Source code for statsmodels.stats.outliers_influence

# -*- coding: utf-8 -*-
"""Influence and Outlier Measures

Created on Sun Jan 29 11:16:09 2012

Author: Josef Perktold
License: BSD-3
from statsmodels.compat.python import lzip
from collections import defaultdict
import numpy as np

from statsmodels.regression.linear_model import OLS
from import cache_readonly
from statsmodels.stats.multitest import multipletests
from import maybe_unwrap_results

# outliers test convenience wrapper

def outlier_test(model_results, method='bonf', alpha=.05, labels=None,
    Outlier Tests for RegressionResults instances.

    model_results : RegressionResults instance
        Linear model results
    method : str
        - `bonferroni` : one-step correction
        - `sidak` : one-step correction
        - `holm-sidak` :
        - `holm` :
        - `simes-hochberg` :
        - `hommel` :
        - `fdr_bh` : Benjamini/Hochberg
        - `fdr_by` : Benjamini/Yekutieli
        See `statsmodels.stats.multitest.multipletests` for details.
    alpha : float
        familywise error rate
    order : bool
        Whether or not to order the results by the absolute value of the
        studentized residuals. If labels are provided they will also be sorted.

    table : ndarray or DataFrame
        Returns either an ndarray or a DataFrame if labels is not None.
        Will attempt to get labels from model_results if available. The
        columns are the Studentized residuals, the unadjusted p-value,
        and the corrected p-value according to method.

    The unadjusted p-value is stats.t.sf(abs(resid), df) where
    df = df_resid - 1.
    from scipy import stats # lazy import
    infl = getattr(model_results, 'get_influence', None)
    if infl is None:
        results = maybe_unwrap_results(model_results)
        raise AttributeError("model_results object %s does not have a "
                "get_influence method." % results.__class__.__name__)
    resid = infl().resid_studentized_external
    if order:
        idx = np.abs(resid).argsort()[::-1]
        resid = resid[idx]
        if labels is not None:
            labels = np.array(labels)[idx].tolist()
    df = model_results.df_resid - 1
    unadj_p = stats.t.sf(np.abs(resid), df) * 2
    adj_p = multipletests(unadj_p, alpha=alpha, method=method)

    data = np.c_[resid, unadj_p, adj_p[1]]
    if labels is None:
        labels = getattr(, 'row_labels', None)
    if labels is not None:
        from pandas import DataFrame
        return DataFrame(data,
                         columns=['student_resid', 'unadj_p', method+"(p)"],
    return data

#influence measures

def reset_ramsey(res, degree=5):
    '''Ramsey's RESET specification test for linear models

    This is a general specification test, for additional non-linear effects
    in a model.

    The test fits an auxiliary OLS regression where the design matrix, exog,
    is augmented by powers 2 to degree of the fitted values. Then it performs
    an F-test whether these additional terms are significant.

    If the p-value of the f-test is below a threshold, e.g. 0.1, then this
    indicates that there might be additional non-linear effects in the model
    and that the linear model is mis-specified.


    order = degree + 1
    k_vars = res.model.exog.shape[1]
    #vander without constant and x:
    y_fitted_vander = np.vander(res.fittedvalues, order)[:, :-2] #drop constant
    exog = np.column_stack((res.model.exog, y_fitted_vander))
    res_aux = OLS(res.model.endog, exog).fit()
    #r_matrix = np.eye(degree, exog.shape[1], k_vars)
    r_matrix = np.eye(degree-1, exog.shape[1], k_vars)
    #df1 = degree - 1
    #df2 = exog.shape[0] - degree - res.df_model  (without constant)
    return res_aux.f_test(r_matrix) #, r_matrix, res_aux

[docs]def variance_inflation_factor(exog, exog_idx): '''variance inflation factor, VIF, for one exogenous variable The variance inflation factor is a measure for the increase of the variance of the parameter estimates if an additional variable, given by exog_idx is added to the linear regression. It is a measure for multicollinearity of the design matrix, exog. One recommendation is that if VIF is greater than 5, then the explanatory variable given by exog_idx is highly collinear with the other explanatory variables, and the parameter estimates will have large standard errors because of this. Parameters ---------- exog : ndarray, (nobs, k_vars) design matrix with all explanatory variables, as for example used in regression exog_idx : int index of the exogenous variable in the columns of exog Returns ------- vif : float variance inflation factor Notes ----- This function does not save the auxiliary regression. See Also -------- xxx : class for regression diagnostics TODO: doesn't exist yet References ---------- ''' k_vars = exog.shape[1] x_i = exog[:, exog_idx] mask = np.arange(k_vars) != exog_idx x_noti = exog[:, mask] r_squared_i = OLS(x_i, x_noti).fit().rsquared vif = 1. / (1. - r_squared_i) return vif
[docs]class OLSInfluence(object): '''class to calculate outlier and influence measures for OLS result Parameters ---------- results : Regression Results instance currently assumes the results are from an OLS regression Notes ----- One part of the results can be calculated without any auxiliary regression (some of which have the `_internal` postfix in the name. Other statistics require leave-one-observation-out (LOOO) auxiliary regression, and will be slower (mainly results with `_external` postfix in the name). The auxiliary LOOO regression only the required results are stored. Using the LOO measures is currently only recommended if the data set is not too large. One possible approach for LOOO measures would be to identify possible problem observations with the _internal measures, and then run the leave-one-observation-out only with observations that are possible outliers. (However, this is not yet available in an automized way.) This should be extended to general least squares. The leave-one-variable-out (LOVO) auxiliary regression are currently not used. ''' def __init__(self, results): #check which model is allowed self.results = maybe_unwrap_results(results) self.nobs, self.k_vars = results.model.exog.shape self.endog = results.model.endog self.exog = results.model.exog self.model_class = results.model.__class__ self.sigma_est = np.sqrt(results.mse_resid) self.aux_regression_exog = {} self.aux_regression_endog = {} @cache_readonly
[docs] def hat_matrix_diag(self): '''(cached attribute) diagonal of the hat_matrix for OLS Notes ----- temporarily calculated here, this should go to model class ''' return (self.exog * self.results.model.pinv_wexog.T).sum(1)
[docs] def resid_press(self): '''(cached attribute) PRESS residuals ''' hii = self.hat_matrix_diag return self.results.resid / (1 - hii)
[docs] def influence(self): '''(cached attribute) influence measure matches the influence measure that gretl reports u * h / (1 - h) where u are the residuals and h is the diagonal of the hat_matrix ''' hii = self.hat_matrix_diag return self.results.resid * hii / (1 - hii)
[docs] def hat_diag_factor(self): '''(cached attribute) factor of diagonal of hat_matrix used in influence this might be useful for internal reuse h / (1 - h) ''' hii = self.hat_matrix_diag return hii / (1 - hii)
[docs] def ess_press(self): '''(cached attribute) error sum of squares of PRESS residuals ''' return, self.resid_press)
[docs] def resid_studentized_internal(self): '''(cached attribute) studentized residuals using variance from OLS this uses sigma from original estimate does not require leave one out loop ''' return self.get_resid_studentized_external(sigma=None) #return self.results.resid / self.sigma_est
[docs] def resid_studentized_external(self): '''(cached attribute) studentized residuals using LOOO variance this uses sigma from leave-one-out estimates requires leave one out loop for observations ''' sigma_looo = np.sqrt(self.sigma2_not_obsi) return self.get_resid_studentized_external(sigma=sigma_looo)
[docs] def get_resid_studentized_external(self, sigma=None): '''calculate studentized residuals Parameters ---------- sigma : None or float estimate of the standard deviation of the residuals. If None, then the estimate from the regression results is used. Returns ------- stzd_resid : ndarray studentized residuals Notes ----- studentized residuals are defined as :: resid / sigma / np.sqrt(1 - hii) where resid are the residuals from the regression, sigma is an estimate of the standard deviation of the residuals, and hii is the diagonal of the hat_matrix. ''' hii = self.hat_matrix_diag if sigma is None: sigma2_est = self.results.mse_resid #can be replace by different estimators of sigma sigma = np.sqrt(sigma2_est) return self.results.resid / sigma / np.sqrt(1 - hii)
[docs] def dffits_internal(self): '''(cached attribute) dffits measure for influence of an observation based on resid_studentized_internal uses original results, no nobs loop ''' #TODO: do I want to use different sigma estimate in # resid_studentized_external # -> move definition of sigma_error to the __init__ hii = self.hat_matrix_diag dffits_ = self.resid_studentized_internal * np.sqrt(hii / (1 - hii)) dffits_threshold = 2 * np.sqrt(self.k_vars * 1. / self.nobs) return dffits_, dffits_threshold
[docs] def dffits(self): '''(cached attribute) dffits measure for influence of an observation based on resid_studentized_external, uses results from leave-one-observation-out loop It is recommended that observations with dffits large than a threshold of 2 sqrt{k / n} where k is the number of parameters, should be investigated. Returns ------- dffits: float dffits_threshold : float References ---------- `Wikipedia <>`_ ''' #TODO: do I want to use different sigma estimate in # resid_studentized_external # -> move definition of sigma_error to the __init__ hii = self.hat_matrix_diag dffits_ = self.resid_studentized_external * np.sqrt(hii / (1 - hii)) dffits_threshold = 2 * np.sqrt(self.k_vars * 1. / self.nobs) return dffits_, dffits_threshold
[docs] def dfbetas(self): '''(cached attribute) dfbetas uses results from leave-one-observation-out loop ''' dfbetas = self.results.params - self.params_not_obsi#[None,:] dfbetas /= np.sqrt(self.sigma2_not_obsi[:,None]) dfbetas /= np.sqrt(np.diag(self.results.normalized_cov_params)) return dfbetas
[docs] def sigma2_not_obsi(self): '''(cached attribute) error variance for all LOOO regressions This is 'mse_resid' from each auxiliary regression. uses results from leave-one-observation-out loop ''' return np.asarray(self._res_looo['mse_resid'])
[docs] def params_not_obsi(self): '''(cached attribute) parameter estimates for all LOOO regressions uses results from leave-one-observation-out loop ''' return np.asarray(self._res_looo['params'])
[docs] def det_cov_params_not_obsi(self): '''(cached attribute) determinant of cov_params of all LOOO regressions uses results from leave-one-observation-out loop ''' return np.asarray(self._res_looo['det_cov_params'])
[docs] def cooks_distance(self): '''(cached attribute) Cooks distance uses original results, no nobs loop ''' hii = self.hat_matrix_diag #Eubank p.93, 94 cooks_d2 = self.resid_studentized_internal**2 / self.k_vars cooks_d2 *= hii / (1 - hii) from scipy import stats #alpha = 0.1 #print stats.f.isf(1-alpha, n_params, res.df_modelwc) pvals = stats.f.sf(cooks_d2, self.k_vars, self.results.df_resid) return cooks_d2, pvals
[docs] def cov_ratio(self): '''(cached attribute) covariance ratio between LOOO and original This uses determinant of the estimate of the parameter covariance from leave-one-out estimates. requires leave one out loop for observations ''' #don't use inplace division / because then we change original cov_ratio = (self.det_cov_params_not_obsi / np.linalg.det(self.results.cov_params())) return cov_ratio
[docs] def resid_var(self): '''(cached attribute) estimate of variance of the residuals :: sigma2 = sigma2_OLS * (1 - hii) where hii is the diagonal of the hat matrix ''' #TODO:check if correct outside of ols return self.results.mse_resid * (1 - self.hat_matrix_diag)
[docs] def resid_std(self): '''(cached attribute) estimate of standard deviation of the residuals See Also -------- resid_var ''' return np.sqrt(self.resid_var)
def _ols_xnoti(self, drop_idx, endog_idx='endog', store=True): '''regression results from LOVO auxiliary regression with cache The result instances are stored, which could use a large amount of memory if the datasets are large. There are too many combinations to store them all, except for small problems. Parameters ---------- drop_idx : int index of exog that is dropped from the regression endog_idx : 'endog' or int If 'endog', then the endogenous variable of the result instance is regressed on the exogenous variables, excluding the one at drop_idx. If endog_idx is an integer, then the exog with that index is regressed with OLS on all other exogenous variables. (The latter is the auxiliary regression for the variance inflation factor.) this needs more thought, memory versus speed not yet used in any other parts, not sufficiently tested ''' #reverse the structure, access store, if fail calculate ? #this creates keys in store even if store = false ! bug if endog_idx == 'endog': stored = self.aux_regression_endog if hasattr(stored, drop_idx): return stored[drop_idx] x_i = self.results.model.endog else: #nested dictionary try: self.aux_regression_exog[endog_idx][drop_idx] except KeyError: pass stored = self.aux_regression_exog[endog_idx] stored = {} x_i = self.exog[:, endog_idx] k_vars = self.exog.shape[1] mask = np.arange(k_vars) != drop_idx x_noti = self.exog[:, mask] res = OLS(x_i, x_noti).fit() if store: stored[drop_idx] = res return res def _get_drop_vari(self, attributes): '''regress endog on exog without one of the variables This uses a k_vars loop, only attributes of the OLS instance are stored. Parameters ---------- attributes : list of strings These are the names of the attributes of the auxiliary OLS results instance that are stored and returned. not yet used ''' from import LeaveOneOut endog = self.results.model.endog exog = self.exog cv_iter = LeaveOneOut(self.k_vars) res_loo = defaultdict(list) for inidx, outidx in cv_iter: for att in attributes: res_i = self.model_class(endog, exog[:,inidx]).fit() res_loo[att].append(getattr(res_i, att)) return res_loo @cache_readonly def _res_looo(self): '''collect required results from the LOOO loop all results will be attached. currently only 'params', 'mse_resid', 'det_cov_params' are stored regresses endog on exog dropping one observation at a time this uses a nobs loop, only attributes of the OLS instance are stored. ''' from import LeaveOneOut get_det_cov_params = lambda res: np.linalg.det(res.cov_params()) endog = self.endog exog = self.exog params = np.zeros_like(exog) mse_resid = np.zeros_like(endog) det_cov_params = np.zeros_like(endog) cv_iter = LeaveOneOut(self.nobs) for inidx, outidx in cv_iter: res_i = self.model_class(endog[inidx], exog[inidx]).fit() params[outidx] = res_i.params mse_resid[outidx] = res_i.mse_resid det_cov_params[outidx] = get_det_cov_params(res_i) return dict(params=params, mse_resid=mse_resid, det_cov_params=det_cov_params)
[docs] def summary_frame(self): """ Creates a DataFrame with all available influence results. Returns ------- frame : DataFrame A DataFrame with all results. Notes ----- The resultant DataFrame contains six variables in addition to the DFBETAS. These are: * cooks_d : Cook's Distance defined in `Influence.cooks_distance` * standard_resid : Standardized residuals defined in `Influence.resid_studentized_internal` * hat_diag : The diagonal of the projection, or hat, matrix defined in `Influence.hat_matrix_diag` * dffits_internal : DFFITS statistics using internally Studentized residuals defined in `Influence.dffits_internal` * dffits : DFFITS statistics using externally Studentized residuals defined in `Influence.dffits` * student_resid : Externally Studentized residuals defined in `Influence.resid_studentized_external` """ from pandas import DataFrame # row and column labels data = row_labels = data.row_labels beta_labels = ['dfb_' + i for i in data.xnames] # grab the results summary_data = DataFrame(dict( cooks_d = self.cooks_distance[0], standard_resid = self.resid_studentized_internal, hat_diag = self.hat_matrix_diag, dffits_internal = self.dffits_internal[0], student_resid = self.resid_studentized_external, dffits = self.dffits[0], ), index = row_labels) #NOTE: if we don't give columns, order of above will be arbitrary dfbeta = DataFrame(self.dfbetas, columns=beta_labels, index=row_labels) return dfbeta.join(summary_data)
[docs] def summary_table(self, float_fmt="%6.3f"): '''create a summary table with all influence and outlier measures This does currently not distinguish between statistics that can be calculated from the original regression results and for which a leave-one-observation-out loop is needed Returns ------- res : SimpleTable instance SimpleTable instance with the results, can be printed Notes ----- This also attaches table_data to the instance. ''' #print self.dfbetas # table_raw = [ np.arange(self.nobs), # self.endog, # self.fittedvalues, # self.cooks_distance(), # self.resid_studentized_internal, # self.hat_matrix_diag, # self.dffits_internal, # self.resid_studentized_external, # self.dffits, # self.dfbetas # ] table_raw = [ ('obs', np.arange(self.nobs)), ('endog', self.endog), ('fitted\nvalue', self.results.fittedvalues), ("Cook's\nd", self.cooks_distance[0]), ("student.\nresidual", self.resid_studentized_internal), ('hat diag', self.hat_matrix_diag), ('dffits \ninternal', self.dffits_internal[0]), ("ext.stud.\nresidual", self.resid_studentized_external), ('dffits', self.dffits[0]) ] colnames, data = lzip(*table_raw) #unzip data = np.column_stack(data) self.table_data = data from statsmodels.iolib.table import SimpleTable, default_html_fmt from statsmodels.iolib.tableformatting import fmt_base from copy import deepcopy fmt = deepcopy(fmt_base) fmt_html = deepcopy(default_html_fmt) fmt['data_fmts'] = ["%4d"] + [float_fmt] * (data.shape[1] - 1) #fmt_html['data_fmts'] = fmt['data_fmts'] return SimpleTable(data, headers=colnames, txt_fmt=fmt, html_fmt=fmt_html)
def summary_table(res, alpha=0.05): '''generate summary table of outlier and influence similar to SAS Parameters ---------- alpha : float significance level for confidence interval Returns ------- st : SimpleTable instance table with results that can be printed data : ndarray calculated measures and statistics for the table ss2 : list of strings column_names for table (Note: rows of table are observations) ''' from scipy import stats from statsmodels.sandbox.regression.predstd import wls_prediction_std infl = OLSInfluence(res) #standard error for predicted mean #Note: using hat_matrix only works for fitted values predict_mean_se = np.sqrt(infl.hat_matrix_diag*res.mse_resid) tppf = stats.t.isf(alpha/2., res.df_resid) predict_mean_ci = np.column_stack([ res.fittedvalues - tppf * predict_mean_se, res.fittedvalues + tppf * predict_mean_se]) #standard error for predicted observation predict_se, predict_ci_low, predict_ci_upp = wls_prediction_std(res) predict_ci = np.column_stack((predict_ci_low, predict_ci_upp)) #standard deviation of residual resid_se = np.sqrt(res.mse_resid * (1 - infl.hat_matrix_diag)) table_sm = np.column_stack([ np.arange(res.nobs) + 1, res.model.endog, res.fittedvalues, predict_mean_se, predict_mean_ci[:,0], predict_mean_ci[:,1], predict_ci[:,0], predict_ci[:,1], res.resid, resid_se, infl.resid_studentized_internal, infl.cooks_distance[0] ]) #colnames, data = lzip(*table_raw) #unzip data = table_sm ss2 = ['Obs', 'Dep Var\nPopulation', 'Predicted\nValue', 'Std Error\nMean Predict', 'Mean ci\n95% low', 'Mean ci\n95% upp', 'Predict ci\n95% low', 'Predict ci\n95% upp', 'Residual', 'Std Error\nResidual', 'Student\nResidual', "Cook's\nD"] colnames = ss2 #self.table_data = data #data = np.column_stack(data) from statsmodels.iolib.table import SimpleTable, default_html_fmt from statsmodels.iolib.tableformatting import fmt_base from copy import deepcopy fmt = deepcopy(fmt_base) fmt_html = deepcopy(default_html_fmt) fmt['data_fmts'] = ["%4d"] + ["%6.3f"] * (data.shape[1] - 1) #fmt_html['data_fmts'] = fmt['data_fmts'] st = SimpleTable(data, headers=colnames, txt_fmt=fmt, html_fmt=fmt_html) return st, data, ss2