Abstract:
This thesis proposes a method for investigating structural breaks in a non-stationary spatial random field and provide a piecewise stationary approximation that best describes the process. Suppose a random field is observed only once over a regular grid. We study the covariance structure of this field in the frequency domain. In the first part the thesis, we define a spectral difference statistic, a spatially varying quantity showing the difference between local spatial spectral density integrated over a range of frequencies. This spectral difference process is expected to be uniformly close to zero if and only if the underlying field is stationary. This intuitive behavior is justified by a rigorous derivation of the asymptotic behavior of this process under an increasing domain asymptotic scheme. This result is then utilized to construct a consistent and asymptotic level alpha test for the hypothesis of stationarity using the maximum of the spectral difference process over locations and range of frequencies. A field plot of this spectral difference process, called discrepancy map, further provides insight to the nature of nonstationarity presents in the observed field. Next, we propose a method to construct a piece-wise approximation of the observed random field, where the pieces are spatial regions with linear boundaries by an iterative search. A hierarchical clustering algorithm is used to appropriately merge the initial partition to produce the final approximation. A computationally efficient implementation of this methodology has been outlined. The accuracy and performance of the proposed methods are demonstrated via extensive simulations and two case studies on real data. The later part of the thesis outlines strategies of extending this methodology for a random field observed at irregularly spaced locations. The efficiency of this extension is investigated by some numerical experiments.
