Spatio-temporal graphical modeling with innovations based on multi-scale diffusion kernel BERNARD CHALMOND University of Cergy-Pontoise, France ∗ and CMLA, Ecole Normale Sup´erieure de Cachan, France

Abstract- A random field of interest is observed on an undirected spatial graph over time, thereby providing a time series of dependent random fields. We propose a general modeling procedure which has the potential to explicitly quantify intrinsic and extrinsic fluctuations of such dynamical system. We adopt a paradigm in which the intrinsic fluctuations correspond to a process of latent diffusion on the graph arising from stochastic interactions within the system, whereas the extrinsic fluctuations correspond to a temporal drift reflecting the effects of the environment on the system. We start with a spatio-temporal diffusion process which gives rise to the latent spatial process. This makes a bridge with the conventional Wold representation, for which the latent process represents the innovation process, and beyond that with the stochastic differential equation associated to the Fokker-Plank dynamic. The innovation process is modeled by a Gaussian distribution whose covariance matrix is defined by a multi-scale diffusion kernel. This model leads to a multi-scale representation of the spatio-temporal process. We propose a statistical procedure to estimate the multi-scale structure and the model parameters in the case of vector autoregressive model with drift. Modeling and estimation tasks are illustrated on simulated and real biological data. Keywords: Spatio-temporal graphical model, Spatial statistic modeling, Multi-scale heat diffusion kernel, Graph Laplacian, Intrinsic and extrinsic stochastic fluctuations, Multi-scale decomposition. DOI : 10.1016/j.spasta.2013.11.004 29 pages.

∗ Author

address : University of Cergy-Pontoise, BP222, 95032, France. E-mail : [email protected]

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