Koopman Spectral Kernels for Comparing Complex Dynamics: Application to Multiagent Sport Plays

Fujii, Keisuke; Inaba, Yuki; Kawahara, Yoshinobu

doi:10.1007/978-3-319-71273-4_11

Cited by 24 publications

(31 citation statements)

References 22 publications

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“…The results of other DMD methods (DMD spectrum and KDMD spectral kernel) suggest that both the reflecting graph structure and the use of appropriate sliding windows were important for recognition. Furthermore, in comparison with KDMD spectral kernel [40] (i.e. decomposition in a feature space), the proposed method has an advantage in physically and semantically interpreting the DMD modes in the observed data space.…”

Section: Discussionmentioning

confidence: 99%

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

Fujii

Takeishi

Hojo

et al. 2020

Sci Rep

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Understanding complex network dynamics is a fundamental issue in various scientific and engineering fields. Network theory is capable of revealing the relationship between elements and their propagation; however, for complex collective motions, the network properties often transiently and complexly change.A fundamental question addressed here pertains to the classification of collective motion network based on physically-interpretable dynamical properties. Here we apply a data-driven spectral analysis called graph dynamic mode decomposition, which obtains the dynamical properties for collective motion classification. Using a ballgame as an example, we classified the strategic collective motions in different global behaviours and discovered that, in addition to the physical properties, the contextual node information was critical for classification. Furthermore, we discovered the label-specific stronger spectra in the relationship among the nearest agents, providing physical and semantic interpretations. Our approach contributes to the understanding of complex networks involving collective motions from the perspective of nonlinear dynamical systems.

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Section: Discussionmentioning

confidence: 99%

“…Previous works [40,38] have predictively classified the time-varying interactions into two group outcomes (i.e. scored or unscored) while reflecting the timevarying interactions among attackers, defenders and the ball.…”

Section: Introductionmentioning

confidence: 99%

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

Fujii

Takeishi

Hojo

et al. 2020

Sci Rep

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“…DMD, originally proposed in fluid physics [6,7], has recently attracted attention also in other areas of science and engineering, including analysis of power systems [30], epidemiology [31], neuroscience [9], image processing [8,32], controlled systems [33], and human behaviors [34,35,36]. Moreover, there are several algorithmic variants to overcome the problem of the original DMD such as the use of nonlinear basis functions [37], a formulation in a reproducing kernel Hilbert space [10], in a supervised learning framework via multitask learning [38], in a Bayesian framework [11], and using a neural network [39].…”

Section: Dynamic Mode Decompositionmentioning

confidence: 99%

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

Fujii

Kawahara

2019

Neural Networks

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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.

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“…The new method can decompose the mode infinite functional space and then can acquire high expressiveness even for the time-varying interaction mode and in low-dimensional sequences through spectral decomposition of the Koopman operator [ 31 ]. We previously developed the classification and prediction methods further using the DMD spectrum and referred to the method as Koopman spectral kernels [ 32 ].…”

Section: Introductionmentioning

confidence: 99%

“…Generally, selection of an appropriate representation of the data to reflect the structure is a fundamental issue in pattern recognition. Among several kernels to reflect time-series data structure, we previously proposed the Koopman spectral kernels [ 32 ] (see Materials and Methods ). For example, our previous work succeeded in the classification into the type of human locomotion from motion capture data [ 30 , 32 ], using feature vectors determined by the Koopman spectral kernels.…”

Section: Introductionmentioning

confidence: 99%

Prediction and classification in equation-free collective motion dynamics

et al. 2018

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Modeling the complex collective behavior is a challenging issue in several material and life sciences. The collective motion has been usually modeled by simple interaction rules and explained by global statistics. However, it remains difficult to bridge the gap between the dynamic properties of the complex interaction and the emerging group-level functions. Here we introduce decomposition methods to directly extract and classify the latent global dynamics of nonlinear dynamical systems in an equation-free manner, even including complex interaction in few data dimensions. We first verified that the basic decomposition method can extract and discriminate the dynamics of a well-known rule-based fish-schooling (or bird-flocking) model. The method extracted different temporal frequency modes with spatial interaction coherence among three distinct emergent motions, whereas these wave properties in multiple spatiotemporal scales showed similar dispersion relations. Second, we extended the basic method to map high-dimensional feature space for application to actual small-dimensional systems complexly changing the interaction rules. Using group sports human data, we classified the dynamics and predicted the group objective achievement. Our methods have a potential for classifying collective motions in various domains which obey in non-trivial dominance law known as active matters.

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Koopman Spectral Kernels for Comparing Complex Dynamics: Application to Multiagent Sport Plays

Cited by 24 publications

References 22 publications

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

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

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

Prediction and classification in equation-free collective motion dynamics

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