# -*- coding: utf-8 -*-
"""Influence and Outlier Measures
Created on Sun Jan 29 11:16:09 2012
Author: Josef Perktold
License: BSD-3
"""
from collections import defaultdict
import numpy as np
from statsmodels.compat.python import lzip
from statsmodels.graphics._regressionplots_doc import _plot_influence_doc
from statsmodels.regression.linear_model import OLS
from statsmodels.stats.multitest import multipletests
from statsmodels.tools.decorators import cache_readonly
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
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, and drop constant
y_fitted_vander = np.vander(res.fittedvalues, order)[:, :-2]
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
class _BaseInfluenceMixin(object):
"""common methods between OLSInfluence and MLE/GLMInfluence
"""
def plot_influence(self, external=None, alpha=.05, criterion="cooks",
size=48, plot_alpha=.75, ax=None, **kwargs):
if external is None:
external = hasattr(self, '_cache') and 'res_looo' in self._cache
from statsmodels.graphics.regressionplots import _influence_plot
res = _influence_plot(self.results, self, external=external,
alpha=alpha,
criterion=criterion, size=size,
plot_alpha=plot_alpha, ax=ax, **kwargs)
return res
plot_influence.__doc__ = _plot_influence_doc.format({
'extra_params_doc': ""})
def _plot_index(self, y, ylabel, threshold=None, title=None, ax=None,
**kwds):
from statsmodels.graphics import utils
fig, ax = utils.create_mpl_ax(ax)
if title is None:
title = "Index Plot"
nobs = len(self.endog)
index = np.arange(nobs)
ax.scatter(index, y, **kwds)
if threshold == 'all':
large_points = np.ones(nobs, np.bool_)
else:
large_points = np.abs(y) > threshold
psize = 3 * np.ones(nobs)
# add point labels
labels = self.results.model.data.row_labels
if labels is None:
labels = np.arange(nobs)
ax = utils.annotate_axes(np.where(large_points)[0], labels,
lzip(index, y),
lzip(-psize, psize), "large",
ax)
font = {"fontsize": 16, "color": "black"}
ax.set_ylabel(ylabel, **font)
ax.set_xlabel("Observation", **font)
ax.set_title(title, **font)
return fig
def plot_index(self, y_var='cooks', threshold=None, title=None, ax=None,
idx=None, **kwds):
"""index plot for influence attributes
Parameters
----------
y_var : string
Name of attribute or shortcut for predefined attributes that will
be plotted on the y-axis.
threshold : None or float
Threshold for adding annotation with observation labels.
Observations for which the absolute value of the y_var is larger
than the threshold will be annotated. Set to a negative number to
label all observations or to a large number to have no annotation.
title : string
If provided, the title will replace the default "Index Plot" title.
ax : matplolib axis instance
The plot will be added to the `ax` if provided, otherwise a new
figure is created.
idx : None or integer
Some attributes require an additional index to select the y-var.
In dfbetas this refers to the column indes.
kwds : optional keywords
Keywords will be used in the call to matplotlib scatter function.
"""
criterion = y_var # alias
if threshold is None:
# TODO: criterion specific defaults
threshold = 'all'
if criterion == 'dfbeta':
y = self.dfbetas[:, idx]
ylabel = 'DFBETA for ' + self.results.model.exog_names[idx]
elif criterion.startswith('cook'):
y = self.cooks_distance[0]
ylabel = "Cook's distance"
elif criterion.startswith('hat') or criterion.startswith('lever'):
y = self.hat_matrix_diag
ylabel = "Leverage (diagonal of hat matrix)"
elif criterion.startswith('cook'):
y = self.cooks_distance[0]
ylabel = "Cook's distance"
elif criterion.startswith('resid'):
y = self.resid_studentized
ylabel = "Internally Studentized Residuals"
else:
# assume we have the name of an attribute
y = getattr(self, y_var)
if idx is not None:
y = y[idx]
ylabel = y_var
fig = self._plot_index(y, ylabel, threshold=threshold, title=title,
ax=ax, **kwds)
return fig
[docs]class MLEInfluence(_BaseInfluenceMixin):
"""Local Influence and outlier measures (experimental)
This currently subclasses GLMInfluence instead of the other way.
No common superclass yet.
This is another version before checking what is common
Parameters
----------
results : instance of results class
This only works for model and results classes that have the necessary
helper methods.
other arguments are only to override default behavior and are used instead
of the corresponding attribute of the results class.
By default resid_pearson is used as resid.
Attributes
----------
hat_matrix_diag (hii) : This is the generalized leverage computed as the
local derivative of fittedvalues (predicted mean) with respect to the
observed response for each observation.
d_params : Change in parameters computed with one Newton step using the
full Hessian corrected by division by (1 - hii).
dbetas : change in parameters divided by the standard error of parameters
from the full model results, ``bse``.
cooks_distance : quadratic form for change in parameters weighted by
``cov_params`` from the full model divided by the number of variables.
It includes p-values based on the F-distribution which are only
approximate outside of linear Gaussian models.
resid_studentized : In the general MLE case resid_studentized are
computed from the score residuals scaled by hessian factor and
leverage. This does not use ``cov_params``.
d_fittedvalues : local change of expected mean given the change in the
parameters as computed in ``d_params``.
d_fittedvalues_scaled : same as d_fittedvalues but scaled by the standard
errors of a predicted mean of the response.
params_one : is the one step parameter estimate computed as ``params``
from the full sample minus ``d_params``.
Notes
-----
MLEInfluence produces the same results as GLMInfluence (verified for GLM
Binomial and Gaussian). There will be some differences for non-canonical
links or if a robust cov_type is used.
Warning: This does currently not work for constrained or penalized models,
e.g. models estimated with fit_constrained or fit_regularized.
This has not yet been tested for correctness when offset or exposure
are used, although they should be supported by the code.
status: experimental,
This class will need changes to support different kinds of models, e.g.
extra parameters in discrete.NegativeBinomial or two-part models like
ZeroInflatedPoisson.
"""
def __init__(self, results, resid=None, endog=None, exog=None,
hat_matrix_diag=None, cov_params=None, scale=None):
# I'm not calling super for now, OLS attributes might not be available
# check which model is allowed
self.results = results = maybe_unwrap_results(results)
# TODO: check for extra params in e.g. NegBin
self.nobs, self.k_vars = results.model.exog.shape
self.endog = endog if endog is not None else results.model.endog
self.exog = exog if exog is not None else results.model.exog
self.resid = resid if resid is not None else results.resid_pearson
self.scale = scale if scale is not None else results.scale
self.cov_params = (cov_params if cov_params is not None
else results.cov_params())
self.model_class = results.model.__class__
self.hessian = self.results.model.hessian(self.results.params)
self.score_obs = self.results.model.score_obs(self.results.params)
if hat_matrix_diag is not None:
self._hat_matrix_diag = hat_matrix_diag
[docs] @cache_readonly
def hat_matrix_diag(self):
"""Diagonal of the generalized leverage
This is the analogue of the hat matrix diagonal for general MLE.
"""
if hasattr(self, '_hat_matrix_diag'):
return self._hat_matrix_diag
dmu_dp = self.results.model._deriv_mean_dparams(self.results.params)
dsdy = self.results.model._deriv_score_obs_dendog(self.results.params)
# dmu_dp = 1 /
# self.results.model.family.link.deriv(self.results.fittedvalues)
h = (dmu_dp * np.linalg.solve(-self.hessian, dsdy.T).T).sum(1)
return h
[docs] @cache_readonly
def d_params(self):
"""Change in parameter estimates
This uses one-step approximation of the parameter change to deleting
one observation.
"""
so_noti = self.score_obs.sum(0) - self.score_obs
beta_i = np.linalg.solve(self.hessian, so_noti.T).T
return beta_i / (1 - self.hat_matrix_diag)[:, None]
[docs] @cache_readonly
def dfbetas(self):
"""Scaled change in parameter estimates
The one-step change of parameters in d_params is rescaled by dividing
by the standard error of the parameter estimate given by results.bse.
"""
beta_i = self.d_params / self.results.bse
return beta_i
[docs] @cache_readonly
def params_one(self):
"""Parameter estimate based on one-step approximation
This the one step parameter estimate computed as
``params`` from the full sample minus ``d_params``.
"""
return self.results.params - self.d_params
[docs] @cache_readonly
def cooks_distance(self):
"""Cook's distance and p-values
Based on one step approximation d_params and on results.cov_params
Cook's distance divides by the number of explanatory variables.
p-values are based on the F-distribution which are only approximate
outside of linear Gaussian models.
Warning: The definition of p-values might change if we switch to using
chi-square distribution instead of F-distribution, or if we make it
dependent on the fit keyword use_t.
"""
cooks_d2 = (self.d_params * np.linalg.solve(self.cov_params,
self.d_params.T).T).sum(1)
cooks_d2 /= self.k_vars
from scipy import stats
# alpha = 0.1
# print stats.f.isf(1-alpha, n_params, res.df_modelwc)
# TODO use chi2 # use_f option
pvals = stats.f.sf(cooks_d2, self.k_vars, self.results.df_resid)
return cooks_d2, pvals
[docs] @cache_readonly
def resid_studentized(self):
"""Score residual divided by sqrt of hessian factor
experimental, agrees with GLMInfluence for Binomial and Gaussian.
no reference for this
"""
sf = self.results.model.score_factor(self.results.params)
hf = self.results.model.hessian_factor(self.results.params)
return sf / np.sqrt(hf) / np.sqrt(1 - self.hat_matrix_diag)
@cache_readonly
def _get_prediction(self):
# TODO: do we cache this or does it need to be a method
# we only need unchanging parts, alpha for confint could change
return self.results.get_prediction()
[docs] @cache_readonly
def d_fittedvalues(self):
"""Change in expected response, fittedvalues
Local change of expected mean given the change in the parameters as
computed in d_params.
Notes
-----
This uses one-step approximation of the parameter change to deleting
one observation ``d_params``.
"""
# results.params might be a pandas.Series
params = np.asarray(self.results.params)
deriv = self.results.model._deriv_mean_dparams(params)
return (deriv * self.d_params).sum(1)
@property
def d_fittedvalues_scaled(self):
"""
Change in fittedvalues scaled by standard errors
This uses one-step approximation of the parameter change to deleting
one observation ``d_params``, and divides by the standard errors
for the predicted mean provided by results.get_prediction.
"""
# Note: this and the previous methods are for the response
# and not for a weighted response, i.e. not the self.exog, self.endog
# this will be relevant for WLS comparing fitted endog versus wendog
return self.d_fittedvalues / self._get_prediction.se_mean
[docs] def summary_frame(self):
"""
Creates a DataFrame with influence results.
Returns
-------
frame : pandas DataFrame
A DataFrame with selected results for each observation.
The index will be the same as provided to the model.
Notes
-----
The resultant DataFrame contains six variables in addition to the
``dfbetas``. These are:
* cooks_d : Cook's Distance defined in ``cooks_distance``
* standard_resid : Standardized residuals defined in
`resid_studentizedl`
* hat_diag : The diagonal of the projection, or hat, matrix defined in
`hat_matrix_diag`
* dffits_internal : DFFITS statistics using internally Studentized
residuals defined in `d_fittedvalues_scaled`
"""
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,
hat_diag=self.hat_matrix_diag,
dffits_internal=self.d_fittedvalues_scaled),
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]class OLSInfluence(_BaseInfluenceMixin):
"""class to calculate outlier and influence measures for OLS result
Parameters
----------
results : RegressionResults
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.resid = results.resid
self.model_class = results.model.__class__
# self.sigma_est = np.sqrt(results.mse_resid)
self.scale = results.mse_resid
self.aux_regression_exog = {}
self.aux_regression_endog = {}
[docs] @cache_readonly
def hat_matrix_diag(self):
"""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):
"""PRESS residuals
"""
hii = self.hat_matrix_diag
return self.resid / (1 - hii)
[docs] @cache_readonly
def influence(self):
"""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.resid * hii / (1 - hii)
[docs] @cache_readonly
def hat_diag_factor(self):
"""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):
"""Error sum of squares of PRESS residuals
"""
return np.dot(self.resid_press, self.resid_press)
[docs] @cache_readonly
def resid_studentized(self):
"""Studentized residuals using variance from OLS
alias for resid_studentized_internal for compatibility with
MLEInfluence this uses sigma from original estimate and does
not require leave one out loop
"""
return self.resid_studentized_internal
[docs] @cache_readonly
def resid_studentized_internal(self):
"""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):
"""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.scale
# can be replace by different estimators of sigma
sigma = np.sqrt(sigma2_est)
return self.resid / sigma / np.sqrt(1 - hii)
# same computation as GLMInfluence
[docs] @cache_readonly
def cooks_distance(self):
"""Cooks distance
uses original results, no nobs loop
"""
hii = self.hat_matrix_diag
# Eubank p.93, 94
cooks_d2 = self.resid_studentized ** 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 dffits_internal(self):
"""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):
"""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):
"""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 dfbeta(self):
"""dfbetas
uses results from leave-one-observation-out loop
"""
dfbeta = self.results.params - self.params_not_obsi
return dfbeta
[docs] @cache_readonly
def sigma2_not_obsi(self):
"""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'])
@property
def params_not_obsi(self):
"""parameter estimates for all LOOO regressions
uses results from leave-one-observation-out loop
"""
return np.asarray(self._res_looo['params'])
@property
def det_cov_params_not_obsi(self):
"""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 cov_ratio(self):
"""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):
"""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.scale * (1 - self.hat_matrix_diag)
[docs] @cache_readonly
def resid_std(self):
"""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
def get_det_cov_params(res):
return np.linalg.det(res.cov_params())
endog = self.results.model.endog
exog = self.results.model.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
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
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
[docs]class GLMInfluence(MLEInfluence):
"""Influence and outlier measures (experimental)
This uses partly formulas specific to GLM, specifically cooks_distance
is based on the hessian, i.e. observed or expected information matrix and
not on cov_params, in contrast to MLEInfluence.
Standardization for changes in parameters, in fittedvalues and in
the linear predictor are based on cov_params.
Parameters
----------
results : instance of results class
This only works for model and results classes that have the necessary
helper methods.
other arguments are only to override default behavior and are used instead
of the corresponding attribute of the results class.
By default resid_pearson is used as resid.
Attributes
----------
dbetas
change in parameters divided by the standard error of parameters from
the full model results, ``bse``.
d_fittedvalues_scaled
same as d_fittedvalues but scaled by the standard errors of a
predicted mean of the response.
d_linpred
local change in linear prediction.
d_linpred_scale
local change in linear prediction scaled by the standard errors for
the prediction based on cov_params.
Notes
-----
This has not yet been tested for correctness when offset or exposure
are used, although they should be supported by the code.
Some GLM specific measures like d_deviance are still missing.
Computing an explicit leave-one-observation-out (LOOO) loop is included
but no influence measures are currently computed from it.
"""
[docs] @cache_readonly
def hat_matrix_diag(self):
"""
Diagonal of the hat_matrix for GLM
Notes
-----
This returns the diagonal of the hat matrix that was provided as
argument to GLMInfluence or computes it using the results method
`get_hat_matrix`.
"""
if hasattr(self, '_hat_matrix_diag'):
return self._hat_matrix_diag
else:
return self.results.get_hat_matrix()
[docs] @cache_readonly
def d_params(self):
"""Change in parameter estimates
Notes
-----
This uses one-step approximation of the parameter change to deleting
one observation.
"""
beta_i = np.linalg.pinv(self.exog) * self.resid_studentized
beta_i /= np.sqrt(1 - self.hat_matrix_diag)
return beta_i.T
# same computation as OLS
[docs] @cache_readonly
def resid_studentized(self):
"""
Internally studentized pearson residuals
Notes
-----
residuals / sqrt( scale * (1 - hii))
where residuals are those provided to GLMInfluence which are
pearson residuals by default, and
hii is the diagonal of the hat matrix.
"""
hii = self.hat_matrix_diag
return self.resid / np.sqrt(self.scale * (1 - hii))
# same computation as OLS
[docs] @cache_readonly
def cooks_distance(self):
"""Cook's distance
Notes
-----
Based on one step approximation using resid_studentized and
hat_matrix_diag for the computation.
Cook's distance divides by the number of explanatory variables.
Computed using formulas for GLM and does not use results.cov_params.
It includes p-values based on the F-distribution which are only
approximate outside of linear Gaussian models.
"""
hii = self.hat_matrix_diag
# Eubank p.93, 94
cooks_d2 = self.resid_studentized ** 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
@property
def d_linpred(self):
"""
Change in linear prediction
This uses one-step approximation of the parameter change to deleting
one observation ``d_params``.
"""
# TODO: This will need adjustment for extra params in Poisson
# use original model exog not transformed influence exog
exog = self.results.model.exog
return (exog * self.d_params).sum(1)
@property
def d_linpred_scaled(self):
"""
Change in linpred scaled by standard errors
This uses one-step approximation of the parameter change to deleting
one observation ``d_params``, and divides by the standard errors
for linpred provided by results.get_prediction.
"""
# Note: this and the previous methods are for the response
# and not for a weighted response, i.e. not the self.exog, self.endog
# this will be relevant for WLS comparing fitted endog versus wendog
return self.d_linpred / self._get_prediction.linpred.se_mean
@property
def _fittedvalues_one(self):
"""experimental code
"""
import warnings
warnings.warn('this ignores offset and exposure', UserWarning)
# TODO: we need to handle offset, exposure and weights
# use original model exog not transformed influence exog
exog = self.results.model.exog
fitted = np.array([self.results.model.predict(pi, exog[i])
for i, pi in enumerate(self.params_one)])
return fitted.squeeze()
@property
def _diff_fittedvalues_one(self):
"""experimental code
"""
# in discrete we cannot reuse results.fittedvalues
return self.results.predict() - self._fittedvalues_one
@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
Reestimates the model with endog and exog dropping one observation
at a time
This uses a nobs loop, only attributes of the results instance are
stored.
Warning: This will need refactoring and API changes to be able to
add options.
"""
from statsmodels.sandbox.tools.cross_val import LeaveOneOut
get_det_cov_params = lambda res: np.linalg.det(res.cov_params())
endog = self.results.model.endog
exog = self.results.model.exog
init_kwds = self.results.model._get_init_kwds()
# We need to drop obs also from extra arrays
freq_weights = init_kwds.pop('freq_weights')
var_weights = init_kwds.pop('var_weights')
offset = offset_ = init_kwds.pop('offset')
exposure = exposure_ = init_kwds.pop('exposure')
n_trials = init_kwds.pop('n_trials', None)
# family Binomial creates `n` i.e. `n_trials`
# we need to reset it
# TODO: figure out how to do this properly
if hasattr(init_kwds['family'], 'initialize'):
# assume we have Binomial
is_binomial = True
else:
is_binomial = False
params = np.zeros(exog.shape, dtype=np.float)
scale = 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:
if offset is not None:
offset_ = offset[inidx]
if exposure is not None:
exposure_ = exposure[inidx]
if n_trials is not None:
init_kwds['n_trials'] = n_trials[inidx]
mod_i = self.model_class(endog[inidx], exog[inidx],
offset=offset_,
exposure=exposure_,
freq_weights=freq_weights[inidx],
var_weights=var_weights[inidx],
**init_kwds)
if is_binomial:
mod_i.family.n = init_kwds['n_trials']
res_i = mod_i.fit(start_params=self.results.params,
method='newton')
params[outidx] = res_i.params.copy()
scale[outidx] = res_i.scale
det_cov_params[outidx] = get_det_cov_params(res_i)
return dict(params=params, scale=scale, mse_resid=scale,
# alias for now
det_cov_params=det_cov_params)
