Pritam Ranjan

A Computationally Stable Approach to Gaussian Process Interpolation of Deterministic Computer Simulation Data

For many expensive deterministic computer simulators, the outputs do not have replication error and the desired metamodel (or statistical emulator) is an interpolator of the observed data. Realizations of Gaussian spatial processes (GP) are commonly used to model such simulator outputs. Fitting a GP model to n data points requires the computation of the inverse and determinant of n×n correlation matrices, R, that are sometimes computationally unstable due to near-singularity of R. This happens if any pair of design points are very close together in the input space. The popular approach to overcome near-singularity is to introduce a small nugget (or jitter) parameter in the model that is estimated along with other model parameters. The inclusion of a nugget in the model often causes unnecessary over-smoothing of the data. In this article, we propose a lower bound on the nugget that minimizes the over-smoothing and an iterative regularization approach to construct a predictor that further improves the interpolation accuracy. We also show that the proposed predictor converges to the GP interpolator.

Modeling an Augmented Lagrangian for Blackbox Constrained Optimization

Constrained blackbox optimization is a difficult problem, with most approaches coming from the mathematical programming literature. The statistical literature is sparse, especially in addressing problems with nontrivial constraints. This situation is unfortunate because statistical methods have many attractive properties: global scope, handling noisy objectives, sensitivity analysis, and so forth. To narrow that gap, we propose a combination of response surface modeling, expected improvement, and the augmented Lagrangian numerical optimization framework. This hybrid approach allows the statistical model to think globally and the augmented Lagrangian to act locally. We focus on problems where the constraints are the primary bottleneck, requiring expensive simulation to evaluate and substantial modeling effort to map out. In that context, our hybridization presents a simple yet effective solution that allows existing objective-oriented statistical approaches, like those based on Gaussian process surrogates and expected improvement heuristics, to be applied to the constrained setting with minor modification. This work is motivated by a challenging, real-data benchmark problem from hydrology where, even with a simple linear objective function, learning a nontrivial valid region complicates the search for a global minimum. Supplementary materials for this article are available online.

A new Bayesian ensemble of trees approach for land cover classification of satellite imagery

Classification of satellite images is a key component of many remote sensing applications. One of the most important products of a raw satellite image is the classification that labels image pixels into meaningful classes. Though several parametric and nonparametric classifiers have been developed thus far, accurate classification still remains a challenge. In this paper, we propose a new reliable multiclass classifier for identifying class labels of a satellite image in remote sensing applications. The proposed multiclass classifier is a generalization of a binary classifier based on the flexible ensemble of regression trees model called Bayesian Additive Regression Trees. We used three small areas from the LANDSAT 5 TM image, acquired on 15 August 2009 (path–row: 08–29, L1T product, UTM map projection) over Kings County, Nova Scotia, Canada, to classify the land cover. Several prediction accuracy and uncertainty measures have been used to compare the reliability of the proposed classifier with the state-of-the-art classifiers in remote sensing.

Compliance Testing for Random Effects Models With Joint Acceptance Criteria

For consumer protection, many governments perform random inspections on goods sold by weight or volume to ensure consistency between actual and labeled net contents. To pass inspection, random samples must jointly comply with restrictions placed on the individual sampled items and on the sample average. In this article, we consider the current United States National Institute of Standards and Technology joint acceptance criteria. Motivated by a problem from a real manufacturing process, we provide an approximation for the probability of sample acceptance that is applicable for processes with one or more known sources of variation via a random effects model. This approach also allows the assessment of the sampling scheme of the items. We use examples and simulations to assess the quality and accuracy of the approximation and illustrate how the methodology can be used to fine-tune process parameters for a prespecified probability of sample acceptance. Simulations are also used for estimating variance components.

Follow-up experimental designs for computer models and physical processes

In many branches of physical science, when the complex physical phenomena are either too expensive or too time consuming to observe, deterministic computer codes are often used to simulate these processes. Nonetheless, true physical processes are also observed in some disciplines. It is preferred to integrate both the true physical process and the computer model data for better understanding of the underlying phenomena. In this paper, we develop a methodology for selecting optimal follow-up designs based on integrated mean squared error that help us capture and reduce prediction uncertainty as much as possible. We also compare the efficiency of the optimal designs with the intuitive choices for the follow-up computer and field trials.

Efficient optimization of the likelihood function in Gaussian process modelling

Gaussian Process (GP) models are popular statistical surrogates used for emulating computationally expensive computer simulators. The quality of a GP model fit can be assessed by a goodness of fit measure based on optimized likelihood. Finding the global maximum of the likelihood function for a GP model is typically challenging, as the likelihood surface often has multiple local optima, and an explicit expression for the gradient of the likelihood function may not be available. Previous methods for optimizing the likelihood function have proven to be robust and accurate, though relatively inefficient. Several likelihood optimization techniques are proposed, including two modified multi-start local search techniques, that are equally as reliable, and significantly more efficient than existing methods. A hybridization of the global search algorithm Dividing Rectangles (DIRECT) with the local optimization algorithm BFGS provides a comparable GP model quality for a fraction of the computational cost, and is the preferred optimization technique when computational resources are limited. Several test functions and an application motivated by oil reservoir development are used to test and compare the performance of the proposed methods with the implementation provided in the R library GPfit. The proposed method is implemented in a Matlab package, GPMfit.

Comment: EI Criteria for Noisy Computer Simulators

The author believes that the methodologies presented by Picheny et al. are innovative and should be useful for computer experiment practitioners.

Determining the magnitude of true analytical-error in geochemical analysis

ABSTRACT Geochemical analysis of geological materials introduces errors at virtually every stage of sample preparation and analysis. Determining the actual analytical error (that error introduced during the analysis of prepared sub-samples of geological materials) is commonly difficult because many forms of analysis destroy the sub-sample. As a result, duplicate analysis cannot be undertaken to measure analytical error directly, and analytical error cannot be isolated from sub-sampling error. However, using replicate analyses of sub-samples of two different masses, and solving a system of three equations in three unknowns, the actual ‘analytical’ error can be deduced and distinguished from the sub-sampling error. This provides a means to estimate sub-sampling and analytical error magnitudes and to determine whether increasing sub-sample mass will result in an efficient reduction in overall error in geochemical analyses. It also provides a means to quantify sub-sampling error in reference materials so that they can be properly used in geochemical analysis to monitor and quantify analytical error.

A Unified Approach to Factorial Designs with Randomization Restrictions

Abtsrcat Factorial designs are commonly used to assess the impact of factors and factor combinations in industrial and agricultural experiments. Though preferred, complete randomization of trials is often infeasible, and randomization restrictions are imposed. In this paper, we discuss a finite projective geometric (PG) approach to unify the existence, construction and analysis of multistage factorial designs with randomization restrictions using randomization defining contrast subspaces (or flats of a PG). Our main focus will be on the construction of such designs, and developing a word length pattern scheme that can be used for generalizing the traditional design rank- ing criteria for factorial designs. We also present a novel isomorphism check algorithm for these designs.