'''
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 warnings
import inspect
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` : Further details on 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 statsmodels.genmod.generalized_linear_model.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 = 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):
if inspect.isclass(link):
warnmssg = "Calling Family(..) with a link class as argument "
warnmssg += "is deprecated.\n"
warnmssg += "Use an instance of a link class instead."
lvl = 2 if type(self) is Family else 3
warnings.warn(warnmssg,
category=DeprecationWarning, stacklevel=lvl)
self.link = link()
else:
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
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, var_weights=1., freq_weights=1., scale=1.):
r"""
The deviance function evaluated at (endog, mu, var_weights,
freq_weights, scale) for the distribution.
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.
var_weights : array_like
1d array of variance (analytic) weights. The default is 1.
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 = 2\sum_i (freq\_weights_i * var\_weights *
(llf(endog_i, endog_i) - llf(endog_i, \mu_i)))
where y is the endogenous variable. The deviance functions are
analytically defined for each family.
Internally, we calculate deviance as:
.. math::
D = \sum_i freq\_weights_i * var\_weights * resid\_dev_i / scale
"""
resid_dev = self._resid_dev(endog, mu)
return np.sum(resid_dev * freq_weights * var_weights / scale)
[docs] def resid_dev(self, endog, mu, var_weights=1., scale=1.):
r"""
The deviance residuals
Parameters
----------
endog : array_like
The endogenous response variable
mu : array_like
The inverse of the link function at the linear predicted values.
var_weights : array_like
1d array of variance (analytic) weights. The default is 1.
scale : float, optional
An optional scale argument. The default is 1.
Returns
-------
resid_dev : float
Deviance residuals as defined below.
Notes
-----
The deviance residuals are defined by the contribution D_i of
observation i to the deviance as
.. math::
resid\_dev_i = sign(y_i-\mu_i) \sqrt{D_i}
D_i is calculated from the _resid_dev method in each family.
Distribution-specific documentation of the calculation is available
there.
"""
resid_dev = self._resid_dev(endog, mu)
resid_dev *= var_weights / scale
return np.sign(endog - mu) * np.sqrt(np.clip(resid_dev, 0., np.inf))
[docs] def fitted(self, lin_pred):
r"""
Fitted values based on linear predictors lin_pred.
Parameters
----------
lin_pred : array
Values of the linear predictor of the model.
:math:`X \cdot \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_obs(self, endog, mu, var_weights=1., scale=1.):
r"""
The log-likelihood function for each observation in terms of the fitted
mean response for the distribution.
Parameters
----------
endog : array
Usually the endogenous response variable.
mu : array
Usually but not always the fitted mean response variable.
var_weights : array_like
1d array of variance (analytic) weights. The default is 1.
scale : float
The scale parameter. The default is 1.
Returns
-------
ll_i : float
The value of the loglikelihood evaluated at
(endog, mu, var_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, you could call
both ``loglike(endog, endog)`` and ``loglike(endog, mu)`` to get the
log-likelihood ratio.
"""
raise NotImplementedError
[docs] def loglike(self, endog, mu, var_weights=1., freq_weights=1., scale=1.):
r"""
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.
var_weights : array_like
1d array of variance (analytic) weights. The default is 1.
freq_weights : array_like
1d array of frequency weights. The default is 1.
scale : float
The scale parameter. The default is 1.
Returns
-------
ll : float
The value of the loglikelihood evaluated at
(endog, mu, var_weights, freq_weights, scale) as defined below.
Notes
-----
Where :math:`ll_i` is the by-observation log-likelihood:
.. math::
ll = \sum(ll_i * freq\_weights_i)
``ll_i`` is defined for each family. endog and mu are not restricted
to ``endog`` and ``mu`` respectively. For instance, you could call
both ``loglike(endog, endog)`` and ``loglike(endog, mu)`` to get the
log-likelihood ratio.
"""
ll_obs = self.loglike_obs(endog, mu, var_weights, scale)
return np.sum(ll_obs * freq_weights)
[docs] def resid_anscombe(self, endog, mu, var_weights=1., scale=1.):
r"""
The Anscombe residuals
Parameters
----------
endog : array
The endogenous response variable
mu : array
The inverse of the link function at the linear predicted values.
var_weights : array_like
1d array of variance (analytic) weights. The default is 1.
scale : float, optional
An optional argument to divide the residuals by sqrt(scale).
The default is 1.
See Also
--------
statsmodels.genmod.families.family.Family : `resid_anscombe` for the
individual families for more information
Notes
-----
Anscombe residuals are defined by
.. math::
resid\_anscombe_i = \frac{A(y)-A(\mu)}{A'(\mu)\sqrt{Var[\mu]}} *
\sqrt(var\_weights)
where :math:`A'(y)=v(y)^{-\frac{1}{3}}` and :math:`v(\mu)` is the
variance function :math:`Var[y]=\frac{\phi}{w}v(mu)`.
The transformation :math:`A(y)` makes the residuals more normal
distributed.
"""
raise NotImplementedError
def _clean(self, x):
"""
Helper function to trim the data so that it 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]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.families.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.varfuncs.mu
See Also
--------
statsmodels.genmod.families.family.Family : Parent class for all links.
:ref:`links` : Further details on links.
"""
links = [L.log, L.identity, L.sqrt]
variance = V.mu
valid = [0, np.inf]
safe_links = [L.Log, ]
def __init__(self, link=None):
if link is None:
link = L.log()
super(Poisson, self).__init__(link=link, variance=Poisson.variance)
def _resid_dev(self, endog, mu):
r"""
Poisson deviance residuals
Parameters
----------
endog : array
The endogenous response variable.
mu : array
The inverse of the link function at the linear predicted values.
Returns
-------
resid_dev : float
Deviance residuals as defined below.
Notes
-----
.. math::
resid\_dev_i = 2 * (endog_i * \ln(endog_i / \mu_i) -
(endog_i - \mu_i))
"""
endog_mu = self._clean(endog / mu)
resid_dev = endog * np.log(endog_mu) - (endog - mu)
return 2 * resid_dev
def loglike_obs(self, endog, mu, var_weights=1., scale=1.):
r"""
The log-likelihood function for each observation in terms of the fitted
mean response for the Poisson distribution.
Parameters
----------
endog : array
Usually the endogenous response variable.
mu : array
Usually but not always the fitted mean response variable.
var_weights : array_like
1d array of variance (analytic) weights. The default is 1.
scale : float
The scale parameter. The default is 1.
Returns
-------
ll_i : float
The value of the loglikelihood evaluated at
(endog, mu, var_weights, scale) as defined below.
Notes
-----
.. math::
ll_i = var\_weights_i / scale * (endog_i * \ln(\mu_i) - \mu_i -
\ln \Gamma(endog_i + 1))
"""
return var_weights / scale * (endog * np.log(mu) - mu -
special.gammaln(endog + 1))
def resid_anscombe(self, endog, mu, var_weights=1., scale=1.):
r"""
The Anscombe residuals
Parameters
----------
endog : array
The endogenous response variable
mu : array
The inverse of the link function at the linear predicted values.
var_weights : array_like
1d array of variance (analytic) weights. The default is 1.
scale : float, optional
An optional argument to divide the residuals by sqrt(scale).
The default is 1.
Returns
-------
resid_anscombe : array
The Anscombe residuals for the Poisson family defined below
Notes
-----
.. math::
resid\_anscombe_i = (3/2) * (endog_i^{2/3} - \mu_i^{2/3}) /
\mu_i^{1/6} * \sqrt(var\_weights)
"""
resid = ((3 / 2.) * (endog**(2 / 3.) - mu**(2 / 3.)) /
(mu ** (1 / 6.) * scale ** 0.5))
resid *= np.sqrt(var_weights)
return resid
[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.genmod.families.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.genmod.families.varfuncs.constant
See Also
--------
statsmodels.genmod.families.family.Family : Parent class for all links.
:ref:`links` : Further details on links.
"""
links = [L.log, L.identity, L.inverse_power]
variance = V.constant
safe_links = links
def __init__(self, link=None):
if link is None:
link = L.identity()
super(Gaussian, self).__init__(link=link, variance=Gaussian.variance)
def _resid_dev(self, endog, mu):
r"""
Gaussian deviance residuals
Parameters
----------
endog : array
The endogenous response variable.
mu : array
The inverse of the link function at the linear predicted values.
Returns
-------
resid_dev : float
Deviance residuals as defined below.
Notes
--------
.. math::
resid\_dev_i = (endog_i - \mu_i) ** 2
"""
return (endog - mu) ** 2
def loglike_obs(self, endog, mu, var_weights=1., scale=1.):
r"""
The log-likelihood function for each observation in terms of the fitted
mean response for the Gaussian distribution.
Parameters
----------
endog : array
Usually the endogenous response variable.
mu : array
Usually but not always the fitted mean response variable.
var_weights : array_like
1d array of variance (analytic) weights. The default is 1.
scale : float
The scale parameter. The default is 1.
Returns
-------
ll_i : float
The value of the loglikelihood evaluated at
(endog, mu, var_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::
ll_i = -1 / 2 \sum_i * var\_weights * ((Y_i - mu_i)^2 / scale +
\log(2 * \pi * scale))
"""
ll_obs = -var_weights * (endog - mu) ** 2 / scale
ll_obs += -np.log(scale / var_weights) - np.log(2 * np.pi)
ll_obs /= 2
return ll_obs
def resid_anscombe(self, endog, mu, var_weights=1., scale=1.):
r"""
The Anscombe residuals
Parameters
----------
endog : array
The endogenous response variable
mu : array
The inverse of the link function at the linear predicted values.
var_weights : array_like
1d array of variance (analytic) weights. The default is 1.
scale : float, optional
An optional argument to divide the residuals by sqrt(scale).
The default is 1.
Returns
-------
resid_anscombe : array
The Anscombe residuals for the Gaussian family defined below
Notes
-----
For the Gaussian distribution, Anscombe residuals are the same as
deviance residuals.
.. math::
resid\_anscombe_i = (Y_i - \mu_i) / \sqrt{scale} *
\sqrt(var\_weights)
"""
resid = (endog - mu) / scale ** 0.5
resid *= np.sqrt(var_weights)
return resid
[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.genmod.families.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.genmod.family.varfuncs.mu_squared
See Also
--------
statsmodels.genmod.families.family.Family : Parent class for all links.
:ref:`links` : Further details on links.
"""
links = [L.log, L.identity, L.inverse_power]
variance = V.mu_squared
safe_links = [L.Log, ]
def __init__(self, link=None):
if link is None:
link = L.inverse_power()
super(Gamma, self).__init__(link=link, variance=Gamma.variance)
def _resid_dev(self, endog, mu):
r"""
Gamma deviance residuals
Parameters
----------
endog : array
The endogenous response variable.
mu : array
The inverse of the link function at the linear predicted values.
Returns
-------
resid_dev : float
Deviance residuals as defined below.
Notes
-----
.. math::
resid\_dev_i = 2 * ((endog_i - \mu_i) / \mu_i -
\log(endog_i / \mu_i))
"""
endog_mu = self._clean(endog / mu)
resid_dev = -np.log(endog_mu) + (endog - mu) / mu
return 2 * resid_dev
def loglike_obs(self, endog, mu, var_weights=1., scale=1.):
r"""
The log-likelihood function for each observation in terms of the fitted
mean response for the Gamma distribution.
Parameters
----------
endog : array
Usually the endogenous response variable.
mu : array
Usually but not always the fitted mean response variable.
var_weights : array_like
1d array of variance (analytic) weights. The default is 1.
scale : float
The scale parameter. The default is 1.
Returns
-------
ll_i : float
The value of the loglikelihood evaluated at
(endog, mu, var_weights, scale) as defined below.
Notes
-----
.. math::
ll_i = var\_weights_i / scale * (\ln(var\_weights_i * endog_i /
(scale * \mu_i)) - (var\_weights_i * endog_i) /
(scale * \mu_i)) - \ln \Gamma(var\_weights_i / scale) - \ln(\mu_i)
"""
endog_mu = self._clean(endog / mu)
weight_scale = var_weights / scale
ll_obs = weight_scale * np.log(weight_scale * endog_mu)
ll_obs -= weight_scale * endog_mu
ll_obs -= special.gammaln(weight_scale) + np.log(endog)
return ll_obs
# 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
def resid_anscombe(self, endog, mu, var_weights=1., scale=1.):
r"""
The Anscombe residuals
Parameters
----------
endog : array
The endogenous response variable
mu : array
The inverse of the link function at the linear predicted values.
var_weights : array_like
1d array of variance (analytic) weights. The default is 1.
scale : float, optional
An optional argument to divide the residuals by sqrt(scale).
The default is 1.
Returns
-------
resid_anscombe : array
The Anscombe residuals for the Gamma family defined below
Notes
-----
.. math::
resid\_anscombe_i = 3 * (endog_i^{1/3} - \mu_i^{1/3}) / \mu_i^{1/3}
/ \sqrt{scale} * \sqrt(var\_weights)
"""
resid = 3 * (endog**(1/3.) - mu**(1/3.)) / mu**(1/3.) / scale ** 0.5
resid *= np.sqrt(var_weights)
return resid
[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.genmod.families.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.genmod.families.varfuncs.binary
See Also
--------
statsmodels.genmod.families.family.Family : Parent class for all links.
:ref:`links` : Further details on links.
Notes
-----
endog for Binomial can be specified in one of three ways:
A 1d array of 0 or 1 values, indicating failure or success
respectively.
A 2d array, with two columns. The first column represents the
success count and the second column represents the failure
count.
A 1d array of proportions, indicating the proportion of
successes, with parameter `var_weights` containing the
number of trials for each row.
"""
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=None): # , n=1.):
if link is None:
link = L.logit()
# 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)
super(Binomial, self).__init__(link=link,
variance=V.Binomial(n=self.n))
[docs] def starting_mu(self, y):
r"""
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
freq_weights : array
1d array of frequency weights
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] > 2:
raise ValueError('endog has more than 2 columns. The Binomial '
'link supports either a single response variable '
'or a paired response variable.')
elif 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])
def _resid_dev(self, endog, mu):
r"""
Binomial deviance residuals
Parameters
----------
endog : array
The endogenous response variable.
mu : array
The inverse of the link function at the linear predicted values.
Returns
-------
resid_dev : float
Deviance residuals as defined below.
Notes
-----
.. math::
resid\_dev_i = 2 * n * (endog_i * \ln(endog_i /\mu_i) +
(1 - endog_i) * \ln((1 - endog_i) / (1 - \mu_i)))
"""
endog_mu = self._clean(endog / mu)
n_endog_mu = self._clean((1. - endog) / (1. - mu))
resid_dev = endog * np.log(endog_mu) + (1 - endog) * np.log(n_endog_mu)
return 2 * self.n * resid_dev
[docs] def loglike_obs(self, endog, mu, var_weights=1., scale=1.):
r"""
The log-likelihood function for each observation in terms of the fitted
mean response for the Binomial distribution.
Parameters
----------
endog : array
Usually the endogenous response variable.
mu : array
Usually but not always the fitted mean response variable.
var_weights : array_like
1d array of variance (analytic) weights. The default is 1.
scale : float
The scale parameter. The default is 1.
Returns
-------
ll_i : float
The value of the loglikelihood evaluated at
(endog, mu, var_weights, scale) as defined below.
Notes
-----
If the endogenous variable is binary:
.. math::
ll_i = \sum_i (y_i * \log(\mu_i/(1-\mu_i)) + \log(1-\mu_i)) *
var\_weights_i
If the endogenous variable is binomial:
.. math::
ll_i = \sum_i var\_weights_i * (\ln \Gamma(n+1) -
\ln \Gamma(y_i + 1) - \ln \Gamma(n_i - y_i +1) + y_i *
\log(\mu_i / (n_i - \mu_i)) + n * \log(1 - \mu_i/n_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.
"""
n = self.n # Number of trials
y = endog * n # Number of successes
# note that mu is still in (0,1), i.e. not converted back
return (special.gammaln(n + 1) - special.gammaln(y + 1) -
special.gammaln(n - y + 1) + y * np.log(mu / (1 - mu)) +
n * np.log(1 - mu)) * var_weights
[docs] def resid_anscombe(self, endog, mu, var_weights=1., scale=1.):
r'''
The Anscombe residuals
Parameters
----------
endog : array
The endogenous response variable
mu : array
The inverse of the link function at the linear predicted values.
var_weights : array_like
1d array of variance (analytic) weights. The default is 1.
scale : float, optional
An optional argument to divide the residuals by sqrt(scale).
The default is 1.
Returns
-------
resid_anscombe : array
The Anscombe residuals as defined below.
Notes
-----
.. math::
n^{2/3}*(cox\_snell(endog)-cox\_snell(mu)) /
(mu*(1-mu/n)*scale^3)^{1/6} * \sqrt(var\_weights)
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 as defined in scipy,
which uses a regularized version (with the unregularized version, one
would just have :math:`cox_snell(x) = Betainc(2/3., 2/3., x)`).
The name 'cox_snell' is idiosyncratic and is simply used for
convenience following the approach suggested in Cox and Snell (1968).
Further note that
:math:`cox\_snell(x) = \frac{3}{2}*x^{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.
'''
endog = endog * self.n # convert back to successes
mu = mu * self.n # convert back to successes
def cox_snell(x):
return special.betainc(2/3., 2/3., x) * special.beta(2/3., 2/3.)
resid = (self.n ** (2/3.) * (cox_snell(endog * 1. / self.n) -
cox_snell(mu * 1. / self.n)) /
(mu * (1 - mu * 1. / self.n) * scale ** 3) ** (1 / 6.))
resid *= np.sqrt(var_weights)
return resid
[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.genmod.families.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.genmod.families.varfuncs.mu_cubed
See Also
--------
statsmodels.genmod.families.family.Family : Parent class for all links.
:ref:`links` : Further details on links.
Notes
-----
The inverse Gaussian 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=None):
if link is None:
link = L.inverse_squared()
super(InverseGaussian, self).__init__(
link=link, variance=InverseGaussian.variance)
def _resid_dev(self, endog, mu):
r"""
Inverse Gaussian deviance residuals
Parameters
----------
endog : array
The endogenous response variable.
mu : array
The inverse of the link function at the linear predicted values.
Returns
-------
resid_dev : float
Deviance residuals as defined below.
Notes
-----
.. math::
resid\_dev_i = 1 / (endog_i * \mu_i^2) * (endog_i - \mu_i)^2
"""
return 1. / (endog * mu ** 2) * (endog - mu) ** 2
def loglike_obs(self, endog, mu, var_weights=1., scale=1.):
r"""
The log-likelihood function for each observation in terms of the fitted
mean response for the Inverse Gaussian distribution.
Parameters
----------
endog : array
Usually the endogenous response variable.
mu : array
Usually but not always the fitted mean response variable.
var_weights : array_like
1d array of variance (analytic) weights. The default is 1.
scale : float
The scale parameter. The default is 1.
Returns
-------
ll_i : float
The value of the loglikelihood evaluated at
(endog, mu, var_weights, scale) as defined below.
Notes
-----
.. math::
ll_i = -1/2 * (var\_weights_i * (endog_i - \mu_i)^2 /
(scale * endog_i * \mu_i^2) + \ln(scale * \endog_i^3 /
var\_weights_i) - \ln(2 * \pi))
"""
ll_obs = -var_weights * (endog - mu) ** 2 / (scale * endog * mu ** 2)
ll_obs += -np.log(scale * endog ** 3 / var_weights) - np.log(2 * np.pi)
ll_obs /= 2
return ll_obs
def resid_anscombe(self, endog, mu, var_weights=1., scale=1.):
r"""
The Anscombe residuals
Parameters
----------
endog : array
The endogenous response variable
mu : array
The inverse of the link function at the linear predicted values.
var_weights : array_like
1d array of variance (analytic) weights. The default is 1.
scale : float, optional
An optional argument to divide the residuals by sqrt(scale).
The default is 1.
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 * scale} *
\sqrt(var\_weights)
"""
resid = np.log(endog / mu) / np.sqrt(mu * scale)
resid *= np.sqrt(var_weights)
return resid
[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.genmod.families.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.genmod.families.varfuncs.nbinom
See Also
--------
statsmodels.genmod.families.family.Family : Parent class for all links.
:ref:`links` : Further details on 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=None, alpha=1.):
self.alpha = 1. * alpha # make it at least float
if link is None:
link = L.log()
super(NegativeBinomial, self).__init__(
link=link, variance=V.NegativeBinomial(alpha=self.alpha))
def _resid_dev(self, endog, mu):
r"""
Negative Binomial deviance residuals
Parameters
----------
endog : array
The endogenous response variable.
mu : array
The inverse of the link function at the linear predicted values.
Returns
-------
resid_dev : float
Deviance residuals as defined below.
Notes
-----
.. math::
resid_dev_i = 2 * (endog_i * \ln(endog_i /
\mu_i) - (endog_i + 1 / \alpha) * \ln((endog_i + 1 / \alpha) /
(\mu_i + 1 / \alpha)))
"""
endog_mu = self._clean(endog / mu)
endog_alpha = endog + 1 / self.alpha
mu_alpha = mu + 1 / self.alpha
resid_dev = endog * np.log(endog_mu)
resid_dev -= endog_alpha * np.log(endog_alpha / mu_alpha)
return 2 * resid_dev
def loglike_obs(self, endog, mu, var_weights=1., scale=1.):
r"""
The log-likelihood function for each observation in terms of the fitted
mean response for the Negative Binomial distribution.
Parameters
----------
endog : array
Usually the endogenous response variable.
mu : array
Usually but not always the fitted mean response variable.
var_weights : array_like
1d array of variance (analytic) weights. The default is 1.
scale : float
The scale parameter. The default is 1.
Returns
-------
ll_i : float
The value of the loglikelihood evaluated at
(endog, mu, var_weights, scale) as defined below.
Notes
-----
Defined as:
.. math::
llf = \sum_i var\_weights_i / scale * (Y_i * \log{(\alpha * \mu_i /
(1 + \alpha * \mu_i))} - \log{(1 + \alpha * \mu_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 )}
constant = (special.gammaln(endog + 1 / self.alpha) -
special.gammaln(endog+1)-special.gammaln(1/self.alpha))
return (endog * np.log(self.alpha * mu / (1 + self.alpha * mu)) -
np.log(1 + self.alpha * mu) / self.alpha +
constant) * var_weights / scale
"""
ll_obs = endog * np.log(self.alpha * mu)
ll_obs -= (endog + 1 / self.alpha) * np.log(1 + self.alpha * mu)
ll_obs += special.gammaln(endog + 1 / self.alpha)
ll_obs -= special.gammaln(1 / self.alpha)
ll_obs -= special.gammaln(endog + 1)
return var_weights / scale * ll_obs
def resid_anscombe(self, endog, mu, var_weights=1., scale=1.):
r"""
The Anscombe residuals
Parameters
----------
endog : array
The endogenous response variable
mu : array
The inverse of the link function at the linear predicted values.
var_weights : array_like
1d array of variance (analytic) weights. The default is 1.
scale : float, optional
An optional argument to divide the residuals by sqrt(scale).
The default is 1.
Returns
-------
resid_anscombe : array
The Anscombe residuals as defined below.
Notes
-----
Anscombe residuals for Negative Binomial are the same as for Binomial
upon setting :math:`n=-\frac{1}{\alpha}`. Due to the negative value of
:math:`-\alpha*Y` the representation with the hypergeometric function
:math:`H2F1(x) = hyp2f1(2/3.,1/3.,5/3.,x)` is advantageous
.. math::
resid\_anscombe_i = \frac{3}{2} *
(Y_i^(2/3)*H2F1(-\alpha*Y_i) - \mu_i^(2/3)*H2F1(-\alpha*\mu_i))
/ (\mu_i * (1+\alpha*\mu_i) * scale^3)^(1/6) * \sqrt(var\_weights)
Note that for the (unregularized) Beta function, one has
:math:`Beta(z,a,b) = z^a/a * H2F1(a,1-b,a+1,z)`
"""
def hyp2f1(x):
return special.hyp2f1(2 / 3., 1 / 3., 5 / 3., x)
resid = (3 / 2. * (endog ** (2 / 3.) * hyp2f1(-self.alpha * endog) -
mu ** (2 / 3.) * hyp2f1(-self.alpha * mu)) /
(mu * (1 + self.alpha * mu) *
scale ** 3) ** (1 / 6.))
resid *= np.sqrt(var_weights)
return resid
[docs]class Tweedie(Family):
"""
Tweedie family.
Parameters
----------
link : a link instance, optional
The default link for the Tweedie family is the log link.
Available links are log and Power.
See statsmodels.genmod.families.links for more information.
var_power : float, optional
The variance power. The default is 1.
eql : bool
If True, the Extended Quasi-Likelihood is used, else the
likelihood is used (however the latter is not implemented).
If eql is True, var_power must be between 1 and 2.
Attributes
----------
Tweedie.link : a link instance
The link function of the Tweedie instance
Tweedie.variance : varfunc instance
``variance`` is an instance of
statsmodels.genmod.families.varfuncs.Power
Tweedie.var_power : float
The power of the variance function.
See Also
--------
statsmodels.genmod.families.family.Family : Parent class for all links.
:ref:`links` : Further details on links.
Notes
-----
Loglikelihood 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
statsmodels.genmod.generalized_linear_model.GLM class.
"""
links = [L.log, L.Power]
variance = V.Power(power=1.5)
safe_links = [L.log, L.Power]
def __init__(self, link=None, var_power=1., eql=False):
self.var_power = var_power
self.eql = eql
if eql and (var_power < 1 or var_power > 2):
raise ValueError("Tweedie: if EQL=True then var_power must fall "
"between 1 and 2")
if link is None:
link = L.log()
super(Tweedie, self).__init__(
link=link, variance=V.Power(power=var_power * 1.))
def _resid_dev(self, endog, mu):
r"""
Tweedie deviance residuals
Parameters
----------
endog : array
The endogenous response variable.
mu : array
The inverse of the link function at the linear predicted values.
Returns
-------
resid_dev : float
Deviance residuals as defined below.
Notes
-----
When :math:`p = 1`,
.. math::
dev_i = \mu_i
when :math:`endog_i = 0` and
.. math::
dev_i = endog_i * \log(endog_i / \mu_i) + (\mu_i - endog_i)
otherwise.
When :math:`p = 2`,
.. math::
dev_i = (endog_i - \mu_i) / \mu_i - \log(endog_i / \mu_i)
For all other p,
.. math::
dev_i = endog_i^{2 - p} / ((1 - p) * (2 - p)) -
endog_i * \mu_i^{1 - p} / (1 - p) + \mu_i^{2 - p} /
(2 - p)
The deviance residual is then
.. math::
resid\_dev_i = 2 * 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 = self._clean(endog)
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 2 * dev
def loglike_obs(self, endog, mu, var_weights=1., scale=1.):
r"""
The log-likelihood function for each observation in terms of the fitted
mean response for the Tweedie distribution.
Parameters
----------
endog : array
Usually the endogenous response variable.
mu : array
Usually but not always the fitted mean response variable.
var_weights : array_like
1d array of variance (analytic) weights. The default is 1.
scale : float
The scale parameter. The default is 1.
Returns
-------
ll_i : float
The value of the loglikelihood evaluated at
(endog, mu, var_weights, scale) as defined below.
Notes
-----
If eql is True, the Extended Quasi-Likelihood is used. At present,
this method returns NaN if eql is False. When the actual likelihood
is implemented, it will be accessible by setting eql to False.
References
----------
JA Nelder, D Pregibon (1987). An extended quasi-likelihood function.
Biometrika 74:2, pp 221-232. https://www.jstor.org/stable/2336136
"""
if not self.eql:
# We have not yet implemented the actual likelihood
return np.nan
# Equations 9-10 or Nelder and Pregibon
p = self.var_power
llf = np.log(2 * np.pi * scale) + p * np.log(mu) - np.log(var_weights)
llf /= -2
if p == 1:
u = endog * np.log(endog / mu) - (endog - mu)
u *= var_weights / scale
elif p == 2:
yr = endog / mu
u = yr - np.log(yr) - 1
u *= var_weights / scale
else:
u = (endog ** (2 - p)
- (2 - p) * endog * mu ** (1 - p)
+ (1 - p) * mu ** (2 - p))
u *= var_weights / (scale * (1 - p) * (2 - p))
llf -= u
return llf
def resid_anscombe(self, endog, mu, var_weights=1., scale=1.):
r"""
The Anscombe residuals
Parameters
----------
endog : array
The endogenous response variable
mu : array
The inverse of the link function at the linear predicted values.
var_weights : array_like
1d array of variance (analytic) weights. The default is 1.
scale : float, optional
An optional argument to divide the residuals by sqrt(scale).
The default is 1.
Returns
-------
resid_anscombe : array
The Anscombe residuals as defined below.
Notes
-----
When :math:`p = 3`, then
.. math::
resid\_anscombe_i = \log(endog_i / \mu_i) / \sqrt{\mu_i * scale} *
\sqrt(var\_weights)
Otherwise,
.. math::
c = (3 - p) / 3
.. math::
resid\_anscombe_i = (1 / c) * (endog_i^c - \mu_i^c) / \mu_i^{p / 6}
/ \sqrt{scale} * \sqrt(var\_weights)
"""
if self.var_power == 3:
resid = np.log(endog / mu) / np.sqrt(mu * scale)
else:
c = (3. - self.var_power) / 3.
resid = ((1. / c) * (endog ** c - mu ** c) /
mu ** (self.var_power / 6.)) / scale ** 0.5
resid *= np.sqrt(var_weights)
return resid