Perron-Frobenius Operator Approximation for Anomaly Detection
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Solution Overview
Problem
Conventional methods for analyzing time-series data with nonlinear behavior and random noise struggle to accurately represent and detect anomalies, as they assume linear relationships and require transfer operators with specific properties, which are not always present in practical models.
Innovation Solution
An anomaly detection apparatus that approximates a Perron-Frobenius operator on a reproducing kernel Hilbert space (RKHS) to predict data points and determine anomalies based on discrepancies, using the approximation unit and detection unit, even when the transfer operator lacks discrete spectrum or boundedness.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If conventional methods using transfer operators are used for anomaly detection, then the analysis can capture nonlinear relationships in time-series data, but the methods fail when the transfer operator does not have good properties such as discrete spectrum or boundedness
Solution Approach 1:
The patent changes the fundamental parameter of the mathematical approach by shifting from eigenvalue-based methods to a residual-based prediction method. Instead of requiring the transfer operator to have discrete spectrum or boundedness properties, the invention uses the transfer operator to generate predictions and detects anomalies through residuals, making the method applicable to general transfer operators without restrictive spectral properties
Solution Approach 2:
The patent extracts the anomaly detection function from the spectral analysis framework. Rather than relying on eigenvalues and eigenfunctions of the transfer operator, the invention extracts anomaly information directly from the discrepancy between predicted and actual values, separating the anomaly detection task from the spectral properties of the operator
2Measurement precision
If classical methods assuming linear relationships are used, then the analysis is computationally simple, but the accuracy falls for data items exhibiting nonlinear behavior
Solution Approach 1:
The patent applies dynamics by using a transfer operator that can represent nonlinear relationships while maintaining a systematic approach. The transfer operator framework allows the model to adapt to nonlinear dynamics in the time-series data, capturing complex behavioral patterns that linear methods cannot, while still providing a structured methodology for anomaly detection
3Adaptability or versatility
If neural networks are used for approximating relationships among data items, then the method can handle nonlinear relationships, but it is difficult to incorporate information on randomness into the approximation
Solution Approach 1:
The patent introduces the transfer operator as an intermediary that bridges the gap between handling nonlinear relationships and preserving randomness information. The transfer operator framework naturally incorporates the probabilistic structure of the underlying dynamical system, allowing randomness to be preserved through the operator's definition while still capturing nonlinear relationships in the data
Data Source
AI summary
An anomaly detection apparatus includes an approximation unit configured to generate, based on observed data, an approximation of a Perron-Frobenius operator on an RKHS that represents a mathematical model to generate the observed data; and a detection unit configured to use the approximation of the Perron-Frobenius operator and an observed data item at time t, to predict a data item at time t+1, and based on a discrepancy between the predicted data item and an observed data item at time t+1, to determine whether the observed data item at time t+1 is anomalous.