# 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 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 """ # 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 :ref: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
[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 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))
[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 Anscome 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 :ref: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
[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 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
[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 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 :ref: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
[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 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
[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 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 :ref: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] > 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 :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=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
[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 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
[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 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 :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=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
[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 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
[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 ----- 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 :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 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
[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 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
[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 ----- 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