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Über dieses Buch

In the past 15 to 20 years, the computer has become a popular tool for exploring the relationship between a measured response and factors thought to affect the response. In many cases, scientific theories exist that implicitly relate the response to the factors by means of systems of mathematical equations. There also exist numerical methods for accurately solving such equations and appropriate computer hardware and software to implement these methods. In many engineering applications, for example, the relationship is described by a dynamical system and the numerical method is a finite element code. In such situations, these numerical methods allow one to produce computer code that can generate the response corresponding to any given set of values of the factors. This allows one to conduct an "experiment" (called a "computer experiment") to explore the relationship between the response and the factors using the code. Indeed, in some cases computer experimentation is feasible when a properly designed physical experiment (the gold standard for establishing cause and effect) is impossible. For example, the number of input variables may be too large to consider performing a physical experiment or it may simply be economically prohibitive to run an experiment on the scale required to gather sufficient information to answer a particular research question. This book describes methods for designing and analyzing experiments conducted using computer code in lieu of a physical experiment. It discusses how to select the values of the factors at which to run the code (the design of the computer experiment) in light of the research objectives of the experimenter. It also provides techniques for analyzing the resulting data so as to achieve these research goals.

Inhaltsverzeichnis

Frontmatter

1. Physical Experiments and Computer Experiments

Abstract
This book describes methods for designing and analyzing research studies that are conducted using computer code in lieu of a physical experiment. Historically, Statistics has been the scientific discipline that creates methodology for conducting empirical research. The process of designing a study to answer a specific research question first addresses the problem of identifying the variables to be observed, i.e., what data to collect. Traditional methods of data collection include retrospective techniques such as cohort studies and the case-control studies used in epidemiology. The gold standard data collection method for establishing cause and effect relationships is the prospective designed experiment. Agricultural field experiments were one of the first subject matter disciplines that used designed experiments. Over time, many other subject matter areas and modes of experimentation have been developed. For example, controlled clinical trials are used extensively in studying medical therapies and simulation experiments are used extensively in operations research to compare the performance of (well) understood physical systems having stochastic components such as the flow of material through a job shop. Once the research study has been designed and executed, Statistics either identifies or develops appropriate methods to analyze the resulting data.
Thomas J. Santner, Brian J. Williams, William I. Notz

2. Preliminaries

Abstract
This chapter outlines the basic considerations in thinking about the design and analysis of computer experiments. This section begins by distinguishing three types of variables that can affect the output of a computer code y(•), depending on the phenomenon being modeled. Using this categorization, we identify some possible experimental goals.
Thomas J. Santner, Brian J. Williams, William I. Notz

3. Predicting Output from Computer Experiments

Abstract
This chapter discusses techniques for predicting the output of a computer model based on training data. A naíve view of this problem might regard it as being point estimation of a fixed population quantity. In contrast, prediction is the problem of providing a point guess of the realization of a random variable. The reason why prediction is the relevant methodology for the computer experiment application will be discussed in Section 3.2.
Thomas J. Santner, Brian J. Williams, William I. Notz

4. Additional Topics in Prediction Methodology

Abstract
A predictive distribution for the random variable Y 0 is meant to capture all the information about Y 0 that is contained in Y n = (Y 1, ..., Y n ). Of course, knowing Y n does not completely specify Y 0 but Y n does provide a probability distribution of more likely and less likely values for Y 0 that is called the predictive distribution of Y 0 given Y n . This section derives predictive distributions useful for computer output based on two hierarchical models for [Y 0, Y n ]. Section 4.2 considers prediction in the case of multiple response models, as described in Section 2.3.
Thomas J. Santner, Brian J. Williams, William I. Notz

5. Space-Filling Designs for Computer Experiments

Abstract
In this chapter and the next, we discuss how to select inputs at which to compute the output of a computer experiment to achieve specific goals. The inputs we select constitute our “experimental design.” The region corresponding to the values of the inputs over which we wish to study or model the response is the experimental region. A point in this region corresponds to a specific set of values of the inputs. Thus, an experimental design is a specification of points in the experimental region at which we wish to compute the response.
Thomas J. Santner, Brian J. Williams, William I. Notz

6. Some Criterion-based Experimental Designs

Abstract
In Chapter 5, we considered designs that attempt to spread observations evenly throughout the experimental region. We called such designs space-filling designs. One rationale for using a space-filling design is the following. If we believe interesting features of the true model are just as likely to be in one part of the experimental region as another, we should take observations in all portions of the experimental region. A space-filling design attempts to do this. One difficulty is deciding exactly what it means for a set of observations to be evenly spread throughout the experimental region. There are many ways in which a design might be considered space-filling, and we discussed several in Chapter 5. Which design is best is not clear.
Thomas J. Santner, Brian J. Williams, William I. Notz

7. Sensitivity Analysis, Validation, and Other Issues

Abstract
In this section we discuss sensitivity analysis. In general, sensitivity analysis is the study of how variation in an observed response can be apportioned to different possible sources or factors. In computer experiments, the “observed response” is the output of the computer code and the “factors” are inputs to the code. In other words, sensitivity analysis tries to determine how variable the output is to changes in the inputs. There are many applications of sensitivity analysis and Saltelli, Chan and Scott (2000) discuss these in detail. One application of sensitivity analysis is the following. Suppose we collect data to determine whether a response depends on any of several factors. We wish to identify which factors are responsible for most of the variation in the response and which produce little variation in the response over some specified range of values of these factors. Sensitivity analysis provides methods that can be used to accomplish this.
Thomas J. Santner, Brian J. Williams, William I. Notz

Backmatter

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