Optimal Sequential Outlier Hypothesis Testing under Universality Constraints

Core Concepts

The authors derive bounds on the achievable error exponents for sequential outlier hypothesis testing under two universality constraints: the error probability universality constraint and the expected stopping time universality constraint.

Abstract

The authors revisit the problem of sequential outlier hypothesis testing, where the goal is to identify a set of outliers among a given number of observed sequences. Most sequences are generated from a nominal distribution, while a few are generated from an anomalous distribution. The authors consider two types of universality constraints: Error Probability Universality Constraint: The test must have the error probability bounded by a tolerable value under each hypothesis. Expected Stopping Time Universality Constraint: The expected stopping time under each hypothesis is bounded. For the case of exactly one outlier, the authors: Derive a matching converse result and a simpler achievability part under the error probability universality constraint, strengthening a previous result. Propose a sequential test that has bounded average sample size under any pair of nominal and anomalous distributions and show that it has better theoretical performance than the fixed-length test. Derive the exact error exponents under the expected stopping time universality constraint. The authors also generalize their results to the case of multiple outliers when the number of outliers is known.

Stats

The authors use the following key metrics and figures: The misclassification error probability under each hypothesis (Eq. 1) The expected stopping time under each hypothesis (Eq. 2) The generalized Jensen-Shannon divergence (Eq. 14) The Rényi divergence (Eq. 20)

Quotes

"For the case of exactly one outlier, we address all above limitations. Furthermore, we generalize our results to the case of multiple outliers when the number of outliers is known." "Our main contribution is summarized in the following subsection."

Key Insights Distilled From

Sequential Outlier Hypothesis Testing under Universality Constraints

by Jun Diao,Lin... at arxiv.org 04-23-2024

https://arxiv.org/pdf/2404.14221.pdf

Deeper Inquiries

How can the proposed sequential tests be extended to the case where the number of outliers is unknown

To extend the proposed sequential tests to the case where the number of outliers is unknown, an additional step to estimate the number of outliers would be necessary. This can be achieved by incorporating a procedure to dynamically adjust the test based on the observed data. One approach could involve iteratively updating the test as more data is collected, allowing for the estimation of the number of outliers. This adaptive framework would involve continuously monitoring the data and adjusting the test parameters accordingly to accommodate the uncertainty in the number of outliers. By incorporating adaptive mechanisms into the sequential testing process, the tests can be extended to handle scenarios where the number of outliers is unknown.

What are the potential applications of the optimal sequential outlier hypothesis testing framework beyond the theoretical analysis presented in this work

The optimal sequential outlier hypothesis testing framework presented in this work has various potential applications beyond the theoretical analysis. One practical application could be in anomaly detection systems, where the framework can be utilized to identify outliers in large datasets in real-time. This can be particularly useful in cybersecurity, fraud detection, and quality control processes where the timely identification of anomalies is crucial. Additionally, the framework can be applied in signal processing for detecting abnormal patterns in data streams, in healthcare for identifying unusual patient conditions, and in finance for detecting fraudulent activities. The robustness and efficiency of the sequential tests make them valuable tools in a wide range of applications where outlier detection is essential.

Can the techniques used in this paper be applied to other sequential hypothesis testing problems with universality constraints

The techniques used in this paper for sequential hypothesis testing with universality constraints can be applied to other related problems with similar constraints. For instance, the framework can be adapted for sequential binary classification tasks where the goal is to classify data into two categories based on sequential observations. By modifying the decision rules and stopping criteria, the methodology can be extended to sequential classification problems with error probability or expected stopping time universality constraints. Furthermore, the techniques can be applied to sequential change-point detection, where the objective is to detect changes in the underlying distribution of sequential data. By leveraging the principles of large deviations and optimal stopping, the framework can be tailored to address a variety of sequential hypothesis testing problems with universality constraints.