یک روش محاسباتی کارآمد برای تجزیه و تحلیل حساسیت جهانی و کاربرد آن در مدلسازی رشد درختان

Global sensitivity analysis has a key role to play in the design and parameterisation of functional–structural plant growth models which combine the description of plant structural development (organogenesis and geometry) and functional growth (biomass accumulation and allocation). We are particularly interested in this study in Sobol's method which decomposes the variance of the output of interest into terms due to individual parameters but also to interactions between parameters. Such information is crucial for systems with potentially high levels of non-linearity and interactions between processes, like plant growth. However, the computation of Sobol's indices relies on Monte Carlo sampling and re-sampling, whose costs can be very high, especially when model evaluation is also expensive, as for tree models. In this paper, we thus propose a new method to compute Sobol's indices inspired by Homma–Saltelli, which improves slightly their use of model evaluations, and then derive for this generic type of computational methods an estimator of the error estimation of sensitivity indices with respect to the sampling size. It allows the detailed control of the balance between accuracy and computing time. Numerical tests on a simple non-linear model are convincing and the method is finally applied to a functional–structural model of tree growth, GreenLab, whose particularity is the strong level of interaction between plant functioning and organogenesis.

Sensitivity analysis (SA) is a fundamental tool in the building, use and understanding of mathematical models [1]. Sampling-based approaches to uncertainty and sensitivity analysis are both effective and widely used [2]. For this purpose, Sobol's method is a key one [3]. Since it is based on variance decomposition, the different types of sensitivity indices that it estimates can fulfill different objectives of sensitivity analysis: factor prioritisation, factor fixing, variance cutting or factor mapping [4]. It is a very informative method but potentially computationally expensive [2]. Besides the first-order effects, Sobol's method also aims at determining the levels of interaction between parameters [5]. In [6], the authors also devised a strategy for sensitivity analysis that could work for correlated input factors, based on the first-order and total-order index from variance decomposition. Such type of global sensitivity analysis method has a key role to play in functional–structural plant growth modelling. In this recent field of research in plant biology [7] models are not yet stable. They aim at combining the description of both plant structural development and eco-physiological functioning at a very detailed scale, typically that of organs (leaves, internodes, etc.). The complexity of the underlying biological processes, especially the interaction between function and structure [8] usually makes parameterisation a complex step in modelling, and the analysis of model sensitivity to parameters and of their levels of interactions provides useful information in this process [9]. Computational methods to evaluate Sobol indices sensitivity rely on Monte Carlo sampling and re-sampling [3] and [10]. For k-dimensional factor of model uncertainty, the k first-order effects and the ‘k’ total-order effects are rather expensive to estimate, needing a number of model evaluations strictly depending on k [11]. Especially, for individual-based tree growth models, at organ level, the cost of model evaluation can be very heavy [7]. Therefore, it is crucial to not only devise efficient computing techniques, in order to make best use of model evaluations [12], but also to have a good control of the estimation accuracy with respect to the number of samples. The objective of this paper is to study these two aspects. First, we propose a computing method inspired by [10], which improves slightly their use of model evaluations, and then derive an estimator of the error of sensitivity indices evaluation with respect to the sampling size for this generic type of computational methods. Numerical tests are then shown to illustrate the results. Finally, the method is applied to a functional–structural model of tree growth, whose particularity is the strong level of interaction between plant functioning and organogenesis [13].