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Plant architecture is commonly defined by the adjacency of organs within the structure and their properties. Few studies consider the effect of endogenous temporal factors, namely phenological factors, on the establishment of plant architecture. This study hypothesized that, in addition to the effect of environmental factors, the observed plant architecture results from both endogenous structural and temporal components, and their interplays. Mango tree, which is characterized by strong phenological asynchronisms within and between trees and by repeated vegetative and reproductive flushes during a growing cycle, was chosen as a plant model. During two consecutive growing cycles, this study described vegetative and reproductive development of 20 trees submitted to the same environmental conditions. Four mango cultivars were considered to assess possible cultivar-specific patterns. Integrative vegetative and reproductive development models incorporating generalized linear models as components were built. These models described the occurrence, intensity, and timing of vegetative and reproductive development at the growth unit scale. This study showed significant interplays between structural and temporal components of plant architectural development at two temporal scales. Within a growing cycle, earliness of bud burst was highly and positively related to earliness of vegetative development and flowering. Between growing cycles, flowering growth units delayed vegetative development compared to growth units that did not flower. These interplays explained how vegetative and reproductive phenological asynchronisms within and between trees were generated and maintained. It is suggested that causation networks involving structural and temporal components may give rise to contrasted tree architectures.

Many regression models for categorical data have been introduced in various applied fields, motivated by different paradigms. But these models are difficult to compare because their specifications are not homogeneous. The first contribution of this paper is to unify the specification of regression models for categorical response variables, whether nominal or ordinal. This unification is based on a decomposition of the link function into an inverse continuous cdf and a ratio of probabilities. This allows us to define the new family of reference models for nominal data, comparable to the adjacent, cumulative and sequential families of models for ordinal data. We introduce the notion of reversible models for ordinal data that enables to distinguish adjacent and cumulative models from sequential ones. Invariances under permutations of categories are then studied for each family. The combination of the proposed specification with the definition of reference and reversible models and the various invariance properties leads to an in-depth renewal of our view of regression models for categorical data. Finally, a family of new supervised classifiers is tested on three benchmark datasets and a biological dataset is investigated with the objective of recovering the order among categories with only partial ordering information. Keywords. invariance under permutation, link function decomposition, models equivalence, nominal variable, ordinal variable, reversibility. arXiv:1404.7331v2 [stat.ME] 12 May 2014Property 4. Let σ J be a permutation of {1, . . . , J} such that σ J (J) = J and let P σ J be the restricted permutation matrix of dimension J − 1 (P σ J ) i,j = 1 if i = σ J (j), 0 otherwise, for i, j ∈ {1, . . . , J − 1}. Then we have (reference, F, Z) σ J = (reference, F, P σ J Z), for any F ∈ F and any Z ∈ Z.Proof. For the reference ratio we havefor j ∈ {1, . . . , J − 1}. Thus we simply need to permute the linear predictors using P σ J and we obtain η = P σ J η = P σ J Zβ.Noting that P σ J is invertible with P −1 σ J = P σ −1 J , we get:Corollary 3. The family of reference models is stable under the (J − 1)! permutations that fix the reference category.Corollary 4. Let F ∈ F. The particular (reference, F , complete) and (reference, F , proportional) models are invariant under the (J − 1)! permutations that fix the reference category.