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=========================================== Robust linear model estimation using RANSAC

In this example, we see how to robustly fit a linear model to faulty data using the :ref:RANSAC <ransac_regression> algorithm.

The ordinary linear regressor is sensitive to outliers, and the fitted line can easily be skewed away from the true underlying relationship of data.

The RANSAC regressor automatically splits the data into inliers and outliers, and the fitted line is determined only by the identified inliers.

Imports for RANSAC Robust Regression

Real-world datasets frequently contain outliers – data points that do not follow the general pattern due to measurement errors, sensor glitches, or data entry mistakes. Ordinary Least Squares (OLS) minimizes the sum of squared residuals, making it extremely sensitive to outliers: a single extreme point can dramatically shift the fitted line away from the true relationship.

RANSAC (RANdom SAmple Consensus) is an iterative algorithm that repeatedly selects random subsets of the data, fits a model to each subset, and classifies all other points as inliers or outliers based on a residual threshold. The final model is fit only on the largest consensus set of inliers. This makes RANSAC highly robust to even a large proportion of outliers – a property critical in applications like computer vision (fitting geometric models to noisy feature matches), autonomous driving, and any domain where data quality cannot be guaranteed.

# Authors: The scikit-learn developers
# SPDX-License-Identifier: BSD-3-Clause

import numpy as np
from matplotlib import pyplot as plt

from sklearn import datasets, linear_model

n_samples = 1000
n_outliers = 50


X, y, coef = datasets.make_regression(
    n_samples=n_samples,
    n_features=1,
    n_informative=1,
    noise=10,
    coef=True,
    random_state=0,
)

# Add outlier data
np.random.seed(0)
X[:n_outliers] = 3 + 0.5 * np.random.normal(size=(n_outliers, 1))
y[:n_outliers] = -3 + 10 * np.random.normal(size=n_outliers)

# Fit line using all data
lr = linear_model.LinearRegression()
lr.fit(X, y)

# Robustly fit linear model with RANSAC algorithm
ransac = linear_model.RANSACRegressor()
ransac.fit(X, y)
inlier_mask = ransac.inlier_mask_
outlier_mask = np.logical_not(inlier_mask)

# Predict data of estimated models
line_X = np.arange(X.min(), X.max())[:, np.newaxis]
line_y = lr.predict(line_X)
line_y_ransac = ransac.predict(line_X)

# Compare estimated coefficients
print("Estimated coefficients (true, linear regression, RANSAC):")
print(coef, lr.coef_, ransac.estimator_.coef_)

lw = 2
plt.scatter(
    X[inlier_mask], y[inlier_mask], color="yellowgreen", marker=".", label="Inliers"
)
plt.scatter(
    X[outlier_mask], y[outlier_mask], color="gold", marker=".", label="Outliers"
)
plt.plot(line_X, line_y, color="navy", linewidth=lw, label="Linear regressor")
plt.plot(
    line_X,
    line_y_ransac,
    color="cornflowerblue",
    linewidth=lw,
    label="RANSAC regressor",
)
plt.legend(loc="lower right")
plt.xlabel("Input")
plt.ylabel("Response")
plt.show()