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=[f"var_{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: Tue, 27 Oct 2026
Time: 21:59:33
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())
[ 4.99615938 0.53236218 -0.01355959]
[0.45658671 0.07049083 0.00623735]
[ 4.65716963 4.9275466 5.19340558 5.45474658 5.7115696 5.96387464
6.2116617 6.45493078 6.69368187 6.92791499 7.15763012 7.38282728
7.60350645 7.81966764 8.03131086 8.23843609 8.44104334 8.63913261
8.83270389 9.0217572 9.20629253 9.38630987 9.56180924 9.73279062
9.89925403 10.06119945 10.21862689 10.37153635 10.51992783 10.66380133
10.80315685 10.93799438 11.06831394 11.19411552 11.31539911 11.43216472
11.54441236 11.65214201 11.75535368 11.85404737 11.94822308 12.03788081
12.12302056 12.20364233 12.27974611 12.35133192 12.41839974 12.48094959
12.53898145 12.59249533]
Estimate RLM:
[9]:
resrlm = sm.RLM(y2, X).fit()
print(resrlm.params)
print(resrlm.bse)
[ 4.91176050e+00 5.19555877e-01 -2.97553573e-03]
[0.1256343 0.01939624 0.00171627]
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 0x7008a97f3770>
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)
[5.54269388 0.39676628]
[0.39565659 0.03409137]
Estimate RLM:
[12]:
resrlm2 = sm.RLM(y2, X2).fit()
print(resrlm2.params)
print(resrlm2.bse)
[5.0139751 0.49210537]
[0.10795157 0.00930154]
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")
