Robust PCA Using MCD and MM Estimators in MARS A Simulation Study

Abstract

Multivariate Adaptive Regression Splines (MARS) models nonlinear relationships through adaptive basis functions but remain sensitive to outliers in the predictor variables. Existing robust extensions of MARS primarily address response outliers, while the few studies integrating Robust Principal Component Analysis (RPCA) with MARS use RPCA only for dimension reduction without comparing robust estimators. This study evaluates RPCA as a robust predictor transformation and systematically compares two robust covariance estimators—the Minimum Covariance Determinant (MCD) and the MM-estimator—within the RPCA-MARS framework. A full factorial simulation with 100 replications per condition covered 45 conditions: five sample sizes (n = 50, 100, 200, 500, 1000), three outlier proportions (5%, 10%, 25%), and three MARS interaction levels (1, 2, 3) with eight predictor variables. Outliers were extreme values in a specified proportion of predictor observations. Performance was measured by Root Mean Square Error (RMSE). For analysis, the 45 conditions were collapsed into 15 scenarios by selecting the interaction level with the minimum RMSE for each sample size and outlier proportion. The MM estimator outperformed the MCD estimator in 8 of 15 scenarios, achieving lower RMSE under moderate-to-high outlier contamination (10%–25%) with moderate sample sizes (n = 100–500). MCD performed better in the remaining 7 scenarios: under low contamination (5%) at n ≤ 200 and n ≥ 1000, and across all contamination levels at n = 1000. MCD showed higher variability at small samples with moderate-to-high contamination, while MM produced tighter confidence intervals and lower standard deviations. Within the RPCA-MARS framework, MM is recommended for moderately sized, highly contaminated data, while MCD is preferable under low contamination or in large-scale settings.