Modelling sequential data is important in many areas of science and engineering. Hidden Markov models (HMMs) and Kalman filter models (KFMs) are popular for this because they are simple and flexible. For example, HMMs have been used for speech recognition and bio-sequence analysis, and KFMs have been used for problems ranging from tracking planes and missiles to predicting the economy. However, HMMs and KFMs are limited in their “expressive power”. Dynamic Bayesian Networks (DBNs) generalize HMMs by allowing the state space to be represented in factored form, instead of as a single discrete random variable. DBNs generalize KFMs by allowing arbitrary probability distributions, not just (unimodal) linear-Gaussian. In this thesis, I will discuss how to represent many different kinds of models as DBNs, how to perform exact and approximate inference in DBNs, and how to learn DBN models from sequential data. In particular, the main novel technical contributions of this thesis are as follows: a way of representing Hierarchical HMMs as DBNs, which enables inference to be done in O(T) time instead of O(T 3), where T is the length of the sequence; an exact smoothing algorithm that takes O(log T) space instead of O(T); a simple way of using the junction tree algorithm for online inference in DBNs; new complexity bounds on exact online inference in DBNs; a new deterministic approximate inference algorithm called factored frontier; an analysis of the relationship between the BK algorithm and loopy belief propagation; a way of applying Rao-Blackwellised particle filtering to DBNs in general, and the SLAM (simultaneous localization and mapping) problem in particular; a way of extending the structural EM algorithm to DBNs; and a variety of different applications of DBNs. However, perhaps the main value of the thesis is its catholic presentation of the field of sequential data modelling.

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.

. As high dimensional data occur as a rule rather than an exception in critical care today, it is of utmost importance to improve acquisition, storage, modelling, and analysis of medical data, which appears feasable only with the help of bedside computers. The use of clinical information systems offers new perspectives of data recording and also causes a new challenge for statistical methodology. A graphical approach for analysing patterns in statistical time series from online monitoring systems in intensive care is proposed here as an example of a simple univariate method, which contains the possibility of a multivariate extension and which can be combined with procedures for dimension reduction. Keywords. Clinical information systems, decision support, high dimensional time series, online monitoring, phase space reconstruction 1 Introduction Increasing technical possibilities in online recording of complex data structures produce manifold challenges for statistical methods. For i...

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.

Abstract. Graphical models provide a framework for exploration of multivariate dependence patterns. The connection between graph and statistical model is made by identifying the vertices of the graph with the observed variables and translating the pattern of edges in the graph into a pattern of conditional independences that is imposed on the variables ’ joint distribution. Focusing on Gaussian models, we review classical graphical models. For these models the defining conditional independences are equivalent to vanishing of certain (partial) correlation coefficients associated with individual edges that are absent from the graph. Hence, Gaussian graphical model selection can be performed by multiple testing of hypotheses about vanishing (partial) correlation coefficients. We show and exemplify how this approach allows one to perform model selection while controlling error rates for incorrect edge inclusion. Key words and phrases: Acyclic directed graph, Bayesian network, bidirected graph, chain graph, concentration graph, covariance graph, DAG, graphical model, multiple testing, undirected graph. 1.

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.

A new class of graphical models capturing the dependence structure of events that occur in time is proposed. The graphs represent so–called local independencies, meaning that the intensities of certain types of events are independent of some (but not necessarilly all) events in the past. This dynamic concept of independence is asymmetric, similar to Granger non–causality, so that the corresponding local independence graphs differ considerably from classical graphical models. Hence a new notion of graph separation, called δ–separation, is introduced and implications for the underlying model as well as for likelihood inference are explored. Benefits regarding facilitation of reasoning about and understanding of dynamic dependencies as well as computational simplifications are discussed.

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.