Dynamic mode decomposition in vector-valued reproducing kernel Hilbert spaces for extracting dynamical structure among observables

Abstract

Understanding nonlinear dynamical systems (NLDSs) is challenging in a variety of engineering and scientific fields. Dynamic mode decomposition (DMD), which is a numerical algorithm for the spectral analysis of Koopman operators, has been attracting attention as a way of obtaining global modal descriptions of NLDSs without requiring explicit prior knowledge. However, since existing DMD algorithms are in principle formulated based on the concatenation of scalar observables, it is not directly applicable to data with dependent structures among observables, which take, for example, the form of a sequence of graphs. In this paper, we formulate Koopman spectral analysis for NLDSs with structures among observables and propose an estimation algorithm for this problem. This method can extract and visualize the underlying low-dimensional global dynamics of NLDSs with structures among observables from data, which can be useful in understanding the underlying dynamics of such NLDSs. To this end, we first formulate the problem of estimating spectra of the Koopman operator defined in vector-valued reproducing kernel Hilbert spaces, and then develop an estimation procedure for this problem by reformulating tensor-based DMD. As a special case of our method, we propose the method named as Graph DMD, which is a numerical algorithm for Koopman spectral analysis of graph dynamical systems, using a sequence of adjacency matrices. We investigate the empirical performance of our method by using synthetic and real-world data.

Physically-interpretable classification of biological network dynamics for complex collective motions

“…For the details of basic DMD and its variants, see Material and Methods. Among several variants of DMDs, Graph DMD [33] can extract and visualise the underlying low-dimensional global dynamics of GDSs with structures among observables from data. Then, we briefly introduce Graph DMD framework, described in Figs.…”

“…For the formulation of Graph DMD, we propose a more straightforward formulation with dependent structure among observables than the previous formulation based on vector-valued reproducing kernel Hilbert spaces (RKHSs) [33].…”

“…Regarding the details of the analysis, the connection with Graph DMD (such as the relation between A t and g c ) and the reason why it can visualise the relation between elements, see Materials and Methods. For the practical implementation of the spectral decomposition, a modified tensor-based DMD [33,42], which is a generalised DMD for the application to tensor data, is applied to the adjacency matrix series.…”

“…DMD has been successfully applied in many real-world problems, such as analyses of epidemiology and electrocorticography [31,32]. Among several variants of DMDs (for details, see Material and Methods), Graph DMD [33] can extract and visualise the underlying low-dimensional global dynamics of GDSs with structures among observables from data. However, for more general complex collective motions including strategic human groups, agents flexibly change the rules of behaviour according to the situation; thus, the computation of a feature for the classifier using the obtained spectra cannot be uniquely determined.…”

Unsupervised and Generic Short-Term Anticipation of Human Body Motions

“…• Can recently developed so called equation free modeling techniques [10][11][12][13][14][15] already explain and predict motions in a short time horizon?…”