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Asymptotically Optimal Search for a Change Point Anomaly under a Composite Hypothesis Model

Liad Lea Didi, Tomer Gafni, Kobi Cohen

TL;DR

This work addresses locating a change point anomaly among $M$ processes where only $K$ can be probed at each time, under a composite parameter model that distinguishes normal and abnormal states. It proposes the deterministic SCPA algorithm, organized into Exploration, Exploitation, and Sequential Testing phases, and proves its asymptotic Bayes-risk optimality as the error probability vanishes. The results cover two scenarios: (i) both normal and abnormal distributions have unknown parameters, yielding $R^* \sim -c \log c / D(\theta^{(1)})$, and (ii) the null parameter is known, yielding $R^* \sim -c \log c / D(\theta^{(1)}||\theta^{(0)})$, with improved detection time; post-change exploration remains $O(1)$ and detection scales with $-\log c$. Extensions include $K>1$ probing, multiple anomalies, and generalized LLR formulations, all supported by simulations. The approach advances sequential design and change-point detection by addressing dynamic, multi-process anomaly localization under composite hypotheses with strong asymptotic guarantees.

Abstract

We address the problem of searching for a change point in an anomalous process among a finite set of M processes. Specifically, we address a composite hypothesis model in which each process generates measurements following a common distribution with an unknown parameter (vector). This parameter belongs to either a normal or abnormal space depending on the current state of the process. Before the change point, all processes, including the anomalous one, are in a normal state; after the change point, the anomalous process transitions to an abnormal state. Our goal is to design a sequential search strategy that minimizes the Bayes risk by balancing sample complexity and detection accuracy. We propose a deterministic search algorithm with the following notable properties. First, we analytically demonstrate that when the distributions of both normal and abnormal processes are unknown, the algorithm is asymptotically optimal in minimizing the Bayes risk as the error probability approaches zero. In the second setting, where the parameter under the null hypothesis is known, the algorithm achieves asymptotic optimality with improved detection time based on the true normal state. Simulation results are presented to validate the theoretical findings.

Asymptotically Optimal Search for a Change Point Anomaly under a Composite Hypothesis Model

TL;DR

This work addresses locating a change point anomaly among processes where only can be probed at each time, under a composite parameter model that distinguishes normal and abnormal states. It proposes the deterministic SCPA algorithm, organized into Exploration, Exploitation, and Sequential Testing phases, and proves its asymptotic Bayes-risk optimality as the error probability vanishes. The results cover two scenarios: (i) both normal and abnormal distributions have unknown parameters, yielding , and (ii) the null parameter is known, yielding , with improved detection time; post-change exploration remains and detection scales with . Extensions include probing, multiple anomalies, and generalized LLR formulations, all supported by simulations. The approach advances sequential design and change-point detection by addressing dynamic, multi-process anomaly localization under composite hypotheses with strong asymptotic guarantees.

Abstract

We address the problem of searching for a change point in an anomalous process among a finite set of M processes. Specifically, we address a composite hypothesis model in which each process generates measurements following a common distribution with an unknown parameter (vector). This parameter belongs to either a normal or abnormal space depending on the current state of the process. Before the change point, all processes, including the anomalous one, are in a normal state; after the change point, the anomalous process transitions to an abnormal state. Our goal is to design a sequential search strategy that minimizes the Bayes risk by balancing sample complexity and detection accuracy. We propose a deterministic search algorithm with the following notable properties. First, we analytically demonstrate that when the distributions of both normal and abnormal processes are unknown, the algorithm is asymptotically optimal in minimizing the Bayes risk as the error probability approaches zero. In the second setting, where the parameter under the null hypothesis is known, the algorithm achieves asymptotic optimality with improved detection time based on the true normal state. Simulation results are presented to validate the theoretical findings.
Paper Structure (20 sections, 6 theorems, 63 equations, 6 figures)

This paper contains 20 sections, 6 theorems, 63 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Related Work
  4. System Model and Problem Formulation
  5. System Model
  6. Objective
  7. The Search Algorithm for Change Point Anomaly (SCPA)
  8. Description of the Algorithm
  9. Insights and Discussion on the SCPA Algorithm
  10. Considerations and Implementation Details
  11. Tackling Multi-Process Probing and Detection of Multiple Anomalies
  12. Using Generalized LLR
  13. Performance Analysis
  14. change point Anomaly Search without Side Information
  15. change point Anomaly Search Under a Known Model of Normality
  16. ...and 5 more sections

Key Result

Theorem 1

Let $R^{*}$ and $R(\Gamma \, ; \,\tau_c)$ be the Bayes risks under the SCPA algorithm and any other policy $\Gamma$, respectively. Then, the Bayes risk satisfies:The notation $f \sim g$ as $c \rightarrow 0$ refers to $\lim_{c \rightarrow 0} f/g=1$. where $D(\theta ^{(1)}) \triangleq \min\nolimits_{\varphi \in \Theta ^{(0)}} D(\theta ^{(1)}||\varphi)$.

Figures (6)

  • Figure 1: The average detection delay as a function of $-\log c$, with $M=5$, $\Theta^{(0)}= \{0.1N , 1\leq N \leq 10 \}$ , $\Theta^{(1)}= \{N , 2\leq N \leq 10 \}$ and $\tau_c=0$.
  • Figure 2: The average detection delay as a function of $-\log c$, with $M=5$, $\Theta^{(0)}= \{0.1N , 1\leq N \leq 10 \}$ , $\Theta^{(1)}= \{N , 2\leq N \leq 10 \}$ and $\tau_c=70$.
  • Figure 3: The error probability as a function of the average detection delay, with $M=5$, $\Theta^{(0)}= \{1+0.1N , 0\leq N \leq 10 \}$ , $\Theta^{(1)}= \{0.5+0.1N , 0\leq N \leq 4 \} \cup \{2.1+0.1N , 0\leq N \leq 4 \}$ and $\tau_c=0$.
  • Figure 4: The error probability as a function of the average detection delay, with $M=5$, $\Theta^{(0)}= \{1+0.1N , 1\leq N \leq 10 \}$ , $\Theta^{(1)}= \{0.5+0.1N , 0\leq N \leq 4 \} \cup \{2.1+0.1N , 0\leq N \leq 4 \}$ and $\tau_c=70$.
  • Figure 5: The error probability as a function of the average detection delay under the SCPA algorithm with side information, the SCPA algorithm without side information and the CUSUM algorithm, with $M=4$, $\Theta^{(0)}= \{0.1N , 1\leq N \leq 9 \}$ , $\Theta^{(1)}= \{N , 1\leq N \leq 30 \}$ and $\tau_c=20$.
  • ...and 1 more figures

Theorems & Definitions (18)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Definition 1
  • Definition 2
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • ...and 8 more