = it.operands[-1]
|
|
return result
|