We study the problem of detecting a change in the mean of one-dimensional Gaussian process data. This problem is investigated in the setting of increasing domain (customarily employed in time series analysis) and in the setting of fixed domain (typically arising in spatial data analysis). We propose a detection method based on the generalized likelihood ratio test (GLRT), and show that our method achieves nearly asymptotically optimal rate in the minimax sense, in both settings. The salient feature of the proposed method is that it exploits in an efficient way the data dependence captured by the Gaussian process covariance structure. When the covariance is not known, we propose the plug-in GLRT method and derive conditions under which the method remains asymptotically near optimal. By contrast, the standard CUSUM method, which does not account for the covariance structure, is shown to be asymptotically optimal only in the increasing domain. Our algorithms and accompanying theory are applicable to a wide variety of covariance structures, including the Matern class, the powered exponential class, and others. The plug-in GLRT method is shown to perform well for maximum likelihood estimators with a dense covariance matrix.
Gaussian random fields are a powerful tool for modeling environmental processes. For high dimensional samples, classical approaches for estimating the covariance parameters require highly challenging and massive computations, such as the evaluation of the Cholesky factorization or solving linear systems. Recently, Anitescu, Chen and Stein [2] proposed a fast and scalable algorithm which does not need such burdensome computations. The main focus of this article is to study the asymptotic behavior of the algorithm of Anitescu et al. (ACS) for regular and irregular grids in the increasing domain setting. Consistency, minimax optimality and asymptotic normality of this algorithm are proved under mild differentiability conditions on the covariance function. Despite the fact that ACS's method entails a non-concave maximization, our results hold for any stationary point of the objective function. A numerical study is presented to evaluate the efficiency of this algorithm for large data sets.
Maximizing the likelihood has been widely used for estimating the unknown covariance parameters of spatial Gaussian processes. However, evaluating and optimizing the likelihood function can be computationally intractable, particularly for large number of (possibly) irregularly spaced observations, due to the need to handle the inverse of ill-conditioned and large covariance matrices. Extending the "inversion-free" method of Anitescu, Chen and Stein [1], we investigate a broad class of covariance parameter estimation based on inversion-free surrogate losses and block diagonal approximation schemes of the covariance structure. This class of estimators yields a spectrum for negotiating the trade-off between statistical accuracy and computational cost. We present fixed-domain asymptotic properties of our proposed method, establishing √ n-consistency and asymptotic normality results for isotropic Matern Gaussian processes observed on a multi-dimensional and irregular lattice. Simulation studies are also presented for assessing the scalability and statistical efficiency of the proposed algorithm for large data sets.MSC 2010 subject classifications: Primary 62M30, 62M40; secondary 60G15. We are grateful to Mihai Anitescu and Michael Stein for valuable discussions and suggestions. 1 arXiv:1811.12602v3 [math.ST] 7 Jul 2019 2. Preconditioning and inversion-free surrogate loss 2.1. Gaussian processes observed on irregular lattices Consider a zero mean, real valued, and stationary Gaussian process G on domain D, where D is a bounded subset of R d such as [0, 1]d . The dependence structure of G is typically parametrized by a variance parameter φ 0 > 0 and a (correlation) range parameter ρ 0 . Specifically, if G is a geometric anisotropic process on D, then there are a fully known covariance function K and a matrix ρ 0 ∈ R d×d such thatThe objective is to estimate the microergodic parameters of the covariance function, given n measurements from one realization of G at locations D n = {s 1 , . . . , s n } ⊂ D. Throughout the paper, we assume that ρ 0 2 2 .However throughout the paper and for simplifying the theoretical analysis, we only consider the case of non-overlapping bins. It will also be assumed that w i = 1 for all i ∈ {1, . . . , b n }.Remark 3.2. It is informative to take an alternative viewpoint of the LIF objective function in (3.2) as corresponding to a block diagonal approximation of the covariance matrix. Interestingly, as a consequence of the asymptotic theory developed in the next section, this approximation does not affect the asymptotic estimation rate, but it can substantially help to speed up the computation. The block diagonal approximation of K n,m (ρ) corresponding to partitioning scheme B, to be denoted by K B n,m (ρ), can be described as follows. Choose any s, s ∈ D n , and let t, t denote the index of the elements in B containing s and s , i.e., s ∈ B t and s ∈ B t . The entries of K B n,m (ρ) can be equivalently represented by K B n,m (ρ) s,s = [K n,m (ρ)] s,s 1 {t=t } .Remark 4.3. It has been discus...
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