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In this paper, we revisit the problem of Bayesian shape-restricted function estimation. The finite-dimensional Gaussian process (GP) approximation proposed by Maatouk and Bay (Math Geosci 49(5): 557–582, 2017) is considered, which admits an equivalent formulation of the shape constraints in terms of basis coefficients. This approximation satisfies a wide variety of shape constraints everywhere, whether applied alone, in combination, or sequentially. We propose a new, efficient, and fast algorithm for sampling from a large Gaussian vector extracted from a stationary GP. The proposed approach significantly improves the novel circulant embedding technique proposed by Ray et al. (Stat Comput 30(4): 839–853, 2020) for efficiently sampling from the resulting posterior constrained distribution. The main idea of the algorithm developed in the present paper is to divide the input domain into smaller subdomains and apply a cross-correlated technique to address the correlation structure in the entire domain. As the number of subdomains increases, the computational complexity is drastically reduced. The developed algorithm is accurate and efficient, as demonstrated through comparisons with competing approaches. The performance of the proposed approach has been evaluated within the context of Bayesian shape-restricted function estimation. [ABSTRACT FROM AUTHOR]

Generating multivariate normal distributions is widely used in various fields, including engineering, statistics, finance and machine learning. In this paper, simulating large multivariate normal distributions truncated on the intersection of a set of hyperplanes is investigated. Specifically, the proposed methodology focuses on cases where the prior multivariate normal is extracted from a stationary Gaussian process (GP). It is based on combining both Karhunen-Loève expansions (KLE) and Matheron's update rules (MUR). The KLE requires the computation of the decomposition of the covariance matrix of the random variables, which can become expensive when the random vector is too large. To address this issue, the input domain is split in smallest subdomains where the eigendecomposition can be computed. Due to the stationary property, only the eigendecomposition of the first subdomain is required. Through this strategy, the computational complexity is drastically reduced. The mean-square truncation and block errors have been calculated. The efficiency of the proposed approach has been demonstrated through both synthetic and real data studies. [ABSTRACT FROM AUTHOR]

In this paper, interpolating curve or surface with linear inequality constraints is considered as a general convex optimization problem in a Reproducing Kernel Hilbert Space. The aim of the present paper is to propose an approximation method in a very general framework based on a discretized optimization problem in a finite-dimensional Hilbert space under the same set of constraints. We prove that the approximate solution converges uniformly to the optimal constrained interpolating function. Numerical examples are provided to illustrate this result in the case of boundedness and monotonicity constraints in one and two dimensions. [ABSTRACT FROM AUTHOR]

Physical phenomena are observed in many fields (science and engineering) and are often studied by time-consuming computer codes. These codes are analyzed with statistical models, often called emulators. In many situations, the physical system (computer model output) may be known to satisfy inequality constraints with respect to some or all input variables. The aim is to build a model capable of incorporating both data interpolation and inequality constraints into a Gaussian process emulator. By using a functional decomposition, a finite-dimensional approximation of Gaussian processes such that all conditional simulations satisfy the inequality constraints in the entire domain is proposed. To show the performance of the proposed model, some conditional simulations with inequality constraints such as boundedness, monotonicity or convexity conditions in one and two dimensions are given. A simulation study to investigate the efficiency of the method in terms of prediction is included. [ABSTRACT FROM AUTHOR]