Statistical analysis of coupled time series with Kernel Cross-Spectral Density operators
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 dependencies when random variables are sampled from stationary time-series of arbitrary objects. To achieve this goal, we study the properties of the Kernel Cross-Spectral Density (KCSD) 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 improvements in terms of detection errors, as well as the suitability of this approach for testing dependency in complex dynamical systems. This similarity measure enables us to identify different types of interactions in electrophysiological neural time series.
The paper considers kernel cross-spectral density 
      (KCSD) as a way to determine interactions among time series in a better 
      way than methods developed under the i.i.d assumption, which is typically 
      violated for time series data. Such a framework was originally proposed in 
      [Besserve et al, ICASSP, 2011]. The main contribution here is to 
      characterize cases where KCSD can be used to test independence, and to 
      propose and study a way to estimate the properties of cross-spectral 
      density operators from finite samples. The method is compared to the 
      Hilbert-Schmidt Independence Criterion test under the same kernels on 
      simulated and real neural data. Developing independence test for 
      time series data that are able to cope with non-linear interactions is an 
      important area of research. The present paper provides a sound and 
      well-motivated approach. Many applications, however, consider a 
      large number of time series data. Even though the proposed approach 
      accommodate very general forms of dependencies, it is a *pair-wise* 
      independence test, and thus suffers from the limitations inherent to 
      pair-wise testing. It would be interesting to discuss whether the proposed 
      method could be extended to testing simultaneously multiple time series 
      (each being potentially multivariate). The methodology and results 
      depend on the choice of kernel, and various kernels might lead to 
      contradictory conclusions. This should be discussed. Kernel 
      methods seem very promising candidates in capturing dependencies in time 
      series, in the present setting and beyond. For instance, it would also be 
      relevant to mention the recent work by Sindhwani et al, Scalable 
      Matrix-valued Kernel Learning for High-dimensional Nonlinear Multivariate 
      Regression and Granger Causality, UAI 2013, which uses kernel methods to 
      capture non-linearity in causal inference, via a generalized form of 
      sparse vector autoregression.