Outlier Detection Using Nonconvex Penalized Regression (1006.2592v3)

Abstract: This paper studies the outlier detection problem from the point of view of penalized regressions. Our regression model adds one mean shift parameter for each of the $n$ data points. We then apply a regularization favoring a sparse vector of mean shift parameters. The usual $L_1$ penalty yields a convex criterion, but we find that it fails to deliver a robust estimator. The $L_1$ penalty corresponds to soft thresholding. We introduce a thresholding (denoted by $\Theta$) based iterative procedure for outlier detection ($\Theta$-IPOD). A version based on hard thresholding correctly identifies outliers on some hard test problems. We find that $\Theta$-IPOD is much faster than iteratively reweighted least squares for large data because each iteration costs at most $O(np)$ (and sometimes much less) avoiding an $O(np^2)$ least squares estimate. We describe the connection between $\Theta$-IPOD and $M$-estimators. Our proposed method has one tuning parameter with which to both identify outliers and estimate regression coefficients. A data-dependent choice can be made based on BIC. The tuned $\Theta$-IPOD shows outstanding performance in identifying outliers in various situations in comparison to other existing approaches. This methodology extends to high-dimensional modeling with $p\gg n$, if both the coefficient vector and the outlier pattern are sparse.

• The paper introduces Θ-IPOD as a robust outlier detection method that employs nonconvex penalized regression to overcome masking and swamping issues.
• It utilizes a unique thresholding operator with BIC-tuned parameters to iteratively refine regression coefficients and isolate anomalies.
• Simulations show that Θ-IPOD outperforms traditional robust methods in accuracy and computational efficiency, especially in high-dimensional settings.

Nonconvex Penalized Regression for Robust Outlier Detection

The paper authored by Yiyuan She and Art B. Owen addresses the problem of outlier detection in regression analysis, specifically through the lens of penalized regression techniques. Traditional approaches such as ordinary least squares (OLS) are notoriously sensitive to outliers, often resulting in unreliable estimates. The authors propose using a nonconvex penalized regression method, which they term Θ-IPOD (Thresholding-based Iterative Procedure for Outlier Detection), to achieve robust outlier detection and coefficient estimation.

Key Contributions

The main contribution of this work is the development of the Θ-IPOD algorithm. This method introduces a mean shift parameter for each datum and applies a regularization technique that encourages sparsity in these parameters. While the traditional L1 penalty (soft thresholding) offers a convex criterion, it lacks robustness in detecting outliers. The authors push beyond convex penalties, employing nonconvex thresholding to improve detection, particularly for multiple outliers which can mask each other (masking) or falsely implicate non-outliers (swamping).

Methodology

Θ-IPOD utilizes a unique thresholding operation (Θ), deviating from the common soft-thresholding to incorporate hard-thresholding and other nonconvex methods. The algorithm iteratively refines estimates of the regression coefficients while simultaneously identifying outliers. A central parameter, tuned using the Bayesian Information Criterion (BIC), facilitates this dual estimation. Importantly, the computational complexity is reduced compared to traditional methods like iteratively reweighted least squares (IRLS), offering significant time savings, especially in high-dimensional datasets.

Numerical Results and Comparisons

The authors demonstrate that Θ-IPOD consistently outperforms existing methods, particularly in scenarios with numerous or high-leverage outliers. Numerical simulations illustrate the method's efficacy in correctly identifying true outliers and minimizing false detections (swamping). The paper provides a thorough comparison with established robust regression techniques like M-estimators, MM-estimators, and Least Trimmed Squares (LTS), with Θ-IPOD showing superior detection rates and lower computational demands.

Theoretical and Practical Implications

Theoretically, this work explores the connections between robust regression and penalized regression frameworks, extending traditional M-estimators with a new characterization that incorporates nonconvex penalties. This methodological advancement potentially impacts econometrics, biostatistics, and any field where robust statistical inference is crucial.

Practically, Θ-IPOD is applicable to high-dimensional settings, where p exceeds n, a common occurrence in modern data analyses. This capability is particularly valuable in fields like genomics or finance, where both the predictor variable space and the potential for outliers are vast.

Future Directions

The exploration into nonconvex penalty functions opens several avenues for future research. There is room for developing adaptive methods to further enhance the robustness and efficiency of the algorithm, possibly integrating machine learning techniques for automated tuning of penalty parameters. Expanding the method to non-linear models and dynamic data settings are also promising directions.

In summary, She and Owen's paper makes a significant contribution to the field of robust statistics by presenting a meticulously designed approach for outlier detection via nonconvex penalized regression. The Θ-IPOD method adeptly handles the challenging issues of masking and swamping, offering researchers a practical and efficient tool for outlier detection in complex datasets.