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: Mon, 23 Dec 2024
Time: 12:04:39
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())
[ 5.11674399 0.52144612 -0.01325776]
[0.47195388 0.07286331 0.00644728]
[ 4.7853 5.05003975 5.31036209 5.56626701 5.81775452 6.06482462
6.3074773 6.54571257 6.77953043 7.00893088 7.23391392 7.45447954
7.67062775 7.88235855 8.08967193 8.2925679 8.49104646 8.68510761
8.87475134 9.05997766 9.24078657 9.41717807 9.58915215 9.75670882
9.91984808 10.07856992 10.23287436 10.38276138 10.52823098 10.66928318
10.80591796 10.93813533 11.06593529 11.18931783 11.30828297 11.42283068
11.53296099 11.63867389 11.73996937 11.83684744 11.92930809 12.01735134
12.10097717 12.18018559 12.25497659 12.32535018 12.39130637 12.45284513
12.50996649 12.56267043]
Estimate RLM:
[9]:
resrlm = sm.RLM(y2, X).fit()
print(resrlm.params)
print(resrlm.bse)
[ 5.04728509e+00 5.02748637e-01 -1.69722699e-03]
[0.13549213 0.02091816 0.00185093]
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 0x7fb3cb37baf0>
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)
[5.65111287 0.38886852]
[0.40700164 0.03506891]
Estimate RLM:
[12]:
resrlm2 = sm.RLM(y2, X2).fit()
print(resrlm2.params)
print(resrlm2.bse)
[5.0830686 0.48992705]
[0.10719644 0.00923648]
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")
Last update:
Dec 23, 2024