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This paper is concerned with the statistical analysis of matrix-valued time series. These are data collected over a network of sensors (typically a set of spatial locations), recording, over time, observations of multiple measurements. From such data, we propose to learn, in an online fashion, a graph that captures two aspects of dependency: one describing the sparse spatial relationship between sensors, and the other characterizing the measurement relationship. To this purpose, we introduce a novel multivariate autoregressive model to infer the graph topology encoded in the coefficient matrix which captures the sparse Granger causality dependency structure present in such matrix-valued time series. We decompose the graph by imposing a Kronecker sum structure on the coefficient matrix. We develop two online approaches to learn the graph in a recursive way. The first one uses Wald test for the projected OLS estimation, where we derive the asymptotic distribution for the estimator. For the second one, we formalize a Lasso-type optimization problem. We rely on homotopy algorithms to derive updating rules for estimating the coefficient matrix. Furthermore, we provide an adaptive tuning procedure for the regularization parameter. Numerical experiments using both synthetic and real data, are performed to support the effectiveness of the proposed learning approaches.
We study anomaly detection and introduce an algorithm that processes variable length, irregularly sampled sequences or sequences with missing values. Our algorithm is fully unsupervised, however, can be readily extended to supervised or semisupervise
Stein variational gradient descent (SVGD) is a particle-based inference algorithm that leverages gradient information for efficient approximate inference. In this work, we enhance SVGD by leveraging preconditioning matrices, such as the Hessian and F
Seglearn is an open-source python package for machine learning time series or sequences using a sliding window segmentation approach. The implementation provides a flexible pipeline for tackling classification, regression, and forecasting problems wi
In online learning from non-stationary data streams, it is both necessary to learn robustly to outliers and to adapt to changes of underlying data generating mechanism quickly. In this paper, we refer to the former nature of online learning algorithm
Many applications require the ability to judge uncertainty of time-series forecasts. Uncertainty is often specified as point-wise error bars around a mean or median forecast. Due to temporal dependencies, such a method obscures some information. We w