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"""Python implementation of chisqprob, to avoid SciPy dependency.
Adapted from SciPy: scipy/special/cephes/{chdtr,igam}.
"""
import math
try:
from math import lgamma as _lgamma
except ImportError:
# Missing in Python 2.6, using Python Python recipe from
# http://code.activestate.com/recipes/576393/
# with PEP8 whitespace and minor import changes
#
# * ====================================================
# * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
# *
# * Developed at SunPro, a Sun Microsystems, Inc. business.
# * Permission to use, copy, modify, and distribute this
# * software is freely granted, provided that this notice
# * is preserved.
# * ====================================================
two52 = 4.50359962737049600000e+15
half = 5.00000000000000000000e-01
one = 1.00000000000000000000e+00
pi = 3.14159265358979311600e+00
a0 = 7.72156649015328655494e-02
a1 = 3.22467033424113591611e-01
a2 = 6.73523010531292681824e-02
a3 = 2.05808084325167332806e-02
a4 = 7.38555086081402883957e-03
a5 = 2.89051383673415629091e-03
a6 = 1.19270763183362067845e-03
a7 = 5.10069792153511336608e-04
a8 = 2.20862790713908385557e-04
a9 = 1.08011567247583939954e-04
a10 = 2.52144565451257326939e-05
a11 = 4.48640949618915160150e-05
tc = 1.46163214496836224576e+00
tf = -1.21486290535849611461e-01
# /* tt = -(tail of tf) */
tt = -3.63867699703950536541e-18
t0 = 4.83836122723810047042e-01
t1 = -1.47587722994593911752e-01
t2 = 6.46249402391333854778e-02
t3 = -3.27885410759859649565e-02
t4 = 1.79706750811820387126e-02
t5 = -1.03142241298341437450e-02
t6 = 6.10053870246291332635e-03
t7 = -3.68452016781138256760e-03
t8 = 2.25964780900612472250e-03
t9 = -1.40346469989232843813e-03
t10 = 8.81081882437654011382e-04
t11 = -5.38595305356740546715e-04
t12 = 3.15632070903625950361e-04
t13 = -3.12754168375120860518e-04
t14 = 3.35529192635519073543e-04
u0 = -7.72156649015328655494e-02
u1 = 6.32827064025093366517e-01
u2 = 1.45492250137234768737e+00
u3 = 9.77717527963372745603e-01
u4 = 2.28963728064692451092e-01
u5 = 1.33810918536787660377e-02
v1 = 2.45597793713041134822e+00
v2 = 2.12848976379893395361e+00
v3 = 7.69285150456672783825e-01
v4 = 1.04222645593369134254e-01
v5 = 3.21709242282423911810e-03
s0 = -7.72156649015328655494e-02
s1 = 2.14982415960608852501e-01
s2 = 3.25778796408930981787e-01
s3 = 1.46350472652464452805e-01
s4 = 2.66422703033638609560e-02
s5 = 1.84028451407337715652e-03
s6 = 3.19475326584100867617e-05
r1 = 1.39200533467621045958e+00
r2 = 7.21935547567138069525e-01
r3 = 1.71933865632803078993e-01
r4 = 1.86459191715652901344e-02
r5 = 7.77942496381893596434e-04
r6 = 7.32668430744625636189e-06
w0 = 4.18938533204672725052e-01
w1 = 8.33333333333329678849e-02
w2 = -2.77777777728775536470e-03
w3 = 7.93650558643019558500e-04
w4 = -5.95187557450339963135e-04
w5 = 8.36339918996282139126e-04
w6 = -1.63092934096575273989e-03
zero = 0.00000000000000000000e+00
# inf = float('inf')
# nan = float('nan')
inf = float(9e999)
def _sin_pi(x):
x = float(x)
e, ix = math.frexp(x)
if(abs(x) < 0.25):
return -math.sin(pi * x)
y = -x # /* x is assume negative */
# * argument reduction, make sure inexact flag not raised if input
# * is an integer
z = floor(y)
if(z != y):
y *= 0.5
y = 2.0 * (y - floor(y)) # /* y = |x| mod 2.0 */
n = int(y * 4.0)
else:
if(abs(ix) >= 53):
y = zero
n = 0 # /* y must be even */
else:
if(abs(ix) < 52):
z = y + two52 # /* exact */
e, n = math.frexp(z)
n &= 1
y = n
n <<= 2
if n == 0:
y = sin(pi * y)
elif (n == 1 or n == 2):
y = cos(pi * (0.5 - y))
elif (n == 3 or n == 4):
y = sin(pi * (one - y))
elif (n == 5 or n == 6):
y = -cos(pi * (y - 1.5))
else:
y = sin(pi * (y - 2.0))
z = cos(pi * (z + 1.0))
return -y * z
def _lgamma(x):
"""Natural logarithm of gamma function of x
raise ValueError if x is negative integer."""
x = float(x)
# /* purge off +-inf, NaN, +-0, and negative arguments */
if ((x == inf) or (x == -inf)):
return inf
# if (x is nan):
# return nan
e, ix = math.frexp(x)
nadj = 0
signgamp = 1
if ((e == 0.0) and (ix == 0)):
return inf
if (ix > 1020):
return inf
if ((e != 0.0) and (ix < -71)):
if (x < 0):
return -math.log(-x)
else:
return -math.log(x)
if e < 0:
if ix > 52:
return inf # one/zero
t = sin_pi(x)
if t == zero:
# return inf
raise ValueError('gamma not defined for negative integer')
nadj = math.log(pi / fabs(t * x))
if t < zero:
signgamp = -1
x = -x
# /* purge off 1 and 2 */
if x == 2.0 or x == 1.0:
r = 0.0
# /* for x < 2.0 */
elif ix < 2:
if x <= 0.9: # /* lgamma(x) = lgamma(x+1)-log(x) */
r = -math.log(x)
if x >= 0.7316:
y = one - x
z = y * y
p1 = a0 + z * (a2 + z * (a4 + z * (a6 + z * (a8 + z * a10))))
p2 = z * (a1 + z * (a3 + z * (a5 + z * (a7 + z * (a9 + z * a11)))))
p = y * p1 + p2
r += (p - 0.5 * y)
elif (x >= 0.23164):
y = x - (tc - one)
z = y * y
w = z * y
p1 = t0 + w * (t3 + w * (t6 + w * (t9 + w * t12))) # /* parallel comp */
p2 = t1 + w * (t4 + w * (t7 + w * (t10 + w * t13)))
p3 = t2 + w * (t5 + w * (t8 + w * (t11 + w * t14)))
p = z * p1 - (tt - w * (p2 + y * p3))
r += (tf + p)
else:
y = x
p1 = y * (u0 + y * (u1 + y * (u2 + y * (u3 + y * (u4 + y * u5)))))
p2 = one + y * (v1 + y * (v2 + y * (v3 + y * (v4 + y * v5))))
r += (-0.5 * y + p1 / p2)
else:
r = zero
if(x >= 1.7316):
y = 2.0 - x # /* [1.7316,2] */
z = y * y
p1 = a0 + z * (a2 + z * (a4 + z * (a6 + z * (a8 + z * a10))))
p2 = z * (a1 + z * (a3 + z * (a5 + z * (a7 + z * (a9 + z * a11)))))
p = y * p1 + p2
r += (p - 0.5 * y)
elif(x >= 1.23164):
y = x - tc # /* [1.23,1.73] */
z = y * y
w = z * y
p1 = t0 + w * (t3 + w * (t6 + w * (t9 + w * t12))) # /* parallel comp */
p2 = t1 + w * (t4 + w * (t7 + w * (t10 + w * t13)))
p3 = t2 + w * (t5 + w * (t8 + w * (t11 + w * t14)))
p = z * p1 - (tt - w * (p2 + y * p3))
r += (tf + p)
else:
y = x - one
p1 = y * (u0 + y * (u1 + y * (u2 + y * (u3 + y * (u4 + y * u5)))))
p2 = one + y * (v1 + y * (v2 + y * (v3 + y * (v4 + y * v5))))
r += (-0.5 * y + p1 / p2)
# /* x < 8.0 */
elif(ix < 4):
i = int(x)
t = zero
y = x - i
p = y * (s0 + y * (s1 + y * (s2 + y * (s3 + y * (s4 + y * (s5 + y * s6))))))
q = one + y * (r1 + y * (r2 + y * (r3 + y * (r4 + y * (r5 + y * r6)))))
r = half * y + p / q
z = one # /* lgamma(1+s) = log(s) + lgamma(s) */
while (i > 2):
i -= 1
z *= (y + i)
r += log(z)
# /* 8.0 <= x < 2**58 */
elif (ix < 58):
t = log(x)
z = one / x
y = z * z
w = w0 + z * (w1 + y * (w2 + y * (w3 + y * (w4 + y * (w5 + y * w6)))))
r = (x - half) * (t - one) + w
# /* 2**58 <= x <= inf */
else:
r = x * (log(x) - one)
if (e < 0):
r = nadj - r
return signgamp * r
# Cephes Math Library Release 2.0: April, 1987
# Copyright 1985, 1987 by Stephen L. Moshier
# Direct inquiries to 30 Frost Street, Cambridge, MA 02140
MACHEP = 0.0000001 # the machine roundoff error / tolerance
BIG = 4.503599627370496e15
BIGINV = 2.22044604925031308085e-16
def chisqprob(x, df):
"""Probability value (1-tail) for the Chi^2 probability distribution.
Broadcasting rules apply.
Parameters
----------
x : array_like or float > 0
df : array_like or float, probably int >= 1
Returns
-------
chisqprob : ndarray
The area from `chisq` to infinity under the Chi^2 probability
distribution with degrees of freedom `df`.
"""
if x <= 0:
return 1.0
if x == 0:
return 0.0
if df <= 0:
raise ValueError("Domain error.")
if x < 1.0 or x < df:
return 1.0 - _igam(0.5 * df, 0.5 * x)
return _igamc(0.5 * df, 0.5 * x)
def _igamc(a, x):
"""Complemented incomplete Gamma integral.
SYNOPSIS:
double a, x, y, igamc();
y = igamc( a, x );
DESCRIPTION:
The function is defined by::
igamc(a,x) = 1 - igam(a,x)
inf.
-
1 | | -t a-1
= ----- | e t dt.
- | |
| (a) -
x
In this implementation both arguments must be positive.
The integral is evaluated by either a power series or
continued fraction expansion, depending on the relative
values of a and x.
"""
# Compute x**a * exp(-x) / Gamma(a)
# TODO: Return this to math.lgamma once drop Python 2.6
ax = math.exp(a * math.log(x) - x - _lgamma(a))
# Continued fraction
y = 1.0 - a
z = x + y + 1.0
c = 0.0
pkm2 = 1.0
qkm2 = x
pkm1 = x + 1.0
qkm1 = z * x
ans = pkm1 / qkm1
while True:
c += 1.0
y += 1.0
z += 2.0
yc = y * c
pk = pkm1 * z - pkm2 * yc
qk = qkm1 * z - qkm2 * yc
if qk != 0:
r = pk / qk
t = abs((ans - r) / r)
ans = r
else:
t = 1.0
pkm2 = pkm1
pkm1 = pk
qkm2 = qkm1
qkm1 = qk
if abs(pk) > BIG:
pkm2 *= BIGINV
pkm1 *= BIGINV
qkm2 *= BIGINV
qkm1 *= BIGINV
if t <= MACHEP:
return ans * ax
def _igam(a, x):
"""Left tail of incomplete Gamma function.
Computes this formula::
inf. k
a -x - x
x e > ----------
- -
k=0 | (a+k+1)
"""
# Compute x**a * exp(-x) / Gamma(a)
ax = math.exp(a * math.log(x) - x - math.lgamma(a))
# Power series
r = a
c = 1.0
ans = 1.0
while True:
r += 1.0
c *= x / r
ans += c
if c / ans <= MACHEP:
return ans * ax / a
# --- Speed ---
# try:
# from scipy.stats import chisqprob
# except ImportError:
# pass
|
