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We develop a statistical method for discriminating and classifying multivariate non- stationary signals. It is assumed that the processes that generate the signals are characterized by their time-evolving spectral matrix—a description of the dynamic connectivity between the time series components. Here, we address two major challenges: first, data massiveness and second, the poor conditioning that leads to numerically unstable estimates of the spectral matrix. We use the SLEX library (a collection of bases functions consisting of localized Fourier waveforms) to extract the set of time–frequency features that best separate classes of time series. The SLEX approach yields readily interpretable results since it is a time-dependent analogue of the Fourier approach to stationary time series. Moreover, it uses computationally efficient algorithms to enable handling of large data sets. We estimate the SLEX spectral matrix by shrinking the initial SLEX periodogram matrix estimator towards the identity matrix. The resulting shrinkage estimator has lower mean-squared error than the classical smoothed periodogram matrix and is more regular. A leave-one out analysis for predicting motor intent (left vs. right movement) using electroencephalograms indicates that the proposed SLEX-shrinkage method gives robust estimates of the evolutionary spectral matrix and good classification results.

In this paper we investigate the performance of periodogram based estimators of the spectral density matrix of possibly high-dimensional time series. We suggest and study shrinkage as a remedy against numerical instabilities due to deteriorating condition numbers of (kernel) smoothed periodogram matrices. Moreover, shrinking the empirical eigenvalues in the frequency domain towards one another also improves at the same time the Mean Squared Error (MSE) of these widely used nonparametric spectral estimators. Compared to some existing time domain approaches, restricted to i.i.d. data, in the frequency domain it is necessary to take the size of the smoothing span as "effective or local sample size" into account. While B\"{o}hm and von Sachs (2007) proposes a multiple of the identity matrix as optimal shrinkage target in the absence of knowledge about the multidimensional structure of the data, here we consider "structural" shrinkage. We assume that the spectral structure of the data is induced by underlying factors. However, in contrast to actual factor modelling suffering from the need to choose the number of factors, we suggest a model-free approach. Our final estimator is the asymptotically MSE-optimal linear combination of the smoothed periodogram and the parametric estimator based on an underfitting (and hence deliberately misspecified) factor model. We complete our theoretical considerations by some extensive simulation studies. In the situation of data generated from a higher-order factor model, we compare all four types of involved estimators (including the one of B\"{o}hm and von Sachs (2007))., Comment: Published in at http://dx.doi.org/10.1214/08-EJS236 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org)

In this paper on developing shrinkage for spectral analysis of multivariate time series of high dimensionality, we propose a new nonparametric estimator of the spectral matrix with two appealing properties. First, compared to the traditional smoothed periodogram our shrinkage estimator has a smaller L2 risk. Second, the proposed shrinkage estimator is numerically more stable due to a smaller condition number. We use the concept of “Kolmogorov” asymptotics where simultaneously the sample size and the dimensionality tend to infinity, to show that the smoothed periodogram is not consistent and to derive the asymptotic properties of our regularized estimator. This estimator is shown to have asymptotically minimal risk among all linear combinations of the identity and the averaged periodogram matrix. Compared to existing work on shrinkage in the time domain, our results show that in the frequency domain it is necessary to take the size of the smoothing span as “effective sample size” into account. Furthermore, we perform extensive Monte Carlo studies showing the overwhelming gain in terms of lower L2 risk of our shrinkage estimator, even in situations of oversmoothing the periodogram by using a large smoothing span.