""" Robust linear models with support for the M-estimators listed under :ref:`norms `. References ---------- PJ Huber. 'Robust Statistics' John Wiley and Sons, Inc., New York. 1981. PJ Huber. 1973, 'The 1972 Wald Memorial Lectures: Robust Regression: Asymptotics, Conjectures, and Monte Carlo.' The Annals of Statistics, 1.5, 799-821. R Venables, B Ripley. 'Modern Applied Statistics in S' Springer, New York, 2002. """ import numpy as np import scipy.stats as stats import statsmodels.base.model as base import statsmodels.base.wrapper as wrap import statsmodels.regression._tools as reg_tools import statsmodels.regression.linear_model as lm import statsmodels.robust.norms as norms import statsmodels.robust.scale as scale from statsmodels.tools.decorators import cache_readonly from statsmodels.tools.sm_exceptions import ConvergenceWarning __all__ = ['RLM'] def _check_convergence(criterion, iteration, tol, maxiter): cond = np.abs(criterion[iteration] - criterion[iteration - 1]) return not (np.any(cond > tol) and iteration < maxiter) class RLM(base.LikelihoodModel): __doc__ = """ Robust Linear Model Estimate a robust linear model via iteratively reweighted least squares given a robust criterion estimator. {params} M : statsmodels.robust.norms.RobustNorm, optional The robust criterion function for downweighting outliers. The current options are LeastSquares, HuberT, RamsayE, AndrewWave, TrimmedMean, Hampel, and TukeyBiweight. The default is HuberT(). See statsmodels.robust.norms for more information. {extra_params} Attributes ---------- df_model : float The degrees of freedom of the model. The number of regressors p less one for the intercept. Note that the reported model degrees of freedom does not count the intercept as a regressor, though the model is assumed to have an intercept. df_resid : float The residual degrees of freedom. The number of observations n less the number of regressors p. Note that here p does include the intercept as using a degree of freedom. endog : ndarray See above. Note that endog is a reference to the data so that if data is already an array and it is changed, then `endog` changes as well. exog : ndarray See above. Note that endog is a reference to the data so that if data is already an array and it is changed, then `endog` changes as well. M : statsmodels.robust.norms.RobustNorm See above. Robust estimator instance instantiated. nobs : float The number of observations n pinv_wexog : ndarray The pseudoinverse of the design / exogenous data array. Note that RLM has no whiten method, so this is just the pseudo inverse of the design. normalized_cov_params : ndarray The p x p normalized covariance of the design / exogenous data. This is approximately equal to (X.T X)^(-1) Examples -------- >>> import statsmodels.api as sm >>> data = sm.datasets.stackloss.load() >>> data.exog = sm.add_constant(data.exog) >>> rlm_model = sm.RLM(data.endog, data.exog, \ M=sm.robust.norms.HuberT()) >>> rlm_results = rlm_model.fit() >>> rlm_results.params array([ 0.82938433, 0.92606597, -0.12784672, -41.02649835]) >>> rlm_results.bse array([ 0.11100521, 0.30293016, 0.12864961, 9.79189854]) >>> rlm_results_HC2 = rlm_model.fit(cov="H2") >>> rlm_results_HC2.params array([ 0.82938433, 0.92606597, -0.12784672, -41.02649835]) >>> rlm_results_HC2.bse array([ 0.11945975, 0.32235497, 0.11796313, 9.08950419]) >>> mod = sm.RLM(data.endog, data.exog, M=sm.robust.norms.Hampel()) >>> rlm_hamp_hub = mod.fit(scale_est=sm.robust.scale.HuberScale()) >>> rlm_hamp_hub.params array([ 0.73175452, 1.25082038, -0.14794399, -40.27122257]) """.format( params=base._model_params_doc, extra_params=base._missing_param_doc ) def __init__(self, endog, exog, M=None, missing='none', **kwargs): self._check_kwargs(kwargs) self.M = M if M is not None else norms.HuberT() super(base.LikelihoodModel, self).__init__(endog, exog, missing=missing, **kwargs) self._initialize() # things to remove_data self._data_attr.extend(['weights', 'pinv_wexog']) def _initialize(self): """ Initializes the model for the IRLS fit. Resets the history and number of iterations. """ self.pinv_wexog = np.linalg.pinv(self.exog) self.normalized_cov_params = np.dot(self.pinv_wexog, np.transpose(self.pinv_wexog)) self.df_resid = (float(self.exog.shape[0] - np.linalg.matrix_rank(self.exog))) self.df_model = float(np.linalg.matrix_rank(self.exog) - 1) self.nobs = float(self.endog.shape[0]) def score(self, params): raise NotImplementedError def information(self, params): raise NotImplementedError def predict(self, params, exog=None): """ Return linear predicted values from a design matrix. Parameters ---------- params : array_like Parameters of a linear model exog : array_like, optional. Design / exogenous data. Model exog is used if None. Returns ------- An array of fitted values """ # copied from linear_model # TODO: then is it needed? if exog is None: exog = self.exog return np.dot(exog, params) def loglike(self, params): raise NotImplementedError def deviance(self, tmp_results): """ Returns the (unnormalized) log-likelihood from the M estimator. """ tmp_resid = self.endog - tmp_results.fittedvalues return self.M(tmp_resid / tmp_results.scale).sum() def _update_history(self, tmp_results, history, conv): history['params'].append(tmp_results.params) history['scale'].append(tmp_results.scale) if conv == 'dev': history['deviance'].append(self.deviance(tmp_results)) elif conv == 'sresid': history['sresid'].append(tmp_results.resid / tmp_results.scale) elif conv == 'weights': history['weights'].append(tmp_results.model.weights) return history def _estimate_scale(self, resid): """ Estimates the scale based on the option provided to the fit method. """ if isinstance(self.scale_est, str): if self.scale_est.lower() == 'mad': return scale.mad(resid, center=0) else: raise ValueError("Option %s for scale_est not understood" % self.scale_est) elif isinstance(self.scale_est, scale.HuberScale): return self.scale_est(self.df_resid, self.nobs, resid) else: return scale.scale_est(self, resid) ** 2 def fit(self, maxiter=50, tol=1e-8, scale_est='mad', init=None, cov='H1', update_scale=True, conv='dev', start_params=None): """ Fits the model using iteratively reweighted least squares. The IRLS routine runs until the specified objective converges to `tol` or `maxiter` has been reached. Parameters ---------- conv : str Indicates the convergence criteria. Available options are "coefs" (the coefficients), "weights" (the weights in the iteration), "sresid" (the standardized residuals), and "dev" (the un-normalized log-likelihood for the M estimator). The default is "dev". cov : str, optional 'H1', 'H2', or 'H3' Indicates how the covariance matrix is estimated. Default is 'H1'. See rlm.RLMResults for more information. init : str Specifies method for the initial estimates of the parameters. Default is None, which means that the least squares estimate is used. Currently it is the only available choice. maxiter : int The maximum number of iterations to try. Default is 50. scale_est : str or HuberScale() 'mad' or HuberScale() Indicates the estimate to use for scaling the weights in the IRLS. The default is 'mad' (median absolute deviation. Other options are 'HuberScale' for Huber's proposal 2. Huber's proposal 2 has optional keyword arguments d, tol, and maxiter for specifying the tuning constant, the convergence tolerance, and the maximum number of iterations. See statsmodels.robust.scale for more information. tol : float The convergence tolerance of the estimate. Default is 1e-8. update_scale : Bool If `update_scale` is False then the scale estimate for the weights is held constant over the iteration. Otherwise, it is updated for each fit in the iteration. Default is True. start_params : array_like, optional Initial guess of the solution of the optimizer. If not provided, the initial parameters are computed using OLS. Returns ------- results : statsmodels.rlm.RLMresults Results instance """ if cov.upper() not in ["H1", "H2", "H3"]: raise ValueError("Covariance matrix %s not understood" % cov) else: self.cov = cov.upper() conv = conv.lower() if conv not in ["weights", "coefs", "dev", "sresid"]: raise ValueError("Convergence argument %s not understood" % conv) self.scale_est = scale_est if start_params is None: wls_results = lm.WLS(self.endog, self.exog).fit() else: start_params = np.asarray(start_params, dtype=np.double).squeeze() if (start_params.shape[0] != self.exog.shape[1] or start_params.ndim != 1): raise ValueError('start_params must by a 1-d array with {} ' 'values'.format(self.exog.shape[1])) fake_wls = reg_tools._MinimalWLS(self.endog, self.exog, weights=np.ones_like(self.endog), check_weights=False) wls_results = fake_wls.results(start_params) if not init: self.scale = self._estimate_scale(wls_results.resid) history = dict(params=[np.inf], scale=[]) if conv == 'coefs': criterion = history['params'] elif conv == 'dev': history.update(dict(deviance=[np.inf])) criterion = history['deviance'] elif conv == 'sresid': history.update(dict(sresid=[np.inf])) criterion = history['sresid'] elif conv == 'weights': history.update(dict(weights=[np.inf])) criterion = history['weights'] # done one iteration so update history = self._update_history(wls_results, history, conv) iteration = 1 converged = 0 while not converged: if self.scale == 0.0: import warnings warnings.warn('Estimated scale is 0.0 indicating that the most' ' last iteration produced a perfect fit of the ' 'weighted data.', ConvergenceWarning) break self.weights = self.M.weights(wls_results.resid / self.scale) wls_results = reg_tools._MinimalWLS(self.endog, self.exog, weights=self.weights, check_weights=True).fit() if update_scale is True: self.scale = self._estimate_scale(wls_results.resid) history = self._update_history(wls_results, history, conv) iteration += 1 converged = _check_convergence(criterion, iteration, tol, maxiter) results = RLMResults(self, wls_results.params, self.normalized_cov_params, self.scale) history['iteration'] = iteration results.fit_history = history results.fit_options = dict(cov=cov.upper(), scale_est=scale_est, norm=self.M.__class__.__name__, conv=conv) # norm is not changed in fit, no old state # doing the next causes exception # self.cov = self.scale_est = None #reset for additional fits # iteration and history could contain wrong state with repeated fit return RLMResultsWrapper(results) class RLMResults(base.LikelihoodModelResults): """ Class to contain RLM results Attributes ---------- bcov_scaled : ndarray p x p scaled covariance matrix specified in the model fit method. The default is H1. H1 is defined as ``k**2 * (1/df_resid*sum(M.psi(sresid)**2)*scale**2)/ ((1/nobs*sum(M.psi_deriv(sresid)))**2) * (X.T X)^(-1)`` where ``k = 1 + (df_model +1)/nobs * var_psiprime/m**2`` where ``m = mean(M.psi_deriv(sresid))`` and ``var_psiprime = var(M.psi_deriv(sresid))`` H2 is defined as ``k * (1/df_resid) * sum(M.psi(sresid)**2) *scale**2/ ((1/nobs)*sum(M.psi_deriv(sresid)))*W_inv`` H3 is defined as ``1/k * (1/df_resid * sum(M.psi(sresid)**2)*scale**2 * (W_inv X.T X W_inv))`` where `k` is defined as above and ``W_inv = (M.psi_deriv(sresid) exog.T exog)^(-1)`` See the technical documentation for cleaner formulae. bcov_unscaled : ndarray The usual p x p covariance matrix with scale set equal to 1. It is then just equivalent to normalized_cov_params. bse : ndarray An array of the standard errors of the parameters. The standard errors are taken from the robust covariance matrix specified in the argument to fit. chisq : ndarray An array of the chi-squared values of the parameter estimates. df_model See RLM.df_model df_resid See RLM.df_resid fit_history : dict Contains information about the iterations. Its keys are `deviance`, `params`, `iteration` and the convergence criteria specified in `RLM.fit`, if different from `deviance` or `params`. fit_options : dict Contains the options given to fit. fittedvalues : ndarray The linear predicted values. dot(exog, params) model : statsmodels.rlm.RLM A reference to the model instance nobs : float The number of observations n normalized_cov_params : ndarray See RLM.normalized_cov_params params : ndarray The coefficients of the fitted model pinv_wexog : ndarray See RLM.pinv_wexog pvalues : ndarray The p values associated with `tvalues`. Note that `tvalues` are assumed to be distributed standard normal rather than Student's t. resid : ndarray The residuals of the fitted model. endog - fittedvalues scale : float The type of scale is determined in the arguments to the fit method in RLM. The reported scale is taken from the residuals of the weighted least squares in the last IRLS iteration if update_scale is True. If update_scale is False, then it is the scale given by the first OLS fit before the IRLS iterations. sresid : ndarray The scaled residuals. tvalues : ndarray The "t-statistics" of params. These are defined as params/bse where bse are taken from the robust covariance matrix specified in the argument to fit. weights : ndarray The reported weights are determined by passing the scaled residuals from the last weighted least squares fit in the IRLS algorithm. See Also -------- statsmodels.base.model.LikelihoodModelResults """ def __init__(self, model, params, normalized_cov_params, scale): super().__init__(model, params, normalized_cov_params, scale) self.model = model self.df_model = model.df_model self.df_resid = model.df_resid self.nobs = model.nobs self._cache = {} # for remove_data self._data_in_cache.extend(['sresid']) self.cov_params_default = self.bcov_scaled # TODO: "pvals" should come from chisq on bse? @cache_readonly def fittedvalues(self): return np.dot(self.model.exog, self.params) @cache_readonly def resid(self): return self.model.endog - self.fittedvalues # before bcov @cache_readonly def sresid(self): if self.scale == 0.0: sresid = self.resid.copy() sresid[:] = 0.0 return sresid return self.resid / self.scale @cache_readonly def bcov_unscaled(self): return self.normalized_cov_params @cache_readonly def weights(self): return self.model.weights @cache_readonly def bcov_scaled(self): model = self.model m = np.mean(model.M.psi_deriv(self.sresid)) var_psiprime = np.var(model.M.psi_deriv(self.sresid)) k = 1 + (self.df_model + 1) / self.nobs * var_psiprime / m ** 2 if model.cov == "H1": ss_psi = np.sum(model.M.psi(self.sresid) ** 2) s_psi_deriv = np.sum(model.M.psi_deriv(self.sresid)) return k ** 2 * (1 / self.df_resid * ss_psi * self.scale ** 2) /\ ((1 / self.nobs * s_psi_deriv) ** 2) *\ model.normalized_cov_params else: W = np.dot(model.M.psi_deriv(self.sresid) * model.exog.T, model.exog) W_inv = np.linalg.inv(W) # [W_jk]^-1 = [SUM(psi_deriv(Sr_i)*x_ij*x_jk)]^-1 # where Sr are the standardized residuals if model.cov == "H2": # These are correct, based on Huber (1973) 8.13 return k * (1 / self.df_resid) * np.sum( model.M.psi(self.sresid) ** 2) * self.scale ** 2 \ / ((1 / self.nobs) * np.sum(model.M.psi_deriv(self.sresid))) * W_inv elif model.cov == "H3": return k ** -1 * 1 / self.df_resid * np.sum( model.M.psi(self.sresid) ** 2) * self.scale ** 2 \ * np.dot( np.dot(W_inv, np.dot(model.exog.T, model.exog)), W_inv) @cache_readonly def pvalues(self): return stats.norm.sf(np.abs(self.tvalues)) * 2 @cache_readonly def bse(self): return np.sqrt(np.diag(self.bcov_scaled)) @cache_readonly def chisq(self): return (self.params / self.bse) ** 2 def summary(self, yname=None, xname=None, title=0, alpha=.05, return_fmt='text'): """ This is for testing the new summary setup """ top_left = [('Dep. Variable:', None), ('Model:', None), ('Method:', ['IRLS']), ('Norm:', [self.fit_options['norm']]), ('Scale Est.:', [self.fit_options['scale_est']]), ('Cov Type:', [self.fit_options['cov']]), ('Date:', None), ('Time:', None), ('No. Iterations:', ["%d" % self.fit_history['iteration']]) ] top_right = [('No. Observations:', None), ('Df Residuals:', None), ('Df Model:', None) ] if title is not None: title = "Robust linear Model Regression Results" # boiler plate from statsmodels.iolib.summary import Summary smry = Summary() smry.add_table_2cols(self, gleft=top_left, gright=top_right, yname=yname, xname=xname, title=title) smry.add_table_params(self, yname=yname, xname=xname, alpha=alpha, use_t=self.use_t) # add warnings/notes, added to text format only etext = [] wstr = ("If the model instance has been used for another fit with " "different fit parameters, then the fit options might not be " "the correct ones anymore .") etext.append(wstr) if etext: smry.add_extra_txt(etext) return smry def summary2(self, xname=None, yname=None, title=None, alpha=.05, float_format="%.4f"): """Experimental summary function for regression results Parameters ---------- yname : str Name of the dependent variable (optional) xname : list[str], optional Names for the exogenous variables. Default is `var_##` for ## in the number of regressors. Must match the number of parameters in the model title : str, optional Title for the top table. If not None, then this replaces the default title alpha : float significance level for the confidence intervals float_format : str print format for floats in parameters summary Returns ------- smry : Summary instance this holds the summary tables and text, which can be printed or converted to various output formats. See Also -------- statsmodels.iolib.summary2.Summary : class to hold summary results """ from statsmodels.iolib import summary2 smry = summary2.Summary() smry.add_base(results=self, alpha=alpha, float_format=float_format, xname=xname, yname=yname, title=title) return smry class RLMResultsWrapper(lm.RegressionResultsWrapper): pass wrap.populate_wrapper(RLMResultsWrapper, RLMResults) # noqa:E305