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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 | #!/usr/bin/env python
import numpy as N
from scipy.special import polygamma,gamma
__doc__ = """This module implements a correction for overdispersion based on a beta-variance model
The procedure is described in more detail in
documents/correction_of_ci.pdf
In short, a beta distribution with the mode given by the fitted psychometric function,
and the number of trials scaled by a input independent factor nu is fitted to the data.
If a point estimate of the psychometric function has already been obtained, the only
free parameter of this model is nu which can be estimated using newton iterations.
The estimated nu can be used to correct the confidence intervals obtained from inference
that was based on a binomial distribution. Furhtermore, nu can be considered a measure of
binomiality of the residuals.
"""
__all__ = ["estimate_nu"]
def l ( nu, psi, k, n ):
"""log likelihood for psi
:Parameters:
*nu* :
scalar value nu, for which the likelihood should be evaluated
*psi* :
m array that gives the psychometric function evaluated at stimulus
levels x_i, i=1,...,m
*k* :
m array that gives the numbers of correct responses (in Yes/No: Yes responses)
at stimulus levels x_i, i=1,...,m
*n* :
m array that gives the numbers of presented trials at stimulus
levels x_i, i=1,...,m
:Example:
>>> psi = [ 0.52370051, 0.58115041, 0.70565915, 0.83343107, 0.89467234, 0.91364765, 0.91867512]
>>> k = [28, 29, 35, 41, 46, 46, 45]
>>> n = [50]*7
>>> l ( 1, psi, k, n )
13.752858759933943
"""
psi = N.array(psi,'d')
k = N.array(k,'d')
n = N.array(n,'d')
p = k/n
return (N.log(gamma(nu*n+2))).sum()-(N.log(gamma(psi*nu*n+1))).sum()-(N.log(gamma((1-psi)*nu*n+1))).sum() \
+ (psi*nu*n*N.log(p)).sum() + ((1-psi)*nu*n*N.log(1-p)).sum()
def dl ( nu, psi, k, n ):
"""first derivative of the likelihood function with respect to nu
:Parameters:
*nu* :
scalar value nu, for which the likelihood should be evaluated
*psi* :
m array that gives the psychometric function evaluated at stimulus
levels x_i, i=1,...,m
*k* :
m array that gives the numbers of correct responses (in Yes/No: Yes responses)
at stimulus levels x_i, i=1,...,m
*n* :
m array that gives the numbers of presented trials at stimulus
levels x_i, i=1,...,m
"""
p = k/n
return (n*polygamma( 0, nu*n+2 )).sum() \
- (psi*n*polygamma(0,nu*psi*n+1)).sum() \
- ((1-psi)*n*polygamma(0,nu*(1-psi)*n+1)).sum() \
+ (psi*n*N.log(p)).sum() \
+ ((1-psi)*n*N.log(1-p)).sum()
def ddl ( nu, psi, k, n ):
"""second derivative of the likelihood function with respect to nu
:Parameters:
*nu* :
scalar value nu, for which the likelihood should be evaluated
*psi* :
m array that gives the psychometric function evaluated at stimulus
levels x_i, i=1,...,m
*k* :
m array that gives the numbers of correct responses (in Yes/No: Yes responses)
at stimulus levels x_i, i=1,...,m
*n* :
m array that gives the numbers of presented trials at stimulus
levels x_i, i=1,...,m
"""
return (n**2*polygamma( 1, nu*n+2)).sum() \
- (psi**2*n**2*polygamma(1,nu*psi*n+1)).sum() \
- ((1-psi)**2*n**2*polygamma(1,nu*(1-psi)*n+1)).sum()
def estimate_nu ( InferenceObject ):
"""Perform a couple of newton iterations to estimate nu
:Parameters:
*InferenceObject* :
An Inference object (either Bayes- or Bootstrap) for which the nu parameter should
be estimated
:Return:
nu,nu_i,l_i
*nu* :
estimated nu parameter
*nu_i* :
sequence of nu values during optimization
*l_i* :
sequence of likelihoods associated with the nu values
"""
psi = InferenceObject.evaluate ( InferenceObject.data[:,0] )
k = InferenceObject.data[:,1].astype('d')
n = InferenceObject.data[:,2].astype('d')
k = N.where ( k==n, k-.01, k )
k = N.where ( k==0, .01, k )
nu = 1.
nu_i = [nu]
l_i = [l(nu,psi,k,n)]
for i in xrange(10):
nu -= dl(nu,psi,k,n)/ddl(nu,psi,k,n)
if nu>1:
nu=1
elif nu<0:
nu=0
nu_i.append ( nu )
l_i.append ( l(nu,psi,k,n) )
return nu, nu_i, l_i
def main ( ):
# An Example of usage
from pypsignifit import BootstrapInference
from pypsignifit.psigobservers import BetaBinomialObserver
import pylab as p
O = BetaBinomialObserver ( 4, 1, .02, 10 )
d = N.array ( O.DoAnExperiment( [1,2,3,4,5,6,7] ) )
B = BootstrapInference ( d, priors=("","","Uniform(0,.1)"))
psi = B.evaluate ( B.data[:,0] )
k = B.data[:,1].astype('d')
n = B.data[:,2].astype('d')
x = N.mgrid[0.001:0.999:100j]
ll = []
for xx in x:
ll.append ( l(xx,psi,k,n) )
print ll[-1]
p.plot(x,ll,'b-')
nu,nu_i,l_i = estimate_nu ( B )
p.plot ( nu_i, l_i, 'ro' )
p.show()
if __name__ == "__main__":
# main()
import doctest
doctest.testmod()
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