تجزیه و تحلیل حساسیت زمانی که خروجی مدل، توابع هستند

When outputs of computational models are time series or functions of other continuous variables like distance, angle, etc., it can be that primary interest is in the general pattern or structure of the curve. In these cases, model sensitivity and uncertainty analysis focuses on the effect of model input choices and uncertainties in the overall shapes of such curves. We explore methods for characterizing a set of functions generated by a series of model runs for the purpose of exploring relationships between these functions and the model inputs.

The outputs of computational models are often time series or functions of other continuous variables like distance, angle, etc. Following Campbell [1], we propose that sensitivity analysis of such outputs be carried out by means of an expansion of the functional output in an appropriate functional coordinate system, i.e., in terms of an appropriate set of basis functions, followed by sensitivity analysis of the coefficients of the expansion using any standard method. The principal new problem, therefore, is choosing an appropriate coordinate system in which to apply the selected sensitivity analysis methods. We consider both predefined basis sets and data-adaptive basis sets, with their associated advantages and disadvantages. We devote only passing mention to some related but important problems, such as increasing the interpretability of the results by appropriate preprocessing of the functional outputs (in particular, alignment or registration of curves), and by enforcing some degree of smoothness when data-adaptive bases are used. We will use a simple made-up example for explaining ideas. Fig. 1 shows a sample of curves generated by varying the four parameters, a, b, c and d in the “model” equation(1) View the MathML sourcef(θ)=10+aexp(-(θ-b)2K1a2+c2)+(b+d)exp(K2aθ). Turn MathJax on Full-size image (27 K) Fig. 1. Functional output from 81 runs of the example model. Figure options We interpret these functions as model output from a problem where the independent variable θ is a polar angle ranging from −90° to 90°. The model was run 81 times, using a complete 34 factorial design for the four input parameters. In analyzing this “model output”, we are typically less interested in what affects the values at, say, 45° than in questions such as: What shifts the curves up and down or moves them left or right? What makes the central peak wider or narrower? What makes the right-hand tail higher or lower? We could, of course, pick some appropriate functionals for answering these questions. The last, for example, we might address by examining the sensitivity of the values at 90° to the four input parameters. In order to address questions such as peak width, we could devise some surrogate measurement that could be computed on each curve and then study its sensitivity to the input parameters. However, such choices are highly problem specific.