The focus of this proposal is the study hitting probabilities for smooth vector-valued random fields with independent, identically distributed Gaussian processes as components. These models can be used to build a variety of non-Gaussian real-valued processes, though they are closely related to Gaussian processes. The smoothness of the processes allows many tools from point processes to be used in studying these hitting probabilities. These point processes, based on critical points of the process yield an explicit representation for the hitting probability as well as an accurate approximation, the so-called expected Euler characteristic approximation. The proposal seeks to extend recent work of the investigator and collaborators on real-valued Gaussian processes to these non-Gaussian models. Insight gained from these models should prove useful in studying other non-Gaussian models.
The practical motivation for this proposal is in its application to estimating the Family Wise Error Rate (FWER) in neuroimaging activation studies. This FWER is important in determining which areas in a neuroimaging study are associated with a given experimental task. In these studies, psychologists are able to collect space-time recordings of activation in the human brain (more precisely, they can record something associated with activation known as the BOLD signal). They are then able to study which areas are activated by their task, which might be a visual task, an auditory task, etc. Having collected the data, the psychologists are faced with the task of determining which perceived activations are true activations. The results of this proposal help psychologists in this decision, by allowing the psychologists to only accept results with a prespecified FWER. The proposal builds on earlier results in the literature, and extends them to more complicated models of activation.
