Anomaly Detection based on Compressed Data: an Information Theoretic Characterization

TL;DR

Numerical evidence confirms that the proposed information-theoretic quantities anticipate the performance of practical compressors and detectors in the case of Gaussian and non-Gaussian signals allowing an assessment of the tradeoff between compression and detection.

Abstract

We analyze the effect of lossy compression in the processing of sensor signals that must be used to detect anomalous events in the system under observation. The intuitive relationship between the quality loss at higher compression and the possibility of telling anomalous behaviours from normal ones is formalized in terms of information-theoretic quantities. Some analytic derivations are made within the Gaussian framework and possibly in the asymptotic regime for what concerns the stretch of signals considered. Analytical conclusions are matched with the performance of practical detectors in a toy case allowing the assessment of different compression/detector configurations.

Figures (7)

• Figure 1: A sensorized plant whose acquisitions are aggregated at the edge before being sent to the cloud.
• Figure 2: The signal chain is tuned on the normal signal $x^{\rm ok}$ to best address the rate-distortion trade-off, guaranteeing a certain quality of service to a given application. An anomalous signal $x^{\rm ko}$ may occur and a detector working on the compressed signal $y$ should be able to detect it.
• Figure 3: Trend of $\Delta_2=\frac{1}{n}\sqrt{\sum_{j,k=0}^{n-1}\left[{\bf \Sigma}^{\rm ko}_{j,k}-(I_n)_{j,k}\right]^2}$ and $\Delta_\infty=\max_{j,k} \left|{\bf \Sigma}^{\rm ko}_{j,k}-(I_n)_{j,k}\right|$ when $n$ increases. Solid lines are mean trends while shaded areas contain $98\%$ of the population.
• Figure 4: Rate distortion curves for the three compression schemes we consider and for different value of the localization of the original signal.
• Figure 5: Distinguishability measures $\zeta$, $\kappa$ and $\psi$ against normalized distortion $d$ in case of RDC.

Theorems & Definitions (12)

• lemma 1
• lemma 2
• lemma 3
• theorem 1
• theorem 2
• proof : Proof of Lemma \ref{['lem:Gfenc']}
• proof : Proof of Lemma \ref{['lem:Gfkoxh']}
• proof : Proof of Lemma \ref{['lem:Gell']}
• proof : Proof of Theorem \ref{['th:Zvanishes']}
• proof : Proof of Theorem \ref{['th:covkoconc']}