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

Estimation

Load data:

In [2]:
data = sm.datasets.stackloss.load()
data.exog = sm.add_constant(data.exog)
/build/statsmodels-0.9.0/.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:                Mon, 29 Apr 2019                                         
Time:                        19:16:46                                         
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())
[ 4.98381041  0.52522734 -0.0125452 ]
[0.4446381  0.06864612 0.00607412]
[ 4.67018035  4.93367377  5.19298719  5.44812062  5.69907406  5.9458475
  6.18844096  6.42685442  6.66108788  6.89114136  7.11701484  7.33870833
  7.55622183  7.76955534  7.97870885  8.18368237  8.3844759   8.58108944
  8.77352298  8.96177654  9.14585009  9.32574366  9.50145724  9.67299082
  9.84034441 10.00351801 10.16251161 10.31732522 10.46795884 10.61441247
 10.75668611 10.89477975 11.0286934  11.15842706 11.28398072 11.4053544
 11.52254808 11.63556177 11.74439546 11.84904916 11.94952288 12.0458166
 12.13793032 12.22586406 12.3096178  12.38919155 12.4645853  12.53579907
 12.60283284 12.66568662]

Estimate RLM:

In [8]:
resrlm = sm.RLM(y2, X).fit()
print(resrlm.params)
print(resrlm.bse)
[ 4.90656276e+00  5.07846047e-01 -1.24511090e-03]
[0.10080179 0.01556244 0.00137704]

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 0x7fb2cc07f518>

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)
[5.48945886 0.39977531]
[0.38358425 0.03305117]

Estimate RLM:

In [11]:
resrlm2 = sm.RLM(y2, X2).fit()
print(resrlm2.params)
print(resrlm2.bse)
[4.94999764 0.496027  ]
[0.08247076 0.00710601]

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")