On the consistency of inversion-free parameter estimation for Gaussian random fields

Keshavarz, Hossein; Scott, Clayton; Nguyen, XuanLong

doi:10.1016/j.jmva.2016.06.003

Cited by 4 publications

(7 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We consider two cases of isotropy and geometric anisotropy for the covariance function. For circumventing the obstacles of computing the Cholesky factorization of the covariance matrix, spectral methods are used for constructing G on D n [11]. We now concisely describe the geometry of D n .…”

Section: Simulation Studiesmentioning

confidence: 99%

“…We then show that the stochastic quadratic quantity P 2 is of order n −1 log n, with high probability. The concentration inequalities involving the quadratic forms (and their supremum over a bounded space) of GPs presented in [11] are crucial for bounding P 2 from above.…”

Section: Proofsmentioning

confidence: 99%

“…We showed in the proof of Theorem 4.1 that M B n,m (ρ) 2→2 / M B n,m (ρ) 2 → 0 when n → ∞ (see (7.13) and (7.14)). Thus applying Lemma A.4 of [11], on asymptotic normality of the normalized generalized χ 2 random variables, leads to ξ n − Eξ n √ var ξ n d → N (0, 1) .…”

Section: →2mentioning

confidence: 99%

“…Indeed, 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 n 2 time. It was established by the authors that when the covariance matrix has a bounded condition number, the resulting estimate possesses consistency and asymptotic normality [11]. It is noted that the boundedness of condition number holds in the increasing domain setting, where the minimum distance among the sampling points is bounded away from zero.…”

Section: Introductionmentioning

confidence: 99%

“…A considerable portion of this article is devoted to the investigation of the asymptotic behavior of the proposed LIF based estimation method in the fixed-domain regime. Theoretical analysis for several specific instances of LIF based estimation have been carried out before, by [1] on his quadratic variation based method on regularly spaced observations in the fixed-domain framework, and in the increasing domain regime by the authors [11]. The asymptotic theory for the fixed-domain regime is considerably more involved than the increasing domain regime, especially for irregularly spaced observations.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Local inversion-free estimation of spatial Gaussian processes

Keshavarz¹,

Nguyen²,

Scott³

2019

Electron. J. Statist.

Self Cite

View full text Add to dashboard Cite

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...

show abstract

Section: Simulation Studiesmentioning

confidence: 99%

Section: Proofsmentioning

confidence: 99%

Section: →2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Local inversion-free estimation of spatial Gaussian processes

Keshavarz¹,

Nguyen²,

Scott³

2019

Electron. J. Statist.

Self Cite

View full text Add to dashboard Cite

show abstract

On the consistency of inversion-free parameter estimation for Gaussian random fields

Keshavarz

Scott

Nguyen

2016

Journal of Multivariate Analysis

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Nonparametric estimation of isotropic covariance function

Wang

Ghosh

2022

Journal of Nonparametric Statistics

View full text Add to dashboard Cite

On the consistency of inversion-free parameter estimation for Gaussian random fields

Cited by 4 publications

References 23 publications

Local inversion-free estimation of spatial Gaussian processes

Local inversion-free estimation of spatial Gaussian processes

On the consistency of inversion-free parameter estimation for Gaussian random fields

Nonparametric estimation of isotropic covariance function

Contact Info

Product

Resources

About