State Classification Using Persistent Homology Attractors
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Solution Overview
Problem
Existing methods for classifying states based on multidimensional time series data, such as invariant analysis and subspace methods, are inadequate for non-linear time series and can lead to false classifications, especially when all variables change simultaneously, making it difficult to sense changes accurately.
Innovation Solution
The proposed solution involves generating an attractor from multidimensional time series data using a persistent homology process to create Betti number sequences, which reflect the topological features of the data, allowing for accurate classification and change detection by comparing these sequences over time.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Ease of operation
If invariant analysis is used to monitor time correlation of multidimensional time series data, then the method can be practiced easily, but it cannot sense changes when all variables change in the same direction simultaneously
Solution Approach 1:
The patent transforms the time series data into a higher-dimensional space by creating delay coordinates (x(t), x(t-τ), x(t-2τ), ...) to form vectors in n-dimensional space. This dimensional transformation allows the system to detect changes that are invisible in the original time domain, particularly when all variables change simultaneously, by observing changes in the geometric structure (attractor) of the data in the expanded phase space.
2Measurement precision
If subspace method is used to generate orthogonal basis from multidimensional time series data, then it enables detection of change of subspace in density, but it is not suitable for non-linear time series
Solution Approach 1:
The patent changes the fundamental parameter of analysis from linear orthogonal bases to topological invariants (Betti numbers) that characterize the geometric structure of the attractor. By using persistent homology to compute Betti numbers across different scales, the method adapts to non-linear time series while maintaining the ability to detect subspace changes, as Betti numbers capture the essential topological features regardless of the underlying dynamics linearity.
3Reliability
If traditional analysis methods are used on multidimensional time series data with unknown properties, then false classification may occur, but checking data properties in advance requires additional work
Solution Approach 1:
The patent creates a universal analysis framework based on topological data analysis that can handle multiple types of time series data (linear, non-linear, periodic, chaotic) with a single methodology. The attractor-based approach with persistent homology serves as a multi-functional tool that automatically adapts to different data characteristics without requiring preliminary property assessment, eliminating the need for separate analysis methods for different data types.
Data Source
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AI summary
A state classifying method includes generating an attractor containing a plurality of points that correspond to a plurality of sets of time series data, coordinate values of each of the plurality of points being values corresponding to the sets of time series data; generating Betti number sequence data by applying a persistent homology process on the attractor; and classifying a state that is represented by the plurality of sets of time series data based on the Betti number sequence data.