# -*- 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 statsmodels.tools.decorators import cache_readonly
from statsmodels.stats.multitest import multipletests
from statsmodels.tools.tools import maybe_unwrap_results
# outliers test convenience wrapper
def outlier_test(model_results, method='bonf', alpha=.05, labels=None,
order=False, cutoff=None):
"""
Outlier Tests for RegressionResults instances.
Parameters
----------
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
labels : None or array_like
If `labels` is not None, then it will be used as index to the
returned pandas DataFrame. See also Returns below
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.
cutoff : None or float in [0, 1]
If cutoff is not None, then the return only includes observations with
multiple testing corrected p-values strictly below the cutoff. The
returned array or dataframe can be empty if there are no outlier
candidates at the specified cutoff.
Returns
-------
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.
Notes
-----
The unadjusted p-value is stats.t.sf(abs(resid), df) where
df = df_resid - 1.
"""
from scipy import stats # lazy import
if labels is None:
labels = getattr(model_results.model.data, 'row_labels', None)
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.asarray(labels)[idx]
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 cutoff is not None:
mask = data[:, -1] < cutoff
data = data[mask]
else:
mask = slice(None)
if labels is not None:
from pandas import DataFrame
return DataFrame(data,
columns=['student_resid', 'unadj_p', method+"(p)"],
index=np.asarray(labels)[mask])
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.
Notes
-----
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.
References
----------
http://en.wikipedia.org/wiki/Ramsey_RESET_test
'''
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
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
----------
http://en.wikipedia.org/wiki/Variance_inflation_factor
'''
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 = {}
[docs] @cache_readonly
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] @cache_readonly
def resid_press(self):
'''(cached attribute) PRESS residuals
'''
hii = self.hat_matrix_diag
return self.results.resid / (1 - hii)
[docs] @cache_readonly
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] @cache_readonly
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] @cache_readonly
def ess_press(self):
'''(cached attribute) error sum of squares of PRESS residuals
'''
return np.dot(self.resid_press, self.resid_press)
[docs] @cache_readonly
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] @cache_readonly
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] @cache_readonly
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] @cache_readonly
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 <http://en.wikipedia.org/wiki/DFFITS>`_
'''
#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] @cache_readonly
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] @cache_readonly
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] @cache_readonly
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] @cache_readonly
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] @cache_readonly
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] @cache_readonly
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] @cache_readonly
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] @cache_readonly
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 statsmodels.sandbox.tools.cross_val 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 statsmodels.sandbox.tools.cross_val import LeaveOneOut
get_det_cov_params = lambda res: np.linalg.det(res.cov_params())
endog = self.endog
exog = self.exog
params = np.zeros(exog.shape, dtype=np.float)
mse_resid = np.zeros(endog.shape, dtype=np.float)
det_cov_params = np.zeros(endog.shape, dtype=np.float)
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 = self.results.model.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
tmp = wls_prediction_std(res, alpha=alpha)
predict_se, predict_ci_low, predict_ci_upp = tmp
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
