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Register a new userMultiscale Information Decomposition: Exact Computation for Multivariate Gaussian Processes
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Added by
Daniele Marinazzo
Publication date
2017
fields
Mathematical Statistics
and research's language is
English
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Exploiting the theory of state space models, we derive the exact expressions of the information transfer, as well as redundant and synergistic transfer, for coupled Gaussian processes observed at multiple temporal scales. All of the terms, constituting the frameworks known as interaction information decomposition and partial information decomposition, can thus be analytically obtained for different time scales from the parameters of the VAR model that fits the processes. We report the application of the proposed methodology firstly to benchmark Gaussian systems, showing that this class of systems may generate patterns of information decomposition characterized by mainly redundant or synergistic information transfer persisting across multiple time scales or even by the alternating prevalence of redundant and synergistic source interaction depending on the time scale. Then, we apply our method to an important topic in neuroscience, i.e., the detection of causal interactions in human epilepsy networks, for which we show the relevance of partial information decomposition to the detection of multiscale information transfer spreading from the seizure onset zone.
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Monitoring several correlated quality characteristics of a process is common in modern manufacturing and service industries. Although a lot of attention has been paid to monitoring the multivariate process mean, not many control charts are available for monitoring the covariance matrix. This paper presents a comprehensive overview of the literature on control charts for monitoring the covariance matrix in a multivariate statistical process monitoring (MSPM) framework. It classifies the research that has previously appeared in the literature. We highlight the challenging areas for research and provide some directions for future research.
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This paper introduces the functional tensor singular value decomposition (FTSVD), a novel dimension reduction framework for tensors with one functional mode and several tabular modes. The problem is motivated by high-order longitudinal data analysis. Our model assumes the observed data to be a random realization of an approximate CP low-rank functional tensor measured on a discrete time grid. Incorporating tensor algebra and the theory of Reproducing Kernel Hilbert Space (RKHS), we propose a novel RKHS-based constrained power iteration with spectral initialization. Our method can successfully estimate both singular vectors and functions of the low-rank structure in the observed data. With mild assumptions, we establish the non-asymptotic contractive error bounds for the proposed algorithm. The superiority of the proposed framework is demonstrated via extensive experiments on both simulated and real data.
Research on Poisson regression analysis for dependent data has been developed rapidly in the last decade. One of difficult problems in a multivariate case is how to construct a cross-correlation structure and at the meantime make sure that the covariance matrix is positive definite. To address the issue, we propose to use convolved Gaussian process (CGP) in this paper. The approach provides a semi-parametric model and offers a natural framework for modeling common mean structure and covariance structure simultaneously. The CGP enables the model to define different covariance structure for each component of the response variables. This flexibility ensures the model to cope with data coming from different resources or having different data structures, and thus to provide accurate estimation and prediction. In addition, the model is able to accommodate large-dimensional covariates. The definition of the model, the inference and the implementation, as well as its asymptotic properties, are discussed. Comprehensive numerical examples with both simulation studies and real data are presented.
The authors have recently defined the Renyi information dimension rate $d({X_t})$ of a stationary stochastic process ${X_t,,tinmathbb{Z}}$ as the entropy rate of the uniformly-quantized process divided by minus the logarithm of the quantizer step size $1/m$ in the limit as $mtoinfty$ (B. Geiger and T. Koch, On the information dimension rate of stochastic processes, in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Aachen, Germany, June 2017). For Gaussian processes with a given spectral distribution function $F_X$, they showed that the information dimension rate equals the Lebesgue measure of the set of harmonics where the derivative of $F_X$ is positive. This paper extends this result to multivariate Gaussian processes with a given matrix-valued spectral distribution function $F_{mathbf{X}}$. It is demonstrated that the information dimension rate equals the average rank of the derivative of $F_{mathbf{X}}$. As a side result, it is shown that the scale and translation invariance of information dimension carries over from random variables to stochastic processes.
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