Sampling large hyperplane‑truncated multivariate normal distributions

This repository includes R code for sampling from a high-dimensional hyperplane-truncated multivariate normal distribution using the Karhunen-Loève expansion (KLE) and Matheron's update rule (MUR), see Maatouk et al. (2024a,b). Additionally, it provides the LS.KLE function for generating a very large Gaussian vector extracted from a stationary Gaussian process. Using this function, a Gaussian vector of size 1,000,000 can be generated in a few seconds.

⚠️ Citation: This repository should be cited as supplementary material to Maatouk et al. (2024a).

Illustrative examples

MyImage: Five GP sample paths when the domain is split into three subdomains. Dashed (resp. solid) curves represent the paths after (resp. before) conditioning.

MyImage: Average time of sampling a MVN over 25 replicates as a function of the dimension N. The proposed approach, LS.KLE, has been compared to the Fast Fourier Transform (FFT) developed in Wood and Chan (1994).

Description of the associated R files

1. 'all_models.R' file contains two models that implement the large hyperplane-truncated multivariate normal distribution with and without hyperparameters updates.
2. 'Illustr_ex_COSTA.R' file includes all synthetic and real data examples featured in Maatouk et al. (2024).
3. 'all-fcts.R' file encompasses all the fundamental functions required to implement our approach. It includes:

i. The naive Matheron's update rule (MUR).

ii. The naive Karhunen-Loève Expansion (KLE).

iii. The proposed LS.KLE for drawing very large Gaussian vector priors.

iv. nu.MH2 for drawing posterior samples of the hyperparameters using the Metropolis–Hastings method and inverse Cholesky factor.

v. LS.KLE_MUR for sampling from large hyperplane-truncated multivariate normal distributions.

vi. samp.WC for generating Gaussian priors using the Fast Fourier Transform method of Wood and Chan (1994).

For more details on the codes or the functions, refer to the associated R files.

Author

Hassan Maatouk (CY-Tech, CY Cergy Paris Université, France).

Maintainer

Hassan Maatouk, hmk@cy-tech.fr

References

Maatouk, H., Rullière, D., and Bay, X. (2024a). "Sampling large hyperplane‐truncated multivariate normal distributions". Computational Statistics, 39: 1779–1806.

Maatouk, H., Rullière, D., Bay, X. (2024b). Large Scale Gaussian Processes with Matheron's Update Rule and Karhunen-Loève Expansion. In: Hinrichs, A., Kritzer, P., Pillichshammer, F. (eds) Monte Carlo and Quasi-Monte Carlo Methods. MCQMC 2022. Springer Proceedings in Mathematics & Statistics, vol 460. Springer, Cham.

Wood, A. and Chan, G. (1994). "Simulation of stationary Gaussian processes in [0,1]^d". Journal of Computational and Graphical Statistics, 3(4): 409-432.