Robust Linear Models

In [1]:
%matplotlib inline
In [2]:
from __future__ import print_function
import numpy as np
import statsmodels.api as sm
import matplotlib.pyplot as plt
from statsmodels.sandbox.regression.predstd import wls_prediction_std

Estimation

Load data:

In [3]:
data = sm.datasets.stackloss.load(as_pandas=False)
data.exog = sm.add_constant(data.exog)

Huber's T norm with the (default) median absolute deviation scaling

In [4]:
huber_t = sm.RLM(data.endog, data.exog, M=sm.robust.norms.HuberT())
hub_results = huber_t.fit()
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))]))

[-41.02649835   0.82938433   0.92606597  -0.12784672]
[9.79189854 0.11100521 0.30293016 0.12864961]
                    Robust linear Model Regression Results                    
==============================================================================
Dep. Variable:                      y   No. Observations:                   21
Model:                            RLM   Df Residuals:                       17
Method:                          IRLS   Df Model:                            3
Norm:                          HuberT                                         
Scale Est.:                       mad                                         
Cov Type:                          H1                                         
Date:                Mon, 24 Jun 2019                                         
Time:                        17:26:53                                         
No. Iterations:                    19                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
var_0        -41.0265      9.792     -4.190      0.000     -60.218     -21.835
var_1          0.8294      0.111      7.472      0.000       0.612       1.047
var_2          0.9261      0.303      3.057      0.002       0.332       1.520
var_3         -0.1278      0.129     -0.994      0.320      -0.380       0.124
==============================================================================

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 .

Huber's T norm with 'H2' covariance matrix

In [5]:
hub_results2 = huber_t.fit(cov="H2")
print(hub_results2.params)
print(hub_results2.bse)

[-41.02649835   0.82938433   0.92606597  -0.12784672]
[9.08950419 0.11945975 0.32235497 0.11796313]

Andrew's Wave norm with Huber's Proposal 2 scaling and 'H3' covariance matrix

In [6]:
andrew_mod = sm.RLM(data.endog, data.exog, M=sm.robust.norms.AndrewWave())
andrew_results = andrew_mod.fit(scale_est=sm.robust.scale.HuberScale(), cov="H3")
print('Parameters: ', andrew_results.params)

Parameters:  [-40.8817957    0.79276138   1.04857556  -0.13360865]

See help(sm.RLM.fit) for more options and module sm.robust.scale for scale options

Comparing OLS and RLM

Artificial data with outliers:

In [7]:
nsample = 50
x1 = np.linspace(0, 20, nsample)
X = np.column_stack((x1, (x1-5)**2))
X = sm.add_constant(X)
sig = 0.3   # smaller error variance makes OLS<->RLM contrast bigger
beta = [5, 0.5, -0.0]
y_true2 = np.dot(X, 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 1: quadratic function with linear truth

Note that the quadratic term in OLS regression will capture outlier effects.

In [8]:
res = sm.OLS(y2, X).fit()
print(res.params)
print(res.bse)
print(res.predict())

[ 5.11315907  0.51233383 -0.01261313]
[0.44651422 0.06893577 0.00609975]
[ 4.79783084  5.05632754  5.31062161  5.56071305  5.80660187  6.04828807
  6.28577164  6.51905258  6.7481309   6.97300659  7.19367966  7.4101501
  7.62241792  7.83048311  8.03434567  8.23400561  8.42946293  8.62071762
  8.80776969  8.99061912  9.16926594  9.34371013  9.51395169  9.67999063
  9.84182694  9.99946063 10.15289169 10.30212013 10.44714594 10.58796913
 10.72458969 10.85700762 10.98522294 11.10923562 11.22904568 11.34465311
 11.45605792 11.56326011 11.66625967 11.7650566  11.85965091 11.95004259
 12.03623165 12.11821808 12.19600189 12.26958307 12.33896162 12.40413755
 12.46511086 12.52188154]

Estimate RLM:

In [9]:
resrlm = sm.RLM(y2, X).fit()
print(resrlm.params)
print(resrlm.bse)

[ 5.05257713e+00  4.97544620e-01 -2.58676193e-03]
[0.14531129 0.0224341  0.00198507]

Draw a plot to compare OLS estimates to the robust estimates:

In [10]:
fig = plt.figure(figsize=(12,8))
ax = fig.add_subplot(111)
ax.plot(x1, y2, 'o',label="data")
ax.plot(x1, y_true2, 'b-', label="True")
prstd, iv_l, iv_u = wls_prediction_std(res)
ax.plot(x1, res.fittedvalues, 'r-', label="OLS")
ax.plot(x1, iv_u, 'r--')
ax.plot(x1, iv_l, 'r--')
ax.plot(x1, resrlm.fittedvalues, 'g.-', label="RLM")
ax.legend(loc="best")
Out[10]:
<matplotlib.legend.Legend at 0x7f4e56395e80>

Example 2: linear function with linear truth

Fit a new OLS model using only the linear term and the constant:

In [11]:
X2 = X[:,[0,1]]
res2 = sm.OLS(y2, X2).fit()
print(res2.params)
print(res2.bse)

[5.62154541 0.38620253]
[0.38524092 0.03319392]

Estimate RLM:

In [12]:
resrlm2 = sm.RLM(y2, X2).fit()
print(resrlm2.params)
print(resrlm2.bse)

[5.13015447 0.47629653]
[0.11499354 0.00990831]

Draw a plot to compare OLS estimates to the robust estimates:

In [13]:
prstd, iv_l, iv_u = wls_prediction_std(res2)

fig, ax = plt.subplots(figsize=(8,6))
ax.plot(x1, y2, 'o', label="data")
ax.plot(x1, y_true2, 'b-', label="True")
ax.plot(x1, res2.fittedvalues, 'r-', label="OLS")
ax.plot(x1, iv_u, 'r--')
ax.plot(x1, iv_l, 'r--')
ax.plot(x1, resrlm2.fittedvalues, 'g.-', label="RLM")
legend = ax.legend(loc="best")