Multivariate sensitivity analysis to measure global contribution of input factors in dynamic models
Introduction
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 . Conducting separate sensitivity analyses on 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 that have a useful interpretation. However, many functions of 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], [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.
Section snippets
Framework
To perform sensitivity analysis, some uncertain parameters and input variables are selected for study, while the others are fixed at given nominal values. The selected parameters and input variables yield the input factors to of the sensitivity analysis. Let denote a scenario, that is, a combination of the levels of the input factors , for in . The model of interest in this paper iswhere is the scalar output at time and, for simplicity,
Description of the model AZODYN
The model AZODYN [28] simulates the wheat crop development, in order to guide farmers in their crop fertilisation strategies. The main goal of the model AZODYN is to develop interesting strategies for fertilisation that meet performance objectives such as grain protein content or preservation of the environment.
AZODYN simulates crop and soil components including yield, biomass, protein content of grains, residual mineral nitrogen in the soil at harvest and the nitrogen nutrition index (INN). In
Discussion
The combination of principal components analysis and global sensitivity analysis leads to indices that provide rich information on dynamic model behaviour. In this paper, we show that the whole output variability can thus be decomposed into meaningful sensitivity indices. In the AZODYN-INN example, the rankings of the factors were quite different between the principal components, suggesting that different structural properties of the model were associated with the different components.
The
Acknowledgements
This work benefited from fruitful discussions with colleagues from the MEXICO and GDR MASCOT-NUM groups, which are both motivated by sensitivity analysis and numerical exploration of model output. We thank the three referees for their useful comments.
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