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Adaptive Out-of-Control Point Pattern Detection in Sequential Random Finite Set Observations

Konstantinos Bourazas, Savvas Papaioannou, Panayiotis Kolios

TL;DR

The paper tackles online anomaly detection for sequential point-pattern data modeled as Poisson Random Finite Sets. It introduces Power Discounting Posteriors to adapt Bayesian parameter estimates over time and derives a posterior predictive density to assess new observations. By separating predictive checks for cardinality and features and combining them with Fisher's method, the approach detects Out-Of-Control patterns with robustness to gradual process shifts. The method demonstrates improved detection performance over offline ranking-based methods, enabling real-time monitoring of dynamic spatio-temporal point processes in applications like manufacturing and surveillance.

Abstract

In this work we introduce a novel adaptive anomaly detection framework specifically designed for monitoring sequential random finite set (RFS) observations. Our approach effectively distinguishes between In-Control data (normal) and Out-Of-Control data (anomalies) by detecting deviations from the expected statistical behavior of the process. The primary contributions of this study include the development of an innovative RFS-based framework that not only learns the normal behavior of the data-generating process online but also dynamically adapts to behavioral shifts to accurately identify abnormal point patterns. To achieve this, we introduce a new class of RFS-based posterior distributions, named Power Discounting Posteriors (PD), which facilitate adaptation to systematic changes in data while enabling anomaly detection of point pattern data through a novel predictive posterior density function. The effectiveness of the proposed approach is demonstrated by extensive qualitative and quantitative simulation experiments.

Adaptive Out-of-Control Point Pattern Detection in Sequential Random Finite Set Observations

TL;DR

The paper tackles online anomaly detection for sequential point-pattern data modeled as Poisson Random Finite Sets. It introduces Power Discounting Posteriors to adapt Bayesian parameter estimates over time and derives a posterior predictive density to assess new observations. By separating predictive checks for cardinality and features and combining them with Fisher's method, the approach detects Out-Of-Control patterns with robustness to gradual process shifts. The method demonstrates improved detection performance over offline ranking-based methods, enabling real-time monitoring of dynamic spatio-temporal point processes in applications like manufacturing and surveillance.

Abstract

In this work we introduce a novel adaptive anomaly detection framework specifically designed for monitoring sequential random finite set (RFS) observations. Our approach effectively distinguishes between In-Control data (normal) and Out-Of-Control data (anomalies) by detecting deviations from the expected statistical behavior of the process. The primary contributions of this study include the development of an innovative RFS-based framework that not only learns the normal behavior of the data-generating process online but also dynamically adapts to behavioral shifts to accurately identify abnormal point patterns. To achieve this, we introduce a new class of RFS-based posterior distributions, named Power Discounting Posteriors (PD), which facilitate adaptation to systematic changes in data while enabling anomaly detection of point pattern data through a novel predictive posterior density function. The effectiveness of the proposed approach is demonstrated by extensive qualitative and quantitative simulation experiments.

Paper Structure

This paper contains 9 sections, 2 theorems, 29 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Random Finite Sets
  4. Problem Formulation
  5. Adaptive Anomaly Detection in Sequential RFS Observations
  6. Power Discounting posteriors for RFS Point-pattern Data
  7. Detecting Anomalies in Sequential RFS Observations
  8. Evaluation
  9. Conclusion

Key Result

Lemma 1

Assuming the prior $\lambda_t \sim G\left(c_0, d_0\right)$, i.e., a Gamma distribution with hyperparameters $c_0$, and $d_0$, then the PD posterior distribution is conjugate, i.e., $\lambda_t | \left( n_{1:t}, \alpha_0 \right) \sim G\left(c_t, d_t\right)$, with $c_t$, and $d_t$, the updated paramet

Figures (3)

  • Figure 1: The figure illustrates the adaptive nature of the proposed approach i.e., how the proposed approach tracks the cardinality, and its posterior mean in time for different values of $\alpha_0 \in \{ 0.8,0.9,1\}$.
  • Figure 2: The figure depicts an illustrative example of the proposed approach. The top panel (a) displays the cardinality and sample mean vector of the observed RFSs over time, the center panel (b) shows scatterplots of the RFSs, and at the lower panel (c) we provide the Fisher score $P_{t+1}$ in Eq. \ref{['eq:fisher']} based on the predictive checks. The Jeffreys' prior is used, while $\alpha_0=1$ for the PD in Eq. \ref{['eq:post']}, and $\alpha=1/100$ for the quantile function in Eq. \ref{['eq:alarm']}. In (c), the red line is the decision limit $q(1-\alpha)$, while the light green area indicates the no-anomaly region for the process. An anomalous RFS observation (marked with red $\diamond$) is detected at time-step 6 as shown in the figure.
  • Figure 3: The $F_1(t)$ scores for $t\in \{2,\ldots,30\}$ of the proposed predictive checks (PC) and ranking function (RF) for 5 scenarios.

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Lemma 2
  • proof