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Source code for scikits.statsmodels.discrete.discrete_model

"""
Limited dependent variable and qualitative variables.

Includes binary outcomes, count data, (ordered) ordinal data and limited
dependent variables.

General References
--------------------

A.C. Cameron and P.K. Trivedi.  `Regression Analysis of Count Data`.  Cambridge,
    1998

G.S. Madalla. `Limited-Dependent and Qualitative Variables in Econometrics`.
    Cambridge, 1983.

W. Greene. `Econometric Analysis`. Prentice Hall, 5th. edition. 2003.
"""

__all__ = ["Poisson","Logit","Probit","MNLogit"]

import numpy as np
from scikits.statsmodels.base.model import (LikelihoodModel,
        LikelihoodModelResults)
import scikits.statsmodels.tools.tools as tools
from scikits.statsmodels.tools.decorators import (resettable_cache,
        cache_readonly)
from scikits.statsmodels.regression.linear_model import OLS
from scipy import stats, special, optimize # opt just for nbin
from scipy.misc import factorial
#import numdifftools as nd #This will be removed when all have analytic hessians

#TODO: add options for the parameter covariance/variance
# ie., OIM, EIM, and BHHH see Green 21.4

def _check_discrete_args(at, method):
    """
    Checks the arguments for margeff if the exogenous variables are discrete.
    """
    if method in ['dyex','eyex']:
        raise ValueError("%s not allowed for discrete variables" % method)
    if at in ['median', 'zero']:
        raise ValueError("%s not allowed for discrete variables" % at)

def _isdummy(X):
    """
    Given an array X, returns a boolean column index for the dummy variables.

    Parameters
    ----------
    X : array-like
        A 1d or 2d array of numbers

    Examples
    --------
    >>> X = np.random.randint(0, 2, size=(15,5)).astype(float)
    >>> X[:,1:3] = np.random.randn(15,2)
    >>> ind = _isdummy(X)
    >>> ind
    array([ True, False, False,  True,  True], dtype=bool)
    """
    X = np.asarray(X)
    if X.ndim > 1:
        ind = np.zeros(X.shape[1]).astype(bool)
    max = (np.max(X, axis=0) == 1)
    min = (np.min(X, axis=0) == 0)
    remainder = np.all(X % 1. == 0, axis=0)
    ind = min & max & remainder
    if X.ndim == 1:
        ind = np.asarray([ind])
    return ind

def _iscount(X):
    """
    Given an array X, returns a boolean column index for count variables.

    Parameters
    ----------
    X : array-like
        A 1d or 2d array of numbers

    Examples
    --------
    >>> X = np.random.randint(0, 10, size=(15,5)).astype(float)
    >>> X[:,1:3] = np.random.randn(15,2)
    >>> ind = _iscount(X)
    >>> ind
    array([ True, False, False,  True,  True], dtype=bool)
    """
    X = np.asarray(X)
    remainder = np.all(X % 1. == 0, axis = 0)
    dummy = _isdummy(X)
    remainder -= dummy
    return remainder

[docs]class DiscreteModel(LikelihoodModel): """ Abstract class for discrete choice models. This class does not do anything itself but lays out the methods and call signature expected of child classes in addition to those of scikits.statsmodels.model.LikelihoodModel. """ def __init___(endog, exog): super(DiscreteModel, self).__init__(endog, exog)
[docs] def initialize(self): """ Initialize is called by scikits.statsmodels.model.LikelihoodModel.__init__ and should contain any preprocessing that needs to be done for a model. """ self.df_model = float(tools.rank(self.exog) - 1) # assumes constant self.df_resid = float(self.exog.shape[0] - tools.rank(self.exog))
[docs] def cdf(self, X): """ The cumulative distribution function of the model. """ raise NotImplementedError
[docs] def pdf(self, X): """ The probability density (mass) function of the model. """ raise NotImplementedError
[docs] def fit(self, start_params=None, method='newton', maxiter=35, full_output=1, disp=1, callback=None, **kwargs): """ Fit the model using maximum likelihood. The rest of the docstring is from scikits.statsmodels.LikelihoodModel.fit """ if start_params is None and isinstance(self, MNLogit): start_params = np.zeros((self.exog.shape[1]*\ (self.wendog.shape[1]-1))) mlefit = super(DiscreteModel, self).fit(start_params=start_params, method=method, maxiter=maxiter, full_output=full_output, disp=disp, callback=callback, **kwargs) if isinstance(self, MNLogit): mlefit.params = mlefit.params.reshape(-1, self.exog.shape[1]) discretefit = DiscreteResults(self, mlefit) return discretefit
fit.__doc__ += LikelihoodModel.fit.__doc__
[docs] def predict(self, exog=None, linear=False): """ Predict response variable of a model given exogenous variables. exog : array-like 1d or 2d array of exogenous values. If not supplied, the whole exog attribute of the model is used. linear : bool, optional If True, returns the linear predictor dot(exog,params). Else, returns the value of the cdf at the linear predictor. Returns ------- array Fitted values at exog. Notes ----- You must fit the model first. """ if self._results is None: raise ValueError("You must fit the model first") if exog is None: exog = self.exog if not linear: return self.cdf(np.dot(exog, self._results.params)) else: return np.dot(exog, self._results.params)
[docs]class Poisson(DiscreteModel): """ Poisson model for count data Parameters ---------- endog : array-like 1-d array of the response variable. exog : array-like `exog` is an n x p array where n is the number of observations and p is the number of regressors including the intercept if one is included in the data. Attributes ----------- endog : array A reference to the endogenous response variable exog : array A reference to the exogenous design. """
[docs] def cdf(self, X): """ Poisson model cumulative distribution function Parameters ----------- X : array-like `X` is the linear predictor of the model. See notes. Returns ------- The value of the Poisson CDF at each point. Notes ----- The CDF is defined as .. math:: \\exp\left(-\\lambda\\right)\\sum_{i=0}^{y}\\frac{\\lambda^{i}}{i!} where :math:`\\lambda` assumes the loglinear model. I.e., .. math:: \\ln\\lambda_{i}=X\\beta The parameter `X` is :math:`X\\beta` in the above formula. """ y = self.endog # xb = np.dot(self.exog, params) return stats.poisson.cdf(y, np.exp(X))
[docs] def pdf(self, X): """ Poisson model probability mass function Parameters ----------- X : array-like `X` is the linear predictor of the model. See notes. Returns ------- The value of the Poisson PMF at each point. Notes -------- The PMF is defined as .. math:: \\frac{e^{-\\lambda_{i}}\\lambda_{i}^{y_{i}}}{y_{i}!} where :math:`\\lambda` assumes the loglinear model. I.e., .. math:: \\ln\\lambda_{i}=X\\beta The parameter `X` is :math:`X\\beta` in the above formula. """ y = self.endog # xb = np.dot(self.exog,params) return stats.poisson.pmf(y, np.exp(X))
[docs] def loglike(self, params): """ Loglikelihood of Poisson model Parameters ---------- params : array-like The parameters of the model. Returns ------- The log likelihood of the model evaluated at `params` Notes -------- .. math :: \\ln L=\\sum_{i=1}^{n}\\left[-\\lambda_{i}+y_{i}x_{i}^{\\prime}\\beta-\\ln y_{i}!\\right] """ XB = np.dot(self.exog, params) endog = self.endog return np.sum(-np.exp(XB) + endog*XB - np.log(factorial(endog)))
[docs] def score(self, params): """ Poisson model score (gradient) vector of the log-likelihood Parameters ---------- params : array-like The parameters of the model Returns ------- The score vector of the model evaluated at `params` Notes ----- .. math:: \\frac{\\partial\\ln L}{\\partial\\beta}=\\sum_{i=1}^{n}\\left(y_{i}-\\lambda_{i}\\right)x_{i} where the loglinear model is assumed .. math:: \\ln\\lambda_{i}=X\\beta """ X = self.exog L = np.exp(np.dot(X,params)) return np.dot(self.endog - L,X)
[docs] def hessian(self, params): """ Poisson model Hessian matrix of the loglikelihood Parameters ---------- params : array-like The parameters of the model Returns ------- The Hessian matrix evaluated at params Notes ----- .. math:: \\frac{\\partial^{2}\\ln L}{\\partial\\beta\\partial\\beta^{\\prime}}=-\\sum_{i=1}^{n}\\lambda_{i}x_{i}x_{i}^{\\prime} where the loglinear model is assumed .. math:: \\ln\\lambda_{i}=X\\beta """ X = self.exog L = np.exp(np.dot(X,params)) return -np.dot(L*X.T, X) # def fit(self, start_params=None, maxiter=35, method='newton', # tol=1e-08): # """ # Fits the Poisson model. # # Parameters # ---------- # start_params : array-like, optional # The default is a 0 vector. # maxiter : int, optional # Maximum number of iterations. The default is 35. # method : str, optional # `method` can be 'newton', 'ncg', 'bfgs'. The default is 'newton'. # tol : float, optional # The convergence tolerance for the solver. The default is # 1e-08. # # Returns # -------- # DiscreteResults object # # See also # -------- # scikits.statsmodels.model.LikelihoodModel # scikits.statsmodels.sandbox.discretemod.DiscreteResults # scipy.optimize # """ # # mlefit = super(Poisson, self).fit(start_params=start_params, # maxiter=maxiter, method=method, tol=tol) # params = mlefit.params # mlefit = DiscreteResults(self, params, self.hessian(params)) # return mlefit
class NbReg(DiscreteModel): pass
[docs]class Logit(DiscreteModel): """ Binary choice logit model Parameters ---------- endog : array-like 1-d array of the response variable. exog : array-like `exog` is an n x p array where n is the number of observations and p is the number of regressors including the intercept if one is included in the data. Attributes ----------- endog : array A reference to the endogenous response variable exog : array A reference to the exogenous design. """
[docs] def cdf(self, X): """ The logistic cumulative distribution function Parameters ---------- X : array-like `X` is the linear predictor of the logit model. See notes. Returns ------- 1/(1 + exp(-X)) Notes ------ In the logit model, .. math:: \\Lambda\\left(x^{\\prime}\\beta\\right)=\\text{Prob}\\left(Y=1|x\\right)=\\frac{e^{x^{\\prime}\\beta}}{1+e^{x^{\\prime}\\beta}} """ X = np.asarray(X) return 1/(1+np.exp(-X))
[docs] def pdf(self, X): """ The logistic probability density function Parameters ----------- X : array-like `X` is the linear predictor of the logit model. See notes. Returns ------- np.exp(-x)/(1+np.exp(-X))**2 Notes ----- In the logit model, .. math:: \\lambda\\left(x^{\\prime}\\beta\\right)=\\frac{e^{-x^{\\prime}\\beta}}{\\left(1+e^{-x^{\\prime}\\beta}\\right)^{2}} """ X = np.asarray(X) return np.exp(-X)/(1+np.exp(-X))**2
[docs] def loglike(self, params): """ Log-likelihood of logit model. Parameters ----------- params : array-like The parameters of the logit model. Returns ------- The log-likelihood function of the logit model. See notes. Notes ------ .. math:: \\ln L=\\sum_{i}\\ln\\Lambda\\left(q_{i}x_{i}^{\\prime}\\beta\\right) Where :math:`q=2y-1`. This simplification comes from the fact that the logistic distribution is symmetric. """ q = 2*self.endog - 1 X = self.exog return np.sum(np.log(self.cdf(q*np.dot(X,params))))
[docs] def score(self, params): """ Logit model score (gradient) vector of the log-likelihood Parameters ---------- params: array-like The parameters of the model Returns ------- The score vector of the model evaluated at `params` Notes ----- .. math:: \\frac{\\partial\\ln L}{\\partial\\beta}=\\sum_{i=1}^{n}\\left(y_{i}-\\Lambda_{i}\\right)x_{i} """ y = self.endog X = self.exog L = self.cdf(np.dot(X,params)) return np.dot(y - L,X)
[docs] def hessian(self, params): """ Logit model Hessian matrix of the log-likelihood Parameters ---------- params : array-like The parameters of the model Returns ------- The Hessian evaluated at `params` Notes ----- .. math:: \\frac{\\partial^{2}\\ln L}{\\partial\\beta\\partial\\beta^{\\prime}}=-\\sum_{i}\\Lambda_{i}\\left(1-\\Lambda_{i}\\right)x_{i}x_{i}^{\\prime} """ X = self.exog L = self.cdf(np.dot(X,params)) return -np.dot(L*(1-L)*X.T,X) # def fit(self, start_params=None, maxiter=35, method='newton', # tol=1e-08): # """ # Fits the binary logit model. # # Parameters # ---------- # start_params : array-like, optional # The default is a 0 vector. # maxiter : int, optional # Maximum number of iterations. The default is 35. # method : str, optional # `method` can be 'newton', 'ncg', 'bfgs'. The default is 'newton'. # tol : float, optional # The convergence tolerance for the solver. The default is # 1e-08. # # Returns # -------- # DiscreteResults object # # See also # -------- # scikits.statsmodels.model.LikelihoodModel # scikits.statsmodels.sandbox.discretemod.DiscreteResults # scipy.optimize # """ # mlefit = super(Logit, self).fit(start_params=start_params, # maxiter=maxiter, method=method, tol=tol) # params = mlefit.params # mlefit = DiscreteResults(self, params, self.hessian(params)) # return mlefit
[docs]class Probit(DiscreteModel): """ Binary choice Probit model Parameters ---------- endog : array-like 1-d array of the response variable. exog : array-like `exog` is an n x p array where n is the number of observations and p is the number of regressors including the intercept if one is included in the data. Attributes ----------- endog : array A reference to the endogenous response variable exog : array A reference to the exogenous design. """
[docs] def cdf(self, X): """ Probit (Normal) cumulative distribution function Parameters ---------- X : array-like The linear predictor of the model (XB). Returns -------- The cdf evaluated at `X`. Notes ----- This function is just an alias for scipy.stats.norm.cdf """ return stats.norm._cdf(X)
[docs] def pdf(self, X): """ Probit (Normal) probability density function Parameters ---------- X : array-like The linear predictor of the model (XB). Returns -------- The pdf evaluated at X. Notes ----- This function is just an alias for scipy.stats.norm.pdf """ X = np.asarray(X) return stats.norm._pdf(X)
[docs] def loglike(self, params): """ Log-likelihood of probit model (i.e., the normal distribution). Parameters ---------- params : array-like The parameters of the model. Returns ------- The log-likelihood evaluated at params Notes ----- .. math:: \\ln L=\\sum_{i}\\ln\\Phi\\left(q_{i}x_{i}^{\\prime}\\beta\\right) Where :math:`q=2y-1`. This simplification comes from the fact that the normal distribution is symmetric. """ q = 2*self.endog - 1 X = self.exog return np.sum(np.log(np.clip(self.cdf(q*np.dot(X,params)),1e-20, 1)))
[docs] def score(self, params): """ Probit model score (gradient) vector Parameters ---------- params : array-like The parameters of the model Returns ------- The score vector of the model evaluated at `params` Notes ----- .. math:: \\frac{\\partial\\ln L}{\\partial\\beta}=\\sum_{i=1}^{n}\\left[\\frac{q_{i}\\phi\\left(q_{i}x_{i}^{\\prime}\\beta\\right)}{\\Phi\\left(q_{i}x_{i}^{\\prime}\\beta\\right)}\\right]x_{i} Where :math:`q=2y-1`. This simplification comes from the fact that the normal distribution is symmetric. """ y = self.endog X = self.exog XB = np.dot(X,params) q = 2*y - 1 # clip to get rid of invalid divide complaint L = q*self.pdf(q*XB)/np.clip(self.cdf(q*XB), 1e-20, 1-1e-20) return np.dot(L,X)
[docs] def hessian(self, params): """ Probit model Hessian matrix of the log-likelihood Parameters ---------- params : array-like The parameters of the model Returns ------- The Hessian evaluated at `params` Notes ----- .. math:: \\frac{\\partial^{2}\\ln L}{\\partial\\beta\\partial\\beta^{\\prime}}=-\lambda_{i}\\left(\\lambda_{i}+x_{i}^{\\prime}\\beta\\right)x_{i}x_{i}^{\\prime} where .. math:: \\lambda_{i}=\\frac{q_{i}\\phi\\left(q_{i}x_{i}^{\\prime}\\beta\\right)}{\\Phi\\left(q_{i}x_{i}^{\\prime}\\beta\\right)} and :math:`q=2y-1` """ X = self.exog XB = np.dot(X,params) q = 2*self.endog - 1 L = q*self.pdf(q*XB)/self.cdf(q*XB) return np.dot(-L*(L+XB)*X.T,X) # def fit(self, start_params=None, maxiter=35, method='newton', # tol=1e-08): # """ # Fits the binary probit model. # # Parameters # ---------- # start_params : array-like, optional # The default is a 0 vector. # maxiter : int, optional # Maximum number of iterations. The default is 35. # method : str, optional # `method` can be 'newton', 'ncg', 'bfgs'. The default is 'newton'. # tol : float, optional # The convergence tolerance for the solver. The default is # 1e-08. # # Returns # -------- # DiscreteResults object # # See also # -------- # scikits.statsmodels.model.LikelihoodModel # scikits.statsmodels.sandbox.discretemod.DiscreteResults # scipy.optimize # """ # mlefit = super(Probit, self).fit(start_params=start_params, # maxiter=maxiter, method=method, tol=tol) # params = mlefit.params # mlefit = DiscreteResults(self, params, self.hessian(params)) # return mlefit
[docs]class MNLogit(DiscreteModel): """ Multinomial logit model Parameters ---------- endog : array-like `endog` is an 1-d vector of the endogenous response. `endog` can contain strings, ints, or floats. Note that if it contains strings, every distinct string will be a category. No stripping of whitespace is done. exog : array-like `exog` is an n x p array where n is the number of observations and p is the number of regressors including the intercept if one is included in the data. Attributes ---------- endog : array A reference to the endogenous response variable exog : array A reference to the exogenous design. J : float The number of choices for the endogenous variable. Note that this is zero-indexed. K : float The actual number of parameters for the exogenous design. Includes the constant if the design has one. names : dict A dictionary mapping the column number in `wendog` to the variables in `endog`. wendog : array An n x j array where j is the number of unique categories in `endog`. Each column of j is a dummy variable indicating the category of each observation. See `names` for a dictionary mapping each column to its category. Notes ----- See developer notes for further information on `MNLogit` internals. """
[docs] def initialize(self): """ Preprocesses the data for MNLogit. Turns the endogenous variable into an array of dummies and assigns J and K. """ super(MNLogit, self).initialize() #This is also a "whiten" method as used in other models (eg regression) wendog, self.names = tools.categorical(self.endog, drop=True, dictnames=True) self.wendog = wendog # don't drop first category self.J = float(wendog.shape[1]) self.K = float(self.exog.shape[1]) self.df_model *= (self.J-1) # for each J - 1 equation. self.df_resid = self.exog.shape[0] - self.df_model - (self.J-1)
def _eXB(self, params, exog=None): """ A private method used by the cdf. Returns ------- :math:`\exp(\beta_{j}^{\prime}x_{i})` where :math:`j = 0,1,...,J` Notes ----- A row of ones is appended for the dropped category. """ if exog == None: exog = self.exog eXB = np.exp(np.dot(params.reshape(-1, exog.shape[1]), exog.T)) eXB = np.vstack((np.ones((1, exog.shape[0])), eXB)) return eXB
[docs] def pdf(self, eXB): """ NotImplemented """ pass
[docs] def cdf(self, eXB): """ Multinomial logit cumulative distribution function. Parameters ---------- eXB : array The exponential predictor of the model exp(XB). Returns -------- The cdf evaluated at `eXB`. Notes ----- In the multinomial logit model. .. math:: \\frac{\\exp\\left(\\beta_{j}^{\\prime}x_{i}\\right)}{\\sum_{k=0}^{J}\\exp\\left(\\beta_{k}^{\\prime}x_{i}\\right)} """ num = eXB denom = eXB.sum(axis=0) return num/denom[None,:]
[docs] def loglike(self, params): """ Log-likelihood of the multinomial logit model. Parameters ---------- params : array-like The parameters of the multinomial logit model. Returns ------- The log-likelihood function of the logit model. See notes. Notes ------ .. math:: \\ln L=\\sum_{i=1}^{n}\\sum_{j=0}^{J}d_{ij}\\ln\\left(\\frac{\\exp\\left(\\beta_{j}^{\\prime}x_{i}\\right)}{\\sum_{k=0}^{J}\\exp\\left(\\beta_{k}^{\\prime}x_{i}\\right)}\\right) where :math:`d_{ij}=1` if individual `i` chose alternative `j` and 0 if not. """ d = self.wendog eXB = self._eXB(params) logprob = np.log(self.cdf(eXB)) return (d.T * logprob).sum()
[docs] def score(self, params): """ Score matrix for multinomial logit model log-likelihood Parameters ---------- params : array The parameters of the multinomial logit model. Returns -------- The 2-d score vector of the multinomial logit model evaluated at `params`. Notes ----- .. math:: \\frac{\\partial\\ln L}{\\partial\\beta_{j}}=\\sum_{i}\\left(d_{ij}-\\frac{\\exp\\left(\\beta_{j}^{\\prime}x_{i}\\right)}{\\sum_{k=0}^{J}\\exp\\left(\\beta_{k}^{\\prime}x_{i}\\right)}\\right)x_{i} for :math:`j=1,...,J` In the multinomial model ths score matrix is K x J-1 but is returned as a flattened array to work with the solvers. """ eXB = self._eXB(params) firstterm = self.wendog[:,1:].T - self.cdf(eXB)[1:,:] return np.dot(firstterm, self.exog).flatten()
[docs] def hessian(self, params): """ Multinomial logit Hessian matrix of the log-likelihood Parameters ----------- params : array-like The parameters of the model Returns ------- The Hessian evaluated at `params` Notes ----- .. math:: \\frac{\\partial^{2}\\ln L}{\\partial\\beta_{j}\\partial\\beta_{l}}=-\\sum_{i=1}^{n}\\frac{\\exp\\left(\\beta_{j}^{\\prime}x_{i}\\right)}{\\sum_{k=0}^{J}\\exp\\left(\\beta_{k}^{\\prime}x_{i}\\right)}\\left[\\boldsymbol{1}\\left(j=l\\right)-\\frac{\\exp\\left(\\beta_{l}^{\\prime}x_{i}\\right)}{\\sum_{k=0}^{J}\\exp\\left(\\beta_{k}^{\\prime}x_{i}\\right)}\\right]x_{i}x_{l}^{\\prime} where :math:`\\boldsymbol{1}\\left(j=l\\right)` equals 1 if `j` = `l` and 0 otherwise. The actual Hessian matrix has J**2 * K x K elements. Our Hessian is reshaped to be square (J*K, J*K) so that the solvers can use it. This implementation does not take advantage of the symmetry of the Hessian and could probably be refactored for speed. """ X = self.exog eXB = self._eXB(params) pr = self.cdf(eXB) partials = [] J = self.wendog.shape[1] - 1 K = self.exog.shape[1] for i in range(J): for j in range(J): # this loop assumes we drop the first col. if i == j: partials.append(\ -np.dot((pr[i+1,:]*(1-pr[j+1,:]))[None,:]*X.T,X)) else: partials.append(-np.dot(pr[i+1,:]*-pr[j+1,:][None,:]*X.T,X)) H = np.array(partials) # the developer's notes on multinomial should clear this math up H = np.transpose(H.reshape(J,J,K,K), (0,2,1,3)).reshape(J*K,J*K) return H # def fit(self, start_params=None, maxiter=35, method='newton', # tol=1e-08): # """ # Fits the multinomial logit model. # # Parameters # ---------- # start_params : array-like, optional # The default is a 0 vector. # maxiter : int, optional # Maximum number of iterations. The default is 35. # method : str, optional # `method` can be 'newton', 'ncg', 'bfgs'. The default is 'newton'. # tol : float, optional # The convergence tolerance for the solver. The default is # 1e-08. # # Notes # ----- # The reference category is always the first column of `wendog` for now. # """ # if start_params == None: # start_params = np.zeros((self.exog.shape[1]*\ # (self.wendog.shape[1]-1))) # mlefit = super(MNLogit, self).fit(start_params=start_params, # maxiter=maxiter, method=method, tol=tol) # params = mlefit.params.reshape(-1, self.exog.shape[1]) # mlefit = DiscreteResults(self, params, self.hessian(params)) # return mlefit #TODO: Weibull can replaced by a survival analsysis function # like stat's streg (The cox model as well) #class Weibull(DiscreteModel): # """ # Binary choice Weibull model # # Notes # ------ # This is unfinished and untested. # """ ##TODO: add analytic hessian for Weibull # def initialize(self): # pass # # def cdf(self, X): # """ # Gumbell (Log Weibull) cumulative distribution function # """ ## return np.exp(-np.exp(-X)) # return stats.gumbel_r.cdf(X) # # these two are equivalent. # # Greene table and discussion is incorrect. # # def pdf(self, X): # """ # Gumbell (LogWeibull) probability distribution function # """ # return stats.gumbel_r.pdf(X) # # def loglike(self, params): # """ # Loglikelihood of Weibull distribution # """ # X = self.exog # cdf = self.cdf(np.dot(X,params)) # y = self.endog # return np.sum(y*np.log(cdf) + (1-y)*np.log(1-cdf)) # # def score(self, params): # y = self.endog # X = self.exog # F = self.cdf(np.dot(X,params)) # f = self.pdf(np.dot(X,params)) # term = (y*f/F + (1 - y)*-f/(1-F)) # return np.dot(term,X) # # def hessian(self, params): # hess = nd.Jacobian(self.score) # return hess(params) # # def fit(self, start_params=None, method='newton', maxiter=35, tol=1e-08): ## The example had problems with all zero start values, Hessian = 0 # if start_params is None: # start_params = OLS(self.endog, self.exog).fit().params # mlefit = super(Weibull, self).fit(start_params=start_params, # method=method, maxiter=maxiter, tol=tol) # return mlefit #
class NBin(DiscreteModel): """ Negative Binomial model. """ # def pdf(self, X, alpha): # a1 = alpha**-1 # term1 = special.gamma(X + a1)/(special.agamma(X+1)*special.gamma(a1)) def loglike(self, params): """ Loglikelihood for negative binomial model Notes ----- The ancillary parameter is assumed to be the last element of the params vector """ lnalpha = params[-1] params = params[:-1] a1 = np.exp(lnalpha)**-1 y = self.endog J = special.gammaln(y+a1) - special.gammaln(a1) - special.gammaln(y+1) mu = np.exp(np.dot(self.exog,params)) pdf = a1*np.log(a1/(a1+mu)) + y*np.log(mu/(mu+a1)) llf = np.sum(J+pdf) return llf def score(self, params, full=False): """ Score vector for NB2 model """ lnalpha = params[-1] params = params[:-1] a1 = np.exp(lnalpha)**-1 y = self.endog[:,None] exog = self.exog mu = np.exp(np.dot(exog,params))[:,None] dparams = exog*a1 * (y-mu)/(mu+a1) da1 = -1*np.exp(lnalpha)**-2 dalpha = (special.digamma(a1+y) - special.digamma(a1) + np.log(a1)\ - np.log(a1+mu) - (a1+y)/(a1+mu) + 1) #multiply above by constant outside of the sum to reduce rounding error if full: return np.column_stack([dparams, dalpha]) return np.r_[dparams.sum(0), da1*dalpha.sum()] def hessian(self, params): """ Hessian of NB2 model. Currently uses numdifftools """ lnalpha = params[-1] params = params[:-1] a1 = np.exp(lnalpha)**-1 exog = self.exog y = self.endog[:,None] mu = np.exp(np.dot(exog,params))[:,None] # for dl/dparams dparams dim = exog.shape[1] hess_arr = np.empty((dim+1,dim+1)) const_arr = a1*mu*(a1+y)/(mu+a1)**2 for i in range(dim): for j in range(dim): if j > i: continue hess_arr[i,j] = np.sum(-exog[:,i,None]*exog[:,j,None] *\ const_arr, axis=0) hess_arr[np.triu_indices(dim, k=1)] = hess_arr.T[np.triu_indices(dim, k =1)] # for dl/dparams dalpha da1 = -1*np.exp(lnalpha)**-2 dldpda = np.sum(mu*exog*(y-mu)*da1/(mu+a1)**2 , axis=0) hess_arr[-1,:-1] = dldpda hess_arr[:-1,-1] = dldpda # for dl/dalpha dalpha #NOTE: polygamma(1,x) is the trigamma function da2 = 2*np.exp(lnalpha)**-3 dalpha = da1 * (special.digamma(a1+y) - special.digamma(a1) + \ np.log(a1) - np.log(a1+mu) - (a1+y)/(a1+mu) + 1) dada = (da2*dalpha/da1 + da1**2 * (special.polygamma(1,a1+y) - \ special.polygamma(1,a1) + 1/a1 -1/(a1+mu) + \ (y-mu)/(mu+a1)**2)).sum() hess_arr[-1,-1] = dada return hess_arr def fit(self, start_params=None, maxiter=35, method='bfgs', tol=1e-08): # start_params = [0]*(self.exog.shape[1])+[1] # Use poisson fit as first guess. start_params = Poisson(self.endog, self.exog).fit(disp=0).params start_params = np.r_[start_params, 0.1] mlefit = super(NegBinTwo, self).fit(start_params=start_params, maxiter=maxiter, method=method, tol=tol) return mlefit ### Results Class ### #class DiscreteResults(object): #TODO: these need to return z scores
[docs]class DiscreteResults(LikelihoodModelResults): """ A results class for the discrete dependent variable models. Parameters ---------- model : A DiscreteModel instance params : array-like The parameters of a fitted model. hessian : array-like The hessian of the fitted model. scale : float A scale parameter for the covariance matrix. Returns ------- *Attributes* aic : float Akaike information criterion. -2*(`llf` - p) where p is the number of regressors including the intercept. bic : float Bayesian information criterion. -2*`llf` + ln(`nobs`)*p where p is the number of regressors including the intercept. bse : array The standard errors of the coefficients. df_resid : float See model definition. df_model : float See model definition. fitted_values : array Linear predictor XB. llf : float Value of the loglikelihood llnull : float Value of the constant-only loglikelihood llr : float Likelihood ratio chi-squared statistic; -2*(`llnull` - `llf`) llr_pvalue : float The chi-squared probability of getting a log-likelihood ratio statistic greater than llr. llr has a chi-squared distribution with degrees of freedom `df_model`. prsquared : float McFadden's pseudo-R-squared. 1 - (`llf`/`llnull`) """ def __init__(self, model, mlefit): # super(DiscreteResults, self).__init__(model, params, # np.linalg.inv(-hessian), scale=1.) self.model = model self.df_model = model.df_model self.df_resid = model.df_resid self._cache = resettable_cache() self.nobs = model.exog.shape[0] self.__dict__.update(mlefit.__dict__) @cache_readonly def bse(self): bse = np.sqrt(np.diag(self.cov_params())) if self.params.ndim == 1 or self.params.shape[1] == 1: return bse else: return bse.reshape(self.params.shape) @cache_readonly def llf(self): model = self.model return model.loglike(self.params) @cache_readonly def prsquared(self): return 1 - self.llf/self.llnull @cache_readonly def llr(self): return -2*(self.llnull - self.llf) @cache_readonly def llr_pvalue(self): return stats.chisqprob(self.llr, self.df_model) @cache_readonly def llnull(self): model = self.model # will this use a new instance? #TODO: what parameters to pass to fit? null = model.__class__(model.endog, np.ones(self.nobs)).fit(disp=0) return null.llf @cache_readonly def resid(self): model = self.model endog = model.endog exog = model.exog # M = # of individuals that share a covariate pattern # so M[i] = 2 for i = the two individuals who share a covariate pattern # use unique row pattern? #TODO: is this common to all models? logit uses Pearson, should have options #These are the deviance residuals M = 1 p = model.predict() Y_0 = np.where(exog==0) Y_M = np.where(exog == M) res = np.zeros_like(endog) res = -(1-endog)*np.sqrt(2*M*np.abs(np.log(1-p))) + \ endog*np.sqrt(2*M*np.abs(np.log(p))) return res @cache_readonly def fittedvalues(self): return np.dot(self.model.exog, self.params) @cache_readonly def aic(self): if hasattr(self.model, "J"): return -2*(self.llf - (self.df_model+self.model.J-1)) else: return -2*(self.llf - (self.df_model+1)) @cache_readonly def bic(self): if hasattr(self.model, "J"): return -2*self.llf + np.log(self.nobs)*\ (self.df_model+self.model.J-1) else: return -2*self.llf + np.log(self.nobs)*(self.df_model+1)
[docs] def conf_int(self, alpha=.05, cols=None): if hasattr(self.model, "J"): confint = super(DiscreteResults, self).conf_int(alpha=alpha, cols=cols) return confint.transpose(0,2,1).reshape(self.model.J-1,self.model.K,2) else: return super(DiscreteResults, self).conf_int(alpha=alpha, cols=cols)
conf_int.__doc__ = LikelihoodModelResults.conf_int.__doc__ @cache_readonly def tvalues(self): if hasattr(self.model, "J"): # for MNLogit column = range(int(self.model.K)) return self.params/self.bse[:,column] else: return super(DiscreteResults, self).tvalues
[docs] def margeff(self, at='overall', method='dydx', atexog=None, dummy=False, count=False): """Get marginal effects of the fitted model. Parameters ---------- at : str, optional Options are: - 'overall', The average of the marginal effects at each observation. - 'mean', The marginal effects at the mean of each regressor. - 'median', The marginal effects at the median of each regressor. - 'zero', The marginal effects at zero for each regressor. - 'all', The marginal effects at each observation. Note that if `exog` is specified, then marginal effects for all variables not specified by `exog` are calculated using the `at` option. method : str, optional - 'dydx' - dy/dx - No transformation is made and marginal effects are returned. This is the default. - 'eyex' - estimate elasticities of variables in `exog` -- d(lny)/d(lnx) - 'dyex' - estimate semielasticity -- dy/d(lnx) - 'eydx' - estimate semeilasticity -- d(lny)/dx Note that tranformations are done after each observation is calculated. Semi-elasticities for binary variables are computed using the midpoint method. 'dyex' and 'eyex' do not make sense for discrete variables. atexog : array-like, optional Optionally, you can provide the exogenous variables over which to get the marginal effects. This should be a dictionary with the key as the zero-indexed column number and the value of the dictionary. Default is None for all independent variables less the constant. dummy : bool, optional If False, treats binary variables (if present) as continuous. This is the default. Else if True, treats binary variables as changing from 0 to 1. Note that any variable that is either 0 or 1 is treated as binary. Each binary variable is treated separately for now. count : bool, optional If False, treats count variables (if present) as continuous. This is the default. Else if True, the marginal effect is the change in probabilities when each observation is increased by one. Returns ------- effects : ndarray the marginal effect corresponding to the input options Notes ----- When using after Poisson, returns the expected number of events per period, assuming that the model is loglinear. """ #TODO: # factor : None or dictionary, optional # If a factor variable is present (it must be an integer, though # of type float), then `factor` may be a dict with the zero-indexed # column of the factor and the value should be the base-outcome. # check arguments if at not in ['overall','mean','median','zero','all']: raise ValueError("%s not a valid option for `at`." % at) if method not in ['dydx','eyex','dyex','eydx']: raise ValueError("method is not understood. Got %s" % method) # get local variables model = self.model params = self.params method = method.lower() at = at.lower() exog = model.exog.copy() # copy because values are changed ind = exog.var(0) != 0 # index for non-constants # handle discrete exogenous variables if dummy: _check_discrete_args(at, method) dummy_ind = _isdummy(exog) if count: _check_discrete_args(at, method) count_ind = _iscount(exog) # get the exogenous variables if atexog is not None: # user supplied if not isinstance(atexog, dict): raise ValueError("atexog should be a dict not %s"\ % type(atexog)) for key in atexog: exog[:,key] = atexog[key] if at == 'mean': exog = np.atleast_2d(exog.mean(0)) elif at == 'median': exog = np.atleast_2d(np.median(exog, axis=0)) elif at == 'zero': exog = np.zeros((1,params.shape[0])) exog[0,~ind] = 1 # get linear fitted values #TODO: just go ahead and get yhat? fittedvalues = np.dot(exog, params) #TODO: add a predict method # that takes an exog kwd # group 1 probit, logit, logistic, cloglog, heckprob, xtprobit if isinstance(model, (Probit, Logit)): effects = np.dot(model.pdf(fittedvalues)[:,None], params[None,:]) # group 2 oprobit, ologit, gologit, mlogit, biprobit #TODO # group 3 poisson, nbreg, zip, zinb elif isinstance(model, (Poisson)): effects = np.exp(fittedvalues)[:,None]*params[None,:] if 'ex' in method: effects *= exog if 'dy' in method: if at == 'all': effects = effects[:,ind] elif at == 'overall': effects = effects.mean(0)[ind] else: effects = effects[0,ind] if 'ey' in method: effects /= model.cdf(fittedvalues[:,None]) if at == 'all': effects = effects[:,ind] elif at == 'overall': effects = effects.mean(0)[ind] else: effects = effects[0,ind] if dummy == True: for i, tf in enumerate(dummy_ind): if tf == True: exog0 = exog.copy() exog0[:,i] = 0 fittedvalues0 = np.dot(exog0,params) exog1 = exog.copy() exog1[:,i] = 1 fittedvalues1 = np.dot(exog1, params) effect0 = model.cdf(fittedvalues0) effect1 = model.cdf(fittedvalues1) if 'ey' in method: effect0 = np.log(effect0) effect1 = np.log(effect1) effects[i] = (effect1 - effect0).mean() # mean for overall if count == True: for i, tf in enumerate(count_ind): if tf == True: exog0 = exog.copy() exog1 = exog.copy() exog1[:,i] += 1 effect0 = model.cdf(np.dot(exog0, params)) effect1 = model.cdf(np.dot(exog1, params)) #TODO: compute discrete elasticity correctly #Stata doesn't use the midpoint method or a weighted average. #Check elsewhere if 'ey' in method: # #TODO: don't know if this is theoretically correct fittedvalues0 = np.dot(exog0,params) fittedvalues1 = np.dot(exog1,params) # weight1 = model.exog[:,i].mean() # weight0 = 1 - weight1 wfv = (.5*model.cdf(fittedvalues1) + \ .5*model.cdf(fittedvalues0)) effects[i] = ((effect1 - effect0)/wfv).mean() effects[i] = (effect1 - effect0).mean() # Set standard error of the marginal effects by Delta method. self.margfx_se = None self.margfx = effects return effects
if __name__=="__main__": import numpy as np import scikits.statsmodels.api as sm # Scratch work for negative binomial models # dvisits was written using an R package, I can provide the dataset # on request until the copyright is cleared up #TODO: request permission to use dvisits data2 = np.genfromtxt('../datasets/dvisits/dvisits.csv', names=True) # note that this has missing values for Accident endog = data2['doctorco'] exog = data2[['sex','age','agesq','income','levyplus','freepoor', 'freerepa','illness','actdays','hscore','chcond1', 'chcond2']].view(float).reshape(len(data2),-1) exog = sm.add_constant(exog, prepend=True) poisson_mod = Poisson(endog, exog) poisson_res = poisson_mod.fit() # nb2_mod = NegBinTwo(endog, exog) # nb2_res = nb2_mod.fit() # solvers hang (with no error and no maxiter warn...) # haven't derived hessian (though it will be block diagonal) to check # newton, note that Lawless (1987) has the derivations # appear to be something wrong with the score? # according to Lawless, traditionally the likelihood is maximized wrt to B # and a gridsearch on a to determin ahat? # or the Breslow approach, which is 2 step iterative. nb2_params = [-2.190,.217,-.216,.609,-.142,.118,-.497,.145,.214,.144, .038,.099,.190,1.077] # alpha is last # taken from Cameron and Trivedi # the below is from Cameron and Trivedi as well # endog2 = np.array(endog>=1, dtype=float) # skipped for now, binary poisson results look off? data = sm.datasets.randhie.load() nbreg = NBin mod = nbreg(data.endog, data.exog.view((float,9))) #FROM STATA: params = np.asarray([-.05654133, -.21214282, .0878311, -.02991813, .22903632, .06210226, .06799715, .08407035, .18532336]) bse = [0.0062541, 0.0231818, 0.0036942, 0.0034796, 0.0305176, 0.0012397, 0.0198008, 0.0368707, 0.0766506] lnalpha = .31221786 mod.loglike(np.r_[params,np.exp(lnalpha)]) poiss_res = Poisson(data.endog, data.exog.view((float,9))).fit() func = lambda x: -mod.loglike(x) grad = lambda x: -mod.score(x) from scipy import optimize # res1 = optimize.fmin_l_bfgs_b(func, np.r_[poiss_res.params,.1], # approx_grad=True) res1 = optimize.fmin_bfgs(func, np.r_[poiss_res.params,.1], fprime=grad) from scikits.statsmodels.sandbox.regression.numdiff import approx_hess_cs # np.sqrt(np.diag(-np.linalg.inv(approx_hess_cs(np.r_[params,lnalpha], mod.loglike)))) #NOTE: this is the hessian in terms of alpha _not_ lnalpha hess_arr = mod.hessian(res1)