Maximin distance optimal designs for computer experiments with time-varying inputs and outputs

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Abstract

Computer models of dynamic systems produce outputs that are functions of time; models that solve systems of differential equations often have this character. Time-indexed inputs, such as the functions that describe time-varying boundary conditions, are also common with such models. Morris (2012) described a generalization of the Gaussian process often used to produce “meta-models” when inputs are finite-dimensional vectors, that can be used in the functional input setting, and showed how the maximin distance design optimality criterion (Johnson et al., 1990) can also be extended to this case. This paper describes an upper bound on the maximin distance criterion for functional inputs. A class of designs that are optimal under certain conditions is also presented; while these designs are of limited practical value, they show that the derived bound cannot be improved in the general case.

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 ϕ

Suppose for simplicity that the problem is scaled so that T=1, and that each input function is constrained so that 0x1, the latter being required so that distance is constrained. Further, require that for each sS0sws(su)du=1so that distances di,js are comparable across all sS.

Result

Restrict n to be an even integer. Then an upper bound on ϕ (as defined in Eq. (1)) is 12n/(n1).

Proof

Define the total distance between the elements of a design at any time s to be ψs=i<jdi,js.

At any particular time u[0,s

Optimal designs

First, consider the case in which there is only one time at which model output is to be predicted, S={s}, and restrict attention to the case of n=0[mod4]. Let Z be the n×(n1) 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: Z=001010100111.(A historic reference for experimental designs of this form is Plackett and Burman, 1946.) Let 0<t1<t2<<tn2<s 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):

  • take only values that are the maximum or minimum allowed at each time (except over a range of zero integration measure), and

  • 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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