Optimal Detection of Abrupt Changes in Gaussian Processes: Fixed and Increasing Domain Analysis

• Activity Number: 602
• Type: Contributed
• Date/Time: Wednesday, August 12, 2015 : 2:00 PM to 3:50 PM
• Sponsor: Section on Statistics and the Environment
• Author(s): Hossein Keshavarz Shenastaghi*
• Companies: University of Michigan
• Keywords: Gaussian processes ; Shift in mean detection ; Fixed domain asymptotics ; increasing domain asymptotics ; minimax optimal detection rate ; GLRT

Abstract

We study minimax optimal detection of shift in mean (SIM) in a univariate Gaussian process. The asymptotic analysis of existing algorithms of SIM detection such as CUSUM mainly adopt an unrealistic assumption about the underlying process (e.g., independent samples). Besides, the majority of former studies on SIM detection in time series with dependent samples, are restricted to the increasing domain asymptotic framework, in which the smallest distance between sampling points is bounded away from zero. Motivated by abrupt change detection in locally stationary processes and change zone detection in spatial processes, we analyze SIM detection of Gaussian processes in fixed domain regime, in which samples gets denser in a bounded domain. To our knowledge, this is the first work on SIM detection in fixed domain regime. We show that despite optimality of CUSUM in increasing domain, it exhibits a poor performance in fixed domain. We also propose a minimax optimal algorithm using exact or approximated generalized likelihood ratio test. Our results demonstrate a strong connection between detection rate and the smoothness of the covariance function of underlying process in fixed domain.