High-Dimensional Time Series Analysis – HIDITSA

Due to the spectacular development of data acquisition devices and sensor networks, it becomes very common to be faced with
high-dimensional time series in various fields such as digital communications, environmental sensing, electroencephalography, analysis of financial datas, industrial monitoring, ....In this context, it is not always possible to collect a large enough number of observations to perform statistical inference because the durations of the signals are limited and/or because their statistics are not time-invariant over large enough temporal windows. As a result, fundamental inference schemes do not behave as in the classical low-dimensional regimes. This stimulated
considerably in the ten past years the development of new statistical approaches aiming at mitigating
the above mentioned difficulties.

In particular, a number of works proposed to use large random matrix theory in the context of high-dimensional statistical signal processing, traditionally modelled by a double asymptotic regime in which the dimension M of the time series and the sample size N both grow towards infinity. These contributions essentially addressed detection or estimation problems in the context of the so-called narrow band array processing model, also known in the statistical community as the linear factor model. In this context, a number of classical statistical inference schemes depend on the behaviour of functionals of the empirical covariance matrix of the observation. Large random matrix theory results were used in the past to evaluate the behaviour of such functionals in the high-dimensional context, and to propose new improved performance approaches.

HIDISTA is focused on statistical inference problems related to the Gaussian wideband array processing model, also known in the statistical community as the Gaussian linear dynamic factor model. The observation is a noisy version (the noise is Gaussian) of a useful M-dimensional Gaussian signal defined as the output of a K < M inputs / M outputs system with rational transfer function driven by a K dimensional Gaussian white noise sequence. In this context, it is often necessary to infer informations on the useful signal using N consecutive observations. Classical estimators often appear to be functionals
of certain fundamental statistics such as empirical spatio-temporal covariance matrices, empirical autocovariance
matrices between past and future observations and estimators of the spectral density of the observation. The goal
of HIDITSA is to develop new large random matrix theory technics in order to study the behaviour of these fundamental statistics, and to use the results in order to address important detection / estimation problems. The behaviour of the
traditional methods, designed in the low dimensional context, is first characterized, and when they
do not provide consistent estimates, the new results of HIDITSA allow to propose improved performance
new schemes. The results are used to design estimators of parameters of state space representation of the useful signal (eigenvalues of the transition matrix, current value of the state from past observations) and, when the noise is
spatially uncorrelated, to develop detection tests of the useful signal checking whether the components of the observation
are spatially correlated (presence of signal) or not (absence of signal).