# -*- coding: utf-8 -*-
"""
Created on Sat Oct 01 20:20:16 2011
Author: Josef Perktold
License: BSD-3
TODO:
check orientation, size and alpha should be increasing for interp1d,
but what is alpha? can be either sf or cdf probability
change it to use one consistent notation
check: instead of bound checking I could use the fill-value of the interpolators
"""
from __future__ import print_function
from statsmodels.compat.python import range
import numpy as np
from scipy.interpolate import interp1d, interp2d, Rbf
from statsmodels.tools.decorators import cache_readonly
[docs]class TableDist(object):
'''Distribution, critical values and p-values from tables
currently only 1 extra parameter, e.g. sample size
Parameters
----------
alpha : array_like, 1d
probabiliy in the table, could be either sf (right tail) or cdf (left
tail)
size : array_like, 1d
second paramater in the table
crit_table : array_like, 2d
array with critical values for sample size in rows and probability in
columns
Notes
-----
size and alpha should be increasing
'''
[docs] def __init__(self, alpha, size, crit_table):
self.alpha = np.asarray(alpha)
self.size = np.asarray(size)
self.crit_table = np.asarray(crit_table)
self.n_alpha = len(alpha)
self.signcrit = np.sign(np.diff(self.crit_table, 1).mean())
if self.signcrit > 0: #increasing
self.critv_bounds = self.crit_table[:,[0,1]]
else:
self.critv_bounds = self.crit_table[:,[1,0]]
@cache_readonly
[docs] def polyn(self):
polyn = [interp1d(self.size, self.crit_table[:,i])
for i in range(self.n_alpha)]
return polyn
@cache_readonly
[docs] def poly2d(self):
#check for monotonicity ?
#fix this, interp needs increasing
poly2d = interp2d(self.size, self.alpha, self.crit_table)
return poly2d
@cache_readonly
[docs] def polyrbf(self):
xs, xa = np.meshgrid(self.size.astype(float), self.alpha)
polyrbf = Rbf(xs.ravel(), xa.ravel(), self.crit_table.T.ravel(),function='linear')
return polyrbf
def _critvals(self, n):
'''rows of the table, linearly interpolated for given sample size
Parameters
----------
n : float
sample size, second parameter of the table
Returns
-------
critv : ndarray, 1d
critical values (ppf) corresponding to a row of the table
Notes
-----
This is used in two step interpolation, or if we want to know the
critical values for all alphas for any sample size that we can obtain
through interpolation
'''
return np.array([p(n) for p in self.polyn])
[docs] def prob(self, x, n):
'''find pvalues by interpolation, eiter cdf(x) or sf(x)
returns extrem probabilities, 0.001 and 0.2, for out of range
Parameters
----------
x : array_like
observed value, assumed to follow the distribution in the table
n : float
sample size, second parameter of the table
Returns
-------
prob : arraylike
This is the probability for each value of x, the p-value in
underlying distribution is for a statistical test.
'''
critv = self._critvals(n)
alpha = self.alpha
# if self.signcrit == 1:
# if x < critv[0]: #generalize: ? np.sign(x - critvals[0]) == self.signcrit:
# return alpha[0]
# elif x > critv[-1]:
# return alpha[-1]
# elif self.signcrit == -1:
# if x > critv[0]:
# return alpha[0]
# elif x < critv[-1]:
# return alpha[-1]
if self.signcrit < 1:
#reverse if critv is decreasing
critv, alpha = critv[::-1], alpha[::-1]
#now critv is increasing
if np.size(x) == 1:
if x < critv[0]:
return alpha[0]
elif x > critv[-1]:
return alpha[-1]
return interp1d(critv, alpha)(x)[()]
else:
#vectorized
cond_low = (x < critv[0])
cond_high = (x > critv[-1])
cond_interior = ~np.logical_or(cond_low, cond_high)
probs = np.nan * np.ones(x.shape) #mistake if nan left
probs[cond_low] = alpha[0]
probs[cond_low] = alpha[-1]
probs[cond_interior] = interp1d(critv, alpha)(x[cond_interior])
return probs
[docs] def crit2(self, prob, n):
'''returns interpolated quantiles, similar to ppf or isf
this can be either cdf or sf depending on the table, twosided?
this doesn't work, no more knots warning
'''
return self.poly2d(n, prob)
[docs] def crit(self, prob, n):
'''returns interpolated quantiles, similar to ppf or isf
use two sequential 1d interpolation, first by n then by prob
Parameters
----------
prob : array_like
probabilities corresponding to the definition of table columns
n : int or float
sample size, second parameter of the table
Returns
-------
ppf : array_like
critical values with same shape as prob
'''
prob = np.asarray(prob)
alpha = self.alpha
critv = self._critvals(n)
#vectorized
cond_ilow = (prob > alpha[0])
cond_ihigh = (prob < alpha[-1])
cond_interior = np.logical_or(cond_ilow, cond_ihigh)
#scalar
if prob.size == 1:
if cond_interior:
return interp1d(alpha, critv)(prob)
else:
return np.nan
#vectorized
quantile = np.nan * np.ones(prob.shape) #nans for outside
quantile[cond_interior] = interp1d(alpha, critv)(prob[cond_interior])
return quantile
[docs] def crit3(self, prob, n):
'''returns interpolated quantiles, similar to ppf or isf
uses Rbf to interpolate critical values as function of `prob` and `n`
Parameters
----------
prob : array_like
probabilities corresponding to the definition of table columns
n : int or float
sample size, second parameter of the table
Returns
-------
ppf : array_like
critical values with same shape as prob, returns nan for arguments
that are outside of the table bounds
'''
prob = np.asarray(prob)
alpha = self.alpha
#vectorized
cond_ilow = (prob > alpha[0])
cond_ihigh = (prob < alpha[-1])
cond_interior = np.logical_or(cond_ilow, cond_ihigh)
#scalar
if prob.size == 1:
if cond_interior:
return self.polyrbf(n, prob)
else:
return np.nan
#vectorized
quantile = np.nan * np.ones(prob.shape) #nans for outside
quantile[cond_interior] = self.polyrbf(n, prob[cond_interior])
return quantile
if __name__ == '__main__':
'''
example Lilliefors test for normality
An Analytic Approximation to the Distribution of Lilliefors's Test Statistic for Normality
Author(s): Gerard E. Dallal and Leland WilkinsonSource: The American Statistician, Vol. 40, No. 4 (Nov., 1986), pp. 294-296Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2684607 .
'''
#for this test alpha is sf probability, i.e. right tail probability
alpha = np.array([ 0.2 , 0.15 , 0.1 , 0.05 , 0.01 , 0.001])[::-1]
size = np.array([ 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19, 20, 25, 30, 40, 100, 400, 900], float)
#critical values, rows are by sample size, columns are by alpha
crit_lf = np.array( [[303, 321, 346, 376, 413, 433],
[289, 303, 319, 343, 397, 439],
[269, 281, 297, 323, 371, 424],
[252, 264, 280, 304, 351, 402],
[239, 250, 265, 288, 333, 384],
[227, 238, 252, 274, 317, 365],
[217, 228, 241, 262, 304, 352],
[208, 218, 231, 251, 291, 338],
[200, 210, 222, 242, 281, 325],
[193, 202, 215, 234, 271, 314],
[187, 196, 208, 226, 262, 305],
[181, 190, 201, 219, 254, 296],
[176, 184, 195, 213, 247, 287],
[171, 179, 190, 207, 240, 279],
[167, 175, 185, 202, 234, 273],
[163, 170, 181, 197, 228, 266],
[159, 166, 176, 192, 223, 260],
[143, 150, 159, 173, 201, 236],
[131, 138, 146, 159, 185, 217],
[115, 120, 128, 139, 162, 189],
[ 74, 77, 82, 89, 104, 122],
[ 37, 39, 41, 45, 52, 61],
[ 25, 26, 28, 30, 35, 42]])[:,::-1] / 1000.
lf = TableDist(alpha, size, crit_lf)
print(lf.prob(0.166, 20), 'should be:', 0.15)
print('')
print(lf.crit2(0.15, 20), 'should be:', 0.166, 'interp2d bad')
print(lf.crit(0.15, 20), 'should be:', 0.166, 'two 1d')
print(lf.crit3(0.15, 20), 'should be:', 0.166, 'Rbf')
print('')
print(lf.crit2(0.17, 20), 'should be in:', (.159, .166), 'interp2d bad')
print(lf.crit(0.17, 20), 'should be in:', (.159, .166), 'two 1d')
print(lf.crit3(0.17, 20), 'should be in:', (.159, .166), 'Rbf')
print('')
print(lf.crit2(0.19, 20), 'should be in:', (.159, .166), 'interp2d bad')
print(lf.crit(0.19, 20), 'should be in:', (.159, .166), 'two 1d')
print(lf.crit3(0.19, 20), 'should be in:', (.159, .166), 'Rbf')
print('')
print(lf.crit2(0.199, 20), 'should be in:', (.159, .166), 'interp2d bad')
print(lf.crit(0.199, 20), 'should be in:', (.159, .166), 'two 1d')
print(lf.crit3(0.199, 20), 'should be in:', (.159, .166), 'Rbf')
#testing
print(np.max(np.abs(np.array([lf.prob(c, size[i]) for i in range(len(size)) for c in crit_lf[i]]).reshape(-1,lf.n_alpha) - lf.alpha)))
#1.6653345369377348e-16
print(np.max(np.abs(np.array([lf.crit(c, size[i]) for i in range(len(size)) for c in lf.alpha]).reshape(-1,lf.n_alpha) - crit_lf)))
#6.9388939039072284e-18)
print(np.max(np.abs(np.array([lf.crit3(c, size[i]) for i in range(len(size)) for c in lf.alpha]).reshape(-1,lf.n_alpha) - crit_lf)))
#4.0615705243496336e-12)
print((np.array([lf.crit3(c, size[i]) for i in range(len(size)) for c in lf.alpha[:-1]*1.1]).reshape(-1,lf.n_alpha-1) < crit_lf[:,:-1]).all())
print((np.array([lf.crit3(c, size[i]) for i in range(len(size)) for c in lf.alpha[:-1]*1.1]).reshape(-1,lf.n_alpha-1) > crit_lf[:,1:]).all())
print((np.array([lf.prob(c*0.9, size[i]) for i in range(len(size)) for c in crit_lf[i,:-1]]).reshape(-1,lf.n_alpha-1) > lf.alpha[:-1]).all())
print((np.array([lf.prob(c*1.1, size[i]) for i in range(len(size)) for c in crit_lf[i,1:]]).reshape(-1,lf.n_alpha-1) < lf.alpha[1:]).all())
#start at size_idx=2 because of non-monotonicity of lf_crit
print((np.array([lf.prob(c, size[i]*0.9) for i in range(2,len(size)) for c in crit_lf[i,:-1]]).reshape(-1,lf.n_alpha-1) > lf.alpha[:-1]).all())