"""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
from statsmodels.base.model import GenericLikelihoodModel
from statsmodels.tsa.arma_mle import Arma
#redefine some shortcuts
np_log = np.log
np_pi = np.pi
sps_gamln = special.gammaln
[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]])
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
