.. _sphx_glr_auto_examples_linear_model_plot_theilsen.py: ==================== Theil-Sen Regression ==================== Computes a Theil-Sen Regression on a synthetic dataset. See :ref:`theil_sen_regression` for more information on the regressor. Compared to the OLS (ordinary least squares) estimator, the Theil-Sen estimator is robust against outliers. It has a breakdown point of about 29.3% in case of a simple linear regression which means that it can tolerate arbitrary corrupted data (outliers) of up to 29.3% in the two-dimensional case. The estimation of the model is done by calculating the slopes and intercepts of a subpopulation of all possible combinations of p subsample points. If an intercept is fitted, p must be greater than or equal to n_features + 1. The final slope and intercept is then defined as the spatial median of these slopes and intercepts. In certain cases Theil-Sen performs better than :ref:`RANSAC ` which is also a robust method. This is illustrated in the second example below where outliers with respect to the x-axis perturb RANSAC. Tuning the ``residual_threshold`` parameter of RANSAC remedies this but in general a priori knowledge about the data and the nature of the outliers is needed. Due to the computational complexity of Theil-Sen it is recommended to use it only for small problems in terms of number of samples and features. For larger problems the ``max_subpopulation`` parameter restricts the magnitude of all possible combinations of p subsample points to a randomly chosen subset and therefore also limits the runtime. Therefore, Theil-Sen is applicable to larger problems with the drawback of losing some of its mathematical properties since it then works on a random subset. .. code-block:: python # Author: Florian Wilhelm -- # License: BSD 3 clause import time import numpy as np import matplotlib.pyplot as plt from sklearn.linear_model import LinearRegression, TheilSenRegressor from sklearn.linear_model import RANSACRegressor print(__doc__) estimators = [('OLS', LinearRegression()), ('Theil-Sen', TheilSenRegressor(random_state=42)), ('RANSAC', RANSACRegressor(random_state=42)), ] colors = {'OLS': 'turquoise', 'Theil-Sen': 'gold', 'RANSAC': 'lightgreen'} lw = 2 Outliers only in the y direction .. code-block:: python np.random.seed(0) n_samples = 200 # Linear model y = 3*x + N(2, 0.1**2) x = np.random.randn(n_samples) w = 3. c = 2. noise = 0.1 * np.random.randn(n_samples) y = w * x + c + noise # 10% outliers y[-20:] += -20 * x[-20:] X = x[:, np.newaxis] plt.scatter(x, y, color='indigo', marker='x', s=40) line_x = np.array([-3, 3]) for name, estimator in estimators: t0 = time.time() estimator.fit(X, y) elapsed_time = time.time() - t0 y_pred = estimator.predict(line_x.reshape(2, 1)) plt.plot(line_x, y_pred, color=colors[name], linewidth=lw, label='%s (fit time: %.2fs)' % (name, elapsed_time)) plt.axis('tight') plt.legend(loc='upper left') plt.title("Corrupt y") .. image:: /auto_examples/linear_model/images/sphx_glr_plot_theilsen_001.png :align: center Outliers in the X direction .. code-block:: python np.random.seed(0) # Linear model y = 3*x + N(2, 0.1**2) x = np.random.randn(n_samples) noise = 0.1 * np.random.randn(n_samples) y = 3 * x + 2 + noise # 10% outliers x[-20:] = 9.9 y[-20:] += 22 X = x[:, np.newaxis] plt.figure() plt.scatter(x, y, color='indigo', marker='x', s=40) line_x = np.array([-3, 10]) for name, estimator in estimators: t0 = time.time() estimator.fit(X, y) elapsed_time = time.time() - t0 y_pred = estimator.predict(line_x.reshape(2, 1)) plt.plot(line_x, y_pred, color=colors[name], linewidth=lw, label='%s (fit time: %.2fs)' % (name, elapsed_time)) plt.axis('tight') plt.legend(loc='upper left') plt.title("Corrupt x") plt.show() .. image:: /auto_examples/linear_model/images/sphx_glr_plot_theilsen_002.png :align: center **Total running time of the script:** (0 minutes 1.785 seconds) .. container:: sphx-glr-download **Download Python source code:** :download:`plot_theilsen.py ` .. container:: sphx-glr-download **Download IPython notebook:** :download:`plot_theilsen.ipynb `