Non-parametric estimation of conditional moments for sensitivity analysis

Abstract

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.

Introduction

In global sensitivity analysis (GSA), the mapping $Y=f(\mathbf{X})$ between an output Y of a computational model and a set of uncertain input factors $\mathbf{X}=(X_1,\dots ,X_k)$ 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], [24]

$$S_{\mathbf{I}}=\frac{\mathrm{Var}(E(Y|{\mathbf{X}}_{\mathbf{I}}))}{\mathrm{Var}(Y)}$$

where ${\mathbf{X}}_{\mathbf{I}}$ denotes a group of factors indexed by $\mathbf{I}=(i_1,\dots ,i_g{)}_{1⩽g⩽k}$, and they tell the portion of variance of $Y$ that is explained by ${\mathbf{X}}_{\mathbf{I}}$.

The two most popular variance-based sensitivity measures are the main effect

$$S_i=\frac{\mathrm{Var}(E(Y|X_i))}{\mathrm{Var}(Y)}$$

and the total effect

$$S_{\mathit{Ti}}=\frac{E(\mathrm{Var}(Y|{\mathbf{X}}_{-i}))}{\mathrm{Var}(Y)}$$

where ${\mathbf{X}}_{-i}$ indicates all input factors except $X_i$.

The main effect measures the singular contribution of the input factor $X_i$ to the uncertainty (variance) of the output Y, while the total effect measures the overall contribution of $X_i$ on Y, including all interaction terms of $X_i$ with all other input factors.

There are clear links between variance-based sensitivity analysis and model emulation. First, a statistical approximation (the emulator) $\widehat{f}(\mathbf{X})$ can be used to compute sensitivity indices in place of the original computational mapping $f(\mathbf{X})$. Second, the variance-based sensitivity measures can be interpreted as the non-parametric $R^2$ or correlation ratio, used in statistics to measure the explanatory power of covariates in regression [2], [3]. In fact, it is well known that the inner argument $E(Y|{\mathbf{X}}_{\mathbf{I}})$ of (1) is the function of the subset of input factors that approximates $f(\mathbf{X})$, by minimizing a quadratic loss (i.e. maximizing the $R^2$). Therefore, estimating $E(Y|{\mathbf{X}}_{\mathbf{I}})$ provides a route for both a model approximation and sensitivity estimation. Smoothing methods that provide more or less accurate and efficient estimations of $E(Y|{\mathbf{X}}_{\mathbf{I}})$ are becoming a popular approach to sensitivity analysis [4], [5], [6], [7], [8]. State-dependent parameter (SDP) modelling is one class of non-parametric smoothing approach first suggested by Young [9], [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], [12].

Variance-based techniques have a quite general applicability, since they apply to a very wide range of non-linear mappings $f(·)$ 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 $X_3$ in Fig. 1), are not accounted for by $E(Y|X_i)$ 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 $X_i$. This lead to the development of a number of sensitivity techniques, such as entropy-based sensitivity measures [13], [14] or moment independent sensitivity measures [15], [16], that provide ‘main effects’ that are able to account for such phenomena.

In this paper, we show how non-parametric techniques can be applied to estimate conditional moments of order larger than one, allowing us to add valuable information to the standard variance-based analysis and, at the same time, avoid the computational load characterizing the latter class of sensitivity measures. In fact, the analysis does not require any additional model evaluation with respect to any standard smoothing method that may be applied to estimate the $E(Y|X_i)$ terms.