Source code for statsmodels.genmod.families.family

'''
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 scipy.stats import ss
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 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.

--------
:ref:links

"""
# TODO: change these class attributes, use valid somewhere...
valid = [-np.inf, np.inf]

"""
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
# for Poisson...
raise TypeError("The input should be a valid Link object.")
errmsg = "Invalid link for family, should be in %s. (got %s)"

"""
Helper method to get the link for a family.
"""

# link property for each family is a pointer to link instance

self.variance = variance

[docs]    def starting_mu(self, y):
"""
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
-----
mu_0 = (endog + mean(endog))/2.

Notes
-----
Only the Binomial family takes a different initial value.
"""
return (y + y.mean())/2.

[docs]    def weights(self, mu):
"""
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
-----
w = 1 / (link'(mu)**2 * variance(mu))
"""
return 1. / (self.link.deriv(mu)**2 * self.variance(mu))

[docs]    def deviance(self, endog, mu, scale=1.):
"""
Deviance of (endog,mu) pair.

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.
scale : float, optional
An optional scale argument

Returns
-------
Deviance : array
The value of deviance function defined below.

Notes
-----
Deviance is defined

.. math::

\sum_i(2 loglike(y_i, y_i) - 2 * loglike(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, 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.
scale : float, optional
An optional argument to divide the residuals by scale

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.
"""

[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.
"""

[docs]    def loglike(self, endog, mu, scale=1.):
"""
The loglikelihood function.

Parameters
----------
endog : array
Usually the endogenous response variable.
mu : array
Usually but not always the fitted mean response variable.

Returns
-------
llf : float
The value of the loglikelihood evaluated at (endog,mu).

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 Anscome residuals.

--------
statsmodels.families.family.Family docstring and the resid_anscombe
"""
raise NotImplementedError

[docs]class Poisson(Family):
"""
Poisson exponential family.

Parameters
----------
The default link for the Poisson family is the log link. Available

Attributes
----------
The link function of the Poisson instance.
Poisson.variance : varfuncs instance
variance is an instance of
statsmodels.genmod.families.family.varfuncs.mu

--------
statsmodels.genmod.families.family.Family
:ref:links

"""

variance = V.mu
valid = [0, np.inf]

self.variance = Poisson.variance

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.):
"""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

Returns
-------
resid_dev : array
Deviance residuals as defined below

Notes
-----
resid_dev = sign(endog-mu)*sqrt(2*endog*log(endog/mu)-2*(endog-mu))
"""
endog_mu = self._clean(endog/mu)
return np.sign(endog - mu) * np.sqrt(2 * endog *
np.log(endog_mu) -
2 * (endog - mu))/scale

[docs]    def deviance(self, endog, mu, scale=1.):
'''
Poisson deviance function

Parameters
----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
scale : float, optional
An optional scale argument

Returns
-------
deviance : float
The deviance function at (endog,mu) as defined below.

Notes
-----
If a constant term is included it is defined as

:math:deviance = 2*\\sum_{i}(Y*\\log(Y/\\mu))
'''
endog_mu = self._clean(endog/mu)
return 2*np.sum(endog*np.log(endog_mu))/scale

[docs]    def loglike(self, endog, mu, scale=1.):
"""
Loglikelihood function for Poisson exponential family distribution.

Parameters
----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
scale : float, optional
The default is 1.

Returns
-------
llf : float
The value of the loglikelihood function evaluated at
(endog,mu,scale) as defined below.

Notes
-----
llf = scale * sum(-mu + endog*log(mu) - gammaln(endog+1))
where gammaln is the log gamma function
"""
return scale * np.sum(-mu + endog*np.log(mu)-special.gammaln(endog+1))

[docs]    def resid_anscombe(self, endog, mu):
"""
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
-----
resid_anscombe is defined

.. math:

(3/2.)*(endog^{2/3.} - \\mu**(2/3.))/\\mu^{1/6.}
"""
return (3/2.)*(endog**(2/3.)-mu**(2/3.))/mu**(1/6.)

[docs]class Gaussian(Family):
"""
Gaussian exponential family distribution.

Parameters
----------
Available links are log, identity, and inverse.

Attributes
----------
The link function of the Gaussian instance
Gaussian.variance : varfunc instance
variance is an instance of statsmodels.family.varfuncs.constant

--------
statsmodels.genmod.families.family.Family
:ref:links

"""

variance = V.constant

self.variance = Gaussian.variance

[docs]    def resid_dev(self, endog, mu, scale=1.):
"""
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

Returns
-------
resid_dev : array
Deviance residuals as defined below

Notes
--------
resid_dev = (endog - mu)/sqrt(variance(mu))
"""

return (endog - mu) / np.sqrt(self.variance(mu))/scale

[docs]    def deviance(self, endog, mu, scale=1.):
"""
Gaussian deviance function

Parameters
----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
scale : float, optional
An optional scale argument

Returns
-------
deviance : float
The deviance function at (endog,mu) as defined below.

Notes
--------
deviance = sum((endog-mu)**2)
"""
return np.sum((endog-mu)**2)/scale

[docs]    def loglike(self, endog, mu, scale=1.):
"""
Loglikelihood function for Gaussian exponential family distribution.

Parameters
----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
scale : float, optional
Scales the loglikelihood function. The default is 1.

Returns
-------
llf : float
The value of the loglikelihood function evaluated at
(endog,mu,scale) as defined below.

Notes
-----
loglikelihood function is the same as the classical OLS model.
llf = -(nobs/2)*(log(SSR) + (1 + log(2*pi/nobs)))

function is defined as
llf = sum((endog*mu-mu**2/2)/scale - endog**2/(2*scale) - \
(1/2.)*log(2*pi*scale))
"""
# This is just the loglikelihood for classical OLS
nobs2 = endog.shape[0]/2.
SSR = ss(endog-self.fitted(mu))
llf = -np.log(SSR) * nobs2
llf -= (1+np.log(np.pi/nobs2))*nobs2
return llf
else:
# Return the loglikelihood for Gaussian GLM
return np.sum((endog * mu - mu**2/2)/scale - endog**2/(2 * scale)
- .5*np.log(2 * np.pi * scale))

[docs]    def resid_anscombe(self, endog, mu):
"""
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
--------
resid_anscombe = endog - mu
"""
return endog-mu

[docs]class Gamma(Family):
"""
Gamma exponential family distribution.

Parameters
----------
Available links are log, identity, and inverse.

Attributes
----------
The link function of the Gamma instance
Gamma.variance : varfunc instance
variance is an instance of statsmodels.family.varfuncs.mu_squared

--------
statsmodels.genmod.families.family.Family
:ref:links

"""

variance = V.mu_squared

self.variance = Gamma.variance

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, scale=1.):
"""
Gamma deviance function

Parameters
-----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
scale : float, optional
An optional scale argument

Returns
-------
deviance : float
Deviance function as defined below

Notes
-----
deviance = 2*sum((endog - mu)/mu - log(endog/mu))
"""
endog_mu = self._clean(endog/mu)
return 2 * np.sum((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

Returns
-------
resid_dev : array
Deviance residuals as defined below

Notes
-----
resid_dev is defined

.. math:

sign(endog - \mu) * \sqrt{-2*(-(endog-\mu)/\mu + \log(endog/\mu))}
"""
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, scale=1.):
"""
Loglikelihood function for Gamma exponential family distribution.

Parameters
----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
scale : float, optional
The default is 1.

Returns
-------
llf : float
The value of the loglikelihood function evaluated at
(endog,mu,scale) as defined below.

Notes
--------
llf = -1/scale * sum(endog/mu + log(mu) + (scale-1)*log(endog) +\
log(scale) + scale*gammaln(1/scale))
where gammaln is the log gamma function.
"""
return - 1./scale * np.sum(endog/mu + np.log(mu) + (scale - 1) *
np.log(endog) + np.log(scale) + scale *
special.gammaln(1./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):
"""
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
-----
resid_anscombe = 3*(endog**(1/3.)-mu**(1/3.))/mu**(1/3.)
"""
return 3*(endog**(1/3.)-mu**(1/3.))/mu**(1/3.)

[docs]class Binomial(Family):
"""
Binomial exponential family distribution.

Parameters
----------
Available links are logit, probit, cauchy, log, and cloglog.

Attributes
----------
The link function of the Binomial instance
Binomial.variance : varfunc instance
variance is an instance of statsmodels.family.varfuncs.binary

--------
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

def __init__(self, link=L.logit):  # , n=1.):
# TODO: it *should* work for a constant n>1 actually, if data_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)

[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

starting_mu = (y + .5)/2
"""
return (y + .5)/2

[docs]    def initialize(self, endog):
'''
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 (endog.ndim > 1 and endog.shape[1] > 1):
y = endog[:, 0]
self.n = endog.sum(1)  # overwrite self.n for deviance below
return y*1./self.n
else:
return endog

[docs]    def deviance(self, endog, mu, scale=1.):
'''
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
scale : float, optional
An optional scale argument

Returns
--------
deviance : float
The deviance function as defined below

Notes
-----
If the endogenous variable is binary:

deviance = -2*sum(I_one * log(mu) + (I_zero)*log(1-mu))

where I_one is an indicator function that evalueates to 1 if
endog_i == 1. and I_zero is an indicator function that evaluates to
1 if endog_i == 0.

If the model is ninomial:

deviance = 2*sum(log(endog/mu) + (n-endog)*log((n-endog)/(n-mu)))
where endog and 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))

else:
return 2 * np.sum(self.n * (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.):
"""
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

Returns
-------
resid_dev : array
Deviance residuals as defined below

Notes
-----
If endog is binary:

resid_dev = sign(endog-mu)*sqrt(-2*log(I_one*mu + I_zero*(1-mu)))

where I_one is an indicator function that evaluates as 1 if endog == 1
and I_zero is an indicator function that evaluates as 1 if endog == 0.

If endog is binomial:

resid_dev = sign(endog - mu) * sqrt(2 * n * (endog * log(endog/mu) +
(1 - endog) * log((1 - endog)/(1 - mu))))

where endog and n are as defined in Binomial.initialize.
"""

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, scale=1.):
"""
Loglikelihood function for Binomial exponential family distribution.

Parameters
----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
scale : float, optional
The default is 1.

Returns
-------
llf : float
The value of the loglikelihood function evaluated at
(endog,mu,scale) as defined below.

Notes
--------
If endog is binary:
llf = scale*sum(endog*log(mu/(1-mu))+log(1-mu))

If endog is binomial:
llf = scale*sum(gammaln(n+1) - gammaln(y+1) - gammaln(n-y+1) +\
y*log(mu/(1-mu)) + n*log(1-mu)

where gammaln is the log gamma function and y = endog*n with endog
and n as defined in Binomial initialize.  This simply makes y 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))
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))

[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
----------
The default link for the inverse Gaussian family is the
Available links are inverse_squared, inverse, log, and identity.

Attributes
----------
The link function of the inverse Gaussian instance
InverseGaussian.variance : varfunc instance
variance is an instance of statsmodels.family.varfuncs.mu_cubed

--------
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

self.variance = InverseGaussian.variance

[docs]    def resid_dev(self, endog, mu, scale=1.):
"""
Returns the deviance residuals for the inverse Gaussian family.

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

Returns
-------
resid_dev : array
Deviance residuals as defined below

Notes
-----
dev_resid = sign(endog-mu)*sqrt((endog-mu)**2/(endog*mu**2))
"""
return np.sign(endog-mu) * np.sqrt((endog-mu)**2/(endog*mu**2))/scale

[docs]    def deviance(self, endog, mu, scale=1.):
"""
Inverse Gaussian deviance function

Parameters
-----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
scale : float, optional
An optional scale argument

Returns
-------
deviance : float
Deviance function as defined below

Notes
-----
deviance = sum((endog=mu)**2/(endog*mu**2))
"""
return np.sum((endog-mu)**2/(endog*mu**2))/scale

[docs]    def loglike(self, endog, mu, scale=1.):
"""
Loglikelihood function for inverse Gaussian distribution.

Parameters
----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
scale : float, optional
The default is 1.

Returns
-------
llf : float
The value of the loglikelihood function evaluated at
(endog,mu,scale) as defined below.

Notes
-----
llf = -(1/2.)*sum((endog-mu)**2/(endog*mu**2*scale)
+ log(scale*endog**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))

[docs]    def resid_anscombe(self, endog, mu):
"""
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
-----
resid_anscombe =  log(endog/mu)/sqrt(mu)
"""
return np.log(endog/mu)/np.sqrt(mu)

[docs]class NegativeBinomial(Family):
"""
Negative Binomial exponential family.

Parameters
----------
The default link for the negative binomial family is the log link.
Available links are log, cloglog, identity, nbinom and power.
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
----------
The link function of the negative binomial instance
NegativeBinomial.variance : varfunc instance
variance is an instance of statsmodels.family.varfuncs.nbinom

--------
statsmodels.genmod.families.family.Family
:ref:links

Notes
-----
Power link functions are not yet supported.

"""
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

self.alpha = alpha
self.variance = V.NegativeBinomial(alpha=self.alpha)
else:

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, scale=1.):
r"""
Returns the value of the deviance function.

Parameters
-----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
scale : float, optional
An optional scale argument

Returns
-------
deviance : float
Deviance function as defined below

Notes
-----
deviance = sum(piecewise)

where piecewise is defined as

If :math:Y_{i} == 0:

.. math::

piecewise_i = 2\log(1+\alpha*\mu)/\alpha

If :math:Y_{i} > 0:

.. math::

piecewise_i = math2 Y \log(Y/\mu)-2/\alpha(1+\alpha Y) * \log((1+\alpha Y)/(1+\alpha\mu))
"""
iszero = np.equal(endog, 0)
notzero = 1 - iszero
tmp = np.zeros(len(endog))
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(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

Returns
--------
resid_dev : array
The array of deviance residuals

Notes
-----
resid_dev = sign(endog-mu) * sqrt(piecewise)

where piecewise is defined as

If :math:Y_i = 0:

.. math::

piecewise_i = 2*log(1+alpha*mu)/alpha

If :math:Y_i > 0:

.. math::

piecewise_i = 2*Y*log(Y/\mu) - 2/\alpha * (1 + \alpha * Y) * \log((1 + \alpha * Y)/(1 + \alpha * \mu))
'''
iszero = np.equal(endog, 0)
notzero = 1 - iszero
tmp = np.zeros(len(endog))
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, lin_pred=None):
"""
The loglikelihood function for the negative binomial family.

Parameters
----------
endog : array-like
Endogenous response variable
lin_pred : array-like
The linear predictor of the model.  This is dot(exog,params),
plus the offset if present.

Returns
-------
llf : float
The value of the loglikelihood function evaluated at
(endog,mu,scale) as defined below.

Notes
-----
sum(endog*log(alpha*exp(lin_pred)/(1+alpha*exp(lin_pred))) -
log(1+alpha*exp(lin_pred))/alpha + constant)

where constant is defined as::

constant = gammaln(endog + 1/alpha) - gammaln(endog + 1) -
gammaln(1/alpha)
"""
# don't need to specify mu
if lin_pred is None:
raise AttributeError('The loglikelihood for the negative binomial'
' requires that the fitted values be '
'provided via the lin_pred keyword '
'argument.')
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))

[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.))