Source code for statsmodels.miscmodels.tmodel

"""Linear Model with Student-t distributed errors

Because the t distribution has fatter tails than the normal distribution, it
can be used to model observations with heavier tails and observations that have
some outliers. For the latter case, the t-distribution provides more robust
estimators for mean or mean parameters (what about var?).



References
----------
Kenneth L. Lange, Roderick J. A. Little, Jeremy M. G. Taylor (1989)
Robust Statistical Modeling Using the t Distribution
Journal of the American Statistical Association
Vol. 84, No. 408 (Dec., 1989), pp. 881-896
Published by: American Statistical Association
Stable URL: http://www.jstor.org/stable/2290063

not read yet


Created on 2010-09-24
Author: josef-pktd
License: BSD

TODO
----
* add starting values based on OLS
* bugs: store_params doesn't seem to be defined, I think this was a module
        global for debugging - commented out
* parameter restriction: check whether version with some fixed parameters works


"""
#mostly copied from the examples directory written for trying out generic mle.

import numpy as np
from scipy import special #, stats
#redefine some shortcuts
np_log = np.log
np_pi = np.pi
sps_gamln = special.gammaln


from statsmodels.base.model import GenericLikelihoodModel

[docs]class TLinearModel(GenericLikelihoodModel): '''Maximum Likelihood Estimation of Linear Model with t-distributed errors This is an example for generic MLE. Except for defining the negative log-likelihood method, all methods and results are generic. Gradients and Hessian and all resulting statistics are based on numerical differentiation. '''
[docs] def initialize(self): # TODO: here or in __init__ self.k_vars = self.exog.shape[1] if not hasattr(self, 'fix_df'): self.fix_df = False if self.fix_df is False: # df will be estimated, no parameter restrictions self.fixed_params = None self.fixed_paramsmask = None self.k_params = self.exog.shape[1] + 2 extra_params_names = ['df', 'scale'] else: # df fixed self.k_params = self.exog.shape[1] + 1 fixdf = np.nan * np.zeros(self.exog.shape[1] + 2) fixdf[-2] = self.fix_df self.fixed_params = fixdf self.fixed_paramsmask = np.isnan(fixdf) extra_params_names = ['scale'] self._set_extra_params_names(extra_params_names) self._set_start_params() super(TLinearModel, self).initialize()
def _set_start_params(self, start_params=None, use_kurtosis=False): if start_params is not None: self.start_params = start_params else: from statsmodels.regression.linear_model import OLS res_ols = OLS(self.endog, self.exog).fit() start_params = 0.1*np.ones(self.k_params) start_params[:self.k_vars] = res_ols.params if self.fix_df is False: if use_kurtosis: kurt = stats.kurtosis(res_ols.resid) df = 6./kurt + 4 else: df = 5 start_params[-2] = df #TODO adjust scale for df start_params[-1] = np.sqrt(res_ols.scale) self.start_params = start_params
[docs] def loglike(self, params): return -self.nloglikeobs(params).sum(0)
[docs] def nloglikeobs(self, params): """ Loglikelihood of linear model with t distributed errors. Parameters ---------- params : array The parameters of the model. The last 2 parameters are degrees of freedom and scale. Returns ------- loglike : array The log likelihood of the model evaluated at `params` for each observation defined by self.endog and self.exog. Notes ----- .. math:: \\ln L=\\sum_{i=1}^{n}\\left[-\\lambda_{i}+y_{i}x_{i}^{\\prime}\\beta-\\ln y_{i}!\\right] The t distribution is the standard t distribution and not a standardized t distribution, which means that the scale parameter is not equal to the standard deviation. self.fixed_params and self.expandparams can be used to fix some parameters. (I doubt this has been tested in this model.) """ #print len(params), #store_params.append(params) if self.fixed_params is not None: #print 'using fixed' params = self.expandparams(params) beta = params[:-2] df = params[-2] scale = np.abs(params[-1]) #TODO check behavior around zero loc = np.dot(self.exog, beta) endog = self.endog x = (endog - loc)/scale #next part is stats.t._logpdf lPx = sps_gamln((df+1)/2) - sps_gamln(df/2.) lPx -= 0.5*np_log(df*np_pi) + (df+1)/2.*np_log(1+(x**2)/df) lPx -= np_log(scale) # correction for scale return -lPx
[docs] def predict(self, params, exog=None): if exog is None: exog = self.exog return np.dot(exog, params[:self.exog.shape[1]])
from scipy import stats from statsmodels.tsa.arma_mle import Arma class TArma(Arma): '''Univariate Arma Model with t-distributed errors This inherit all methods except loglike from tsa.arma_mle.Arma This uses the standard t-distribution, the implied variance of the error is not equal to scale, but :: error_variance = df/(df-2)*scale**2 Notes ----- This might be replaced by a standardized t-distribution with scale**2 equal to variance ''' def loglike(self, params): return -self.nloglikeobs(params).sum(0) #add for Jacobian calculation bsejac in GenericMLE, copied from loglike def nloglikeobs(self, params): """ Loglikelihood for arma model for each observation, t-distribute Notes ----- The ancillary parameter is assumed to be the last element of the params vector """ errorsest = self.geterrors(params[:-2]) #sigma2 = np.maximum(params[-1]**2, 1e-6) #do I need this #axis = 0 #nobs = len(errorsest) df = params[-2] scale = np.abs(params[-1]) llike = - stats.t._logpdf(errorsest/scale, df) + np_log(scale) return llike #TODO rename fit_mle -> fit, fit -> fit_ls def fit_mle(self, order, start_params=None, method='nm', maxiter=5000, tol=1e-08, **kwds): nar, nma = order if start_params is not None: if len(start_params) != nar + nma + 2: raise ValueError('start_param need sum(order) + 2 elements') else: start_params = np.concatenate((0.05*np.ones(nar + nma), [5, 1])) res = super(TArma, self).fit_mle(order=order, start_params=start_params, method=method, maxiter=maxiter, tol=tol, **kwds) return res