Covariance approximation for large multivariate spatial data sets with an application to multiple climate model errors

Abstract

This paper investigates the cross-correlations across multiple climate model errors. We build a Bayesian hierarchical model that accounts for the spatial dependence of individual models as well as cross-covariances across different climate models. Our method allows for a nonseparable and nonstationary cross-covariance structure. We also present a covariance approximation approach to facilitate the computation in the modeling and analysis of very large multivariate spatial data sets. The covariance approximation consists of two parts: a reduced-rank part to capture the large-scale spatial dependence, and a sparse covariance matrix to correct the small-scale dependence error induced by the reduced rank approximation. We pay special attention to the case that the second part of the approximation has a block-diagonal structure. Simulation results of model fitting and prediction show substantial improvement of the proposed approximation over the predictive process approximation and the independent blocks analysis. We then apply our computational approach to the joint statistical modeling of multiple climate model errors.1. Introduction. This paper addresses the problem of combining multiple climate model outputs while accounting for dependence across different models as well as spatial dependence within each individual model. To study the impact of human activity on climate change, the Intergovernmental Panel on Climate Change (IPCC) is coordinating efforts worldwide to develop coupled atmosphere-ocean general circulation models (AOGCMs).

Geostatistics for Large Datasets

“…They decomposed the spatial Gaussian process into two parts: a reduced rank process to characterize the large scale dependence and a residual process to capture the small scale spatial dependence that is unexplained by the reduced rank process. This idea was then extended to the multivariate setting by Sang and Jun (2010). However, the application of tapering techniques to multivariate random fields remains to be explored due to the lack of flexible compactly supported cross-covariance functions.…”

Absolute versus Relative Entropy Parameter Estimation in a Coarse-Grain Model of DNA

“…Notice that µ y ∈ R 6n and K y ∈ R 6n×6n , whereas µ x ∈ R 18(n−1) and K x ∈ R 18(n−1)×18(n−1) . By separately comparing the coefficients of the quadratic, linear and constant terms in w, we find that the coefficients in (27) can be expressed in terms of the coefficients in (33). Specifically, from the quadratic and linear terms we get…”

“…While we describe our results within the specific context of the cgDNA model, the maximum entropy approach to enforcing a prescribed sparsity pattern in the stiffness (or precision) matrix in a Gaussian is potentially also of wider interest [3,6,10,36,38]. For example, in the specific application fields of numerical weather prediction and data assimilation, both sparse covariance and sparse inverse covariance (or precision) matrix estimates are adopted using other techniques such as tapering [2,5,11,33,37].…”

Full scale multi-output Gaussian process emulator with nonseparable auto-covariance functions

“…In this work we propose to use the Full-Scale Approximation with Block modulating function (FSA-Block) to speed up computations [14,15]. It consists of a summation of a reduced rank covariance and a sparse covariance with the block diagonal structure.…”

“…the separability, the computations for the emulator with nonseparable auto-covariance models can be prohibitive when n is large. To overcome the computational bottleneck, we introduced the Full-Scale approximation (FSA) approach to reduce computations [14,15], which applies to both separable and nonseparable covariance structure. The FSA approach combines the ideas of the Gaussian predictive process [16] and covariance tapering [17] to provide a satisfactory approximation of the original covariance, under both large and small dependence scales of the data.…”