"""
Robust linear models with support for the M-estimators listed under
:ref:`norms <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.
"""
from statsmodels.compat.python import string_types
import numpy as np
import scipy.stats as stats
from statsmodels.tools.decorators import (cache_readonly,
resettable_cache)
import statsmodels.regression.linear_model as lm
import statsmodels.robust.norms as norms
import statsmodels.robust.scale as scale
import statsmodels.base.model as base
import statsmodels.base.wrapper as wrap
from statsmodels.compat.numpy import np_matrix_rank
__all__ = ['RLM']
def _check_convergence(criterion, iteration, tol, maxiter):
return not (np.any(np.fabs(criterion[iteration] -
criterion[iteration-1]) > tol) and iteration < maxiter)
[docs]class RLM(base.LikelihoodModel):
__doc__ = """
Robust Linear Models
Estimate a robust linear model via iteratively reweighted least squares
given a robust criterion estimator.
%(params)s
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)s
Notes
-----
**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 : array
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 : array
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 : array
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 : array
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])
>>>
>>> rlm_hamp_hub = sm.RLM(data.endog, data.exog,
M=sm.robust.norms.Hampel()).fit(
sm.robust.scale.HuberScale())
>>> rlm_hamp_hub.params
array([ 0.73175452, 1.25082038, -0.14794399, -40.27122257])
""" % {'params' : base._model_params_doc,
'extra_params' : base._missing_param_doc}
def __init__(self, endog, exog, M=norms.HuberT(), missing='none',
**kwargs):
self.M = M
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 = (np.float(self.exog.shape[0] -
np_matrix_rank(self.exog)))
self.df_model = np.float(np_matrix_rank(self.exog)-1)
self.nobs = float(self.endog.shape[0])
[docs] def score(self, params):
raise NotImplementedError
[docs] def predict(self, params, exog=None):
"""
Return linear predicted values from a design matrix.
Parameters
----------
params : array-like, optional after fit has been called
Parameters of a linear model
exog : array-like, optional.
Design / exogenous data. Model exog is used if None.
Returns
-------
An array of fitted values
Notes
-----
If the model as not yet been fit, params is not optional.
"""
#copied from linear_model
if exog is None:
exog = self.exog
return np.dot(exog, params)
[docs] def loglike(self, params):
raise NotImplementedError
[docs] def deviance(self, tmp_results):
"""
Returns the (unnormalized) log-likelihood from the M estimator.
"""
return self.M((self.endog - tmp_results.fittedvalues) /
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)
if self.scale_est.lower() == 'stand_mad':
return scale.mad(resid)
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
[docs] def fit(self, maxiter=50, tol=1e-8, scale_est='mad', init=None, cov='H1',
update_scale=True, conv='dev'):
"""
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 : string
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 : string, optional
'H1', 'H2', or 'H3'
Indicates how the covariance matrix is estimated. Default is 'H1'.
See rlm.RLMResults for more information.
init : string
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 : string 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.
Returns
-------
results : object
statsmodels.rlm.RLMresults
"""
if not cov.upper() in ["H1","H2","H3"]:
raise ValueError("Covariance matrix %s not understood" % cov)
else:
self.cov = cov.upper()
conv = conv.lower()
if not conv in ["weights","coefs","dev","sresid"]:
raise ValueError("Convergence argument %s not understood" \
% conv)
self.scale_est = scale_est
if (isinstance(scale_est,
string_types) and scale_est.lower() == "stand_mad"):
from warnings import warn
warn("stand_mad is deprecated and will be removed in 0.7.0",
FutureWarning)
wls_results = lm.WLS(self.endog, self.exog).fit()
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:
self.weights = self.M.weights(wls_results.resid/self.scale)
wls_results = lm.WLS(self.endog, self.exog,
weights=self.weights).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)
[docs]class RLMResults(base.LikelihoodModelResults):
"""
Class to contain RLM results
Returns
-------
**Attributes**
bcov_scaled : array
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 : array
The usual p x p covariance matrix with scale set equal to 1. It
is then just equivalent to normalized_cov_params.
bse : array
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 : array
An array of the chi-squared values of the paramter 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 : array
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 : array
See RLM.normalized_cov_params
params : array
The coefficients of the fitted model
pinv_wexog : array
See RLM.pinv_wexog
pvalues : array
The p values associated with `tvalues`. Note that `tvalues` are assumed to be distributed
standard normal rather than Student's t.
resid : array
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 : array
The scaled residuals.
tvalues : array
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 : array
The reported weights are determined by passing the scaled residuals
from the last weighted least squares fit in the IRLS algortihm.
See also
--------
statsmodels.model.LikelihoodModelResults
"""
def __init__(self, model, params, normalized_cov_params, scale):
super(RLMResults, self).__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 = resettable_cache()
#for remove_data
self.data_in_cache = ['sresid']
self.cov_params_default = self.bcov_scaled
#TODO: "pvals" should come from chisq on bse?
@cache_readonly
[docs] def fittedvalues(self):
return np.dot(self.model.exog, self.params)
@cache_readonly
[docs] def resid(self):
return self.model.endog - self.fittedvalues # before bcov
@cache_readonly
[docs] def sresid(self):
return self.resid/self.scale
@cache_readonly
[docs] def bcov_unscaled(self):
return self.normalized_cov_params
@cache_readonly
[docs] def weights(self):
return self.model.weights
@cache_readonly
[docs] 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":
return k**2 * (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)))**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
[docs] def pvalues(self):
return stats.norm.sf(np.abs(self.tvalues))*2
@cache_readonly
[docs] def bse(self):
return np.sqrt(np.diag(self.bcov_scaled))
@cache_readonly
[docs] def chisq(self):
return (self.params/self.bse)**2
[docs] def remove_data(self):
super(self.__class__, self).remove_data()
#self.model.history['sresid'] = None
#self.model.history['weights'] = None
remove_data.__doc__ = base.LikelihoodModelResults.remove_data.__doc__
[docs] def summary(self, yname=None, xname=None, title=0, alpha=.05,
return_fmt='text'):
"""
This is for testing the new summary setup
"""
from statsmodels.iolib.summary import (summary_top,
summary_params, summary_return)
## left = [(i, None) for i in (
## 'Dependent Variable:',
## 'Model type:',
## 'Method:',
## 'Date:',
## 'Time:',
## 'Number of Obs:',
## 'df resid',
## 'df model',
## )]
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 not title is 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)
#diagnostic table is not used yet
# smry.add_table_2cols(self, gleft=diagn_left, gright=diagn_right,
# yname=yname, xname=xname,
# title="")
#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
[docs] def summary2(self, xname=None, yname=None, title=None, alpha=.05,
float_format="%.4f"):
"""Experimental summary function for regression results
Parameters
-----------
xname : List of strings of length equal to the number of parameters
Names of the independent variables (optional)
yname : string
Name of the dependent variable (optional)
title : string, 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: string
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.summary.Summary : class to hold summary
results
"""
# Summary
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)
if __name__=="__main__":
#NOTE: This is to be removed
#Delivery Time Data is taken from Montgomery and Peck
import statsmodels.api as sm
#delivery time(minutes)
endog = np.array([16.68, 11.50, 12.03, 14.88, 13.75, 18.11, 8.00, 17.83,
79.24, 21.50, 40.33, 21.00, 13.50, 19.75, 24.00, 29.00, 15.35, 19.00,
9.50, 35.10, 17.90, 52.32, 18.75, 19.83, 10.75])
#number of cases, distance (Feet)
exog = np.array([[7, 3, 3, 4, 6, 7, 2, 7, 30, 5, 16, 10, 4, 6, 9, 10, 6,
7, 3, 17, 10, 26, 9, 8, 4], [560, 220, 340, 80, 150, 330, 110, 210, 1460,
605, 688, 215, 255, 462, 448, 776, 200, 132, 36, 770, 140, 810, 450, 635,
150]])
exog = exog.T
exog = sm.add_constant(exog)
# model_ols = models.regression.OLS(endog, exog)
# results_ols = model_ols.fit()
# model_ramsaysE = RLM(endog, exog, M=norms.RamsayE())
# results_ramsaysE = model_ramsaysE.fit(update_scale=False)
# model_andrewWave = RLM(endog, exog, M=norms.AndrewWave())
# results_andrewWave = model_andrewWave.fit(update_scale=False)
# model_hampel = RLM(endog, exog, M=norms.Hampel(a=1.7,b=3.4,c=8.5)) # convergence problems with scale changed, not with 2,4,8 though?
# results_hampel = model_hampel.fit(update_scale=False)
#######################
### Stack Loss Data ###
#######################
from statsmodels.datasets.stackloss import load
data = load()
data.exog = sm.add_constant(data.exog)
#############
### Huber ###
#############
# m1_Huber = RLM(data.endog, data.exog, M=norms.HuberT())
# results_Huber1 = m1_Huber.fit()
# m2_Huber = RLM(data.endog, data.exog, M=norms.HuberT())
# results_Huber2 = m2_Huber.fit(cov="H2")
# m3_Huber = RLM(data.endog, data.exog, M=norms.HuberT())
# results_Huber3 = m3_Huber.fit(cov="H3")
##############
### Hampel ###
##############
# m1_Hampel = RLM(data.endog, data.exog, M=norms.Hampel())
# results_Hampel1 = m1_Hampel.fit()
# m2_Hampel = RLM(data.endog, data.exog, M=norms.Hampel())
# results_Hampel2 = m2_Hampel.fit(cov="H2")
# m3_Hampel = RLM(data.endog, data.exog, M=norms.Hampel())
# results_Hampel3 = m3_Hampel.fit(cov="H3")
################
### Bisquare ###
################
# m1_Bisquare = RLM(data.endog, data.exog, M=norms.TukeyBiweight())
# results_Bisquare1 = m1_Bisquare.fit()
# m2_Bisquare = RLM(data.endog, data.exog, M=norms.TukeyBiweight())
# results_Bisquare2 = m2_Bisquare.fit(cov="H2")
# m3_Bisquare = RLM(data.endog, data.exog, M=norms.TukeyBiweight())
# results_Bisquare3 = m3_Bisquare.fit(cov="H3")
##############################################
# Huber's Proposal 2 scaling #
##############################################
################
### Huber'sT ###
################
m1_Huber_H = RLM(data.endog, data.exog, M=norms.HuberT())
results_Huber1_H = m1_Huber_H.fit(scale_est=scale.HuberScale())
# m2_Huber_H
# m3_Huber_H
# m4 = RLM(data.endog, data.exog, M=norms.HuberT())
# results4 = m1.fit(scale_est="Huber")
# m5 = RLM(data.endog, data.exog, M=norms.Hampel())
# results5 = m2.fit(scale_est="Huber")
# m6 = RLM(data.endog, data.exog, M=norms.TukeyBiweight())
# results6 = m3.fit(scale_est="Huber")
# print """Least squares fit
#%s
#Huber Params, t = 2.
#%s
#Ramsay's E Params
#%s
#Andrew's Wave Params
#%s
#Hampel's 17A Function
#%s
#""" % (results_ols.params, results_huber.params, results_ramsaysE.params,
# results_andrewWave.params, results_hampel.params)
