Computing the moments of probability distributions for branching processes

This document is a comparative study of four methods (Markov chains, compound processes, substructure factorization and generating functions) to compute the moments of probability distributions associated to homogeneous branching processes. Although all these methods have their own interests, the generating functions seem to be the most appropriate tools for this kind of probability model. In order to compare them, each method is first described and then applied to an example of multitype branching processes: the evolution of the number of active buds for a particular GreenLab plant growth model. Finally, the methods are listed according to their effectiveness

Using a hierarchical segmented model to assess the dynamics of leaf appearance in plant populations.

International audienceModeling inter-individual variability in plant populations is a key issue to enhance the predictive capacity of plant growth models at field level. In sugar beet, this variability is well illustrated by the phyllochron (thermal time elapsing between two successive leaf appearances): even if the mean phyllochron remains stable within a given variety, there is a high heterogeneity between individuals. When considering the dynamics of leaf appearance as a function of thermal time in sugar beet, two linear phases can be observed, leading to the definition of a hierarchical segmented model with four random parameters varying from one individual to another: thermal time of initiation, first phyllochron, rupture thermal time and second phyllochron. The SAEM-MCMC algorithm is used to estimate the model parameters.L'amélioration des capacités prédictives des modèles de croissance de plantes passe par la modélisation de la variabilité inter-individus au sein de la population de plantes. Dans le cas de la betterave à sucre, cette variabilité se retrouve dans le phyllochrone (temps thermique nécessaire à l'élaboration d'une feuille): si le phyllochrone moyen reste stable pour une variété donnée, de fortes variations existent d'une plante à l'autre. Deux phases linéaires peuvent être observées dans la dynamique d'apparition des feuilles en fonction du temps thermique, nous amenant à considérer un modèle hiérarchique segmenté à quatre paramètres aléatoires: le temps thermique d'initiation, le premier phyllochrone, le temps thermique de rupture, et le second phyllochrone. Les paramètres du modèle ont été estimés à l'aide de l'algorithme SAEM-MCMC

The use of Sensitivity Analysis for the design of Functional Structural Plant Models

International audienceDeveloped recently, Functional Structural Models of Plant Growth (FSPM) aim at describing plant structural development (organogenesis and geometry), functional growth (biomass accumulation and allocation) and the complex interactions between both. They serve as a framework to integrate complex biological and biophysical processes in interaction with the environment, at different scales. The resulting complexity of such models regarding the dimensionalities of the parameter space and state space often makes them difficult to parameterize. There is usually no systematic model identification from experimental data and such models still remain ill-adapted for applicative purposes. The objective of this study is to explore how global sensitivity analysis can help for the parameterization of FSPM, by quantifying the driving forces during plant growth and the relative importance of the described biophysical processes regarding the outputs of interest. The tests are performed on the GreenLab model. Its particularity is that both structural development and functional growth are described mathematically as a dynamical system (Cournède et al., 2006). Its parameterization relies on parameter estimation from experimental data. Sensitivity analysis may help to optimize the trade-off between experimental cost and accuracy. This is crucial to develop a predictive capacity that scales from genotype to phenotype for FSPM

Assessment of Parameter Uncertainty in Plant Growth Model Identification

International audienceFor the parametric identification of plant growth models, we generally face limited or uneven experimental data, and complex nonlinear dynamics. Both aspects make model parametrization and uncertainty analysis a difficult task. The Generalized Least Squares (GLS) estimator is often used since it can provide estimations rather rapidly with an appropriate goodness-of-fit. However, the confidence intervals are generally calculated based on linear approximations which make the uncertainty evaluation unreliable in the case of strong nonlinearity. A Bayesian approach, the Convolution Particle Filtering (CPF), can thus be applied to estimate the unknown parameters along with the hidden states. In this case, the posterior distribution obtained can be used to evaluate the uncertainty of the estimates. In order to improve its performance especially with stochastic models and in the case of rare or irregular experimental data, a conditional iterative version of the Convolution Particle Filtering (ICPF) is proposed. When applied to the Log Normal Allocation and Senescence model (LNAS) with sugar beet data, the two CPF related approaches showed better performance compared to the GLS method. The ICPF approach provided the most reliable estimations. Meanwhile, two sources of the estimation uncertainty were identified: the variance generated by the stochastic nature of the algorithm (relatively small for the ICPF approach) and the residual variance partly due to the noise models

A Markovian framework to formalize stochastic L-systems and application to models of plant development

This document is an extension of the article written by \textit{Loi and Cournède} (DMTCS, 2008). This article shows the relationship between stochastic L-Systems and a simplified GreenLab growth model with only branching and differentiation. By writing the probability generating function corresponding to each phenomenon and by compounding them, we get the expected values of the numbers of metamers of each type in the whole plant. In this report, we recall the main results of this article. In addition, we show how to derive the generating function in the general case when growth units contain a random number of metamers. We also get a recursive equation to compute the variance of the numbers of metamers of each type in the plant. Finally, we illustrate the results throughout Monte-Carlo simulations in four cases

Generating Functions of Stochastic L-Systems and Application to Models of Plant Development

International audienceIf the interest of stochastic L-systems for plant growth simulation and visualization is broadly acknowledged, their full mathematical potential has not been taken advantage of. In this article, we show how to link stochastic L-systems to multitype branching processes, in order to characterize the probability distributions and moments of the numbers of organs in plant structure. Plant architectural development can be seen as the combination of two subprocesses driving the bud population dynamics, branching and differentiation. By writing the stochastic L-system associated to each subprocess, we get the generating function associated to the whole system by compounding the associated generating functions. The modelling of stochastic branching is classical, but to model differentiation, we introduce a new framework based on multivariate phase-type random vectors

Sensitivity analysis of GreenLab model for maize

International audienceAs one necessary step for a good modeling, this study was aimed at analyzing the sensitivity of GreenLab model for maize. When instantaneous value of biomass generation is considered as the output, the system tends to be linear, the level is above 94% in SRC(Standardized Regression coefficients)study. Conversion efficiency and characteristic surface are proved to be the most sensitive factors. In Sobol's measure, we excluded the two most sensitive factors in the analysis, then the system linearity tends to be weaker and we got the detailed sensitivity indexes for the other uncertain parameters, by which we get the clearer driven force of maize growth in different stages

Quantitative Genetics and Functional-Structural Plant Growth Models: Simulation of Quantitative Trait Loci Detection for Model Parameters and Application to Potential Yield Optimization

Background and Aims: Prediction of phenotypic traits from new genotypes under untested environmental conditions is crucial to build simulations of breeding strategies to improve target traits. Although the plant response to environmental stresses is characterized by both architectural and functional plasticity, recent attempts to integrate biological knowledge into genetics models have mainly concerned specific physiological processes or crop models without architecture, and thus may prove limited when studying genotype x environment interactions. Consequently, this paper presents a simulation study introducing genetics into a functional-structural growth model, which gives access to more fundamental traits for quantitative trait loci (QTL) detection and thus to promising tools for yield optimization. Methods: The GreenLab model was selected as a reasonable choice to link growth model parameters to QTL. Virtual genes and virtual chromosomes were defined to build a simple genetic model that drove the settings of the species-specific parameters of the model. The QTL Cartographer software was used to study QTL detection of simulated plant traits. A genetic algorithm was implemented to define the ideotype for yield maximization based on the model parameters and the associated allelic combination. Key Results and Conclusions: By keeping the environmental factors constant and using a virtual population with a large number of individuals generated by a Mendelian genetic model, results for an ideal case could be simulated. Virtual QTL detection was compared in the case of phenotypic traits - such as cob weight - and when traits were model parameters, and was found to be more accurate in the latter case. The practical interest of this approach is illustrated by calculating the parameters (and the corresponding genotype) associated with yield optimization of a GreenLab maize model. The paper discusses the potentials of GreenLab to represent environment x genotype interactions, in particular through its main state variable, the ratio of biomass supply over demand