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:                Sun, 24 Nov 2019                                         
Time:                        07:50:45                                         
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())

[ 5.08835141  0.5166759  -0.01241802]
[0.4458426  0.06883208 0.00609058]
[ 4.77790102  5.03740611  5.29277359  5.54400346  5.7910957   6.03405034
  6.27286736  6.50754676  6.73808856  6.96449273  7.18675929  7.40488824
  7.61887957  7.82873329  8.03444939  8.23602788  8.43346876  8.62677202
  8.81593766  9.00096569  9.18185611  9.35860891  9.5312241   9.69970167
  9.86404163 10.02424397 10.1803087  10.33223581 10.48002531 10.6236772
 10.76319147 10.89856812 11.02980716 11.15690859 11.2798724  11.3986986
 11.51338718 11.62393815 11.7303515  11.83262724 11.93076537 12.02476588
 12.11462877 12.20035405 12.28194172 12.35939177 12.43270421 12.50187903
 12.56691624 12.62781583]

Estimate RLM:

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

[ 5.03201868e+00  4.94326407e-01 -1.13736379e-03]
[0.1279254  0.01974996 0.00174757]

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 0x7f97af67cf10>

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)

[5.58887348 0.39249574]
[0.38421572 0.03310558]

Estimate RLM:

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

[5.06431439 0.48503838]
[0.10140737 0.00873767]

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")