Developments in forest dynamics modelling have led to complex indivi- dual-tree distance-dependent models. Questions that arise are: how might such com- plex models be approximated (possibly through aggregation techniques) to obtain distribution-based models; and, when are such approximated models to the complex models from which they are derived. The approximation process usu- ally involves replacing distance-dependent tree interactions by distance-independent interactions. We used a that is derived from an individual-based distance- dependent model of a forest in French Guiana to address these methodological ques- tions. This toy model (model I) was defined in two forms; first a short-range tree interaction (1 m); second for a long-range tree interaction (30 m). Mean-field approx- imations were used to convert model I into an individual-tree distance-independent model (model II) and into a distribution-based model (model III). If the starting model is model I(long-range), models II and III are shown to be equivalent to each other. Model II is also found to be consistent with model I(long-range). Hence, for the toy model considered, long-range interactions can be disregarded, and have no influence on forest growth predictions. On the other hand, model I(short-range) is not equivalent with models II or III. Even so, it was found to be possible to obtain, by ad hoc methods, a distribution-based model (model IV) which, by taking into account the spatial structure generated by short-range interactions, is consistent with model I(short range).
This paper presents an extension of Correspondence Analysis (CA) to tensors through High Order Singular Value Decomposition (HOSVD) from a geometric viewpoint. Correspondence analysis is a well-known tool, developed from principal component analysis, for studying contingency tables. Different algebraic extensions of CA to multi-way tables have been proposed over the years, nevertheless neglecting its geometric meaning. Relying on the Tucker model and the HOSVD, we propose a direct way to associate with each tensor mode a point cloud. We prove that the point clouds are related to each other. Specifically using the CA metrics we show that the barycentric relation is still true in the tensor framework. Finally two data sets are used to underline the advantages and the drawbacks of our strategy with respect to the classical matrix approaches.
In the post-genomics era, non-model species like most Fagaceae still lack operational diversity resources for population genomics studies. Sequence data were produced from over 800 gene fragments covering ~530 kb across the genic partition of European oaks, in a discovery panel of 25 individuals from western and central Europe (11 Quercus petraea, 13 Q. robur, one Q. ilex as an outgroup). Regions targeted represented broad functional categories potentially involved in species ecological preferences, and a random set of genes. Using a high-quality dedicated pipeline, we provide a detailed characterization of these genic regions, which included over 14500 polymorphisms, with ~12500 SNPs -218 being triallelic-, over 1500 insertion-deletions, and ~200 novel di- and tri-nucleotide SSR loci. This catalog also provides various summary statistics within and among species, gene ontology information, and standard formats to assist loci choice for genotyping projects. The distribution of nucleotide diversity (theta.pi)and differentiation (Fst) across genic regions are also described for the first time in those species, with a mean theta.pi close to ~0.0049 in Q. petraea and to ~0.0045 in Q. robur across random regions, and a mean Fst ~0.13 across SNPs. The magnitude of diversity across genes is within the range estimated for long-term perennial outcrossers, and can be considered relatively high in the plant kingdom, with an estimate across the genome of 41 to 51 million SNPs expected in both species. Individuals with typical species morphology were more easily assigned to their corresponding genetic cluster for Q. robur than for Q. petraea, revealing higher or more recent introgression in Q. petraea and a stronger species integration in Q. robur in this particular discovery panel. We also observed robust patterns of a slightly but significantly higher diversity in Q. petraea, across a random gene set and in the abiotic stress functional category, and a heterogeneous landscape of both diversity and differentiation. To explain these patterns, we discuss an alternative and non-exclusive hypothesis of stronger selective constraints in Q. robur, the most pioneering species in oak forest stand dynamics, additionally to the recognized and documented introgression history in both species despite their strong reproductive barriers. The quality of the data provided here and their representativity in terms of species genomic diversity make them useful for possible applications in medium-scale landscape and molecular ecology projects. Moreover, they can serve as reference resources for validation purposes in larger-scale resequencing projects. This type of project is preferentially recommended in oaks in contrast to SNP array development, given the large nucleotide variation and the low levels of linkage disequilibrium revealed.
We consider the problem of specifying the joint distribution of a collection of variables with maximum entropy when a set of marginals are fixed. One can easily derive that the structure of the solution joint distribution is that of a graphical model. The potential functions are then marginals at some power. We address the following question, Under which conditions on the set of constraints is it possible to fully identify the canonical exponents in the maximum entropy solution as functions of the problem structure? Literature related to this topic is somewhat scattered in disciplines such as statistical mechanics, information theory, graph theory, and inference in graphical models. In this article we gather and link results from these different fields. From this, we show that for a particular class of constraints set on marginals, the chordal maximal coherent sets of constraints, it is possible to derive the canonical exponents of the graphical model solution of the maximum entropy problem as the numbers of occurrences of separators in an associated join tree. Conversely, we present sufficient conditions to ensure that a graphical model is a solution of a maximum entropy problem.
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