# Robust linear estimator fitting¶

Here a sine function is fit with a polynomial of order 3, for values close to zero.

Robust fitting is demoed in different situations:

• No measurement errors, only modelling errors (fitting a sine with a polynomial)

• Measurement errors in X

• Measurement errors in y

The median absolute deviation to non corrupt new data is used to judge the quality of the prediction.

What we can see that:

• RANSAC is good for strong outliers in the y direction

• TheilSen is good for small outliers, both in direction X and y, but has a break point above which it performs worse than OLS.

• The scores of HuberRegressor may not be compared directly to both TheilSen and RANSAC because it does not attempt to completely filter the outliers but lessen their effect.

• • • • • from matplotlib import pyplot as plt
import numpy as np

from sklearn.linear_model import (
LinearRegression, TheilSenRegressor, RANSACRegressor, HuberRegressor)
from sklearn.metrics import mean_squared_error
from sklearn.preprocessing import PolynomialFeatures
from sklearn.pipeline import make_pipeline

np.random.seed(42)

X = np.random.normal(size=400)
y = np.sin(X)
# Make sure that it X is 2D
X = X[:, np.newaxis]

X_test = np.random.normal(size=200)
y_test = np.sin(X_test)
X_test = X_test[:, np.newaxis]

y_errors = y.copy()
y_errors[::3] = 3

X_errors = X.copy()
X_errors[::3] = 3

y_errors_large = y.copy()
y_errors_large[::3] = 10

X_errors_large = X.copy()
X_errors_large[::3] = 10

estimators = [('OLS', LinearRegression()),
('Theil-Sen', TheilSenRegressor(random_state=42)),
('RANSAC', RANSACRegressor(random_state=42)),
('HuberRegressor', HuberRegressor())]
colors = {'OLS': 'turquoise', 'Theil-Sen': 'gold', 'RANSAC': 'lightgreen', 'HuberRegressor': 'black'}
linestyle = {'OLS': '-', 'Theil-Sen': '-.', 'RANSAC': '--', 'HuberRegressor': '--'}
lw = 3

x_plot = np.linspace(X.min(), X.max())
for title, this_X, this_y in [
('Modeling Errors Only', X, y),
('Corrupt X, Small Deviants', X_errors, y),
('Corrupt y, Small Deviants', X, y_errors),
('Corrupt X, Large Deviants', X_errors_large, y),
('Corrupt y, Large Deviants', X, y_errors_large)]:
plt.figure(figsize=(5, 4))
plt.plot(this_X[:, 0], this_y, 'b+')

for name, estimator in estimators:
model = make_pipeline(PolynomialFeatures(3), estimator)
model.fit(this_X, this_y)
mse = mean_squared_error(model.predict(X_test), y_test)
y_plot = model.predict(x_plot[:, np.newaxis])
plt.plot(x_plot, y_plot, color=colors[name], linestyle=linestyle[name],
linewidth=lw, label='%s: error = %.3f' % (name, mse))

legend_title = 'Error of Mean\nAbsolute Deviation\nto Non-corrupt Data'
legend = plt.legend(loc='upper right', frameon=False, title=legend_title,
prop=dict(size='x-small'))
plt.xlim(-4, 10.2)
plt.ylim(-2, 10.2)
plt.title(title)
plt.show()


Total running time of the script: ( 0 minutes 2.588 seconds)

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