DOC: add an example of robust fitting

This compares RANSAC and ThielSen

examples/linear_model/plot_robust_fit.py

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+"""
+Demo robust 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 worst than OLS.
+
+"""
+
+from matplotlib import pyplot as plt
+import numpy as np
+
+from sklearn import linear_model, metrics
+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', linear_model.LinearRegression()),
+              ('Theil-Sen', linear_model.TheilSen(random_state=42)),
+              ('RANSAC', linear_model.RANSACRegressor(random_state=42)), ]
+
+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()
+    plt.plot(this_X[:, 0], this_y, 'k+')
+
+    for name, estimator in estimators:
+        model = make_pipeline(PolynomialFeatures(3), estimator)
+        model.fit(this_X, this_y)
+        mse = metrics.mean_squared_error(model.predict(X_test), y_test)
+        y_plot = model.predict(x_plot[:, np.newaxis])
+        plt.plot(x_plot, y_plot,
+                    label='%s: error = %.3f' % (name, mse))
+
+    plt.legend(loc='best', frameon=False,
+            title='Error: mean absolute deviation to non corrupt data')
+    plt.xlim(-4, 10.2)
+    plt.ylim(-2, 10.2)
+    plt.title(title)
+plt.show()