'''
The one parameter exponential family distributions used by GLM.
'''
# TODO: quasi, quasibinomial, quasipoisson
# see http://www.biostat.jhsph.edu/~qli/biostatistics_r_doc/library/stats/html/family.html
# for comparison to R, and McCullagh and Nelder
import numpy as np
from scipy import special
from . import links as L
from . import varfuncs as V
FLOAT_EPS = np.finfo(float).eps
[docs]class Family(object):
"""
The parent class for one-parameter exponential families.
Parameters
----------
link : a link function instance
Link is the linear transformation function.
See the individual families for available links.
variance : a variance function
Measures the variance as a function of the mean probabilities.
See the individual families for the default variance function.
See Also
--------
:ref:`links`
"""
# TODO: change these class attributes, use valid somewhere...
valid = [-np.inf, np.inf]
links = []
def _setlink(self, link):
"""
Helper method to set the link for a family.
Raises a ValueError exception if the link is not available. Note that
the error message might not be that informative because it tells you
that the link should be in the base class for the link function.
See glm.GLM for a list of appropriate links for each family but note
that not all of these are currently available.
"""
# TODO: change the links class attribute in the families to hold
# meaningful information instead of a list of links instances such as
# [<statsmodels.family.links.Log object at 0x9a4240c>,
# <statsmodels.family.links.Power object at 0x9a423ec>,
# <statsmodels.family.links.Power object at 0x9a4236c>]
# for Poisson...
self._link = link
if not isinstance(link, L.Link):
raise TypeError("The input should be a valid Link object.")
if hasattr(self, "links"):
validlink = link in self.links
validlink = max([isinstance(link, _) for _ in self.links])
if not validlink:
errmsg = "Invalid link for family, should be in %s. (got %s)"
raise ValueError(errmsg % (repr(self.links), link))
def _getlink(self):
"""
Helper method to get the link for a family.
"""
return self._link
# link property for each family is a pointer to link instance
link = property(_getlink, _setlink, doc="Link function for family")
def __init__(self, link, variance):
self.link = link()
self.variance = variance
[docs] def starting_mu(self, y):
r"""
Starting value for mu in the IRLS algorithm.
Parameters
----------
y : array
The untransformed response variable.
Returns
-------
mu_0 : array
The first guess on the transformed response variable.
Notes
-----
.. math::
\mu_0 = (Y + \overline{Y})/2
Notes
-----
Only the Binomial family takes a different initial value.
"""
return (y + y.mean())/2.
[docs] def weights(self, mu):
r"""
Weights for IRLS steps
Parameters
----------
mu : array-like
The transformed mean response variable in the exponential family
Returns
-------
w : array
The weights for the IRLS steps
Notes
-----
.. math::
w = 1 / (g'(\mu)^2 * Var(\mu))
"""
return 1. / (self.link.deriv(mu)**2 * self.variance(mu))
[docs] def deviance(self, endog, mu, freq_weights=1., scale=1.):
r"""
The deviance function evaluated at (endog,mu,freq_weights,mu).
Deviance is usually defined as twice the loglikelihood ratio.
Parameters
----------
endog : array-like
The endogenous response variable
mu : array-like
The inverse of the link function at the linear predicted values.
freq_weights : array-like
1d array of frequency weights. The default is 1.
scale : float, optional
An optional scale argument. The default is 1.
Returns
-------
Deviance : array
The value of deviance function defined below.
Notes
-----
Deviance is defined
.. math::
D = \sum_i (2 * freq\_weights_i * llf(Y_i, Y_i) - 2 *
llf(Y_i, \mu_i)) / scale
where y is the endogenous variable. The deviance functions are
analytically defined for each family.
"""
raise NotImplementedError
[docs] def resid_dev(self, endog, mu, freq_weights=1., scale=1.):
"""
The deviance residuals
Parameters
----------
endog : array
The endogenous response variable
mu : array
The inverse of the link function at the linear predicted values.
freq_weights : array-like
1d array of frequency weights. The default is 1.
scale : float, optional
An optional argument to divide the residuals by scale. The default
is 1.
Returns
-------
Deviance residuals.
Notes
-----
The deviance residuals are defined for each family.
"""
raise NotImplementedError
[docs] def fitted(self, lin_pred):
"""
Fitted values based on linear predictors lin_pred.
Parameters
-----------
lin_pred : array
Values of the linear predictor of the model.
dot(X,beta) in a classical linear model.
Returns
--------
mu : array
The mean response variables given by the inverse of the link
function.
"""
fits = self.link.inverse(lin_pred)
return fits
[docs] def predict(self, mu):
"""
Linear predictors based on given mu values.
Parameters
----------
mu : array
The mean response variables
Returns
-------
lin_pred : array
Linear predictors based on the mean response variables. The value
of the link function at the given mu.
"""
return self.link(mu)
[docs] def loglike(self, endog, mu, freq_weights=1., scale=1.):
"""
The log-likelihood function in terms of the fitted mean response.
Parameters
----------
endog : array
Usually the endogenous response variable.
mu : array
Usually but not always the fitted mean response variable.
freq_weights : array-like
1d array of frequency weights. The default is 1.
scale : float
The scale parameter. The default is 1.
Returns
-------
llf : float
The value of the loglikelihood evaluated at
(endog,mu,freq_weights,scale) as defined below.
Notes
-----
This is defined for each family. endog and mu are not restricted to
`endog` and `mu` respectively. For instance, the deviance function
calls both loglike(endog,endog) and loglike(endog,mu) to get the
likelihood ratio.
"""
raise NotImplementedError
[docs] def resid_anscombe(self, endog, mu):
"""
The Anscombe residuals
See Also
--------
statsmodels.genmod.families.family.Family : `resid_anscombe` for the
individual families for more information
"""
raise NotImplementedError
[docs]class Poisson(Family):
"""
Poisson exponential family.
Parameters
----------
link : a link instance, optional
The default link for the Poisson family is the log link. Available
links are log, identity, and sqrt. See statsmodels.family.links for
more information.
Attributes
----------
Poisson.link : a link instance
The link function of the Poisson instance.
Poisson.variance : varfuncs instance
`variance` is an instance of
statsmodels.genmod.families.family.varfuncs.mu
See also
--------
statsmodels.genmod.families.family.Family
:ref:`links`
"""
links = [L.log, L.identity, L.sqrt]
variance = V.mu
valid = [0, np.inf]
safe_links = [L.Log, ]
def __init__(self, link=L.log):
self.variance = Poisson.variance
self.link = link()
def _clean(self, x):
"""
Helper function to trim the data so that is in (0,inf)
Notes
-----
The need for this function was discovered through usage and its
possible that other families might need a check for validity of the
domain.
"""
return np.clip(x, FLOAT_EPS, np.inf)
[docs] def resid_dev(self, endog, mu, scale=1.):
r"""Poisson deviance residual
Parameters
----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
scale : float, optional
An optional argument to divide the residuals by scale. The default
is 1.
Returns
-------
resid_dev : array
Deviance residuals as defined below
Notes
-----
.. math::
resid\_dev_i = sign(Y_i - \mu_i) * \sqrt{2 *
(Y_i * \log(Y_i / \mu_i) - (Y_i - \mu_i))} / scale
"""
endog_mu = self._clean(endog / mu)
return (np.sign(endog - mu) *
np.sqrt(2 * (endog * np.log(endog_mu) - (endog - mu))) / scale)
[docs] def deviance(self, endog, mu, freq_weights=1., scale=1.):
r'''
Poisson deviance function
Parameters
----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
freq_weights : array-like
1d array of frequency weights. The default is 1.
scale : float, optional
An optional scale argument. The default is 1.
Returns
-------
deviance : float
The deviance function at (endog,mu,freq_weights,scale) as defined
below.
Notes
-----
If a constant term is included it is defined as
.. math::
D = 2 * \sum_i (freq\_weights_i * Y_i * \log(Y_i / \mu_i))/ scale
'''
endog_mu = self._clean(endog / mu)
return 2 * np.sum(freq_weights * (endog * np.log(endog_mu) -
(endog - mu))) / scale
[docs] def loglike(self, endog, mu, freq_weights=1., scale=1.):
r"""
The log-likelihood function in terms of the fitted mean response.
Parameters
----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
freq_weights : array-like
1d array of frequency weights. The default is 1.
scale : float, optional
The scale parameter, defaults to 1.
Returns
-------
llf : float
The value of the loglikelihood function evaluated at
(endog,mu,freq_weights,scale) as defined below.
Notes
-----
.. math::
llf = scale * \sum_i freq\_weights_i * (Y_i * \log(\mu_i) - \mu_i -
\ln \Gamma(Y_i + 1))
"""
loglike = np.sum(freq_weights * (endog * np.log(mu) - mu -
special.gammaln(endog + 1)))
return scale * loglike
[docs] def resid_anscombe(self, endog, mu):
r"""
Anscombe residuals for the Poisson exponential family distribution
Parameters
----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
Returns
-------
resid_anscombe : array
The Anscome residuals for the Poisson family defined below
Notes
-----
.. math::
resid\_anscombe_i = (3/2) * (Y_i^{2/3} - \mu_i^{2/3}) / \mu_i^{1/6}
"""
return (3 / 2.) * (endog**(2/3.) - mu**(2 / 3.)) / mu**(1 / 6.)
[docs]class Gaussian(Family):
"""
Gaussian exponential family distribution.
Parameters
----------
link : a link instance, optional
The default link for the Gaussian family is the identity link.
Available links are log, identity, and inverse.
See statsmodels.family.links for more information.
Attributes
----------
Gaussian.link : a link instance
The link function of the Gaussian instance
Gaussian.variance : varfunc instance
`variance` is an instance of statsmodels.family.varfuncs.constant
See also
--------
statsmodels.genmod.families.family.Family
:ref:`links`
"""
links = [L.log, L.identity, L.inverse_power]
variance = V.constant
safe_links = links
def __init__(self, link=L.identity):
self.variance = Gaussian.variance
self.link = link()
[docs] def resid_dev(self, endog, mu, scale=1.):
r"""
Gaussian deviance residuals
Parameters
-----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
scale : float, optional
An optional argument to divide the residuals by scale. The default
is 1.
Returns
-------
resid_dev : array
Deviance residuals as defined below
Notes
--------
.. math::
resid\_dev_i = (Y_i - \mu_i) / \sqrt{Var(\mu_i)} / scale
"""
return (endog - mu) / np.sqrt(self.variance(mu)) / scale
[docs] def deviance(self, endog, mu, freq_weights=1., scale=1.):
r"""
Gaussian deviance function
Parameters
----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
freq_weights : array-like
1d array of frequency weights. The default is 1.
scale : float, optional
An optional scale argument. The default is 1.
Returns
-------
deviance : float
The deviance function at (endog,mu,freq_weights,scale)
as defined below.
Notes
--------
.. math::
D = \sum_i freq\_weights_i * (Y_i - \mu_i)^2 / scale
"""
return np.sum((freq_weights * (endog - mu)**2)) / scale
[docs] def loglike(self, endog, mu, freq_weights=1., scale=1.):
r"""
The log-likelihood in terms of the fitted mean response.
Parameters
----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
freq_weights : array-like
1d array of frequency weights. The default is 1.
scale : float, optional
Scales the loglikelihood function. The default is 1.
Returns
-------
llf : float
The value of the loglikelihood function evaluated at
(endog,mu,freq_weights,scale) as defined below.
Notes
-----
If the link is the identity link function then the
loglikelihood function is the same as the classical OLS model.
.. math::
llf = -nobs / 2 * (\log(SSR) + (1 + \log(2 \pi / nobs)))
where
.. math::
SSR = \sum_i (Y_i - g^{-1}(\mu_i))^2
If the links is not the identity link then the loglikelihood
function is defined as
.. math::
llf = \sum_i freq\_weights_i * ((Y_i * \mu_i - \mu_i^2 / 2) / scale-
Y^2 / (2 * scale) - (1/2) * \log(2 * \pi * scale))
"""
if isinstance(self.link, L.Power) and self.link.power == 1:
# This is just the loglikelihood for classical OLS
nobs2 = np.sum(freq_weights, axis=0) / 2.
SSR = np.sum((endog-self.fitted(mu))**2, axis=0)
llf = -np.log(SSR) * nobs2
llf -= (1+np.log(np.pi/nobs2))*nobs2
return llf
else:
return np.sum(freq_weights * ((endog * mu - mu**2/2)/scale -
endog**2/(2 * scale) - .5*np.log(2 * np.pi * scale)))
[docs] def resid_anscombe(self, endog, mu):
r"""
The Anscombe residuals for the Gaussian exponential family distribution
Parameters
----------
endog : array
Endogenous response variable
mu : array
Fitted mean response variable
Returns
-------
resid_anscombe : array
The Anscombe residuals for the Gaussian family defined below
Notes
--------
.. math::
resid\_anscombe_i = Y_i - \mu_i
"""
return endog - mu
[docs]class Gamma(Family):
"""
Gamma exponential family distribution.
Parameters
----------
link : a link instance, optional
The default link for the Gamma family is the inverse link.
Available links are log, identity, and inverse.
See statsmodels.family.links for more information.
Attributes
----------
Gamma.link : a link instance
The link function of the Gamma instance
Gamma.variance : varfunc instance
`variance` is an instance of statsmodels.family.varfuncs.mu_squared
See also
--------
statsmodels.genmod.families.family.Family
:ref:`links`
"""
links = [L.log, L.identity, L.inverse_power]
variance = V.mu_squared
safe_links = [L.Log, ]
def __init__(self, link=L.inverse_power):
self.variance = Gamma.variance
self.link = link()
def _clean(self, x):
"""
Helper function to trim the data so that is in (0,inf)
Notes
-----
The need for this function was discovered through usage and its
possible that other families might need a check for validity of the
domain.
"""
return np.clip(x, FLOAT_EPS, np.inf)
[docs] def deviance(self, endog, mu, freq_weights=1., scale=1.):
r"""
Gamma deviance function
Parameters
-----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
freq_weights : array-like
1d array of frequency weights. The default is 1.
scale : float, optional
An optional scale argument. The default is 1.
Returns
-------
deviance : float
Deviance function as defined below
Notes
-----
.. math::
D = 2 * \sum_i freq\_weights_i * ((Y_i - \mu_i)/\mu_i - \log(Y_i /
\mu_i))
"""
endog_mu = self._clean(endog/mu)
return 2*np.sum(freq_weights*((endog-mu)/mu-np.log(endog_mu)))
[docs] def resid_dev(self, endog, mu, scale=1.):
r"""
Gamma deviance residuals
Parameters
-----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
scale : float, optional
An optional argument to divide the residuals by scale. The default
is 1.
Returns
-------
resid_dev : array
Deviance residuals as defined below
Notes
-----
.. math::
resid\_dev_i = sign(Y_i - \mu_i) \sqrt{-2 *
(-(Y_i - \mu_i) / \mu_i + \log(Y_i / \mu_i))}
"""
endog_mu = self._clean(endog / mu)
return np.sign(endog - mu) * np.sqrt(-2 * (-(endog - mu)/mu +
np.log(endog_mu)))
[docs] def loglike(self, endog, mu, freq_weights=1., scale=1.):
r"""
The log-likelihood function in terms of the fitted mean response.
Parameters
----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
freq_weights : array-like
1d array of frequency weights. The default is 1.
scale : float, optional
The default is 1.
Returns
-------
llf : float
The value of the loglikelihood function evaluated at
(endog,mu,freq_weights,scale) as defined below.
Notes
--------
.. math::
llf = -1 / scale * \sum_i *(Y_i / \mu_i+ \log(\mu_i)+
(scale -1) * \log(Y) + \log(scale) + scale *
\ln \Gamma(1 / scale))
"""
endog_mu = self._clean(endog / mu)
return - np.sum((endog_mu - np.log(endog_mu) + scale *
np.log(endog) + np.log(scale) + scale *
special.gammaln(1./scale)) * freq_weights) / scale
# in Stata scale is set to equal 1 for reporting llf
# in R it's the dispersion, though there is a loss of precision vs.
# our results due to an assumed difference in implementation
[docs] def resid_anscombe(self, endog, mu):
r"""
The Anscombe residuals for Gamma exponential family distribution
Parameters
----------
endog : array
Endogenous response variable
mu : array
Fitted mean response variable
Returns
-------
resid_anscombe : array
The Anscombe residuals for the Gamma family defined below
Notes
-----
.. math::
resid\_anscombe_i = 3 * (Y_i^{1/3} - \mu_i^{1/3}) / \mu_i^{1/3}
"""
return 3 * (endog**(1/3.) - mu**(1/3.)) / mu**(1/3.)
[docs]class Binomial(Family):
"""
Binomial exponential family distribution.
Parameters
----------
link : a link instance, optional
The default link for the Binomial family is the logit link.
Available links are logit, probit, cauchy, log, and cloglog.
See statsmodels.family.links for more information.
Attributes
----------
Binomial.link : a link instance
The link function of the Binomial instance
Binomial.variance : varfunc instance
`variance` is an instance of statsmodels.family.varfuncs.binary
See also
--------
statsmodels.genmod.families.family.Family
:ref:`links`
Notes
-----
endog for Binomial can be specified in one of three ways.
"""
links = [L.logit, L.probit, L.cauchy, L.log, L.cloglog, L.identity]
variance = V.binary # this is not used below in an effort to include n
# Other safe links, e.g. cloglog and probit are subclasses
safe_links = [L.Logit, L.CDFLink]
def __init__(self, link=L.logit): # , n=1.):
# TODO: it *should* work for a constant n>1 actually, if freq_weights
# is equal to n
self.n = 1
# overwritten by initialize if needed but always used to initialize
# variance since endog is assumed/forced to be (0,1)
self.variance = V.Binomial(n=self.n)
self.link = link()
[docs] def starting_mu(self, y):
"""
The starting values for the IRLS algorithm for the Binomial family.
A good choice for the binomial family is :math:`\mu_0 = (Y_i + 0.5)/2`
"""
return (y + .5)/2
[docs] def initialize(self, endog, freq_weights):
'''
Initialize the response variable.
Parameters
----------
endog : array
Endogenous response variable
Returns
--------
If `endog` is binary, returns `endog`
If `endog` is a 2d array, then the input is assumed to be in the format
(successes, failures) and
successes/(success + failures) is returned. And n is set to
successes + failures.
'''
# if not np.all(np.asarray(freq_weights) == 1):
# self.variance = V.Binomial(n=freq_weights)
if (endog.ndim > 1 and endog.shape[1] > 1):
y = endog[:, 0]
# overwrite self.freq_weights for deviance below
self.n = endog.sum(1)
return y*1./self.n, self.n
else:
return endog, np.ones(endog.shape[0])
[docs] def deviance(self, endog, mu, freq_weights=1, scale=1.):
r'''
Deviance function for either Bernoulli or Binomial data.
Parameters
----------
endog : array-like
Endogenous response variable (already transformed to a probability
if appropriate).
mu : array
Fitted mean response variable
freq_weights : array-like
1d array of frequency weights. The default is 1.
scale : float, optional
An optional scale argument. The default is 1.
Returns
--------
deviance : float
The deviance function as defined below
Notes
-----
If the endogenous variable is binary:
.. math::
D = -2 * \sum_i freq\_weights * (I_{1,i} * \log(\mu_i) + I_{0,i} *
\log(1 - \mu_i))
where :math:`I_{1,i}` is an indicator function that evalueates to 1 if
:math:`Y_i = 1`. and :math:`I_{0,i}` is an indicator function that
evaluates to 1 if :math:`Y_i = 0`.
If the model is ninomial:
.. math::
D = 2 * \sum_i freq\_weights * (\log(Y_i / \mu_i) + (n_i - Y_i) *
\log((n_i - Y_i) / n_i - \mu_i))
where :math:`Y_i` and :math:`n` are as defined in Binomial.initialize.
'''
if np.shape(self.n) == () and self.n == 1:
one = np.equal(endog, 1)
return -2 * np.sum((one * np.log(mu + 1e-200) + (1-one) *
np.log(1 - mu + 1e-200)) * freq_weights)
else:
return 2 * np.sum(self.n * freq_weights *
(endog * np.log(endog/mu + 1e-200) +
(1 - endog) * np.log((1 - endog) /
(1 - mu) + 1e-200)))
[docs] def resid_dev(self, endog, mu, scale=1.):
r"""
Binomial deviance residuals
Parameters
-----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
scale : float, optional
An optional argument to divide the residuals by scale. The default
is 1.
Returns
-------
resid_dev : array
Deviance residuals as defined below
Notes
-----
If the endogenous variable is binary:
.. math::
resid\_dev_i = sign(Y_i - \mu_i) * \sqrt{-2 *
\log(I_{1,i} * \mu_i + I_{0,i} * (1 - \mu_i))}
where :math:`I_{1,i}` is an indicator function that evalueates to 1 if
:math:`Y_i = 1`. and :math:`I_{0,i}` is an indicator function that
evaluates to 1 if :math:`Y_i = 0`.
If the endogenous variable is binomial:
.. math::
resid\_dev_i = sign(Y_i - \mu_i) \sqrt{2 * n_i *
(Y_i * \log(Y_i / \mu_i) + (1 - Y_i) *
\log(1 - Y_i)/(1 - \mu_i))}
where :math:`Y_i` and :math:`n` are as defined in Binomial.initialize.
"""
mu = self.link._clean(mu)
if np.shape(self.n) == () and self.n == 1:
one = np.equal(endog, 1)
return np.sign(endog-mu)*np.sqrt(-2 *
np.log(one * mu + (1 - one) *
(1 - mu)))/scale
else:
return (np.sign(endog - mu) *
np.sqrt(2 * self.n *
(endog * np.log(endog/mu + 1e-200) +
(1 - endog) * np.log((1 - endog)/(1 - mu) +
1e-200)))/scale)
[docs] def loglike(self, endog, mu, freq_weights=1, scale=1.):
r"""
The log-likelihood function in terms of the fitted mean response.
Parameters
----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
freq_weights : array-like
1d array of frequency weights. The default is 1.
scale : float, optional
Not used for the Binomial GLM.
Returns
-------
llf : float
The value of the loglikelihood function evaluated at
(endog,mu,freq_weights,scale) as defined below.
Notes
--------
If the endogenous variable is binary:
.. math::
llf = scale * \sum_i (y_i * \log(\mu_i/(1-\mu_i)) + \log(1-\mu_i)) *
freq\_weights_i
If the endogenous variable is binomial:
.. math::
llf = scale * \sum_i freq\_weights_i * (\ln \Gamma(n+1) -
\ln \Gamma(y_i + 1) - \ln \Gamma(n_i - y_i +1) + y_i *
\log(\mu_i / (1 - \mu_i)) + n * \log(1 - \mu_i))
where :math:`y_i = Y_i * n_i` with :math:`Y_i` and :math:`n_i` as
defined in Binomial initialize. This simply makes :math:`y_i` the
original number of successes.
"""
if np.shape(self.n) == () and self.n == 1:
return scale * np.sum((endog * np.log(mu/(1 - mu) + 1e-200) +
np.log(1 - mu)) * freq_weights)
else:
y = endog * self.n # convert back to successes
return scale * np.sum((special.gammaln(self.n + 1) -
special.gammaln(y + 1) -
special.gammaln(self.n - y + 1) + y *
np.log(mu/(1 - mu)) + self.n *
np.log(1 - mu)) * freq_weights)
[docs] def resid_anscombe(self, endog, mu):
'''
The Anscombe residuals
Parameters
----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
Returns
-------
resid_anscombe : array
The Anscombe residuals as defined below.
Notes
-----
sqrt(n)*(cox_snell(endog)-cox_snell(mu))/(mu**(1/6.)*(1-mu)**(1/6.))
where cox_snell is defined as
cox_snell(x) = betainc(2/3., 2/3., x)*betainc(2/3.,2/3.)
where betainc is the incomplete beta function
The name 'cox_snell' is idiosyncratic and is simply used for
convenience following the approach suggested in Cox and Snell (1968).
Further note that
cox_snell(x) = x**(2/3.)/(2/3.)*hyp2f1(2/3.,1/3.,5/3.,x)
where hyp2f1 is the hypergeometric 2f1 function. The Anscombe
residuals are sometimes defined in the literature using the
hyp2f1 formulation. Both betainc and hyp2f1 can be found in scipy.
References
----------
Anscombe, FJ. (1953) "Contribution to the discussion of H. Hotelling's
paper." Journal of the Royal Statistical Society B. 15, 229-30.
Cox, DR and Snell, EJ. (1968) "A General Definition of Residuals."
Journal of the Royal Statistical Society B. 30, 248-75.
'''
cox_snell = lambda x: (special.betainc(2/3., 2/3., x)
* special.beta(2/3., 2/3.))
return np.sqrt(self.n) * ((cox_snell(endog) - cox_snell(mu)) /
(mu**(1/6.) * (1 - mu)**(1/6.)))
[docs]class InverseGaussian(Family):
"""
InverseGaussian exponential family.
Parameters
----------
link : a link instance, optional
The default link for the inverse Gaussian family is the
inverse squared link.
Available links are inverse_squared, inverse, log, and identity.
See statsmodels.family.links for more information.
Attributes
----------
InverseGaussian.link : a link instance
The link function of the inverse Gaussian instance
InverseGaussian.variance : varfunc instance
`variance` is an instance of statsmodels.family.varfuncs.mu_cubed
See also
--------
statsmodels.genmod.families.family.Family
:ref:`links`
Notes
-----
The inverse Guassian distribution is sometimes referred to in the
literature as the Wald distribution.
"""
links = [L.inverse_squared, L.inverse_power, L.identity, L.log]
variance = V.mu_cubed
safe_links = [L.inverse_squared, L.Log, ]
def __init__(self, link=L.inverse_squared):
self.variance = InverseGaussian.variance
self.link = link()
[docs] def resid_dev(self, endog, mu, scale=1.):
r"""
Returns the deviance residuals for the inverse Gaussian family.
Parameters
-----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
freq_weights : array-like
1d array of frequency weights. The default is 1.
scale : float, optional
An optional argument to divide the residuals by scale. The default
is 1.
Returns
-------
resid_dev : array
Deviance residuals as defined below
Notes
-----
.. math::
resid\_dev_i = sign(Y_i - \mu_i) *
\sqrt {(Y_i - \mu_i)^2 / (Y_i * \mu_i^2)} / scale
"""
return np.sign(endog-mu) * np.sqrt((endog-mu)**2/(endog*mu**2))/scale
[docs] def deviance(self, endog, mu, freq_weights=1., scale=1.):
r"""
Inverse Gaussian deviance function
Parameters
-----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
freq_weights : array-like
1d array of frequency weights. The default is 1.
scale : float, optional
An optional scale argument. The default is 1.
Returns
-------
deviance : float
Deviance function as defined below
Notes
-----
.. math::
D = \sum_i freq\_weights_i * ((Y_i - \mu_i)^2 / (Y_i *\mu_i^2)) /
scale
"""
return np.sum(freq_weights*(endog-mu)**2/(endog*mu**2))/scale
[docs] def loglike(self, endog, mu, freq_weights=1., scale=1.):
r"""
The log-likelihood function in terms of the fitted mean response.
Parameters
----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
freq_weights : array-like
1d array of frequency weights. The default is 1.
scale : float, optional
The default is 1.
Returns
-------
llf : float
The value of the loglikelihood function evaluated at
(endog,mu,freq_weights,scale) as defined below.
Notes
-----
.. math::
llf = -1/2 * \sum_i freq\_weights_i * ((Y_i - \mu_i)^2 / (Y_i *
\mu_i * scale) + \log(scale * Y_i^3) + \log(2 * \pi))
"""
return -.5 * np.sum(((endog - mu)**2/(endog * mu**2 * scale) +
np.log(scale * endog**3) + np.log(2 * np.pi)) *
freq_weights)
[docs] def resid_anscombe(self, endog, mu):
r"""
The Anscombe residuals for the inverse Gaussian distribution
Parameters
----------
endog : array
Endogenous response variable
mu : array
Fitted mean response variable
Returns
-------
resid_anscombe : array
The Anscombe residuals for the inverse Gaussian distribution as
defined below
Notes
-----
.. math::
resid\_anscombe_i = \log(Y_i / \mu_i) / \sqrt{\mu_i}
"""
return np.log(endog / mu) / np.sqrt(mu)
[docs]class NegativeBinomial(Family):
r"""
Negative Binomial exponential family.
Parameters
----------
link : a link instance, optional
The default link for the negative binomial family is the log link.
Available links are log, cloglog, identity, nbinom and power.
See statsmodels.family.links for more information.
alpha : float, optional
The ancillary parameter for the negative binomial distribution.
For now `alpha` is assumed to be nonstochastic. The default value
is 1. Permissible values are usually assumed to be between .01 and 2.
Attributes
----------
NegativeBinomial.link : a link instance
The link function of the negative binomial instance
NegativeBinomial.variance : varfunc instance
`variance` is an instance of statsmodels.family.varfuncs.nbinom
See also
--------
statsmodels.genmod.families.family.Family
:ref:`links`
Notes
-----
Power link functions are not yet supported.
Parameterization for :math:`y=0,1,2,\ldots` is
:math:`f(y) = \frac{\Gamma(y+\frac{1}{\alpha})}{y!\Gamma(\frac{1}{\alpha})}
\left(\frac{1}{1+\alpha\mu}\right)^{\frac{1}{\alpha}}
\left(\frac{\alpha\mu}{1+\alpha\mu}\right)^y`
with :math:`E[Y]=\mu\,` and :math:`Var[Y]=\mu+\alpha\mu^2`.
"""
links = [L.log, L.cloglog, L.identity, L.nbinom, L.Power]
# TODO: add the ability to use the power links with an if test
# similar to below
variance = V.nbinom
safe_links = [L.Log, ]
def __init__(self, link=L.log, alpha=1.):
self.alpha = 1. * alpha # make it at least float
self.variance = V.NegativeBinomial(alpha=self.alpha)
if isinstance(link, L.NegativeBinomial):
self.link = link(alpha=self.alpha)
else:
self.link = link()
def _clean(self, x):
"""
Helper function to trim the data so that is in (0,inf)
Notes
-----
The need for this function was discovered through usage and its
possible that other families might need a check for validity of the
domain.
"""
return np.clip(x, FLOAT_EPS, np.inf)
[docs] def deviance(self, endog, mu, freq_weights=1., scale=1.):
r"""
Returns the value of the deviance function.
Parameters
-----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
freq_weights : array-like
1d array of frequency weights. The default is 1.
scale : float, optional
An optional scale argument. The default is 1.
Returns
-------
deviance : float
Deviance function as defined below
Notes
-----
:math:`D = \sum_i piecewise_i` where :math:`piecewise_i` is defined as:
If :math:`Y_{i} = 0`:
:math:`piecewise_i = 2* \log(1 + \alpha * \mu_i) / \alpha`
If :math:`Y_{i} > 0`:
.. math:
piecewise_i = 2 * Y_i * \log(Y_i / \mu_i) - (2 / \alpha) *
(1 + \alpha * Y_i) * \ln(1 + \alpha * Y_i) / (1 + \alpha * \mu_i)
"""
iszero = np.equal(endog, 0)
notzero = 1 - iszero
endog_mu = self._clean(endog/mu)
tmp = iszero * 2 * np.log(1 + self.alpha * mu) / self.alpha
tmp += notzero * (2 * endog * np.log(endog_mu) - 2 / self.alpha *
(1 + self.alpha * endog) *
np.log((1 + self.alpha * endog) /
(1 + self.alpha * mu)))
return np.sum(freq_weights * tmp) / scale
[docs] def resid_dev(self, endog, mu, scale=1.):
r"""
Negative Binomial Deviance Residual
Parameters
----------
endog : array-like
`endog` is the response variable
mu : array-like
`mu` is the fitted value of the model
scale : float, optional
An optional argument to divide the residuals by scale. The default
is 1.
Returns
--------
resid_dev : array
The array of deviance residuals
Notes
-----
:math:`resid\_dev_i = sign(Y_i-\mu_i) * \sqrt{piecewise_i}`
where :math:`piecewise_i` is defined as
If :math:`Y_i = 0`:
:math:`piecewise_i = 2 * \log(1 + \alpha * \mu_i)/ \alpha`
If :math:`Y_i > 0`:
:math:`piecewise_i = 2 * Y_i * \log(Y_i / \mu_i) - (2 / \alpha) *
(1 + \alpha * Y_i) * \log((1 + \alpha * Y_i) / (1 + \alpha * \mu_i))`
"""
iszero = np.equal(endog, 0)
notzero = 1 - iszero
endog_mu = self._clean(endog / mu)
tmp = iszero * 2 * np.log(1 + self.alpha * mu) / self.alpha
tmp += notzero * (2 * endog * np.log(endog_mu) - 2 / self.alpha *
(1 + self.alpha * endog) *
np.log((1 + self.alpha * endog) /
(1 + self.alpha * mu)))
return np.sign(endog - mu) * np.sqrt(tmp) / scale
[docs] def loglike(self, endog, mu, freq_weights=1., scale=1.):
r"""
The log-likelihood function in terms of the fitted mean response.
Parameters
----------
endog : array-like
Endogenous response variable
mu : array-like
The fitted mean response values
freq_weights : array-like
1d array of frequency weights. The default is 1.
scale : float
The scale parameter. The default is 1.
Returns
-------
llf : float
The value of the loglikelihood function evaluated at
(endog,mu,freq_weights,scale) as defined below.
Notes
-----
Defined as:
.. math::
llf = \sum_i freq\_weights_i * (Y_i * \log{(\alpha * e^{\eta_i} /
(1 + \alpha * e^{\eta_i}))} - \log{(1 + \alpha * e^{\eta_i})}/
\alpha + Constant)
where :math:`Constant` is defined as:
.. math::
Constant = \ln \Gamma{(Y_i + 1/ \alpha )} - \ln \Gamma(Y_i + 1) -
\ln \Gamma{(1/ \alpha )}
"""
lin_pred = self._link(mu)
constant = (special.gammaln(endog + 1 / self.alpha) -
special.gammaln(endog+1)-special.gammaln(1/self.alpha))
exp_lin_pred = np.exp(lin_pred)
return np.sum((endog * np.log(self.alpha * exp_lin_pred /
(1 + self.alpha * exp_lin_pred)) -
np.log(1 + self.alpha * exp_lin_pred) /
self.alpha + constant) * freq_weights)
[docs] def resid_anscombe(self, endog, mu):
"""
The Anscombe residuals for the negative binomial family
Parameters
----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
Returns
-------
resid_anscombe : array
The Anscombe residuals as defined below.
Notes
-----
`resid_anscombe` = (hyp2f1(-alpha*endog)-hyp2f1(-alpha*mu)+\
1.5*(endog**(2/3.)-mu**(2/3.)))/(mu+alpha*mu**2)**(1/6.)
where hyp2f1 is the hypergeometric 2f1 function parameterized as
hyp2f1(x) = hyp2f1(2/3.,1/3.,5/3.,x)
"""
hyp2f1 = lambda x: special.hyp2f1(2 / 3., 1 / 3., 5 / 3., x)
return ((hyp2f1(-self.alpha * endog) - hyp2f1(-self.alpha * mu) +
1.5 * (endog**(2 / 3.) - mu**(2 / 3.))) /
(mu + self.alpha * mu**2)**(1 / 6.))
[docs]class Tweedie(Family):
"""
Tweedie family.
Parameters
----------
link : a link instance, optional
The default link for the Tweedie family is the log link when the
link_power is 0. Otherwise, the power link is default.
Available links are log and Power.
var_power : float, optional
The variance power.
link_power : float, optional
The link power.
Attributes
----------
Tweedie.link : a link instance
The link function of the Tweedie instance
Tweedie.variance : varfunc instance
`variance` is an instance of statsmodels.family.varfuncs.Power
Tweedie.link_power : float
The power of the link function, or 0 if its a log link.
Tweedie.var_power : float
The power of the variance function.
See also
--------
statsmodels.genmod.families.family.Family
:ref:`links`
Notes
-----
Logliklihood function not implemented because of the complexity of
calculating an infinite series of summations. The variance power can be
estimated using the `estimate_tweedie_power` function that is part of the
`GLM` class.
"""
links = [L.log, L.Power]
variance = V.Power
safe_links = [L.log, L.Power]
def __init__(self, link=None, var_power=1., link_power=0):
self.var_power = var_power
self.link_power = link_power
self.variance = V.Power(power=var_power * 1.)
if link_power != 0 and not ((link is L.Power) or (link is None)):
msg = 'link_power of {} not supported specified link'
msg = msg.format(link_power)
raise ValueError(msg)
if (link_power == 0) and ((link is None) or (link is L.Log)):
self.link = L.log()
elif link_power != 0:
self.link = L.Power(power=link_power * 1.)
else:
self.link = link()
def _clean(self, x):
"""
Helper function to trim the data so that is in (0,inf)
Notes
-----
The need for this function was discovered through usage and its
possible that other families might need a check for validity of the
domain.
"""
return np.clip(x, 0, np.inf)
[docs] def deviance(self, endog, mu, freq_weights=1., scale=1.):
r"""
Returns the value of the deviance function.
Parameters
-----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
freq_weights : array-like
1d array of frequency weights. The default is 1.
scale : float, optional
An optional scale argument. The default is 1.
Returns
-------
deviance : float
Deviance function as defined below
Notes
-----
When :math:`p = 1`,
.. math::
resid\_dev_i = \mu
when :math:`endog = 0` and
.. math::
resid\_dev_i = endog * \log(endog / \mu) + (\mu - endog)
otherwise.
When :math:`p = 2`,
.. math::
resid\_dev_i = (endog - \mu) / \mu - \log(endog / \mu)
For all other p,
.. math::
resid\_dev_i = endog ^{2 - p} / ((1 - p) * (2 - p)) -
endog * \mu ^{1 - p} / (1 - p) + \mu ^{2 - p} /
(2 - p)
Once :math:`resid\_dev_i` is calculated, then calculate deviance as
.. math::
D = \sum{2 * freq\_weights * resid\_dev_i}
"""
p = self.var_power
if p == 1:
dev = np.where(endog == 0,
mu,
endog * np.log(endog / mu) + (mu - endog))
elif p == 2:
endog1 = np.clip(endog, FLOAT_EPS, np.inf)
dev = ((endog - mu) / mu) - np.log(endog1 / mu)
else:
dev = (endog ** (2 - p) / ((1 - p) * (2 - p)) -
endog * mu ** (1-p) / (1 - p) + mu ** (2 - p) / (2 - p))
return np.sum(2 * freq_weights * dev)
[docs] def resid_dev(self, endog, mu, scale=1.):
r"""
Tweedie Deviance Residual
Parameters
----------
endog : array-like
`endog` is the response variable
mu : array-like
`mu` is the fitted value of the model
scale : float, optional
An optional argument to divide the residuals by scale. The default
is 1.
Returns
--------
resid_dev : array
The array of deviance residuals
Notes
-----
When :math:`p = 1`,
.. math::
resid\_dev_i = \mu
when :math:`endog = 0` and
.. math::
resid\_dev_i = endog * \log(endog / \mu) + (\mu - endog)
otherwise.
When :math:`p = 2`,
.. math::
resid\_dev_i = (endog - \mu) / \mu - \log(endog / \mu)
For all other p,
.. math::
resid\_dev_i = endog ^{2 - p} / ((1 - p) * (2 - p)) -
endog * \mu ^{1 - p} / (1 - p) + \mu ^{2 - p} /
(2 - p)
"""
p = self.var_power
if p == 1:
dev = np.where(endog == 0,
mu,
endog * np.log(endog / mu) + (mu - endog))
elif p == 2:
endog1 = np.clip(endog, FLOAT_EPS, np.inf)
dev = ((endog - mu) / mu) - np.log(endog1 / mu)
else:
dev = (endog ** (2 - p) / ((1 - p) * (2 - p)) -
endog * mu ** (1-p) / (1 - p) + mu ** (2 - p) / (2 - p))
return np.sign(endog - mu) * np.sqrt(2 * dev)
[docs] def loglike(self, endog, mu, freq_weights=1., scale=1.):
r"""
The log-likelihood function in terms of the fitted mean response.
Parameters
----------
endog : array-like
Endogenous response variable
mu : array-like
The fitted mean response values
freq_weights : array-like
1d array of frequency weights. The default is 1.
scale : float
The scale parameter. The default is 1.
Returns
-------
llf : float
The value of the loglikelihood function evaluated at
(endog,mu,freq_weights,scale) as defined below.
Notes
-----
This is not implemented because of the complexity of calculating an
infinite series of sums.
"""
return np.nan
[docs] def resid_anscombe(self, endog, mu):
"""
The Anscombe residuals for the Tweedie family
Parameters
----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
Returns
-------
resid_anscombe : array
The Anscombe residuals as defined below.
Notes
-----
When :math:`p = 3`, then
.. math::
resid\_anscombe_i = (\log(endog) - \log(\mu)) / \sqrt{mu}
Otherwise,
.. math::
c = (3 - p) / 3
.. math::
resid\_anscombe_i = (1 / c) * (endog ^ c - \mu ^ c) / \mu ^{p / 6}
"""
if self.var_power == 3:
return (np.log(endog) - np.log(mu)) / np.sqrt(mu)
else:
c = (3. - self.var_power) / 3.
return ((1. / c) * (endog ** c - mu ** c) /
mu ** (self.var_power / 6.))
