On the smallest eigenvalues of covariance matrices of multivariate spatial processes

Bachoc, François; Furrer, Reinhard

doi:10.1002/sta4.107

Cited by 14 publications

(24 citation statements)

References 13 publications

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“…Since the Fourier transform of k is strictly positive everywhere, then also the Fourier transform of k is strictly positive everywhere. Hence, from Theorem 4 in [11], k satisfies Condition 2 ii).…”

Section: A2 Proofs Of the Main Resultsmentioning

confidence: 86%

Asymptotic properties of the maximum likelihood and cross validation estimators for transformed Gaussian processes

Bachoc¹,

Betancourt²,

Furrer

et al. 2020

Electron. J. Statist.

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The asymptotic analysis of covariance parameter estimation of Gaussian processes has been subject to intensive investigation. However, this asymptotic analysis is very scarce for non-Gaussian processes. In this paper, we study a class of non-Gaussian processes obtained by regular nonlinear transformations of Gaussian processes. We provide the increasingdomain asymptotic properties of the (Gaussian) maximum likelihood and cross validation estimators of the covariance parameters of a non-Gaussian process of this class. We show that these estimators are consistent and asymptotically normal, although they are defined as if the process was Gaussian. They do not need to model or estimate the non-linear transformation. Our results can thus be interpreted as a robustness of (Gaussian) maximum likelihood and cross validation towards non-Gaussianity. Our proofs rely on two technical results that are of independent interest for the increasing-domain asymptotic literature of spatial processes. First, we show that, under mild assumptions, coefficients of inverses of large covariance matrices decay at an inverse polynomial rate as a function of the corresponding observation location distances. Second, we provide a general central limit theorem for quadratic forms obtained from transformed Gaussian processes. Finally, our asymptotic results are illustrated by numerical simulations.

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Section: A2 Proofs Of the Main Resultsmentioning

confidence: 86%

Asymptotic properties of the maximum likelihood and cross validation estimators for transformed Gaussian processes

Bachoc¹,

Betancourt²,

Furrer

et al. 2020

Electron. J. Statist.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The procedure is shown as follows. with respect to β. Denote the estimate byβ (1) ; 4. With β =β (1) , estimate θ by maximizing Q(θ,β (1) ; Z) in (3.1) with respect to θ. Denote the estimate byθ (1) .…”

Section: The Penalized Maximum Likelihood Estimation (Pmle)mentioning

confidence: 99%

“…Thenθ ose =θ (1) andβ ose =β (1) are the obtained estimates. We callθ ose andβ ose as the one-step PMLE.…”

Section: One-step Pmle (Pmlementioning

confidence: 99%

“…Assumption 1 assumes increasing domain framework. Bachoc and Furrer [1] showed that under Assumption 1 and some weak assumptions on the matrix covariance function, if the spectral density of the covariance function is positive, the smallest eigenvalue of the covariance matrix is asymptotically bounded away from zero. Most standard covariance function such as Matérn covariance function satisfy those assumptions hence satisfy the second part of Assumption 3.…”

Section: Remarks On Assumptionmentioning

confidence: 99%

“…p depends on n. Assume that the two classes have equal prior probabilities, i.e. both the probabilities that a new observation comes from C 1 and C 2 are 1 2 . Then we obtain the classification ruleδ:…”

Section: Introductionmentioning

confidence: 99%

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High dimensional classification for spatially dependent data with application to neuroimaging

Li¹,

Zhang²,

Maiti³

2020

Electron. J. Statist.

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Discriminating patients with Alzheimer's disease (AD) from healthy subjects is a crucial task in the research of Alzheimer's disease. The task can be potentially achieved by linear discriminant analysis (LDA), which is one of the most classical and popular classification techniques. However, the classification problem becomes challenging for LDA because of the high-dimensionality and the spatial dependency of the brain imaging data. To address the challenges, researchers have proposed various ways to generalize LDA into high-dimensional context in recent years. However, these existing methods did not reach any consensus on how to incorporate spatially dependent structure. In light of the current needs and limitations, we propose a new classification method, named as Penalized Maximum Likelihood Estimation LDA (PMLE-LDA). The proposed method uses Matérn covariance function to describe the spatial correlation of brain regions. Additionally, PMLE is designed to model the sparsity of highdimensional features. The spatial location information is used to address the singularity of the covariance. Tapering technique is introduced to reduce computational burden. We show in theory that the proposed method can not only provide consistent results of parameter estimation and feature selection, but also generate an asymptotically optimal classifier driven by high dimensional data with specific spatially dependent structure. Finally, the method is validated through simulations and an application into ADNI data for classifying Alzheimer's patients.

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Asymptotic Analysis of Maximum Likelihood Estimation of Covariance Parameters for Gaussian Processes: An Introduction with Proofs

Bachoc

2021

Advances in Contemporary Statistics and Econometrics

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On the smallest eigenvalues of covariance matrices of multivariate spatial processes

Cited by 14 publications

References 13 publications

Asymptotic properties of the maximum likelihood and cross validation estimators for transformed Gaussian processes

Asymptotic properties of the maximum likelihood and cross validation estimators for transformed Gaussian processes

High dimensional classification for spatially dependent data with application to neuroimaging

Asymptotic Analysis of Maximum Likelihood Estimation of Covariance Parameters for Gaussian Processes: An Introduction with Proofs

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