In this paper, we consider the non-parametric estimation of conditional moments, which is useful for applications in global sensitivity analysis (GSA) and in the more general emulation framework. The estimation is based on the state-dependent parameter (SDP) estimation approach and allows for the estimation of conditional moments of order larger than unity. This allows one to identify a wider spectrum of parameter sensitivities with respect to the variance-based main effects, like shifts in the variance, skewness or kurtosis of the model output, so adding valuable information for the analyst, at a small computational cost.
In global sensitivity analysis (GSA), the mapping Y=f(X)Y=f(X) between an output Y of a computational model and a set of uncertain input factors X=(X1,…,Xk)X=(X1,…,Xk) is analyzed in order to quantify the relative contribution of each input factor to the uncertainty of Y . Variance-based analysis is the most popular method in GSA. Variance-based sensitivity indices of single factors or of groups of them are defined as [1] and [24]
equation(1)
View the MathML sourceSI=Var(E(Y|XI))Var(Y)
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where XIXI denotes a group of factors indexed by I=(i1,…,ig)1⩽g⩽kI=(i1,…,ig)1⩽g⩽k, and they tell the portion of variance of YY that is explained by XIXI.
The two most popular variance-based sensitivity measures are the main effect
equation(2)
View the MathML sourceSi=Var(E(Y|Xi))Var(Y)
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and the total effect
equation(3)
View the MathML sourceSTi=E(Var(Y|X-i))Var(Y)
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where X-iX-i indicates all input factors except XiXi.
The main effect measures the singular contribution of the input factor XiXi to the uncertainty (variance) of the output Y , while the total effect measures the overall contribution of XiXi on Y , including all interaction terms of XiXi with all other input factors.
There are clear links between variance-based sensitivity analysis and model emulation. First, a statistical approximation (the emulator) View the MathML sourcef^(X) can be used to compute sensitivity indices in place of the original computational mapping f(X)f(X). Second, the variance-based sensitivity measures can be interpreted as the non-parametric R2R2 or correlation ratio, used in statistics to measure the explanatory power of covariates in regression [2] and [3]. In fact, it is well known that the inner argument E(Y|XI)E(Y|XI) of (1) is the function of the subset of input factors that approximates f(X)f(X), by minimizing a quadratic loss (i.e. maximizing the R2R2). Therefore, estimating E(Y|XI)E(Y|XI) provides a route for both a model approximation and sensitivity estimation. Smoothing methods that provide more or less accurate and efficient estimations of E(Y|XI)E(Y|XI) are becoming a popular approach to sensitivity analysis [4], [5], [6], [7] and [8]. State-dependent parameter (SDP) modelling is one class of non-parametric smoothing approach first suggested by Young [9] and [10]. The estimation is performed with the help of the ‘classical’ recursive (numerically non-intensive) Kalman filter (KF) and associated fixed interval smoothing (FIS) algorithms: it has been applied for sensitivity analysis by Ratto et al. in [11] and [12].
Variance-based techniques have a quite general applicability, since they apply to a very wide range of non-linear mappings f(·)f(·) and rely on only a few assumptions, namely Y has to be square integrable and the variance is an adequate measure of the uncertainty of Y . Nonetheless, these techniques are sometimes criticized, since all kinds of sensitivity patterns that cannot be attributed to shifts in the mean (the first moment—see factor X3X3 in Fig. 1), are not accounted for by E(Y|Xi)E(Y|Xi) and the related variance-based sensitivity index. Such sensitivity patterns can be characterized by a shift in higher order moments: the simplest example of which is the heteroscedastic process, where the variance of Y changes along the conditioning term XiXi. This lead to the development of a number of sensitivity techniques, such as entropy-based sensitivity measures [13] and [14] or moment independent sensitivity measures [15] and [16], that provide ‘main effects’ that are able to account for such phenomena.
In this paper, we have discussed an extended use of smoothing procedures for the estimation of conditional moments in sensitivity analysis. This analysis can be performed at no additional cost with respect to standard smoothing analysis for the estimate of E(Y|Xi)E(Y|Xi) and provides a very useful completion of the standard variance-based sensitivity analysis. In particular, it generates:
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estimates of significant patterns in the conditional variance that are an indication of an interaction structure and, when such estimated patterns are fed back to the KF and FIS algorithms, provide a more precise estimation of E(Y|Xi)E(Y|Xi) and of SiSi;
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additional analysis of conditional moments that provide further insight into the interaction and non-linearity of the model under analysis, such as the asymmetry of the distribution of Y and the presence of fat/thin tails;
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an ensemble of information that provides the key to interpreting sensitivity patterns obtained using techniques like the moment-independent measure δiδi of Borgonovo [15];
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proxies of the δiδi using the truncated Edgeworth series, based on the smoothed estimates of conditional moments.
