seMCD: Sequentially implemented Monte Carlo depth computation with statistical guarantees

TL;DR

seMCD introduces a sequential Monte Carlo method for efficiently computing statistical depth functions with probabilistic guarantees, applicable to multivariate and functional data, and useful in outlier detection and classification.

Contribution

It provides a novel sequential Monte Carlo approach with statistical guarantees for depth function computation, reducing sample size requirements compared to traditional methods.

Findings

Efficient depth computation with fewer samples.

High-probability interval estimates for depth functions.

Applicable to multivariate and functional spaces.

Abstract

Statistical depth functions provide center-outward orderings in spaces of dimension larger than one, where a natural ordering does not exist. The numerical evaluation of such depth functions can be computationally prohibitive, even for relatively low dimensions. We present a novel sequentially implemented Monte Carlo methodology for the computation of, theoretical and empirical, depth functions and related quantities (seMCD), that outputs an interval, a so-called seMCD-bucket, to which the quantity of interest belongs with a high probability prespecified by the user. For specific classes of depth functions, we adapt algorithms from sequential testing, providing finite-sample guarantees. For depth functions dependent on unknown distributions, we offer asymptotic guarantees using non-parametric statistical methods. In contrast to plain-vanilla Monte Carlo methodology the number of samples required in the algorithm is random but typically much smaller than standard choices suggested in the literature. The seMCD method can be applied to various depth functions, covering multivariate and functional spaces. We demonstrate the efficiency and reliability of our approach through empirical studies, highlighting its applicability in outlier or anomaly detection, classification, and depth region computation. In conclusion, the seMCD-algorithm can achieve accurate depth approximations with few Monte Carlo samples while maintaining rigorous statistical guarantees.