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Statistical analysis of coupled time series with Kernel Cross-Spectral Density operators

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Besserve,  M
Department Physiology of Cognitive Processes, Max Planck Institute for Biological Cybernetics, Max Planck Society;
Max Planck Institute for Biological Cybernetics, Max Planck Society;

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Logothetis,  NK
Department Physiology of Cognitive Processes, Max Planck Institute for Biological Cybernetics, Max Planck Society;
Max Planck Institute for Biological Cybernetics, Max Planck Society;

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Schölkopf,  B
Dept. Empirical Inference, Max Planck Institute for Intelligent Systems, Max Planck Society;

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Citation

Besserve, M., Logothetis, N., & Schölkopf, B. (2014). Statistical analysis of coupled time series with Kernel Cross-Spectral Density operators. In C. Burges, L. Bottou, M. Welling, Z. Ghahramani, & K. Weinberger (Eds.), Advances in Neural Information Processing Systems 26 (pp. 2535-2543). Red Hook, NY, USA: Curran.


Cite as: https://hdl.handle.net/11858/00-001M-0000-001A-1269-2
Abstract
Many applications require the analysis of complex interactions between time series. These interactions can be non-linear and involve vector valued as well as complex data structures such as graphs or strings. Here we provide a general framework for the statistical analysis of these interactions when random variables are sampled from stationary time-series of arbitrary objects. To achieve this goal we analyze the properties of the kernel cross-spectral density operator induced by positive definite kernels on arbitrary input domains. This framework enables us to develop an independence test between time series as well as a similarity measure to compare different types of coupling. The performance of our test is compared to the HSIC test using i.i.d. assumptions, showing improvement in terms of detection errors as well as the suitability of this approach for testing dependency in complex dynamical systems. Finally, we use this approach to characterize complex interactions in electrophysiological neural time series.