One of the major settings of global sensitivity analysis is that of fixing non-influential factors, in order to reduce the dimensionality of a model. However, this is often done without knowing the magnitude of the approximation error being produced. This paper presents a new theorem for the estimation of the average approximation error generated when fixing a group of non-influential factors. A simple function where analytical solutions are available is used to illustrate the theorem. The numerical estimation of small sensitivity indices is discussed.
This work is related to global sensitivity analysis based on the use of ANOVA decomposition and global sensitivity indices (see [1], [6], [7] and [8] for theory, and [3] and [4] for applications). Definitions of the sensitivity indices can be found in Section 2.
Different settings for sensitivity analysis are available, depending on the modeler's needs. One of these is the factors fixing setting. It is used for identifying non-influential factors in the model (those factors that can be fixed at any value in their domains without significantly reducing the output variance). A limit with factor fixing is that of fixing unessential factors without knowing the magnitude of the approximation error that is being produced. In Section 2, we prove one new theorem which quantifies this approximation error of the model output when one factor or a group of factors is fixed. So, once we know from total indices that a factor is unessential, we will also have an estimate of the error that is generated by fixing it.
In this paper we study a model function f(x1,…,xn)f(x1,…,xn), where the factors x1,…,xnx1,…,xn are non-random independent scaled variables: 0⩽x1⩽1,…,0⩽xn⩽10⩽x1⩽1,…,0⩽xn⩽1. Thus the point x=(x1,…,xn)x=(x1,…,xn) is defined in the n -dimensional unit hypercube with Lebesgue measure. Clearly the factors x1,…,xnx1,…,xn can be regarded as independent random variables uniformly distributed in the unit interval [0,1]. In this case the quantities that are called variances are real variances of certain random variables.
The sensitivity analysis based on ANOVA decomposition and global sensitivity indices can be easily (mutatis mutandis) generalized to independent random factors x1,…,xnx1,…,xn with arbitrary distribution functions F1(x1),…,Fn(xn)F1(x1),…,Fn(xn) (e.g., [8]). However, the requirement of independence is important.
Section 2 contains a new theorem, Section 3—an illustration of the theorem, and in Section 4 numerical estimation of small sensitivity indices is discussed.
This work shows how to estimate the approximation error committed when fixing non-important factors.
In our example the sensitivity indices were estimated both analytically and numerically; in general, they can be computed numerically.
The proposed theorem can be easily applied to global sensitivity methods that provide estimates of total indices; we have shown the applicability of the procedure also in cases where factors are treated by groups.
For numerical computation of small sensitivity indices a modified Monte Carlo algorithm was studied that reduces the loss of accuracy.
