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Robust Linear ModelsΒΆ

Link to Notebook GitHub

In [1]:
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)

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))]))
[-41.0265   0.8294   0.9261  -0.1278]
[ 9.7919  0.111   0.3029  0.1286]
                    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:                Wed, 24 Dec 2014
Time:                        13:10:49
No. Iterations:                    19
==============================================================================
                 coef    std err          z      P>|z|      [95.0% Conf. Int.]
------------------------------------------------------------------------------
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)
[-41.0265   0.8294   0.9261  -0.1278]
[ 9.0895  0.1195  0.3224  0.118 ]

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:  [-40.8818   0.7928   1.0486  -0.1336]

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())
[ 4.9943  0.5347 -0.0147]
[ 0.4543  0.0701  0.0062]
[  4.6277   4.9034   5.1741   5.44     5.701    5.9571   6.2084   6.4547
   6.6962   6.9327   7.1644   7.3912   7.6131   7.8301   8.0423   8.2495
   8.4519   8.6494   8.842    9.0297   9.2125   9.3904   9.5635   9.7316
   9.8949  10.0533  10.2068  10.3554  10.4991  10.638   10.7719  10.901
  11.0252  11.1445  11.2589  11.3684  11.4731  11.5728  11.6677  11.7577
  11.8428  11.923   11.9983  12.0688  12.1343  12.195   12.2508  12.3017
  12.3477  12.3888]

Estimate RLM:

In [8]:
resrlm = sm.RLM(y2, X).fit()
print(resrlm.params)
print(resrlm.bse)
[ 4.9029  0.5256 -0.0053]
[ 0.1289  0.0199  0.0018]

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 0x7f0091cd2d10>
<matplotlib.figure.Figure at 0x7f0091c0fed0>

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)
[ 5.5854  0.3881]
[ 0.3969  0.0342]

Estimate RLM:

In [11]:
resrlm2 = sm.RLM(y2, X2).fit()
print(resrlm2.params)
print(resrlm2.bse)
[ 5.0572  0.4834]
[ 0.1141  0.0098]

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")
<matplotlib.figure.Figure at 0x7f009230b9d0>

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