Robust Linear Models

In [1]:
%matplotlib inline

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

/build/statsmodels-763zc8/statsmodels-0.8.0/.pybuild/cpython3_3.7_statsmodels/build/statsmodels/compat/pandas.py:56: FutureWarning: The pandas.core.datetools module is deprecated and will be removed in a future version. Please use the pandas.tseries module instead.
  from pandas.core import datetools

Estimation

Load data:

In [2]:
data = sm.datasets.stackloss.load()
data.exog = sm.add_constant(data.exog)

/build/statsmodels-763zc8/statsmodels-0.8.0/.pybuild/cpython3_3.7_statsmodels/build/statsmodels/datasets/utils.py:100: FutureWarning: arrays to stack must be passed as a "sequence" type such as list or tuple. Support for non-sequence iterables such as generators is deprecated as of NumPy 1.16 and will raise an error in the future.
  exog = np.column_stack(data[field] for field in exog_name)

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

In [3]:
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:                Sat, 02 Mar 2019                                         
Time:                        00:23:12                                         
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 [4]:
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 [5]:
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 [6]:
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 [7]:
res = sm.OLS(y2, X).fit()
print(res.params)
print(res.bse)
print(res.predict())

[ 5.19715596  0.4903268  -0.00980344]
[0.4732082  0.07305696 0.00646441]
[ 4.95207005  5.19058424  5.42583197  5.65781326  5.8865281   6.11197648
  6.33415841  6.55307389  6.76872293  6.9811055   7.19022163  7.39607131
  7.59865454  7.79797131  7.99402163  8.18680551  8.37632293  8.5625739
  8.74555842  8.92527648  9.1017281   9.27491327  9.44483198  9.61148424
  9.77487005  9.93498941 10.09184232 10.24542878 10.39574879 10.54280234
 10.68658945 10.8271101  10.9643643  11.09835205 11.22907335 11.3565282
 11.4807166  11.60163854 11.71929404 11.83368308 11.94480567 12.05266181
 12.1572515  12.25857474 12.35663153 12.45142187 12.54294575 12.63120319
 12.71619417 12.7979187 ]

Estimate RLM:

In [8]:
resrlm = sm.RLM(y2, X).fit()
print(resrlm.params)
print(resrlm.bse)

[5.14432844e+00 4.68042119e-01 2.30396626e-03]
[0.1378037  0.02127503 0.00188251]

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

In [9]:
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[9]:
<matplotlib.legend.Legend at 0x7fa5971fc710>

Example 2: linear function with linear truth

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

In [10]:
X2 = X[:,[0,1]] 
res2 = sm.OLS(y2, X2).fit()
print(res2.params)
print(res2.bse)

[5.59229448 0.39229243]
[0.40032782 0.03449386]

Estimate RLM:

In [11]:
resrlm2 = sm.RLM(y2, X2).fit()
print(resrlm2.params)
print(resrlm2.bse)

[5.07972569 0.48595129]
[0.12730822 0.01096939]

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

In [12]:
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")