100 research outputs found
Sequential Design for Computer Experiments with a Flexible Bayesian Additive Model
In computer experiments, a mathematical model implemented on a computer is
used to represent complex physical phenomena. These models, known as computer
simulators, enable experimental study of a virtual representation of the
complex phenomena. Simulators can be thought of as complex functions that take
many inputs and provide an output. Often these simulators are themselves
expensive to compute, and may be approximated by "surrogate models" such as
statistical regression models. In this paper we consider a new kind of
surrogate model, a Bayesian ensemble of trees (Chipman et al. 2010), with the
specific goal of learning enough about the simulator that a particular feature
of the simulator can be estimated. We focus on identifying the simulator's
global minimum. Utilizing the Bayesian version of the Expected Improvement
criterion (Jones et al. 1998), we show that this ensemble is particularly
effective when the simulator is ill-behaved, exhibiting nonstationarity or
abrupt changes in the response. A number of illustrations of the approach are
given, including a tidal power application.Comment: 21 page
GPfit: An R package for Gaussian Process Model Fitting using a New Optimization Algorithm
Gaussian process (GP) models are commonly used statistical metamodels for
emulating expensive computer simulators. Fitting a GP model can be numerically
unstable if any pair of design points in the input space are close together.
Ranjan, Haynes, and Karsten (2011) proposed a computationally stable approach
for fitting GP models to deterministic computer simulators. They used a genetic
algorithm based approach that is robust but computationally intensive for
maximizing the likelihood. This paper implements a slightly modified version of
the model proposed by Ranjan et al. (2011), as the new R package GPfit. A novel
parameterization of the spatial correlation function and a new multi-start
gradient based optimization algorithm yield optimization that is robust and
typically faster than the genetic algorithm based approach. We present two
examples with R codes to illustrate the usage of the main functions in GPfit.
Several test functions are used for performance comparison with a popular R
package mlegp. GPfit is a free software and distributed under the general
public license, as part of the R software project (R Development Core Team
2012).Comment: 20 pages, 17 image
Bayesian Variable Selection with Related Predictors
In data sets with many predictors, algorithms for identifying a good subset
of predictors are often used. Most such algorithms do not account for any
relationships between predictors. For example, stepwise regression might select
a model containing an interaction AB but neither main effect A or B. This paper
develops mathematical representations of this and other relations between
predictors, which may then be incorporated in a model selection procedure. A
Bayesian approach that goes beyond the standard independence prior for variable
selection is adopted, and preference for certain models is interpreted as prior
information. Priors relevant to arbitrary interactions and polynomials, dummy
variables for categorical factors, competing predictors, and restrictions on
the size of the models are developed. Since the relations developed are for
priors, they may be incorporated in any Bayesian variable selection algorithm
for any type of linear model. The application of the methods here is
illustrated via the Stochastic Search Variable Selection algorithm of George
and McCulloch (1993), which is modified to utilize the new priors. The
performance of the approach is illustrated with two constructed examples and a
computer performance dataset. Keywords: Model Selection, Prior Distributions,
Interaction, Dummy VariableComment: uuencoded, gzipped postscript file, 24 pages including graphics and
tables. Revised version includes new example and improved plot. Paper also
available at http://gsbhac.uchicago.edu/techreports/ Author has web page at
http://www-gsb.uchicago.edu
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