forked from 170010011/fr
597 lines
20 KiB
Python
597 lines
20 KiB
Python
# Compute the two-sided one-sample Kolmogorov-Smirnov Prob(Dn <= d) where:
|
|
# D_n = sup_x{|F_n(x) - F(x)|},
|
|
# F_n(x) is the empirical CDF for a sample of size n {x_i: i=1,...,n},
|
|
# F(x) is the CDF of a probability distribution.
|
|
#
|
|
# Exact methods:
|
|
# Prob(D_n >= d) can be computed via a matrix algorithm of Durbin[1]
|
|
# or a recursion algorithm due to Pomeranz[2].
|
|
# Marsaglia, Tsang & Wang[3] gave a computation-efficient way to perform
|
|
# the Durbin algorithm.
|
|
# D_n >= d <==> D_n+ >= d or D_n- >= d (the one-sided K-S statistics), hence
|
|
# Prob(D_n >= d) = 2*Prob(D_n+ >= d) - Prob(D_n+ >= d and D_n- >= d).
|
|
# For d > 0.5, the latter intersection probability is 0.
|
|
#
|
|
# Approximate methods:
|
|
# For d close to 0.5, ignoring that intersection term may still give a
|
|
# reasonable approximation.
|
|
# Li-Chien[4] and Korolyuk[5] gave an asymptotic formula extending
|
|
# Kolmogorov's initial asymptotic, suitable for large d. (See
|
|
# scipy.special.kolmogorov for that asymptotic)
|
|
# Pelz-Good[6] used the functional equation for Jacobi theta functions to
|
|
# transform the Li-Chien/Korolyuk formula produce a computational formula
|
|
# suitable for small d.
|
|
#
|
|
# Simard and L'Ecuyer[7] provided an algorithm to decide when to use each of
|
|
# the above approaches and it is that which is used here.
|
|
#
|
|
# Other approaches:
|
|
# Carvalho[8] optimizes Durbin's matrix algorithm for large values of d.
|
|
# Moscovich and Nadler[9] use FFTs to compute the convolutions.
|
|
|
|
# References:
|
|
# [1] Durbin J (1968).
|
|
# "The Probability that the Sample Distribution Function Lies Between Two
|
|
# Parallel Straight Lines."
|
|
# Annals of Mathematical Statistics, 39, 398-411.
|
|
# [2] Pomeranz J (1974).
|
|
# "Exact Cumulative Distribution of the Kolmogorov-Smirnov Statistic for
|
|
# Small Samples (Algorithm 487)."
|
|
# Communications of the ACM, 17(12), 703-704.
|
|
# [3] Marsaglia G, Tsang WW, Wang J (2003).
|
|
# "Evaluating Kolmogorov's Distribution."
|
|
# Journal of Statistical Software, 8(18), 1-4.
|
|
# [4] LI-CHIEN, C. (1956).
|
|
# "On the exact distribution of the statistics of A. N. Kolmogorov and
|
|
# their asymptotic expansion."
|
|
# Acta Matematica Sinica, 6, 55-81.
|
|
# [5] KOROLYUK, V. S. (1960).
|
|
# "Asymptotic analysis of the distribution of the maximum deviation in
|
|
# the Bernoulli scheme."
|
|
# Theor. Probability Appl., 4, 339-366.
|
|
# [6] Pelz W, Good IJ (1976).
|
|
# "Approximating the Lower Tail-areas of the Kolmogorov-Smirnov One-sample
|
|
# Statistic."
|
|
# Journal of the Royal Statistical Society, Series B, 38(2), 152-156.
|
|
# [7] Simard, R., L'Ecuyer, P. (2011)
|
|
# "Computing the Two-Sided Kolmogorov-Smirnov Distribution",
|
|
# Journal of Statistical Software, Vol 39, 11, 1-18.
|
|
# [8] Carvalho, Luis (2015)
|
|
# "An Improved Evaluation of Kolmogorov's Distribution"
|
|
# Journal of Statistical Software, Code Snippets; Vol 65(3), 1-8.
|
|
# [9] Amit Moscovich, Boaz Nadler (2017)
|
|
# "Fast calculation of boundary crossing probabilities for Poisson
|
|
# processes",
|
|
# Statistics & Probability Letters, Vol 123, 177-182.
|
|
|
|
|
|
import numpy as np
|
|
import scipy.special
|
|
import scipy.special._ufuncs as scu
|
|
import scipy.misc
|
|
|
|
_E128 = 128
|
|
_EP128 = np.ldexp(np.longdouble(1), _E128)
|
|
_EM128 = np.ldexp(np.longdouble(1), -_E128)
|
|
|
|
_SQRT2PI = np.sqrt(2 * np.pi)
|
|
_LOG_2PI = np.log(2 * np.pi)
|
|
_MIN_LOG = -708
|
|
_SQRT3 = np.sqrt(3)
|
|
_PI_SQUARED = np.pi ** 2
|
|
_PI_FOUR = np.pi ** 4
|
|
_PI_SIX = np.pi ** 6
|
|
|
|
# [Lifted from _loggamma.pxd.] If B_m are the Bernoulli numbers,
|
|
# then Stirling coeffs are B_{2j}/(2j)/(2j-1) for j=8,...1.
|
|
_STIRLING_COEFFS = [-2.955065359477124183e-2, 6.4102564102564102564e-3,
|
|
-1.9175269175269175269e-3, 8.4175084175084175084e-4,
|
|
-5.952380952380952381e-4, 7.9365079365079365079e-4,
|
|
-2.7777777777777777778e-3, 8.3333333333333333333e-2]
|
|
|
|
def _log_nfactorial_div_n_pow_n(n):
|
|
# Computes n! / n**n
|
|
# = (n-1)! / n**(n-1)
|
|
# Uses Stirling's approximation, but removes n*log(n) up-front to
|
|
# avoid subtractive cancellation.
|
|
# = log(n)/2 - n + log(sqrt(2pi)) + sum B_{2j}/(2j)/(2j-1)/n**(2j-1)
|
|
rn = 1.0/n
|
|
return np.log(n)/2 - n + _LOG_2PI/2 + rn * np.polyval(_STIRLING_COEFFS, rn/n)
|
|
|
|
|
|
def _clip_prob(p):
|
|
"""clips a probability to range 0<=p<=1."""
|
|
return np.clip(p, 0.0, 1.0)
|
|
|
|
|
|
def _select_and_clip_prob(cdfprob, sfprob, cdf=True):
|
|
"""Selects either the CDF or SF, and then clips to range 0<=p<=1."""
|
|
p = np.where(cdf, cdfprob, sfprob)
|
|
return _clip_prob(p)
|
|
|
|
|
|
def _kolmogn_DMTW(n, d, cdf=True):
|
|
r"""Computes the Kolmogorov CDF: Pr(D_n <= d) using the MTW approach to
|
|
the Durbin matrix algorithm.
|
|
|
|
Durbin (1968); Marsaglia, Tsang, Wang (2003). [1], [3].
|
|
"""
|
|
# Write d = (k-h)/n, where k is positive integer and 0 <= h < 1
|
|
# Generate initial matrix H of size m*m where m=(2k-1)
|
|
# Compute k-th row of (n!/n^n) * H^n, scaling intermediate results.
|
|
# Requires memory O(m^2) and computation O(m^2 log(n)).
|
|
# Most suitable for small m.
|
|
|
|
if d >= 1.0:
|
|
return _select_and_clip_prob(1.0, 0.0, cdf)
|
|
nd = n * d
|
|
if nd <= 0.5:
|
|
return _select_and_clip_prob(0.0, 1.0, cdf)
|
|
k = int(np.ceil(nd))
|
|
h = k - nd
|
|
m = 2 * k - 1
|
|
|
|
H = np.zeros([m, m])
|
|
|
|
# Initialize: v is first column (and last row) of H
|
|
# v[j] = (1-h^(j+1)/(j+1)! (except for v[-1])
|
|
# w[j] = 1/(j)!
|
|
# q = k-th row of H (actually i!/n^i*H^i)
|
|
intm = np.arange(1, m + 1)
|
|
v = 1.0 - h ** intm
|
|
w = np.empty(m)
|
|
fac = 1.0
|
|
for j in intm:
|
|
w[j - 1] = fac
|
|
fac /= j # This might underflow. Isn't a problem.
|
|
v[j - 1] *= fac
|
|
tt = max(2 * h - 1.0, 0)**m - 2*h**m
|
|
v[-1] = (1.0 + tt) * fac
|
|
|
|
for i in range(1, m):
|
|
H[i - 1:, i] = w[:m - i + 1]
|
|
H[:, 0] = v
|
|
H[-1, :] = np.flip(v, axis=0)
|
|
|
|
Hpwr = np.eye(np.shape(H)[0]) # Holds intermediate powers of H
|
|
nn = n
|
|
expnt = 0 # Scaling of Hpwr
|
|
Hexpnt = 0 # Scaling of H
|
|
while nn > 0:
|
|
if nn % 2:
|
|
Hpwr = np.matmul(Hpwr, H)
|
|
expnt += Hexpnt
|
|
H = np.matmul(H, H)
|
|
Hexpnt *= 2
|
|
# Scale as needed.
|
|
if np.abs(H[k - 1, k - 1]) > _EP128:
|
|
H /= _EP128
|
|
Hexpnt += _E128
|
|
nn = nn // 2
|
|
|
|
p = Hpwr[k - 1, k - 1]
|
|
|
|
# Multiply by n!/n^n
|
|
for i in range(1, n + 1):
|
|
p = i * p / n
|
|
if np.abs(p) < _EM128:
|
|
p *= _EP128
|
|
expnt -= _E128
|
|
|
|
# unscale
|
|
if expnt != 0:
|
|
p = np.ldexp(p, expnt)
|
|
|
|
return _select_and_clip_prob(p, 1.0-p, cdf)
|
|
|
|
|
|
def _pomeranz_compute_j1j2(i, n, ll, ceilf, roundf):
|
|
"""Compute the endpoints of the interval for row i."""
|
|
if i == 0:
|
|
j1, j2 = -ll - ceilf - 1, ll + ceilf - 1
|
|
else:
|
|
# i + 1 = 2*ip1div2 + ip1mod2
|
|
ip1div2, ip1mod2 = divmod(i + 1, 2)
|
|
if ip1mod2 == 0: # i is odd
|
|
if ip1div2 == n + 1:
|
|
j1, j2 = n - ll - ceilf - 1, n + ll + ceilf - 1
|
|
else:
|
|
j1, j2 = ip1div2 - 1 - ll - roundf - 1, ip1div2 + ll - 1 + ceilf - 1
|
|
else:
|
|
j1, j2 = ip1div2 - 1 - ll - 1, ip1div2 + ll + roundf - 1
|
|
|
|
return max(j1 + 2, 0), min(j2, n)
|
|
|
|
|
|
def _kolmogn_Pomeranz(n, x, cdf=True):
|
|
r"""Computes Pr(D_n <= d) using the Pomeranz recursion algorithm.
|
|
|
|
Pomeranz (1974) [2]
|
|
"""
|
|
|
|
# V is n*(2n+2) matrix.
|
|
# Each row is convolution of the previous row and probabilities from a
|
|
# Poisson distribution.
|
|
# Desired CDF probability is n! V[n-1, 2n+1] (final entry in final row).
|
|
# Only two rows are needed at any given stage:
|
|
# - Call them V0 and V1.
|
|
# - Swap each iteration
|
|
# Only a few (contiguous) entries in each row can be non-zero.
|
|
# - Keep track of start and end (j1 and j2 below)
|
|
# - V0s and V1s track the start in the two rows
|
|
# Scale intermediate results as needed.
|
|
# Only a few different Poisson distributions can occur
|
|
t = n * x
|
|
ll = int(np.floor(t))
|
|
f = 1.0 * (t - ll) # fractional part of t
|
|
g = min(f, 1.0 - f)
|
|
ceilf = (1 if f > 0 else 0)
|
|
roundf = (1 if f > 0.5 else 0)
|
|
npwrs = 2 * (ll + 1) # Maximum number of powers needed in convolutions
|
|
gpower = np.empty(npwrs) # gpower = (g/n)^m/m!
|
|
twogpower = np.empty(npwrs) # twogpower = (2g/n)^m/m!
|
|
onem2gpower = np.empty(npwrs) # onem2gpower = ((1-2g)/n)^m/m!
|
|
# gpower etc are *almost* Poisson probs, just missing normalizing factor.
|
|
|
|
gpower[0] = 1.0
|
|
twogpower[0] = 1.0
|
|
onem2gpower[0] = 1.0
|
|
expnt = 0
|
|
g_over_n, two_g_over_n, one_minus_two_g_over_n = g/n, 2*g/n, (1 - 2*g)/n
|
|
for m in range(1, npwrs):
|
|
gpower[m] = gpower[m - 1] * g_over_n / m
|
|
twogpower[m] = twogpower[m - 1] * two_g_over_n / m
|
|
onem2gpower[m] = onem2gpower[m - 1] * one_minus_two_g_over_n / m
|
|
|
|
V0 = np.zeros([npwrs])
|
|
V1 = np.zeros([npwrs])
|
|
V1[0] = 1 # first row
|
|
V0s, V1s = 0, 0 # start indices of the two rows
|
|
|
|
j1, j2 = _pomeranz_compute_j1j2(0, n, ll, ceilf, roundf)
|
|
for i in range(1, 2 * n + 2):
|
|
# Preserve j1, V1, V1s, V0s from last iteration
|
|
k1 = j1
|
|
V0, V1 = V1, V0
|
|
V0s, V1s = V1s, V0s
|
|
V1.fill(0.0)
|
|
j1, j2 = _pomeranz_compute_j1j2(i, n, ll, ceilf, roundf)
|
|
if i == 1 or i == 2 * n + 1:
|
|
pwrs = gpower
|
|
else:
|
|
pwrs = (twogpower if i % 2 else onem2gpower)
|
|
ln2 = j2 - k1 + 1
|
|
if ln2 > 0:
|
|
conv = np.convolve(V0[k1 - V0s:k1 - V0s + ln2], pwrs[:ln2])
|
|
conv_start = j1 - k1 # First index to use from conv
|
|
conv_len = j2 - j1 + 1 # Number of entries to use from conv
|
|
V1[:conv_len] = conv[conv_start:conv_start + conv_len]
|
|
# Scale to avoid underflow.
|
|
if 0 < np.max(V1) < _EM128:
|
|
V1 *= _EP128
|
|
expnt -= _E128
|
|
V1s = V0s + j1 - k1
|
|
|
|
# multiply by n!
|
|
ans = V1[n - V1s]
|
|
for m in range(1, n + 1):
|
|
if np.abs(ans) > _EP128:
|
|
ans *= _EM128
|
|
expnt += _E128
|
|
ans *= m
|
|
|
|
# Undo any intermediate scaling
|
|
if expnt != 0:
|
|
ans = np.ldexp(ans, expnt)
|
|
ans = _select_and_clip_prob(ans, 1.0 - ans, cdf)
|
|
return ans
|
|
|
|
|
|
def _kolmogn_PelzGood(n, x, cdf=True):
|
|
"""Computes 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]
|
|
"""
|
|
if x <= 0.0:
|
|
return _select_and_clip_prob(0.0, 1.0, cdf=cdf)
|
|
if x >= 1.0:
|
|
return _select_and_clip_prob(1.0, 0.0, cdf=cdf)
|
|
|
|
z = np.sqrt(n) * x
|
|
zsquared, zthree, zfour, zsix = z**2, z**3, z**4, z**6
|
|
|
|
qlog = -_PI_SQUARED / 8 / zsquared
|
|
if qlog < _MIN_LOG: # z ~ 0.041743441416853426
|
|
return _select_and_clip_prob(0.0, 1.0, cdf=cdf)
|
|
|
|
q = np.exp(qlog)
|
|
|
|
# Coefficients of terms in the sums for K1, K2 and K3
|
|
k1a = -zsquared
|
|
k1b = _PI_SQUARED / 4
|
|
|
|
k2a = 6 * zsix + 2 * zfour
|
|
k2b = (2 * zfour - 5 * zsquared) * _PI_SQUARED / 4
|
|
k2c = _PI_FOUR * (1 - 2 * zsquared) / 16
|
|
|
|
k3d = _PI_SIX * (5 - 30 * zsquared) / 64
|
|
k3c = _PI_FOUR * (-60 * zsquared + 212 * zfour) / 16
|
|
k3b = _PI_SQUARED * (135 * zfour - 96 * zsix) / 4
|
|
k3a = -30 * zsix - 90 * z**8
|
|
|
|
K0to3 = np.zeros(4)
|
|
# Use a Horner scheme to evaluate sum c_i q^(i^2)
|
|
# Reduces to a sum over odd integers.
|
|
maxk = int(np.ceil(16 * z / np.pi))
|
|
for k in range(maxk, 0, -1):
|
|
m = 2 * k - 1
|
|
msquared, mfour, msix = m**2, m**4, m**6
|
|
qpower = np.power(q, 8 * k)
|
|
coeffs = np.array([1.0,
|
|
k1a + k1b*msquared,
|
|
k2a + k2b*msquared + k2c*mfour,
|
|
k3a + k3b*msquared + k3c*mfour + k3d*msix])
|
|
K0to3 *= qpower
|
|
K0to3 += coeffs
|
|
K0to3 *= q
|
|
K0to3 *= _SQRT2PI
|
|
# z**10 > 0 as z > 0.04
|
|
K0to3 /= np.array([z, 6 * zfour, 72 * z**7, 6480 * z**10])
|
|
|
|
# Now do the other sum over the other terms, all integers k
|
|
# K_2: (pi^2 k^2) q^(k^2),
|
|
# K_3: (3pi^2 k^2 z^2 - pi^4 k^4)*q^(k^2)
|
|
# Don't expect much subtractive cancellation so use direct calculation
|
|
q = np.exp(-_PI_SQUARED / 2 / zsquared)
|
|
ks = np.arange(maxk, 0, -1)
|
|
ksquared = ks ** 2
|
|
sqrt3z = _SQRT3 * z
|
|
kspi = np.pi * ks
|
|
qpwers = q ** ksquared
|
|
k2extra = np.sum(ksquared * qpwers)
|
|
k2extra *= _PI_SQUARED * _SQRT2PI/(-36 * zthree)
|
|
K0to3[2] += k2extra
|
|
k3extra = np.sum((sqrt3z + kspi) * (sqrt3z - kspi) * ksquared * qpwers)
|
|
k3extra *= _PI_SQUARED * _SQRT2PI/(216 * zsix)
|
|
K0to3[3] += k3extra
|
|
powers_of_n = np.power(n * 1.0, np.arange(len(K0to3)) / 2.0)
|
|
K0to3 /= powers_of_n
|
|
|
|
if not cdf:
|
|
K0to3 *= -1
|
|
K0to3[0] += 1
|
|
|
|
Ksum = sum(K0to3)
|
|
return Ksum
|
|
|
|
|
|
def _kolmogn(n, x, cdf=True):
|
|
"""Computes 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].
|
|
"""
|
|
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:
|
|
return _select_and_clip_prob(1.0, 0.0, cdf=cdf)
|
|
if x <= 0.0:
|
|
return _select_and_clip_prob(0.0, 1.0, cdf=cdf)
|
|
t = n * x
|
|
if t <= 1.0: # Ruben-Gambino: 1/2n <= x <= 1/n
|
|
if t <= 0.5:
|
|
return _select_and_clip_prob(0.0, 1.0, cdf=cdf)
|
|
if n <= 140:
|
|
prob = np.prod(np.arange(1, n+1) * (1.0/n) * (2*t - 1))
|
|
else:
|
|
prob = np.exp(_log_nfactorial_div_n_pow_n(n) + n * np.log(2*t-1))
|
|
return _select_and_clip_prob(prob, 1.0 - prob, cdf=cdf)
|
|
if t >= n - 1: # Ruben-Gambino
|
|
prob = 2 * (1.0 - x)**n
|
|
return _select_and_clip_prob(1 - prob, prob, cdf=cdf)
|
|
if x >= 0.5: # Exact: 2 * smirnov
|
|
prob = 2 * scipy.special.smirnov(n, x)
|
|
return _select_and_clip_prob(1.0 - prob, prob, cdf=cdf)
|
|
|
|
nxsquared = t * x
|
|
if n <= 140:
|
|
if nxsquared <= 0.754693:
|
|
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
|