Maximin distance optimal designs for computer experiments with time-varying inputs and outputs
Section snippets
Introduction and notation
The statistical treatment of computer experiments, in which a computer model of a physical system is the object of experimental focus, has received substantial attention in the statistical literature over the last two decades. The computer model is often an implementation of a complicated deterministic function for which evaluation requires the specification of quantities that complete the definition of a problem to be solved. Because the function is complex, it is often difficult or impossible
An upper bound for
Suppose for simplicity that the problem is scaled so that T=1, and that each input function is constrained so that , the latter being required so that distance is constrained. Further, require that for each so that distances are comparable across all . Result Restrict n to be an even integer. Then an upper bound on (as defined in Eq. (1)) is . Proof Define the total distance between the elements of a design at any time s to be At any particular time
Optimal designs
First, consider the case in which there is only one time at which model output is to be predicted, , and restrict attention to the case of . Let be the design matrix for any balanced, orthogonal, main-effects-saturated, 2-level design, with coding levels 0 and 1 in each column, e.g. for n=4: (A historic reference for experimental designs of this form is Plackett and Burman, 1946.) Let be the set of values that divides the integral of ws
Practical limitations and conclusion
The construction outlined in Section 3 demonstrates that there are designs for which the maximin distance bound of Section 2 is reached, proving that no tighter general bound exists. However, the optimal designs constructed here would often be of limited practical value in many cases because they require that each x(t):
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take only values that are the maximum or minimum allowed at each time (except over a range of zero integration measure), and
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oscillate between these two extreme values
Acknowledgments
The author gratefully acknowledges the contributions of the Associate Editor and Referees; their comments have led to significant improvements in this paper.
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