We present a generalization of independent component analysis (ICA), where instead of looking for a linear transform that makes the data components independent, we look for a transform that makes the data components well fit by a tree-structured graphical model. This tree-dependent component analysis (TCA) provides a tractable and flexible approach to weakening the assumption of independence in ICA. In particular, TCA allows the underlying graph to have multiple connected components, and thus the method is able to find “clusters ” of components such that components are dependent within a cluster and independent between clusters. Finally, we make use of a notion of graphical models for time series due to Brillinger (1996) to extend these ideas to the temporal setting. In particular, we are able to fit models that incorporate tree-structured dependencies among multiple time series.

We present a class of algorithms that find clusters in independent component analysis: the data are linearly transformed so that the resulting components can be grouped into clusters, such that components are dependent within clusters and independent between clusters. In order to find such clusters, we look for a transform that fits the estimated sources to a forest-structured graphical model. In the non-Gaussian, temporally independent case, the optimal transform is found by minimizing a contrast function based on mutual information that directly extends the contrast function used for classical ICA. We also derive a contrast function in the Gaussian stationary case that is based on spectral densities and generalizes the contrast function of Pham [22] to richer classes of dependency.

In critical care extremely high dimensional time series are generated by clinical information systems. This yields new perspectives of data recording and also causes a new challenge for statistical methodology. Recently graphical correlation models have been developed for analysing the partial associations between the components of multivariate time series. We apply this technique to the hemodynamic system of critically ill patients monitored in intensive care. We appraise the practical value of the procedure by reidentifying known associations between the variables. From separate analyses for different pathophysiological states we conclude that distinct clinical states can be characterised by distinct partial correlation structures.

Probabilistic graphical models can be extended to time series by considering probabilistic dependencies between entire time series. For stationary Gaussian time series, the graphical model semantics can be expressed naturally in the frequency domain, leading to interesting families of structured time series models that are complementary to families defined in the time domain. In this paper, we present an algorithm to learn the structure from data for directed graphical models for stationary Gaussian time series. We describe an algorithm for efficient forecasting for stationary Gaussian time series whose spectral densities factorize in a graphical model. We also explore the relationships between graphical model structure and sparsity, comparing and contrasting the notions of sparsity in the time domain and the frequency domain. Finally, we show how to make use of Mercer kernels in this setting, allowing our ideas to be extended to nonlinear models.

We discuss cross-spectral analysis and report applications for the investigation of human tremors. For the physiological tremor in healthy subjects, the analysis enables to determine the resonant contribution to the oscillation and allows to test for a contribution of reflexes to this tremor. Comparing the analysis of the relation between the tremor of both hands in normal subjects and subjects with a rare abnormal organization of certain neural pathways proves the involvement of central structures in enhanced physiological tremor. The relation between the left and the right side of the body in pathological tremor shows a specific difference between orthostatic and all other forms of tremor. An investigation of EEG and tremor in patients suffering from Parkinson’s disease reveals the tremor-correlated cortical activity. Finally, the general issue of interpreting the results of methods designed for the analysis of bivariate processes when applied to multivariate processes is considered. We discuss and apply partial cross-spectral analysis in the frame of graphical models as an extention of bivariate cross-spectral analysis for the multivariate case.

In modern statistical process control, intelligent alarm systems have to be constructed which extract the important information from multivariate time series and detect critical "out-of control " states of the underlying mechanism quickly and reliably. Regarding high-dimensional time series, statistical methods for dimension reduction can help to compress the data into a few relevant variables before characteristic patterns in the data are searched for. In this paper we apply graphical models as a preliminary step preceding a factor analysis of the vital signs of critically ill patients in intensive care. Then a procedure for the online-detection of change points in univariate time series is applied to the original series and to each of the factors and the results are compared to the judgment of an experienced physician.

In this work spike trains of firing times of neurons recorded from various locations in the cat's auditory thalamus are studied. A goal is making inferences concerning connections amongst different regions of the thalamus in both the presence and the absence of a stimulus. Both second-order moment (frequency domain) and full likelihood analyses (a threshold crossing model), are carried through. 1 Introduction The sequence of spikes of a neuron, referred to as a "spike train", may carry important information processed by the brain and thus may underlie cognitive functions and sensory perception [1]. The data studied are recorded stretches of point processes corresponding to the firing times of Statistics Department, University of California, Berkeley y Institute of Physiology, University of Lausanne, Switzerland Pars dorsalis (D) Pars lateralis (PL) Pars magnocellularis (M) Auditory Cortex RE Input Figure 1: A block diagram of the auditory regions of the cat's brain. neurons mea...

In this paper, we discuss the properties of mixed graphs which visualize causal relationships between the components of multivariate time series. In these Granger-causality graphs, the vertices, representing the components of the time series, are connected by arrows according to the Granger-causality relations between the variables whereas lines correspond to contemporaneous conditional association. We show that the concept of Granger-causality graphs provides a framework for the derivation of general noncausality relations relative to reduced information sets by performing sequences of simple operations on the graphs. We briefly discuss the implications for the identification of causal relationships. Finally we provide an extension of the linear concept to strong Granger-causality. JEL classification: C320 Keywords: Granger-causality, graphical models, spurious causality, multivariate

Abstract. Latent variable techniques are helpful to reduce high-dimensional time series to a few relevant variables that are easier to model and analyze. An inherent problem is the identifiability of the model and the interpretation of the latent variables. We apply graphical models to find the essential relations in the data and to deduce suitable assumptions leading to meaningful latent variables. 1

One major challenge in neuroscience is the identification of interrelations between signals reflecting neural activity. When applying multivariate time series analysis techniques to neural signals, detection of directed relationships, which can be described in terms of Granger-causality, is of particular interest. Partial directed coherence has been introduced for a frequency domain analysis of linear Granger-causality based on modeling the underlying dynamics by vector autoregressive processes. We discuss the statistical properties of estimates for partial directed coherence and propose a significance level for testing for nonzero partial directed coherence at a given frequency. The performance of this test is illustrated by means of linear and non-linear model systems and in an application to electroencephalography and electromyography data recorded from a patient suffering from essential tremor. © 2005 Elsevier B.V. All rights reserved.