Mathematical Sciences: Time Series and Point Processes: Networks and Wavelets

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DMS 9625774 Brillinger This project has two parts. A first is to develop statistical techniques for dealing with graphical models whose vertex variates are time series or point processes. Part of the motivation comes from problems in the neurosciences and in system sciences where one sets down box and arrow diagrams meant to provide the flow of information between interconnected units. In the neuroscience case the units will be neurons, the data will be the firing times and models will developed and fit for the interconnections of single units or of groups of units. Analyses will be carried out for experimental data including some collected from neurons distributed through the auditory system of the cat. In various other substantive fields one is also interested in detecting the presence of connections and estimating the strength of those connections. The second part of the project concerns the development of wavelet variants, for random processes, of some existing ordinary statistical techniques. Wavelets are of use in eliciting signals of nonstationary character from time series and images for example. Both parts of the project will require development of complex stochastic models and their properties, having in mind practical implementation. A time series is a wiggly line meant to represent a phenomenon fluctuating in time, such as a seismogram, an electrocardiogram or an electroencephalogram. A point process is a collection of times at which an event of interest occurs, such as an earthquake or a nerve cell firing or changes in mean level (for example of a river). A network is a flow or wiring diagram of interacting entities such as nerve cells or cities or computers. A focus of the research is with the case of time series or point processes recorded at each of the units of a network. The time series and point processes are interacting and it is desired to learn the strengths and directions of those interactions. Wavelet analysis is a novel method for elicitilg information concerning phenomena that may be changing, from time series or image records. The problem of detecting and estimating change arises often in problems of the environment and in particular of global warming. It is intended to extend the wavelet technique to other types of evolving and distributed data such as point processes or categorical-valued processes.