Modelling Vast Time Series: Sparsity and Segmentation

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In this modern information age the availability of large or vast time series data brings opportunities with challenges to time series analysts. The demand for modelling and forecasting high-dimensional time series arises from various practical problems such as panel study of economic, social and natural phenomena (such as weather), financial market analysis and communications engineering. We propose two new approaches for analyzing high-dimensional time series data when the dimension is as large as, or even greater than, the length of observed time series. The first approach is to fit the data with sparse vector auto-regressive models (VAR). For some applications when the components are ordered, we will further explore the sparsity due to a band structure. Note that we impose sparsity or banding directly on the coefficient matrices in VAR models. Hence, the relevant inference methods and the associated theory are different from those for the estimation of large covariance matrices. Our second approach is segmentation via transformation. We seek for a contemporaneous linear transformation such that the transformed time series is divided into several sub-vectors, and those sub-vectors are both contemporaneously and serially uncorrelated. Therefore, they can be modelled separately.The challenges of our proposal are two-fold: First we need to develop the statistical inference methods and the associated theory for identifying the sparse structure and for fitting sparse VAR models with large dimensions. Let p denote the dimension of the time series. We aim to reduce the number of model parameters from the order of the square of p to the order of p, and to develop the valid inference methods when log(p)= o(n). Secondly, we need to identify the linear transformation to identify the latent segmentation structure, i.e. the block-diagonal autocovariance structure when such a structure exists.High-dimensional data analysis (i.e. 'big data') is one of the most vibrant research areas in statistics in the last decade. Most work to date concentrates on linear regression with a large number of candidate regressors (i.e. the so-called 'large p small n' paradigm). Another stream of the research is on the inference of large covariance matrices. Though bearing a similar banner, the problems addressed in the proposal are different, as we deal with high-dimensional time series and we need to estimate large transformation or coefficient matrices that are not positive semi-definite. We aim for simple and effective inference methods so that they can be implemented with ordinary PCs for the data of dimensions in the order of thousands.