Data Fusion via Intrinsic Dynamic Variables: An Application of Data-Driven Koopman Spectral Analysis

TL;DR

This paper introduces a data-driven method for nonlinear data fusion using Koopman eigenfunctions, enabling the integration of heterogeneous measurements through intrinsic coordinates derived from spectral analysis.

Contribution

It presents a novel framework leveraging Koopman spectral analysis and Extended Dynamic Mode Decomposition for data fusion in nonlinear systems, demonstrated on PDE measurements.

Findings

Koopman eigenfunctions serve as intrinsic coordinates for data fusion.

The method successfully merges different types of measurements in nonlinear systems.

Eigenvalues are invariant under invertible transformations, aiding in data alignment.

Abstract

We demonstrate that numerically computed approximations of Koopman eigenfunctions and eigenvalues create a natural framework for data fusion in applications governed by nonlinear evolution laws. This is possible because the eigenvalues of the Koopman operator are invariant to invertible transformations of the system state, so that the values of the Koopman eigenfunctions serve as a set of intrinsic coordinates that can be used to map between different observations (e.g., measurements obtained through different sets of sensors) of the same fundamental behavior. The measurements we wish to merge can also be nonlinear, but must be "rich enough" to allow (an effective approximation of) the state to be reconstructed from a single set of measurements. This approach requires independently obtained sets of data that capture the evolution of the heterogeneous measurements and a single pair of "joint" measurements taken at one instance in time. Computational approximations of eigenfunctions and their corresponding eigenvalues from data are accomplished using Extended Dynamic Mode Decomposition. We illustrate this approach on measurements of spatio-temporal oscillations of the FitzHugh-Nagumo PDE, and show how to fuse point measurements with principal component measurements, after which either set of measurements can be used to estimate the other set.