'''
Defines the link functions to be used with GLM and GEE families.
'''
import numpy as np
import scipy.stats
FLOAT_EPS = np.finfo(float).eps
[docs]class Link(object):
"""
A generic link function for one-parameter exponential family.
`Link` does nothing, but lays out the methods expected of any subclass.
"""
def __call__(self, p):
"""
Return the value of the link function. This is just a placeholder.
Parameters
----------
p : array_like
Probabilities
Returns
-------
g(p) : array_like
The value of the link function g(p) = z
"""
return NotImplementedError
def inverse(self, z):
"""
Inverse of the link function. Just a placeholder.
Parameters
----------
z : array_like
`z` is usually the linear predictor of the transformed variable
in the IRLS algorithm for GLM.
Returns
-------
g^(-1)(z) : array
The value of the inverse of the link function g^(-1)(z) = p
"""
return NotImplementedError
def deriv(self, p):
"""
Derivative of the link function g'(p). Just a placeholder.
Parameters
----------
p : array_like
Returns
-------
g'(p) : array
The value of the derivative of the link function g'(p)
"""
return NotImplementedError
def deriv2(self, p):
"""Second derivative of the link function g''(p)
implemented through numerical differentiation
"""
from statsmodels.tools.numdiff import approx_fprime_cs
# TODO: workaround proplem with numdiff for 1d
return np.diag(approx_fprime_cs(p, self.deriv))
def inverse_deriv(self, z):
"""
Derivative of the inverse link function g^(-1)(z).
Parameters
----------
z : array_like
`z` is usually the linear predictor for a GLM or GEE model.
Returns
-------
g'^(-1)(z) : array
The value of the derivative of the inverse of the link function
Notes
-----
This reference implementation gives the correct result but is
inefficient, so it can be overridden in subclasses.
"""
return 1 / self.deriv(self.inverse(z))
def inverse_deriv2(self, z):
"""
Second derivative of the inverse link function g^(-1)(z).
Parameters
----------
z : array_like
`z` is usually the linear predictor for a GLM or GEE model.
Returns
-------
g'^(-1)(z) : array
The value of the second derivative of the inverse of the link
function
Notes
-----
This reference implementation gives the correct result but is
inefficient, so it can be overridden in subclasses.
"""
iz = self.inverse(z)
return -self.deriv2(iz) / self.deriv(iz)**3
[docs]class Logit(Link):
"""
The logit transform
Notes
-----
call and derivative use a private method _clean to make trim p by
machine epsilon so that p is in (0,1)
Alias of Logit:
logit = Logit()
"""
def _clean(self, p):
"""
Clip logistic values to range (eps, 1-eps)
Parameters
----------
p : array_like
Probabilities
Returns
-------
pclip : array
Clipped probabilities
"""
return np.clip(p, FLOAT_EPS, 1. - FLOAT_EPS)
def __call__(self, p):
"""
The logit transform
Parameters
----------
p : array_like
Probabilities
Returns
-------
z : array
Logit transform of `p`
Notes
-----
g(p) = log(p / (1 - p))
"""
p = self._clean(p)
return np.log(p / (1. - p))
def inverse(self, z):
"""
Inverse of the logit transform
Parameters
----------
z : array_like
The value of the logit transform at `p`
Returns
-------
p : array
Probabilities
Notes
-----
g^(-1)(z) = exp(z)/(1+exp(z))
"""
z = np.asarray(z)
t = np.exp(-z)
return 1. / (1. + t)
def deriv(self, p):
"""
Derivative of the logit transform
Parameters
----------
p: array_like
Probabilities
Returns
-------
g'(p) : array
Value of the derivative of logit transform at `p`
Notes
-----
g'(p) = 1 / (p * (1 - p))
Alias for `Logit`:
logit = Logit()
"""
p = self._clean(p)
return 1. / (p * (1 - p))
def inverse_deriv(self, z):
"""
Derivative of the inverse of the logit transform
Parameters
----------
z : array_like
`z` is usually the linear predictor for a GLM or GEE model.
Returns
-------
g'^(-1)(z) : array
The value of the derivative of the inverse of the logit function
"""
t = np.exp(z)
return t/(1 + t)**2
def deriv2(self, p):
"""
Second derivative of the logit function.
Parameters
----------
p : array_like
probabilities
Returns
-------
g''(z) : array
The value of the second derivative of the logit function
"""
v = p * (1 - p)
return (2*p - 1) / v**2
[docs]class logit(Logit):
pass
[docs]class Power(Link):
"""
The power transform
Parameters
----------
power : float
The exponent of the power transform
Notes
-----
Aliases of Power:
inverse = Power(power=-1)
sqrt = Power(power=.5)
inverse_squared = Power(power=-2.)
identity = Power(power=1.)
"""
def __init__(self, power=1.):
self.power = power
def __call__(self, p):
"""
Power transform link function
Parameters
----------
p : array_like
Mean parameters
Returns
-------
z : array_like
Power transform of x
Notes
-----
g(p) = x**self.power
"""
if self.power == 1:
return p
else:
return np.power(p, self.power)
def inverse(self, z):
"""
Inverse of the power transform link function
Parameters
----------
`z` : array_like
Value of the transformed mean parameters at `p`
Returns
-------
`p` : array
Mean parameters
Notes
-----
g^(-1)(z`) = `z`**(1/`power`)
"""
if self.power == 1:
return z
else:
return np.power(z, 1. / self.power)
def deriv(self, p):
"""
Derivative of the power transform
Parameters
----------
p : array_like
Mean parameters
Returns
-------
g'(p) : array
Derivative of power transform of `p`
Notes
-----
g'(`p`) = `power` * `p`**(`power` - 1)
"""
if self.power == 1:
return np.ones_like(p)
else:
return self.power * np.power(p, self.power - 1)
def deriv2(self, p):
"""
Second derivative of the power transform
Parameters
----------
p : array_like
Mean parameters
Returns
-------
g''(p) : array
Second derivative of the power transform of `p`
Notes
-----
g''(`p`) = `power` * (`power` - 1) * `p`**(`power` - 2)
"""
if self.power == 1:
return np.zeros_like(p)
else:
return self.power * (self.power - 1) * np.power(p, self.power - 2)
def inverse_deriv(self, z):
"""
Derivative of the inverse of the power transform
Parameters
----------
z : array_like
`z` is usually the linear predictor for a GLM or GEE model.
Returns
-------
g^(-1)'(z) : array
The value of the derivative of the inverse of the power transform
function
"""
if self.power == 1:
return np.ones_like(z)
else:
return np.power(z, (1 - self.power)/self.power) / self.power
def inverse_deriv2(self, z):
"""
Second derivative of the inverse of the power transform
Parameters
----------
z : array_like
`z` is usually the linear predictor for a GLM or GEE model.
Returns
-------
g^(-1)'(z) : array
The value of the derivative of the inverse of the power transform
function
"""
if self.power == 1:
return np.zeros_like(z)
else:
return ((1 - self.power) *
np.power(z, (1 - 2*self.power)/self.power) / self.power**2)
[docs]class inverse_power(Power):
"""
The inverse transform
Notes
-----
g(p) = 1/p
Alias of statsmodels.family.links.Power(power=-1.)
"""
def __init__(self):
super(inverse_power, self).__init__(power=-1.)
class sqrt(Power):
"""
The square-root transform
Notes
-----
g(`p`) = sqrt(`p`)
Alias of statsmodels.family.links.Power(power=.5)
"""
def __init__(self):
super(sqrt, self).__init__(power=.5)
[docs]class inverse_squared(Power):
r"""
The inverse squared transform
Notes
-----
g(`p`) = 1/(`p`\*\*2)
Alias of statsmodels.family.links.Power(power=2.)
"""
def __init__(self):
super(inverse_squared, self).__init__(power=-2.)
[docs]class identity(Power):
"""
The identity transform
Notes
-----
g(`p`) = `p`
Alias of statsmodels.family.links.Power(power=1.)
"""
def __init__(self):
super(identity, self).__init__(power=1.)
[docs]class Log(Link):
"""
The log transform
Notes
-----
call and derivative call a private method _clean to trim the data by
machine epsilon so that p is in (0,1). log is an alias of Log.
"""
def _clean(self, x):
return np.clip(x, FLOAT_EPS, np.inf)
def __call__(self, p, **extra):
"""
Log transform link function
Parameters
----------
x : array_like
Mean parameters
Returns
-------
z : array
log(x)
Notes
-----
g(p) = log(p)
"""
x = self._clean(p)
return np.log(x)
def inverse(self, z):
"""
Inverse of log transform link function
Parameters
----------
z : array
The inverse of the link function at `p`
Returns
-------
p : array
The mean probabilities given the value of the inverse `z`
Notes
-----
g^{-1}(z) = exp(z)
"""
return np.exp(z)
def deriv(self, p):
"""
Derivative of log transform link function
Parameters
----------
p : array_like
Mean parameters
Returns
-------
g'(p) : array
derivative of log transform of x
Notes
-----
g'(x) = 1/x
"""
p = self._clean(p)
return 1. / p
def deriv2(self, p):
"""
Second derivative of the log transform link function
Parameters
----------
p : array_like
Mean parameters
Returns
-------
g''(p) : array
Second derivative of log transform of x
Notes
-----
g''(x) = -1/x^2
"""
p = self._clean(p)
return -1. / p**2
def inverse_deriv(self, z):
"""
Derivative of the inverse of the log transform link function
Parameters
----------
z : array
The inverse of the link function at `p`
Returns
-------
g^(-1)'(z) : array
The value of the derivative of the inverse of the log function,
the exponential function
"""
return np.exp(z)
[docs]class log(Log):
"""
The log transform
Notes
-----
log is a an alias of Log.
"""
pass
# TODO: the CDFLink is untested
[docs]class CDFLink(Logit):
"""
The use the CDF of a scipy.stats distribution
CDFLink is a subclass of logit in order to use its _clean method
for the link and its derivative.
Parameters
----------
dbn : scipy.stats distribution
Default is dbn=scipy.stats.norm
Notes
-----
The CDF link is untested.
"""
def __init__(self, dbn=scipy.stats.norm):
self.dbn = dbn
def __call__(self, p):
"""
CDF link function
Parameters
----------
p : array_like
Mean parameters
Returns
-------
z : array
(ppf) inverse of CDF transform of p
Notes
-----
g(`p`) = `dbn`.ppf(`p`)
"""
p = self._clean(p)
return self.dbn.ppf(p)
def inverse(self, z):
"""
The inverse of the CDF link
Parameters
----------
z : array_like
The value of the inverse of the link function at `p`
Returns
-------
p : array
Mean probabilities. The value of the inverse of CDF link of `z`
Notes
-----
g^(-1)(`z`) = `dbn`.cdf(`z`)
"""
return self.dbn.cdf(z)
def deriv(self, p):
"""
Derivative of CDF link
Parameters
----------
p : array_like
mean parameters
Returns
-------
g'(p) : array
The derivative of CDF transform at `p`
Notes
-----
g'(`p`) = 1./ `dbn`.pdf(`dbn`.ppf(`p`))
"""
p = self._clean(p)
return 1. / self.dbn.pdf(self.dbn.ppf(p))
def deriv2(self, p):
"""
Second derivative of the link function g''(p)
implemented through numerical differentiation
"""
from statsmodels.tools.numdiff import approx_fprime
p = np.atleast_1d(p)
# Note: special function for norm.ppf does not support complex
return np.diag(approx_fprime(p, self.deriv, centered=True))
def inverse_deriv(self, z):
"""
Derivative of the inverse of the CDF transformation link function
Parameters
----------
z : array
The inverse of the link function at `p`
Returns
-------
g^(-1)'(z) : array
The value of the derivative of the inverse of the logit function
"""
return 1/self.deriv(self.inverse(z))
[docs]class probit(CDFLink):
"""
The probit (standard normal CDF) transform
Notes
-----
g(p) = scipy.stats.norm.ppf(p)
probit is an alias of CDFLink.
"""
pass
[docs]class cauchy(CDFLink):
"""
The Cauchy (standard Cauchy CDF) transform
Notes
-----
g(p) = scipy.stats.cauchy.ppf(p)
cauchy is an alias of CDFLink with dbn=scipy.stats.cauchy
"""
def __init__(self):
super(cauchy, self).__init__(dbn=scipy.stats.cauchy)
def deriv2(self, p):
"""
Second derivative of the Cauchy link function.
Parameters
----------
p: array_like
Probabilities
Returns
-------
g''(p) : array
Value of the second derivative of Cauchy link function at `p`
"""
a = np.pi * (p - 0.5)
d2 = 2 * np.pi**2 * np.sin(a) / np.cos(a)**3
return d2
[docs]class CLogLog(Logit):
"""
The complementary log-log transform
CLogLog inherits from Logit in order to have access to its _clean method
for the link and its derivative.
Notes
-----
CLogLog is untested.
"""
def __call__(self, p):
"""
C-Log-Log transform link function
Parameters
----------
p : array
Mean parameters
Returns
-------
z : array
The CLogLog transform of `p`
Notes
-----
g(p) = log(-log(1-p))
"""
p = self._clean(p)
return np.log(-np.log(1 - p))
def inverse(self, z):
"""
Inverse of C-Log-Log transform link function
Parameters
----------
z : array_like
The value of the inverse of the CLogLog link function at `p`
Returns
-------
p : array
Mean parameters
Notes
-----
g^(-1)(`z`) = 1-exp(-exp(`z`))
"""
return 1 - np.exp(-np.exp(z))
def deriv(self, p):
"""
Derivative of C-Log-Log transform link function
Parameters
----------
p : array_like
Mean parameters
Returns
-------
g'(p) : array
The derivative of the CLogLog transform link function
Notes
-----
g'(p) = - 1 / ((p-1)*log(1-p))
"""
p = self._clean(p)
return 1. / ((p - 1) * (np.log(1 - p)))
def deriv2(self, p):
"""
Second derivative of the C-Log-Log ink function
Parameters
----------
p : array_like
Mean parameters
Returns
-------
g''(p) : array
The second derivative of the CLogLog link function
"""
p = self._clean(p)
fl = np.log(1 - p)
d2 = -1 / ((1 - p)**2 * fl)
d2 *= 1 + 1 / fl
return d2
def inverse_deriv(self, z):
"""
Derivative of the inverse of the C-Log-Log transform link function
Parameters
----------
z : array_like
The value of the inverse of the CLogLog link function at `p`
Returns
-------
g^(-1)'(z) : array
The derivative of the inverse of the CLogLog link function
"""
return np.exp(z - np.exp(z))
[docs]class cloglog(CLogLog):
"""
The CLogLog transform link function.
Notes
-----
g(`p`) = log(-log(1-`p`))
cloglog is an alias for CLogLog
cloglog = CLogLog()
"""
pass
[docs]class NegativeBinomial(Link):
'''
The negative binomial link function
Parameters
----------
alpha : float, optional
Alpha is the ancillary parameter of the Negative Binomial link
function. It is assumed to be nonstochastic. The default value is 1.
Permissible values are usually assumed to be in (.01, 2).
'''
def __init__(self, alpha=1.):
self.alpha = alpha
def _clean(self, x):
return np.clip(x, FLOAT_EPS, np.inf)
def __call__(self, p):
'''
Negative Binomial transform link function
Parameters
----------
p : array_like
Mean parameters
Returns
-------
z : array
The negative binomial transform of `p`
Notes
-----
g(p) = log(p/(p + 1/alpha))
'''
p = self._clean(p)
return np.log(p/(p + 1/self.alpha))
def inverse(self, z):
'''
Inverse of the negative binomial transform
Parameters
----------
z : array_like
The value of the inverse of the negative binomial link at `p`.
Returns
-------
p : array
Mean parameters
Notes
-----
g^(-1)(z) = exp(z)/(alpha*(1-exp(z)))
'''
return -1/(self.alpha * (1 - np.exp(-z)))
def deriv(self, p):
'''
Derivative of the negative binomial transform
Parameters
----------
p : array_like
Mean parameters
Returns
-------
g'(p) : array
The derivative of the negative binomial transform link function
Notes
-----
g'(x) = 1/(x+alpha*x^2)
'''
return 1/(p + self.alpha * p**2)
def deriv2(self, p):
'''
Second derivative of the negative binomial link function.
Parameters
----------
p : array_like
Mean parameters
Returns
-------
g''(p) : array
The second derivative of the negative binomial transform link
function
Notes
-----
g''(x) = -(1+2*alpha*x)/(x+alpha*x^2)^2
'''
numer = -(1 + 2 * self.alpha * p)
denom = (p + self.alpha * p**2)**2
return numer / denom
def inverse_deriv(self, z):
'''
Derivative of the inverse of the negative binomial transform
Parameters
----------
z : array_like
Usually the linear predictor for a GLM or GEE model
Returns
-------
g^(-1)'(z) : array
The value of the derivative of the inverse of the negative
binomial link
'''
t = np.exp(z)
return t / (self.alpha * (1-t)**2)
[docs]class nbinom(NegativeBinomial):
"""
The negative binomial link function.
Notes
-----
g(p) = log(p/(p + 1/alpha))
nbinom is an alias of NegativeBinomial.
nbinom = NegativeBinomial(alpha=1.)
"""
pass