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A0452
Title: Random topology in soft-thresholded Gaussian models Authors:  Jian Kang - University of Michigan (United States)
Paul Marriott - University of Waterloo (Canada)
Weinan Qi - University of Waterloo (Canada)
Yi Shen - University of Waterloo (Canada) [presenting]
Abstract: The soft-thresholded Gaussian model has been developed in biostatistics with applications in brain imaging. It has a Bayesian structure and hence requires a rule to choose an appropriate prior distribution. This often means choosing the height of the threshold according to known information, for example, the number of active areas, which corresponds to the number of connected components of the excursion set above the threshold. We discuss the recent results that we obtained concerning the distribution of such a number. More precisely, we show that for certain Gaussian random fields, when the threshold tends to infinity and the searching area expands with a matching speed, both the location of the excursion sets and the location of the local maxima above the threshold will converge weakly to a Poisson point process. Moreover, when the threshold is high but not tending to infinity, the distribution of these locations can be satisfactorily approximated by a Poisson process plus a correction term. We provide theoretical support to predict the number of connecting components when performing topological data analysis to the extreme values of a random field model.