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: Wed, 29 Jul 2026
Time: 17:37:32
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())
[ 4.9792621 0.55240743 -0.01585546]
[0.4808475 0.07423636 0.00656877]
[ 4.58287565 4.87042275 5.1526869 5.4296681 5.70136635 5.96778164
6.22891399 6.48476338 6.73532982 6.98061331 7.22061385 7.45533144
7.68476607 7.90891776 8.12778649 8.34137227 8.5496751 8.75269498
8.95043191 9.14288589 9.33005691 9.51194499 9.68855011 9.85987228
10.0259115 10.18666777 10.34214109 10.49233145 10.63723887 10.77686333
10.91120484 11.0402634 11.16403901 11.28253167 11.39574137 11.50366813
11.60631193 11.70367279 11.79575069 11.88254564 11.96405763 12.04028668
12.11123278 12.17689592 12.23727611 12.29237335 12.34218765 12.38671898
12.42596737 12.45993281]
Estimate RLM:
[9]:
resrlm = sm.RLM(y2, X).fit()
print(resrlm.params)
print(resrlm.bse)
[ 4.86806645 0.54461762 -0.0056461 ]
[0.14692 0.02268246 0.00200705]
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 0x7f288fae2900>
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)
[5.61833412 0.39385286]
[0.42108812 0.03628266]
Estimate RLM:
[12]:
resrlm2 = sm.RLM(y2, X2).fit()
print(resrlm2.params)
print(resrlm2.bse)
[5.05964567 0.49699963]
[0.11713858 0.01009313]
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")
