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

  • Keisuke Fujii
  • Yuki Inaba
  • Yoshinobu Kawahara
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10536)

Abstract

Understanding the complex dynamics in the real-world such as in multi-agent behaviors is a challenge in numerous engineering and scientific fields. Spectral analysis using Koopman operators has been attracting attention as a way of obtaining a global modal description of a nonlinear dynamical system, without requiring explicit prior knowledge. However, when applying this to the comparison or classification of complex dynamics, it is necessary to incorporate the Koopman spectra of the dynamics into an appropriate metric. One way of implementing this is to design a kernel that reflects the dynamics via the spectra. In this paper, we introduced Koopman spectral kernels to compare the complex dynamics by generalizing the Binet-Cauchy kernel to nonlinear dynamical systems without specifying an underlying model. We applied this to strategic multiagent sport plays wherein the dynamics can be classified, e.g., by the success or failure of the shot. We mapped the latent dynamic characteristics of multiple attacker-defender distances to the feature space using our kernels and then evaluated the scorability of the play by using the features in different classification models.

Notes

Acknowledgements

We would like to thank Charlie Rohlf and the STATS team for their help and support for this work. This work was supported by JSPS KAKENHI Grant Numbers 16H01548.

References

  1. 1.
    Bonnet, J., Cole, D., Delville, J., Glauser, M., Ukeiley, L.: Stochastic estimation and proper orthogonal decomposition: complementary techniques for identifying structure. Exp. Fluids 17(5), 307–314 (1994)CrossRefGoogle Scholar
  2. 2.
    Brunton, B.W., Johnson, L.A., Ojemann, J.G., Kutz, J.N.: Extracting spatial-temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition. J. Neurosci. Methods 258, 1–15 (2016)CrossRefGoogle Scholar
  3. 3.
    Chang, J.M., Beveridge, J.R., Draper, B.A., Kirby, M., Kley, H., Peterson, C.: Illumination face spaces are idiosyncratic. In: Proceedings of International Conference on Image Processing, Computer Vision, & Pattern Recognition, vol. 2, pp. 390–396 (2006)Google Scholar
  4. 4.
    Chen, K.K., Tu, J.H., Rowley, C.W.: Variants of dynamic mode decomposition: boundary condition, Koopman, and Fourier analyses. J. Nonlinear Sci. 22(6), 887–915 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Couzin, I.D., Krause, J., Franks, N.R., Levin, S.A.: Effective leadership and decision-making in animal groups on the move. Nature 433(7025), 513–516 (2005)CrossRefGoogle Scholar
  6. 6.
    De Cock, K., De Moor, B.: Subspace angles between ARMA models. Syst. Control Lett. 46(4), 265–270 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fodor, E., Nardini, C., Cates, M.E., Tailleur, J., Visco, P., van Wijland, F.: How far from equilibrium is active matter? Phys. Rev. Lett. 117(3), 038103 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Fujii, K., Yokoyama, K., Koyama, T., Rikukawa, A., Yamada, H., Yamamoto, Y.: Resilient help to switch and overlap hierarchical subsystems in a small human group. Scientific Reports 6 (2016)Google Scholar
  9. 9.
    Fujii, K., Isaka, T., Kouzaki, M., Yamamoto, Y.: Mutual and asynchronous anticipation and action in sports as globally competitive and locally coordinative dynamics. Scientific Reports 5 (2015)Google Scholar
  10. 10.
    Ghahramani, Z., Roweis, S.T.: Learning nonlinear dynamical systems using an EM algorithm. In: Advances in Neural Information Processing Systems, pp. 431–437 (1999)Google Scholar
  11. 11.
    Goldman, M., Rao, J.M.: Live by the three, die by the three? The price of risk in the NBA. In: Proceedings of MIT Sloan Sports Analytics Conference (2013)Google Scholar
  12. 12.
    Hamm, J., Lee, D.D.: Grassmann discriminant analysis: a unifying view on subspace-based learning. In: Proceedings of International Conference on Machine Learning, pp. 376–383 (2008)Google Scholar
  13. 13.
    Hutchins, E.: The technology of team navigation. In: Intellectual Teamwork: Social and Technological Foundations of Cooperative Work, vol. 1, pp. 191–220 (1990)Google Scholar
  14. 14.
    Jovanović, M.R., Schmid, P.J., Nichols, J.W.: Sparsity-promoting dynamic mode decomposition. Phys. Fluids 26(2), 024103 (2014)CrossRefGoogle Scholar
  15. 15.
    Kashima, H., Tsuda, K., Inokuchi, A.: Kernels for graphs. Kernel Methods Comput. Biol. 39(1), 101–113 (2004)Google Scholar
  16. 16.
    Kawahara, Y.: Dynamic mode decomposition with reproducing kernels for Koopman spectral analysis. In: Proceedings of Advances in Neural Information Processing Systems, pp. 911–919 (2016)Google Scholar
  17. 17.
    Kondor, R.I., Lafferty, J.: Diffusion kernels on graphs and other discrete input spaces. In: Proceedings of International Conference on Machine Learning, vol. 2, pp. 315–322 (2002)Google Scholar
  18. 18.
    Koopman, B.O.: Hamiltonian systems and transformation in Hilbert space. Proc. Natl. Acad. Sci. 17(5), 315–318 (1931)CrossRefzbMATHGoogle Scholar
  19. 19.
    Kulesza, A., Jiang, N., Singh, S.P.: Spectral learning of predictive state representations with insufficient statistics. In: Proceedings of Association for the Advancement of Artificial Intelligence, pp. 2715–2721 (2015)Google Scholar
  20. 20.
    Loan, C.V., Golub, G.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)zbMATHGoogle Scholar
  21. 21.
    Martin, R.J.: A metric for ARMA processes. IEEE Trans. Signal Process. 48(4), 1164–1170 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Miller, A.C., Bornn, L.: Possession sketches: mapping NBA strategies (2017)Google Scholar
  23. 23.
    Rowley, C.W., Mezić, I., Bagheri, S., Schlatter, P., Henningson, D.S.: Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115–127 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Schmid, P.J.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Sirovich, L.: Turbulence and the dynamics of coherent structures. I. Coherent structures. Q. Appl. Math. 45(3), 561–571 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Susuki, Y., Mezić, I.: Nonlinear koopman modes and power system stability assessment without models. IEEE Trans. Power Syst. 29(2), 899–907 (2014)CrossRefGoogle Scholar
  27. 27.
    Takeishi, N., Kawahara, Y., Tabei, Y., Yairi, T.: Bayesian dynamic mode decomposition. In: Proceedings of the International Joint Conference on Artificial Intelligence (2017)Google Scholar
  28. 28.
    Tomasello, M., Carpenter, M.: Shared intentionality. Dev. Sci. 10(1), 121–125 (2007)CrossRefGoogle Scholar
  29. 29.
    Tu, J.H., Rowley, C.W., Luchtenburg, D.M., Brunton, S.L., Kutz, J.N.: On dynamic mode decomposition: theory and applications. J. Comput. Dyn. 1(2), 391–421 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Vishwanathan, S., Smola, A.J., Vidal, R.: Binet-Cauchy kernels on dynamical systems and its application to the analysis of dynamic scenes. Int. J. Comput. Vis. 73(1), 95–119 (2007)CrossRefGoogle Scholar
  31. 31.
    Wang, K.C., Zemel, R.: Classifying NBA offensive plays using neural networks. In: Proceedings of MIT Sloan Sports Analytics Conference (2016)Google Scholar
  32. 32.
    Williams, M.O., Kevrekidis, I.G., Rowley, C.W.: A data-driven approximation of the Koopman operator: extending dynamic mode decomposition. J. Nonlinear Sci. 25(6), 1307–1346 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Wolf, L., Shashua, A.: Learning over sets using kernel principal angles. J. Mach. Learn. Res. 4, 913–931 (2003)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Keisuke Fujii
    • 1
  • Yuki Inaba
    • 2
  • Yoshinobu Kawahara
    • 1
    • 3
  1. 1.Center for Advanced Intelligence Project, RIKENOsakaJapan
  2. 2.Japanese Institute of Sports SciencesTokyoJapan
  3. 3.The Institute of Scientific and Industrial Research, Osaka UniversityOsakaJapan

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