Interactions and computer experiments

Abstract

Identifying interactions and understanding the underlying generating mechanism is essential for interpreting the response of black-box models. We offer a systematic analysis of interaction types and corresponding sources, merging results of the broad statistical literature with findings developed within the computer experiment literature. Piecewise-definiteness emerges a self-standing interaction mechanism, alternative to the presence of interaction terms. We find that the scale of the analysis is essential for interpretation, and that no single method is capable of providing the correct identification of the underlying interaction generating mechanisms; conversely a combined approach involving indicators at difference scales is required. We propose a graphical tool called Mikado plot that exploits the link between interaction indicators at the finite scale and global scales to ease the regional visualization of two-factor interactions. The findings are illustrated via numerical experiments with three well-known computer models of different dimensionality and structure.

Elementary effects for models with dimensional inputs of arbitrary type and range: Scaling and trajectory generation

“…Again let Y = X 1 + X 2 , with X 1 ∈ [0, 20] and X 2 ∈ [ 9 , 11 ] uniformly. Take p 1 = p 2 = 4 and | δ 1 | = | δ 2 | = 2/3 and consider the 4 trajectories as depicted in Fig 3 .…”

“…mm), indicating the parameter is in fact the most important one. Secondly, let Y = X 1 + X 2 , with X 1 2 [0, 20] and X 2 2 [9,11], so that the inputs have equal mean but different standard deviation. Clearly X 1 contributes most significantly to the variability in the output.…”

“…Again let Y = X 1 + X 2 , with X 1 2 [0, 20] and X 2 2 [9,11] uniformly. Take p 1 = p 2 = 4 and |δ 1 | = |δ 2 | = 2/3 and consider the 4 trajectories as depicted in Fig 3 . As is shown in Table 3, scaling by the distributional mean of the input and the distributional standard deviation of the input or parameter range results in different rankings.…”

“…Local interaction effects are typically defined in a similar way to first-order local sensitivity, by considering mixed higher order partial derivatives. In general, interaction effects can be defined at different scales, and no single method or scale is capable of fully characterizing interactions in numerical simulators [ 9 ].…”