Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2003.11183

Exploiting Low Rank Covariance Structures for Computing High-Dimensional Normal and Student-$t$ Probabilities

Jian Cao·Marc G. Genton·David E. Keyes·George M. Turkiyyah
View original source ↗Read source document ↗Export metadata ↓

Abstract

We present a preconditioned Monte Carlo method for computing high-dimensional multivariate normal and Student-$t$ probabilities arising in spatial statistics. The approach combines a tile-low-rank representation of covariance matrices with a block-reordering scheme for efficient Quasi-Monte Carlo simulation. The tile-low-rank representation decomposes the high-dimensional problem into many diagonal-block-size problems and low-rank connections. The block-reordering scheme reorders between and within the diagonal blocks to reduce the impact of integration variables from right to left, thus improving the Monte Carlo convergence rate. Simulations up to dimension $65{,}536$ suggest that the new method can improve the run time by an order of magnitude compared with the non-reordered tile-low-rank Quasi-Monte Carlo method and two orders of magnitude compared with the dense Quasi-Monte Carlo method. Our method also forms a strong substitute for the approximate conditioning methods as a more robust estimation with error guarantees. An application study to wind stochastic generators is provided to illustrate that the new computational method makes the maximum likelihood estimation feasible for high-dimensional skew-normal random fields.

Explore related subjects

stat.CO

Keep this discovery

BibTeXRIS

Jian Cao, Marc G. Genton, David E. Keyes, George M. Turkiyyah. 2020-03-25. Exploiting Low Rank Covariance Structures for Computing High-Dimensional Normal and Student-$t$ Probabilities. https://arxiv.org/abs/2003.11183

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Estimating Hierarchically Rank Structured Covariance Matrices

We consider the problem of estimating a high-dimensional covariance matrix from a very limited number of samples. This problem is ubiquitous in computational fluid dynamics, where a small number of fluid snapshots must be used to construct a Gramian matrix determining a reduced-order model, as well as in computational geoscience, where a small ensemble of Earth system forecasts must be used to estimate the covariance matrix associated with the forecast uncertainty. It is common practice to regularize the small-sample covariance by imposing a "localization" structure that enforces a physically realistic correlation length scale, imposing a sparsity constraint, "shrinking" towards a prescribed target, or attenuating small correlations. We propose an alternate technique that regularizes the small-sample covariance by imposing hierarchical rank structure. Compared to regularization methods that assume sparsity such as spatial localization, hierarchical rank structure accommodates a wider range of covariance matrices, roughly corresponding to situations where long-range correlations vary more smoothly than short-range ones. It also results in a data-sparse matrix format that permits highly efficient matrix-vector products. We present theory and algorithms which show how to efficiently estimate a high-dimensional, hierarchically rank structured covariance matrix from limited samples. Through an error analysis and numerical experiments with a variety of model problems, we demonstrate that these techniques are effective at reducing sampling errors, and that in many cases they achieve smaller estimation error than conventional techniques.

stat.CO↗
arXiv2026

Optimal Slice-Adaptive Tuning of Hybrid Slice Sampling

Slice sampling is a Markov chain Monte Carlo algorithm that draws its next state uniformly from a "slice"---a super-level set of the target density function---at each iteration, thereby providing automatic local adaptivity to the scale of the target. In practice the exact slice is not known, so general-purpose implementations use an approximate slice that is grown from a starting interval of length $w>0$, with a computational cost that depends on $w$. This work presents an analysis of the average per-iteration number of target density evaluations, as a function of $w$, of hybrid slice sampling with various slice-finding schemes for targets with contiguous slices. The paper uses the results of the analysis to develop automated, slice-adaptive tuning schemes along with suboptimality bounds and asymptotic convergence guarantees. Simulations demonstrate that the tuning schemes reliably yield near-optimal slice-adaptive tuning with essentially no dependence on the initial setting of $w$.

stat.CO↗