About the use of rank transformation in sensitivity analysis of model output

Abstract

Rank transformations are frequently employed in numerical experiments involving a computational model, especially in the context of sensitivity and uncertainty analyses. Response surface replacement and parameter screening are tasks which may benefit from a rank transformation. Ranks can cope with nonlinear (albeit monotonic) input-output distributions, allowing the use of linear regression techniques. Rank transformed statistics are more robust, and provide a useful solution in the presence of long tailed input and output distributions. As is known to practitioners, care must be employed when interpreting the results of such analyses, as any conclusion drawn using ranks does not translate easily to the original model. In the present note an heuristic approach is taken, to explore, by way of practical examples, the effect of a rank transformation on the outcome of a sensitivity analysis. An attempt is made to identify trends, and to correlate these effects to a model taxonomy. Employing sensitivity indices, whereby the total variance of the model output is decomposed into a sum of terms of increasing dimensionality, we show that the main effect of the rank transformation is to increase the relative weight of the first order terms (the ‘main effects’), at the expense of the ‘interactions’ and ‘higher order interactions’. As a result the influence of those parameters which influence the output mostly by way of interactions may be overlooked in an analysis based on the ranks. This difficulty increases with the dimensionality of the problem, and may lead to the failure of a rank based sensitivity analysis. We suggest that the models can be ranked, with respect to the complexity of their input-output relationship, by mean of an ‘Association’ index I y. I y may complement the usual model coefficient of determination R y 2 as a measure of model complexity for the purpose of uncertainty and sensitivity analysis.