Abstract
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 $\sqrt{n}$-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.
Highlights
Gaussian processes (GPs) are one of the most common modelling tools for the analysis of spatiotemporal data
In this paper we have introduced a family of scalable covariance estimation algorithms, called the local inversion-free (LIF) algorithm, by amalgamating the ideas of the inversion-free estimation procedure in [2] a√nd a block diagonal approximation of the covariance matrix of the preconditioned data
It had only been asserted that the inversionfree estimator is statistically comparable to the MLE, when there exists a linear transformation to uniformly control the condition number of the covariance matrix below some constant, independent of the sample size [2]
Summary
Gaussian processes (GPs) are one of the most common modelling tools for the analysis of spatiotemporal data (see e.g., [6, 8]). Itescu, Chen and Stein [2] proposed one such surrogate loss based method for covariance estimation, and showed that it is considerably computationally more efficient than the standard MLE, especially for irregularly spaced observations. Their loss function, which we call inversion-free (IF) in this manuscript, does not require computing the precision matrix (covariance inverse), and so it can be evaluated in O n2 time. In this new regime, which is referred to as fixed-domain (or infill ) setting, because of strong spatial correlation the condition number often grows without bound with n This points to an unresolved question regarding the statistical efficiency of the inversion-free algorithm in the fixed-domain setting, including the situation of irregularly spaced observations. Kν (·) and Γ (·) respectively represent the modified Bessel function of the second kind of order ν and the Gamma function
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