Detection of spatial change points in the mean and covariances of multivariate simultaneous autoregressive models

Abstract

In this paper, we propose a test procedure to detect change points of multidimensional autoregressive processes. The considered process differs from typical applied spatial autoregressive processes in that it is assumed to evolve from a predefined center into every dimension. Additionally, structural breaks in the process can occur at a certain distance from the predefined center. The main aim of this paper is to detect such spatial changes. In particular, we focus on shifts in the mean and the autoregressive parameter. The proposed test procedure is based on the likelihood-ratio approach. Eventually, the goodness-of-fit values of the estimators are compared for different shifts. Moreover, the empirical distribution of the test statistic of the likelihood-ratio test is obtained via Monte Carlo simulations. We show that the generalized Gumbel distribution seems to be a suitable limiting distribution of the proposed test statistic. Finally, we discuss the detection of lung cancer in computed tomography scans and illustrate the proposed test procedure.

Discussion of “Statistical methods for network surveillance” by Daniel Jeske, Nathaniel Stevens, Alexander Tartakovsky, and James Wilson

“…Diggle et al additionally supposed that it is isotropic. While these approaches are suitable to detect changes in the mean, Otto and Schmid and Garthoff and Otto additionally discussed methods for the detection of changes in the spatial covariance. In particular, they focused on random processes with a multidimensional support, such that purely spatial and spatiotemporal processes are covered.…”

Inference for Structural Breaks in Spatial Models

“…When µ(s) = c 1 I(s ∈ B) + c 2 I(s ∈ B c ) and the noise {ε i } is independent, issue 2 is equivalent to edge estimation in image processing or the detection of disease outbreaks in public health, which have been actively pursued in the literature. See, for example, Song, Zhan, Long, Zhang and Yao (2011), Müller and Song (1994), and Otto and Schmid (2016) for edge estimation; Kulldorff (2001), Huang, Kulldorff and Gregorio (2007), and Neill (2012) for detection of disease outbreaks.…”

“…Issues 1 and 2 have been extensively studied in a time series context; for examples, see Bickel and Rosenblatt (1973), Perron (1999, 2003), Davis, Lee and Rodriguez-Yam (2006), Wu and Zhao (2007), Harchaoui and Lévy-Leduc (2010), Chen and Hong (2012), Chan, Yau and Zhang (2014), and Aue, Rice and Sönmez (2018). Structural break phenomena of spatial data may also be found in many practical problems: the detection of tumors in computed tomography scans described in Otto and Schmid (2016); the estimation of the change boundary for the support of a multivariate probability density, as given in Hall, Peng and Rau (2001); and the estimation of edges in image processing studied in Tsybakov (1994). Some procedures for detecting change-boundaries have also been developed.…”

“…For example, Müller and Song (1994) assume that the edges divide the region into two subregions and estimate the change curve over candidate plateau sets based on cumulative sum (CUSUM) type test statistic, and Otto and Schmid (2016) assume that the means of the observations to be different between the interior and the exterior of a circle of radius δ, and then estimate the circular change boundary based on the maximum likelihood function under the Gaussian assumption. These methods fail when changes occur in more than two regions.…”

Estimating seasonal onsets and peaks of bronchiolitis with spatially and temporally uncertain data

“…To accomplish these goals, we develop a novel temporal change point regression model where the change points are estimated under a Bayesian framework. Change point models are used in a variety of settings such as genetics, ecology, and economics to model inflection points in time as well as in space . Because the onset and peak times of bronchiolitis seasons vary by location, as a novel statistical contribution, we develop a change point model where the change points and associated parameters of the regression model vary over space which, unlike previous studies, allows us to make direct inference on the spatial variability of temporal parameters.…”