Computing High-Dimensional Normal and Student-T Probabilities with Tile-Low-Rank Quasi-Monte Carlo and Block Reordering

• Activity Number: 19 - Statistical Computing and Statistical Graphics: Student Paper Award and Chambers Statistical Software Award
• Type: Topic Contributed
• Date/Time: Sunday, July 28, 2019 : 2:00 PM to 3:50 PM
• Sponsor: Section on Statistical Computing
• Author(s): Jian Cao* and Marc Genton and David Keyes and George Turkiyyah
• Companies: King Abdullah University of Science and Technology and King Abdullah University of Science and Technology and King Abdullah University of Science and Technology and American University of Beirut
• Keywords: Block reordering; Hierarchical matrix; Tile-low-rank matrix; Adaptive cross approximation; Skewed Gaussian random field

Abstract

We present a preconditioned Monte Carlo method for computing large-scale multivariate normal and Student-t probabilities. The approach combines a block-reordering scheme with the tile-low-rank Quasi-Monte Carlo simulation, which distributes a 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 improve the Monte Carlo convergence rate. Simulations up to dimension 65,536 suggests that the new method can improve the run time by one factor of magnitude compared with the hierarchical Quasi-Monte Carlo method and two orders of magnitude compared with the dense Monte Carlo method. Our method also forms a strong substitute for the conditioning methods as the estimation error can be provided within time on par. An application study is provided to illustrate that the new computational method makes the maximum likelihood estimation feasible for high-dimensional skewed Gaussian random fields.