Section: New Results

Model Evaluation and Parameter Estimation

Maximum Likelihood Estimation

In [18] , a first approach for parameter estimation was introduced based on the assumption of an underlying deterministic model of biomass production and uncorrelated errors in the mass measurements of different types of organs in the plant structure. A novel idea is developed on the modeling plant growh in the framework of non-homogeneous hidden Markov models, for a certain class of plants with known organogenesis (structural development). Unknown parameters of the models are estimated via a stochastic variant of a generalised EM (Expectation-Maximization) algorithm where both steps (E-step, M-step) are non-explicit. For this reason, the E-step is approximated via a sequential Monte-Carlo procedure (sequential importance sampling with resampling) and the M-step is separated into two steps (Conditional-Maximization), where before applying a numerical maximization procedure (quasi-Newton type), a large subset of unknown parameters is updated explicitly conditioned on the other subset. The model is tested with real data and the results are satisfying. Further work is in progress, including MCMC techniques for parameter estimation (with the collaboration of Dr. Sonia Malefaki from the University of Patras, Greece) and Bayesian type estimation, see [33] .

Convolution Particle Filter for parameter estimation

Although Kalman filter is applied to various fields and dominated for decades, it is limited by its assumptions of linearity.

Particle filter, which combines Bayesian inference with Monte Carlo sequential sampling approach, is a method using different combinations of random variables sampled directly from the parameter space (or the state space) to estimate parameters and states of a complex system. These combinations, generally called particles, propagate by introducing new observations and provide updated posterior distributions by taking into account their weights. Meanwhile, a resampling procedure is used to prevent the degeneracy problem.

Since classical filtering methods are generally not able to estimate the dynamical state vector along with the unknown parameters, the convolution particle filter is implemented based on convolution kernel approximation to meet the need while modelling with Markovian dynamical system.

Several tests are carried out to examine the performance of the Convolution Particle Filtering method [42] , [36] , and efforts are made to find the optimal perturbation parameters. The applications of the method rely on the Lotka-Volterra model and the sugar beet model.

Since the quality of the estimations is limited by the number of the observations, the Conditional Iterative Bayesian Filtering method is applied. The principal is simply to use the posteriori distributions as the a priori information to re-perform over and over again the estimation algorithm and each time we introduce the same sequence of observations. This approach helps us to improve significantly the final estimation of the hidden states and the unknown parameters while testing with the virtual data. The bootstrapping method is implemented in order to compare with the results from different methods.

In the case of applications based on dynamical stochastic systems, two types of noise are introduced, one is involved in the modelization technique and the other is attached to the observation procedure. An alternation of deterministic parameter estimation and stochastic parameter estimation is proposed (in progress) which allows us to estimate these two kinds of parameters at the same time.

Modelling the inter-individual variability of organogenesis in sugar beet populations

Modelling the inter-individual variability in plant populations is a very important issue to enhance the predictive capacity of plant growth models at the field scale. In the case of sugar beet, this variability is well illustrated by the phyllochron (the thermal time elapsing between the appearance of two successive leaves): if the mean phyllochron remains very stable across seasons, there is a high heterogeneity among individuals. Likewise, seedling emergence may strongly vary within a population, potentially inducing important variations in individual plant productions.

A hierarchical segmented model was used to describe and study the variability of the dynamics of leaf appearance in sugar beet crops. The use of this nonlinear mixed model allows for a better handling of the heterogeneity in the plant population, and gives estimates of this variability: each model's parameter is considered as a random variable, varying from one plant to another around a mean population value, with a given variance.

These mean population values and inter-individual variability can be used as input of functional-structural models, the main issue being then to compute the propagation of these sources of probabilistic uncertainty in the dynamic system of Greenlab.