Robust Linear Models

[1]:
%matplotlib inline
[2]:
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:

[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

[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:                Thu, 05 Nov 2020
Time:                        07:28:38
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

[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

[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:

[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.

[8]:
res = sm.OLS(y2, X).fit()
print(res.params)
print(res.bse)
print(res.predict())
[ 5.1409435   0.51155665 -0.01167533]
[0.44050267 0.06800767 0.00601763]
[ 4.8490602   5.10356818  5.35418599  5.60091365  5.84375115  6.0826985
  6.31775569  6.54892272  6.7761996   6.99958632  7.21908288  7.43468929
  7.64640554  7.85423163  8.05816757  8.25821335  8.45436898  8.64663445
  8.83500976  9.01949491  9.20008991  9.37679476  9.54960944  9.71853397
  9.88356835 10.04471256 10.20196663 10.35533053 10.50480428 10.65038787
 10.79208131 10.92988459 11.06379771 11.19382067 11.31995348 11.44219614
 11.56054863 11.67501098 11.78558316 11.89226519 11.99505706 12.09395877
 12.18897033 12.28009173 12.36732298 12.45066407 12.530115   12.60567578
 12.6773464  12.74512686]

Estimate RLM:

[9]:
resrlm = sm.RLM(y2, X).fit()
print(resrlm.params)
print(resrlm.bse)
[ 5.10410129e+00  4.91683010e-01 -1.04482612e-03]
[0.15423383 0.02381162 0.00210696]

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

[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")
[10]:
<matplotlib.legend.Legend at 0x7f7b49051460>
../../../_images/examples_notebooks_generated_robust_models_0_18_1.png

Example 2: linear function with linear truth

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

[11]:
X2 = X[:,[0,1]]
res2 = sm.OLS(y2, X2).fit()
print(res2.params)
print(res2.bse)
[5.61153086 0.39480333]
[0.37815392 0.03258327]

Estimate RLM:

[12]:
resrlm2 = sm.RLM(y2, X2).fit()
print(resrlm2.params)
print(resrlm2.bse)
[5.13611868 0.4824832 ]
[0.12046995 0.01038018]

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

[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")
../../../_images/examples_notebooks_generated_robust_models_0_24_0.png