Many dynamic models are used for risk assessment and decision support in ecology and crop science. Such models generate time-dependent model predictions, with time either discretised or continuous. Their global sensitivity analysis is usually applied separately on each time output, but Campbell et al. (2006 [1]) advocated global sensitivity analyses on the expansion of the dynamics in a well-chosen functional basis. This paper focuses on the particular case when principal components analysis is combined with analysis of variance. In addition to the indices associated with the principal components, generalised sensitivity indices are proposed to synthesize the influence of each parameter on the whole time series output. Index definitions are given when the uncertainty on the input factors is either discrete or continuous and when the dynamic model is either discrete or functional. A general estimation algorithm is proposed, based on classical methods of global sensitivity analysis.
The method is applied to a dynamic wheat crop model with 13 uncertain parameters. Three methods of global sensitivity analysis are compared: the Sobol'–Saltelli method, the extended FAST method, and the fractional factorial design of resolution 6.
Global sensitivity analysis is frequently applied to models with multivariate or functional output. For example, many dynamic models used for risk assessment and decision support in ecology and crop science generate time-dependent model predictions, with time being either discretised in a finite number of time steps or considered as continuous. In such situations, as mentioned by Campbell et al. [1], it may be unsufficiently informative to perform sensitivity analyses on each output separately or on a few context-specific scalar functions of the output. Indeed, it may be more interesting to apply sensitivity analysis to the multivariate output as a whole. Consequently, there is a need to define criteria and to develop methods specifically adapted to the sensitivity analysis of multivariate or functional outputs.
In particular, consider a model with dynamic output or multi-outputs y(1),…,y(T)y(1),…,y(T). Conducting separate sensitivity analyses on y(1),…,y(T)y(1),…,y(T) gives information on how the sensitivity of y(t) evolves over time. This is interesting, but it leads to much redundancy because of the strong relationship between responses from one time step to the next one. It may also miss important features of the y(t) dynamics because many features cannot be efficiently detected through single-time measurements.
To improve relevance, sensitivity analysis can be applied to pre-defined scalar functions h(y(1),…,y(T))h(y(1),…,y(T)) that have a useful interpretation. However, many functions of y(1),…,y(T)y(1),…,y(T) are potentially interesting to look at. A general and more sophisticated approach consists in modelling the output as a joint function of time and of the input variables and uncertain parameters. Several examples are illustrated in Chapter 7 of Fang et al. [2], based on spatio-temporal, functional or semiparametric modelling tools.
However there is also a need to apply data-driven methods that can identify the most interesting features in the y(t) dynamics and perform sensitivity analyses on these features. Campbell et al. [1] proposed a simple and very useful approach to do so. It consists in (i) performing an orthogonal decomposition of the multivariate output, and (ii) applying sensitivity analysis to the most informative components individually. There is a large collection of available methods for the first step: it can be based either on a data driven method such as principal component analysis, or on the projections of output on a polynomial, spline, or Fourier basis defined by the user. The second step can also be performed by several different methods of sensitivity analysis, such as factorial design, FAST, or Sobol' and its most recent versions developed by Saltelli et al. [3] and [4].
The method proposed by Campbell et al. [1] allows to restrict attention to a few components rather than a whole dynamic. However, there is a need also for a synthetic criterion to summarise the sensitivity over the whole dynamic. This criterion must be adapted to discrete or continuous uncertainty distributions, whereas the examples in [1] are restricted to the first case. In this paper, we first show that there is a full “factorial by component” decomposition of the output variability or inertia, as illustrated in Lurette et al. [5] and Lamboni et al. [6]. Based on this decomposition, we propose a new synthetic sensitivity criterion for discrete factors first. We then extend this criterion to the cases when the input factors and output are continuous, and estimation methods are proposed and compared through simulations on a crop model.
Section 2 presents the general framework (Section 2.1), and then three special cases: (i) the number of model output variables is finite and a complete or fractional factorial design is used to explore the input domain (Section 2.2); (ii) the number of model output variables is finite but the input domain is continuous (Section 2.3); (iii) the model outputs are defined as a continuous function over time (Section 2.4). In Section 3, the methods are illustrated on a crop model with 13 parameters. The main results are discussed in Section 4.
