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Robust Linear Models
The syntax for the arguments will be shortened to accept string arguments
in the future.
import numpy as np
import statsmodels.api as sm
import matplotlib.pyplot as plt
from statsmodels.sandbox.regression.predstd import wls_prediction_std
#Estimating RLM
# Load data
data = sm.datasets.stackloss.load()
data.exog = sm.add_constant(data.exog)
# Huber's T norm with the (default) median absolute deviation scaling
huber_t = sm.RLM(data.endog, data.exog, M=sm.robust.norms.HuberT())
hub_results =
print hub_results.params
print hub_results.bse
print hub_results.summary(yname='y',
xname=['var_%d' % i for i in range(len(hub_results.params))])
# Huber's T norm with 'H2' covariance matrix
hub_results2 ="H2")
print hub_results2.params
print hub_results2.bse
# Andrew's Wave norm with Huber's Proposal 2 scaling and 'H3' covariance matrix
andrew_mod = sm.RLM(data.endog, data.exog, M=sm.robust.norms.AndrewWave())
andrew_results =, cov="H3")
print andrew_results.params
# See ``help(`` for more options and ``module sm.robust.scale`` for
# scale options
#Comparing OLS and RLM
#Artificial data
nsample = 50
x1 = np.linspace(0, 20, nsample)
X = np.c_[x1, (x1-5)**2, np.ones(nsample)]
sig = 0.3 # smaller error variance makes OLS<->RLM contrast bigger
beta = [0.5, -0.0, 5.]
y_true2 =, beta)
y2 = y_true2 + sig*1. * np.random.normal(size=nsample)
y2[[39,41,43,45,48]] -= 5 # add some outliers (10% of nsample)
#Example: quadratic function with linear truth
# Note that the quadratic term in OLS regression will capture outlier effects.
res = sm.OLS(y2, X).fit()
print res.params
print res.bse
print res.predict
# Estimate RLM
resrlm = sm.RLM(y2, X).fit()
print resrlm.params
print resrlm.bse
# Draw a plot to compare OLS estimates to the robust estimates
plt.plot(x1, y2, 'o', x1, y_true2, 'b-')
prstd, iv_l, iv_u = wls_prediction_std(res)
plt.plot(x1, res.fittedvalues, 'r-')
plt.plot(x1, iv_u, 'r--')
plt.plot(x1, iv_l, 'r--')
plt.plot(x1, resrlm.fittedvalues, 'g.-')
#@savefig rlm_ols_0.png
plt.title('blue: true, red: OLS, green: RLM')
#Example: linear function with linear truth
# Fit a new OLS model using only the linear term and the constant
X2 = X[:,[0,2]]
res2 = sm.OLS(y2, X2).fit()
print res2.params
print res2.bse
# Estimate RLM
resrlm2 = sm.RLM(y2, X2).fit()
print resrlm2.params
print resrlm2.bse
# Draw a plot to compare OLS estimates to the robust estimates
prstd, iv_l, iv_u = wls_prediction_std(res2)
plt.plot(x1, y2, 'o', x1, y_true2, 'b-')
plt.plot(x1, res2.fittedvalues, 'r-')
plt.plot(x1, iv_u, 'r--')
plt.plot(x1, iv_l, 'r--')
plt.plot(x1, resrlm2.fittedvalues, 'g.-')
#@savefig rlm_ols_1.png
plt.title('blue: true, red: OLS, green: RLM')
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