A statistical model for analyzing jointly growth phases, the influence of environmental factors and inter-individual heterogeneity : Applications to forest trees

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Multivariate dynamic model for ordinal outcomes

Individual or stand-level biomass is not easy to measure. The current methods employed, based on cutting down a representative sample of plantations, make it possible to assess the biomasses for various compartments (bark, dead branches, leaves, . . .). However, this felling makes individual longitudinal follow-up impossible. In this context, we propose a method to evaluate individual biomasses by compartments when these biomasses are taken as ordinals. Biomass is measured visually and observations are therefore not destructive. The technique is based on a probit model redefined in terms of latent variables. A generalization of the univariate case to the multivariate case is then natural and takes into account the dependency between compartment biomasses. These models are then extended to the longitudinal case by developing a Dynamic Multivariate Ordinal Probit Model. The performance of the MCMC algorithm used for the estimation is illustrated by means of simulations built from known biomass models. The quality of the estimates and the impact of certain parameters, are then discussed

Markov and semi-Markov switching linear mixed models used to identify forest tree growth components.

International audienceTree growth is assumed to be mainly the result of three components: (i) an endogenous component assumed to be structured as a succession of roughly stationary phases separated by marked change points that are asynchronous between individuals, (ii) a time-varying environmental component assumed to take the form of synchronous fluctuations between individuals, (iii) an individual component corresponding mainly to the local environment of each tree. In order to identify and characterize these three components, we propose to use semi-Markov switching linear mixed models, i.e. models that combine linear mixed models in a semi-markovian manner. The underlying semi-Markov chain represents the succession of growth phases and their lengths (endogenous component) while the linear mixed models attached to each state of the underlying semi-Markov chain represent -in the corresponding growth phase- both the influence of time-varying climatic covariates (environmental component) as fixed effects, and inter-individual heterogeneity (individual component) as random effects. In this paper we address the estimation of Markov and semi-Markov switching linear mixed models in a general framework. We propose a MCEM-like algorithm whose iterations decompose into three steps: (i) sampling of state sequences given random effects, (ii) prediction of random effects given state sequences, (iii) maximization. The proposed statistical modeling approach is illustrated by the analysis of successive annual shoots along Corsican pine trunks influenced by climatic covariates

Markov and semi-Markov switching linear mixed models for identifying forest tree growth components.

Observed tree growth is the result of three components: (i) an endogenous component which is assumed to be structured as a succession of roughly stationary phases separated by marked change points asynchronous between individuals, (ii) a time-varying environmental component which is assumed to take the form of local fluctuations synchronous between individuals, (iii) an individual component which corresponds to the local environmental of each tree. In order to identify and to characterize these three omponents, we propose to use semi-Markov switching linear mixed models, i.e. models that combine linear mixed models in a semi-markovian manner. The underlying semi-Markov chain represents the succession of growth phases (endogenous component) while the linear mixed models attached to each state of the underlying semi-Markov chain represent in the corresponding growth phase both the influence of time-varying environmental covariates (environmental component) as fixed effects and inter-individual heterogeneity (individual component) as random effects. In this paper, we address the estimation of Markov and semi-Markov switching linear mixed models in a general framework. We propose a MCEM-like algorithm whose iterations decompose into three steps (sampling of state sequences given random effects, prediction of random effects given the state sequence and maximization). The proposed statistical modeling approach is illustrated by the analysis of successive annual shoots along Corsican pine trunks influenced by climatic covariates

Deciphering the developmental plasticity of walnut saplings in relation to climatic factors and light environment

International audienceDevelopmental plasticity, the acclimation of plants to their local environment, is known to be crucial for the fitness of perennial organisms such as trees. However, deciphering the many possible developmental and environmental influences involved in such plasticity in natural conditions requires dedicated statistical models integrating developmental phases, environmental factors and inter-individual heterogeneity. These models should be able to analyze retrospective data (number of leaves or length of annual shoots along the main stem in our case). In this study Markov switching linear mixed models were applied to the analysis of the developmental plasticity of walnut saplings during the establishment phase in a mixed Mediterranean forest. In the Markov switching linear mixed models estimated from walnut data sets, the underlying Markov chain represents both the succession and lengths of growth phases, while the linear mixed models represent both the influence of climatic factors and inter-individual heterogeneity within each growth phase. On the basis of these integrative statistical models, it is shown that walnut saplings have an opportunistic mode of development that is primarily driven by the changing light environment. In particular, light availability explains the ability of a tree to reach a phase of strong growth where the first branches can appear. It is also shown that growth fluctuation amplitudes in response to climatic factors increased while inter-individual heterogeneity decreased along tree development

Markov and semi-Markov switching linear mixed models for identifying forest tree growth components.

Observed tree growth is the result of three components: (i) an endogenous component which is assumed to be structured as a succession of roughly stationary phases separated by marked change points asynchronous between individuals, (ii) a time-varying environmental component which is assumed to take the form of local fluctuations synchronous between individuals, (iii) an individual component which corresponds to the local environmental of each tree. In order to identify and to characterize these three omponents, we propose to use semi-Markov switching linear mixed models, i.e. models that combine linear mixed models in a semi-markovian manner. The underlying semi-Markov chain represents the succession of growth phases (endogenous component) while the linear mixed models attached to each state of the underlying semi-Markov chain represent in the corresponding growth phase both the influence of time-varying environmental covariates (environmental component) as fixed effects and inter-individual heterogeneity (individual component) as random effects. In this paper, we address the estimation of Markov and semi-Markov switching linear mixed models in a general framework. We propose a MCEM-like algorithm whose iterations decompose into three steps (sampling of state sequences given random effects, prediction of random effects given the state sequence and maximization). The proposed statistical modeling approach is illustrated by the analysis of successive annual shoots along Corsican pine trunks influenced by climatic covariates

Estimating Markov and semi-Markov switching linear mixed models with individual-wise random effects

International audienceWe address the estimation of Markov (and semi-Markov) switching linear mixed models i.e. models that combine linear mixed models with individual-wise random effects in a (semi-)Markovian manner. A MCEM-like algorithm whose iterations decompose into three steps (sampling of state sequences given random effects, prediction of random effects given the state sequence and maximization) is proposed. This statistical modeling approach is illustrated by the analysis of successive annual shoots along Corsican pine trunks

Combinaisons markoviennes et semi-markoviennes de modèles de régression. Application à la croissance d'arbres forestiers.

This work focuses on Markov and semi-Markov switching regression models, i.e. finite mixtures of regression models with (semi-)Markovian dependencies. These statistical models enable to analyse data structured as a succession of stationary phases that are asynchronous between individuals, influenced by time-varying covariates and which present inter-individual heterogeneity. The proposed inference algorithm for (semi-)Markov switching generalized linear models is a gradient EM algorithm. For (semi-)Markov switching linear mixed models, we propose MCEM-like algorithms whose E-step decomposes into two conditional restoration steps: one for the random effects given the state sequences (and the observed data) and one for the state sequences given the random effects (and the observed data). Various conditional restoration steps are presented. We study two types of random effects: individual-wise random effects and environmental random effects. The relevance of these models isillustrated by the analysis of forest tree growth influenced by climatic covariates. These models allow us to identify and characterize the three main growth components (ontogenetic component, environmental component and individual component). We show that the weight of each component varies according to speciesand silvicultural interventions.Ce travail est consacré à l'étude des combinaisons markoviennes et semi-markoviennes de modèles de régression, i.e. des mélanges finis de modèles de régression avec dépendances (semi-)markoviennes. Cette famille de modèles statistiques permet l'analyse de données structurées en phases successives synchrones entre individus, influencées par des covariables pouvant varier dans le temps et présentant une hétérogénéité inter-individuelle. L'algorithme d'inférence proposé pour les combinaisons (semi-)markoviennes de modèles linéaires généralisés est un algorithme du gradient EM. Pour les combinaisons (semi-)markoviennes de modèles linéaires mixtes, nous proposons des algorithmes de type MCEM où l'étape E se décompose en deux étapes de restauration conditionnelle: une pour les séquences d'états sachant les effets aléatoires (et les données observées) et une pour les effets aléatoires sachant les séquences d'états (et les données observées). Différentes méthodes de restauration conditionnelle sont présentées. Nous étudions deux types d'effets aléatoires: des effets aléatoires individuels et des effets aléatoires temporels. L'intérêt de cette famille de modèles est illustré par l'analyse de la croissance d'arbres forestiers en fonctions de facteurs climatiques. Ces modèles nous permettent d'identifier et de caractériser les trois principales composantes de la croissance (la composante ontogénique, la composante environnementale et la composante individuelle). Nous montrons que le poids de chaque composante varie en fonction de l'espèce et des interventions sylvicoles

Modèle probit multivarié ordinal dynamique. Application à l'estimation de la biomasse d'un peuplement forestier d'eucalyptus. [Cd-Rom]

International audienceLa biomasse d'un individu ou d'un peuplement est difficilement mesurable. Les méthodes de mesures actuelles, basées sur l'abattage d'un échantillon représentatif du peuple-ment, permettent d'évaluer les biomasses pour différents compartiments (feuille, tronc,...). Cependant, cet abattage rend impossible un suivi longitudinal des individus: les arbres abattus à l'âge t + 1 sont ils bien représentatifs des arbres abattus à l'âge t? Dans ce contexte, nous proposons une méthode, permettant d'estimer les biomasses par compartiment d'un individu lorsque celles-ci sont classées de façon discrètes ordinales. Fondée sur les Modèles Probit redéfinis en terme de variables latentes gaussiennes, une généralisation du cas univarié au cas multivarié à un temps donné est naturelle. Par suite, ces modèles sont étendus au cas longitudinal en développant un Modèle Probit Multivarié Ordinal Dynamique. Pour finir, les performances des algorithmes MCMC d'estimation sont illustrées sur des simulations construites à partir de modèles de biomasse. On discutera de la qualité des estimations et de l'impact de certains paramètres sur ces dernières