A Deep Dive into RANSAC and MLESAC for Robust Regression Analysis

Regression analysis is one of the most fundamental and widely used techniques in data science and machine learning. At its core, regression aims to model the relationship between input features and a continuous output variable. This allows us to make predictions, understand feature importance, and extract insights from data.

However, real-world data is messy. It often contains noise, outliers, and missing values that can severely degrade the performance of traditional regression methods like ordinary least squares (OLS). In the presence of outliers, OLS estimates can be heavily biased, as it blindly tries to minimize the sum of squared residuals.

Clearly, we need regression techniques that are robust to outliers and noise if we want to glean accurate insights from imperfect real-world data. This is where algorithms like RANSAC (RANdom SAmple Consensus) and MLESAC (Maximum Likelihood Estimation SAmple Consensus) come into play.

In this post, we‘ll take a deep dive into these two powerful tools for robust regression. We‘ll explore how they work, compare their strengths and weaknesses, walk through their implementation, and discuss best practices for application. Let‘s get started!

Random Sample Consensus (RANSAC)

RANSAC is a general approach for fitting models to data that contains a significant percentage of outliers. It was first introduced by Fischler and Bolles in 1981 and has since become a go-to algorithm in computer vision and robust statistics.

The key idea behind RANSAC is simple but powerful: instead of using all the data at once to fit a model (like OLS does), RANSAC repeatedly samples small subsets of the data, fits a model to each subset, and chooses the model that best explains the rest of the data. More specifically, the RANSAC algorithm works as follows:

1. Randomly sample a subset of the data with the minimum number of points needed to fit the model (e.g. 2 points for a line).

2. Fit a model to this subset.

3. Determine the set of "inliers" for this model – the data points that fall within a predefined error tolerance of the model.

4. If the fraction of inliers exceeds a predefined threshold, re-estimate the model using all identified inliers and terminate.

5. If the fraction of inliers is too low, repeat steps 1-4 (up to a predefined maximum number of iterations).

6. After reaching the maximum number of iterations, return the model with the largest inlier set.

The power of RANSAC lies in its ability to robustly estimate model parameters even when a significant portion of the data consists of outliers. By only considering the set of inliers for each model, RANSAC avoids being unduly influenced by erroneous data points.

However, RANSAC also has some limitations. It requires the user to specify error thresholds and the number of iterations, which can significantly impact performance if not tuned properly. RANSAC also does not guarantee finding the optimal model, only a model that is good enough to explain a sufficient fraction of the data. Finally, RANSAC can struggle when the percentage of outliers is very high (>50%) or when the data contains multiple valid models.

Despite these limitations, RANSAC remains a powerful and widely used tool for robust model fitting. It has been successfully applied to problems like line and plane fitting, homography estimation, and motion segmentation.

Maximum Likelihood Estimation Sample Consensus (MLESAC)

MLESAC is a generalization of RANSAC proposed by Torr and Zisserman in 2000. It aims to address some of the limitations of RANSAC by reformulating the robust estimation problem in a probabilistic framework.

The key insight behind MLESAC is that inliers and outliers can be modeled using different error distributions. Inliers are assumed to be normally distributed around the true model, while outliers are modeled using a uniform distribution.

Given this setup, MLESAC seeks the model that maximizes the likelihood of observing the data. This is in contrast to RANSAC, which simply maximizes the number of inliers.

Mathematically, if we let $\theta$ denote the model parameters, $p(e|\theta)$ the probability of an error $e$ given model $\theta$, and $\gamma$ the mixing parameter representing the fraction of inliers, then the log likelihood of the data under the MLESAC model is:

$$\log L(\theta,\gamma) = \sum_{i=1}^n \log (\gamma p(e_i|\theta) + (1-\gamma)p(e_i|outlier))$$

MLESAC uses the same sampling strategy as RANSAC to generate candidate models, but chooses the model that maximizes this log likelihood rather than the raw inlier count.

The advantages of MLESAC over RANSAC are several-fold. By explicitly modeling the error distributions of inliers and outliers, MLESAC provides a more principled way to score models. The mixing parameter $\gamma$ also provides a way to automatically adapt to different levels of contamination in the data. Finally, the probabilistic formulation allows for more nuanced decision making and easier integration with other probabilistic techniques.

However, MLESAC does require specifying the error distributions, which may not always be known a priori. It also tends to be more computationally expensive than RANSAC due to the need to evaluate likelihoods. Nevertheless, in many scenarios the improved performance is worth the added complexity.

Implementing RANSAC and MLESAC

Now that we understand how RANSAC and MLESAC work in principle, let‘s see how they can be implemented in practice. We‘ll use Python and the numpy library for demonstration purposes.

First, let‘s define a simple linear regression problem with outliers:

import numpy as np
import matplotlib.pyplot as plt

# Generate data
n_samples = 100
n_outliers = 25

X = np.random.rand(n_samples, 1)
y = 3 * X.squeeze() + 2 + np.random.randn(n_samples) * 0.5

# Add outliers
outlier_indices = np.random.choice(np.arange(n_samples), size=n_outliers, replace=False)
y[outlier_indices] += np.random.randint(low=-10, high=10, size=n_outliers)

plt.figure(figsize=(12,8))
plt.scatter(X, y, color=‘blue‘, marker=‘o‘, label=‘Data‘)
plt.xlabel(‘X‘)
plt.ylabel(‘y‘)
plt.legend()
plt.show()

This code generates a simple linear dataset with Gaussian noise and adds some outliers. The result looks like this:

[Generated data with outliers]

Now let‘s implement RANSAC to robustly fit a line to this data:

def ransac(X, y, n_iterations=100, threshold=1, min_inliers=10):
    best_inliers = np.zeros(len(X), dtype=bool)

    for i in range(n_iterations):
        # Randomly sample 2 points
        sample_indices = np.random.choice(np.arange(len(X)), size=2, replace=False)
        sample_X = X[sample_indices]
        sample_y = y[sample_indices]

        # Fit line to points
        line_params = np.polyfit(sample_X.squeeze(), sample_y, deg=1)

        # Compute distances to line
        distances = np.abs(np.polyval(line_params, X.squeeze()) - y)

        # Compute inliers
        inliers = distances < threshold

        # Update best model if necessary
        if np.sum(inliers) > np.sum(best_inliers):
            best_inliers = inliers
            best_line_params = line_params

            if np.sum(inliers) > min_inliers:
                break

    return best_line_params, best_inliers

ransac_line, ransac_inliers = ransac(X, y)

plt.figure(figsize=(12,8))
plt.scatter(X[ransac_inliers], y[ransac_inliers], color=‘blue‘, marker=‘o‘, label=‘Inliers‘)
plt.scatter(X[~ransac_inliers], y[~ransac_inliers], color=‘red‘, marker=‘x‘, label=‘Outliers‘) 
x_plot = np.linspace(0, 1, 100)
plt.plot(x_plot, np.polyval(ransac_line, x_plot), color=‘green‘, label=‘RANSAC Line‘)
plt.xlabel(‘X‘)
plt.ylabel(‘y‘) 
plt.legend()
plt.show()

This RANSAC implementation repeatedly samples subsets of 2 points, fits a line to each subset, and chooses the line with the most inliers (within a predefined threshold). The result looks like this:

[RANSAC fitted line]

As we can see, RANSAC is able to find a line that fits the underlying linear trend well, despite the presence of significant outliers.

Now let‘s implement MLESAC:

from scipy.stats import norm

def mlesac(X, y, n_iterations=100, threshold=1, min_inliers=10, sigma=1, p_outlier=0.5):
    best_likelihood = -np.inf

    def likelihood(distances, line_params):
        gamma = 0.5
        inlier_dist = norm(loc=0, scale=sigma)
        p_inliers = gamma * inlier_dist.pdf(distances)
        p_outliers = (1 - gamma) * p_outlier
        return np.sum(np.log(p_inliers + p_outliers))

    for i in range(n_iterations):
        # Randomly sample 2 points
        sample_indices = np.random.choice(np.arange(len(X)), size=2, replace=False)
        sample_X = X[sample_indices]
        sample_y = y[sample_indices]

        # Fit line to points
        line_params = np.polyfit(sample_X.squeeze(), sample_y, deg=1)

        # Compute distances to line
        distances = np.abs(np.polyval(line_params, X.squeeze()) - y)

        # Compute likelihood of distances
        current_likelihood = likelihood(distances, line_params)

        # Update best model if necessary
        if current_likelihood > best_likelihood:
            best_likelihood = current_likelihood
            best_line_params = line_params

    return best_line_params

mlesac_line = mlesac(X, y)

plt.figure(figsize=(12,8))
plt.scatter(X, y, color=‘blue‘, marker=‘o‘, label=‘Data‘)
x_plot = np.linspace(0, 1, 100)
plt.plot(x_plot, np.polyval(mlesac_line, x_plot), color=‘red‘, label=‘MLESAC Line‘)  
plt.xlabel(‘X‘)
plt.ylabel(‘y‘)
plt.legend()
plt.show()

The MLESAC implementation is similar to RANSAC, but instead of counting inliers, it computes the likelihood of the observed distances under a mixture model of inliers and outliers. The model with the maximum likelihood is chosen as the best fit. The result looks like this:

[MLESAC fitted line]

We can see that MLESAC also finds a good line fit, similar to RANSAC in this case. However, the probabilistic formulation of MLESAC provides more flexibility and robustness in general.

Tuning and Best Practices

While RANSAC and MLESAC are powerful algorithms, their performance can heavily depend on the choice of hyperparameters like the error threshold, number of iterations, and assumed inlier/outlier distributions.

In practice, it‘s important to tune these parameters for each specific problem. Some general guidelines include:

• Set the error threshold based on the expected level of noise in the inliers. Too high a threshold will cause the model to fit noise, while too low a threshold will reject valid data points.

• Choose the number of iterations based on the expected fraction of outliers and desired probability of finding a good model. More iterations increase the chances of finding a good fit but also increase computation time.

• For MLESAC, choose the inlier/outlier error distributions based on domain knowledge or empirical analysis of the data. Misspecified distributions can lead to poor results.

• Preprocess the data to remove obvious outliers before applying RANSAC or MLESAC. This can improve efficiency and reduce the number of iterations needed.

• Validate the final model on a held-out test set to check for overfitting.

Applications and Use Cases

RANSAC and MLESAC have found wide application in domains where data is prone to outliers and noise. Some common use cases include:

• Computer Vision: Fitting fundamental matrices, homographies, and 3D transformations in the presence of mismatched feature points.

• Robotics: Estimating sensor models and robot poses from noisy sensor data.

• Image Processing: Fitting parametric models to noisy edge or line detections.

• Time Series Analysis: Robustly estimating trends and change points in sensor data.

The general principle of outlier-robust model fitting embodied by RANSAC and MLESAC can be extended to many other types of models beyond just linear regression, making these algorithms versatile tools in the data scientist‘s toolkit.

Conclusion

In this post, we took a deep dive into the RANSAC and MLESAC algorithms for robust regression in the presence of outliers and noise. We saw how these techniques use random sampling and inlier scoring to find models that best explain the underlying trends in data without being unduly influenced by erroneous points.

We walked through practical implementations of both algorithms, discussed best practices for tuning and application, and highlighted some common use cases.

The key takeaway is that thoughtful model fitting is crucial when working with real-world, imperfect data. Algorithms like RANSAC and MLESAC provide principled ways to handle outliers and extract accurate insights from noisy data.

However, it‘s important to remember that these techniques are not silver bullets. They require careful parameter tuning and can be computationally expensive for large datasets. Moreover, they assume that the data contains a single dominant model plus outliers – if there are multiple valid models or structures in the data, more sophisticated techniques may be needed.

Nonetheless, RANSAC and MLESAC remain important tools that every data scientist should be familiar with. By adding robustness to the core task of regression analysis, they expand the range and reliability of models that can be learned from real-world data.