232 lines
7.1 KiB
Python
232 lines
7.1 KiB
Python
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"""Linear Model with Student-t distributed errors
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Because the t distribution has fatter tails than the normal distribution, it
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can be used to model observations with heavier tails and observations that have
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some outliers. For the latter case, the t-distribution provides more robust
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estimators for mean or mean parameters (what about var?).
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References
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----------
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Kenneth L. Lange, Roderick J. A. Little, Jeremy M. G. Taylor (1989)
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Robust Statistical Modeling Using the t Distribution
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Journal of the American Statistical Association
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Vol. 84, No. 408 (Dec., 1989), pp. 881-896
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Published by: American Statistical Association
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Stable URL: http://www.jstor.org/stable/2290063
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not read yet
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Created on 2010-09-24
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Author: josef-pktd
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License: BSD
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TODO
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----
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* add starting values based on OLS
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* bugs: store_params does not seem to be defined, I think this was a module
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global for debugging - commented out
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* parameter restriction: check whether version with some fixed parameters works
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"""
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#mostly copied from the examples directory written for trying out generic mle.
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import numpy as np
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from scipy import special, stats
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from statsmodels.base.model import GenericLikelihoodModel
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from statsmodels.tsa.arma_mle import Arma
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#redefine some shortcuts
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np_log = np.log
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np_pi = np.pi
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sps_gamln = special.gammaln
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class TLinearModel(GenericLikelihoodModel):
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'''Maximum Likelihood Estimation of Linear Model with t-distributed errors
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This is an example for generic MLE.
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Except for defining the negative log-likelihood method, all
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methods and results are generic. Gradients and Hessian
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and all resulting statistics are based on numerical
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differentiation.
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'''
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def initialize(self):
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print("running Tmodel initialize")
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# TODO: here or in __init__
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self.k_vars = self.exog.shape[1]
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if not hasattr(self, 'fix_df'):
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self.fix_df = False
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if self.fix_df is False:
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# df will be estimated, no parameter restrictions
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self.fixed_params = None
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self.fixed_paramsmask = None
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self.k_params = self.exog.shape[1] + 2
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extra_params_names = ['df', 'scale']
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else:
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# df fixed
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self.k_params = self.exog.shape[1] + 1
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fixdf = np.nan * np.zeros(self.exog.shape[1] + 2)
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fixdf[-2] = self.fix_df
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self.fixed_params = fixdf
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self.fixed_paramsmask = np.isnan(fixdf)
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extra_params_names = ['scale']
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super().initialize()
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# Note: this needs to be after super initialize
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# super initialize sets default df_resid,
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#_set_extra_params_names adjusts it
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self._set_extra_params_names(extra_params_names)
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self._set_start_params()
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def _set_start_params(self, start_params=None, use_kurtosis=False):
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if start_params is not None:
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self.start_params = start_params
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else:
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from statsmodels.regression.linear_model import OLS
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res_ols = OLS(self.endog, self.exog).fit()
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start_params = 0.1*np.ones(self.k_params)
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start_params[:self.k_vars] = res_ols.params
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if self.fix_df is False:
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if use_kurtosis:
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kurt = stats.kurtosis(res_ols.resid)
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df = 6./kurt + 4
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else:
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df = 5
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start_params[-2] = df
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#TODO adjust scale for df
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start_params[-1] = np.sqrt(res_ols.scale)
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self.start_params = start_params
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def loglike(self, params):
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return -self.nloglikeobs(params).sum(0)
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def nloglikeobs(self, params):
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"""
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Loglikelihood of linear model with t distributed errors.
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Parameters
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----------
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params : ndarray
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The parameters of the model. The last 2 parameters are degrees of
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freedom and scale.
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Returns
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-------
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loglike : ndarray
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The log likelihood of the model evaluated at `params` for each
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observation defined by self.endog and self.exog.
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Notes
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-----
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.. math:: \\ln L=\\sum_{i=1}^{n}\\left[-\\lambda_{i}+y_{i}x_{i}^{\\prime}\\beta-\\ln y_{i}!\\right]
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The t distribution is the standard t distribution and not a standardized
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t distribution, which means that the scale parameter is not equal to the
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standard deviation.
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self.fixed_params and self.expandparams can be used to fix some
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parameters. (I doubt this has been tested in this model.)
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"""
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#print len(params),
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#store_params.append(params)
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if self.fixed_params is not None:
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#print 'using fixed'
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params = self.expandparams(params)
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beta = params[:-2]
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df = params[-2]
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scale = np.abs(params[-1]) #TODO check behavior around zero
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loc = np.dot(self.exog, beta)
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endog = self.endog
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x = (endog - loc)/scale
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#next part is stats.t._logpdf
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lPx = sps_gamln((df+1)/2) - sps_gamln(df/2.)
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lPx -= 0.5*np_log(df*np_pi) + (df+1)/2.*np_log(1+(x**2)/df)
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lPx -= np_log(scale) # correction for scale
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return -lPx
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def predict(self, params, exog=None):
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if exog is None:
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exog = self.exog
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return np.dot(exog, params[:self.exog.shape[1]])
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class TArma(Arma):
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'''Univariate Arma Model with t-distributed errors
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This inherit all methods except loglike from tsa.arma_mle.Arma
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This uses the standard t-distribution, the implied variance of
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the error is not equal to scale, but ::
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error_variance = df/(df-2)*scale**2
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Notes
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-----
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This might be replaced by a standardized t-distribution with scale**2
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equal to variance
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'''
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def loglike(self, params):
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return -self.nloglikeobs(params).sum(0)
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#add for Jacobian calculation bsejac in GenericMLE, copied from loglike
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def nloglikeobs(self, params):
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"""
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Loglikelihood for arma model for each observation, t-distribute
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Notes
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-----
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The ancillary parameter is assumed to be the last element of
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the params vector
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"""
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errorsest = self.geterrors(params[:-2])
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#sigma2 = np.maximum(params[-1]**2, 1e-6) #do I need this
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#axis = 0
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#nobs = len(errorsest)
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df = params[-2]
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scale = np.abs(params[-1])
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llike = - stats.t._logpdf(errorsest/scale, df) + np_log(scale)
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return llike
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#TODO rename fit_mle -> fit, fit -> fit_ls
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def fit_mle(self, order, start_params=None, method='nm', maxiter=5000,
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tol=1e-08, **kwds):
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nar, nma = order
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if start_params is not None:
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if len(start_params) != nar + nma + 2:
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raise ValueError('start_param need sum(order) + 2 elements')
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else:
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start_params = np.concatenate((0.05*np.ones(nar + nma), [5, 1]))
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res = super().fit_mle(order=order,
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start_params=start_params,
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method=method, maxiter=maxiter,
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tol=tol, **kwds)
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return res
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