Robust Linear Models¶
[1]:
%matplotlib inline
[2]:
import matplotlib.pyplot as plt
import numpy as np
import statsmodels.api as sm
Estimation¶
Load data:
[3]:
data = sm.datasets.stackloss.load()
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))]
)
)
const -41.026498
AIRFLOW 0.829384
WATERTEMP 0.926066
ACIDCONC -0.127847
dtype: float64
const 9.791899
AIRFLOW 0.111005
WATERTEMP 0.302930
ACIDCONC 0.128650
dtype: float64
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: Fri, 05 Dec 2025
Time: 18:07:55
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)
const -41.026498
AIRFLOW 0.829384
WATERTEMP 0.926066
ACIDCONC -0.127847
dtype: float64
const 9.089504
AIRFLOW 0.119460
WATERTEMP 0.322355
ACIDCONC 0.117963
dtype: float64
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: const -40.881796
AIRFLOW 0.792761
WATERTEMP 1.048576
ACIDCONC -0.133609
dtype: float64
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 * 1.0 * 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())
[ 5.09209616 0.52225984 -0.0135906 ]
[0.45484388 0.07022176 0.00621354]
[ 4.75233103 5.01870602 5.28055268 5.53787104 5.79066108 6.0389228
6.28265621 6.5218613 6.75653808 6.98668655 7.2123067 7.43339853
7.64996205 7.86199726 8.06950415 8.27248272 8.47093298 8.66485493
8.85424856 9.03911387 9.21945087 9.39525956 9.56653993 9.73329199
9.89551573 10.05321116 10.20637827 10.35501707 10.49912755 10.63870971
10.77376357 10.9042891 11.03028633 11.15175524 11.26869583 11.38110811
11.48899207 11.59234772 11.69117505 11.78547407 11.87524477 11.96048716
12.04120124 12.117387 12.18904444 12.25617357 12.31877439 12.37684689
12.43039107 12.47940694]
Estimate RLM:
[9]:
resrlm = sm.RLM(y2, X).fit()
print(resrlm.params)
print(resrlm.bse)
[ 5.04838226e+00 4.95072966e-01 -1.49849195e-03]
[0.1279066 0.01974705 0.00174731]
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")
pred_ols = res.get_prediction()
iv_l = pred_ols.summary_frame()["obs_ci_lower"]
iv_u = pred_ols.summary_frame()["obs_ci_upper"]
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 0x7f6c02f652a0>
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)
[5.63988074 0.3863538 ]
[0.39436763 0.03398031]
Estimate RLM:
[12]:
resrlm2 = sm.RLM(y2, X2).fit()
print(resrlm2.params)
print(resrlm2.bse)
[5.09970311 0.48215037]
[0.09935267 0.00856063]
Draw a plot to compare OLS estimates to the robust estimates:
[13]:
pred_ols = res2.get_prediction()
iv_l = pred_ols.summary_frame()["obs_ci_lower"]
iv_u = pred_ols.summary_frame()["obs_ci_upper"]
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")
