Entropy: The Markov Ordering Approach

The focus of this article is on entropy and Markov processes. We study the properties of functionals which are invariant with respect to monotonic transformations and analyze two invariant "additivity" properties: (i) existence of a monotonic transformation which makes the functional additive with respect to the joining of independent systems and (ii) existence of a monotonic transformation which makes the functional additive with respect to the partitioning of the space of states. All Lyapunov functionals for Markov chains which have properties (i) and (ii) are derived. We describe the most general ordering of the distribution space, with respect to which all continuous-time Markov processes are monotonic (the {\em Markov order}). The solution differs significantly from the ordering given by the inequality of entropy growth. For inference, this approach results in a convex compact set of conditionally "most random" distributions.Comment: 50 pages, 4 figures, Postprint version. More detailed discussion of the various entropy additivity properties and separation of variables for independent subsystems in MaxEnt problem is added in Section 4.2. Bibliography is extende

Accuracy and responses of genomic selection on key traits in apple breeding

open13siThe application of genomic selection in fruit tree crops is expected to enhance breeding efficiency by increasing prediction accuracy, increasing selection intensity and decreasing generation interval. The objectives of this study were to assess the accuracy of prediction and selection response in commercial apple breeding programmes for key traits. The training population comprised 977 individuals derived from 20 pedigreed full-sib families. Historic phenotypic data were available on 10 traits related to productivity and fruit external appearance and genotypic data for 7829 SNPs obtained with an Illumina 20K SNP array. From these data, a genome-wide prediction model was built and subsequently used to calculate genomic breeding values of five application full-sib families. The application families had genotypes at 364 SNPs from a dedicated 512 SNP array, and these genotypic data were extended to the high-density level by imputation. These five families were phenotyped for 1 year and their phenotypes were compared to the predicted breeding values. Accuracy of genomic prediction across the 10 traits reached a maximum value of 0.5 and had a median value of 0.19. The accuracies were strongly affected by the phenotypic distribution and heritability of traits. In the largest family, significant selection response was observed for traits with high heritability and symmetric phenotypic distribution. Traits that showed non-significant response often had reduced and skewed phenotypic variation or low heritability. Among the five application families the accuracies were uncorrelated to the degree of relatedness to the training population. The results underline the potential of genomic prediction to accelerate breeding progress in outbred fruit tree crops that still need to overcome long generation intervals and extensive phenotyping costs.openMuranty, H.; Troggio, M.; Sadok, I.B.; Mehdi A.R.; Auwerkerken, A.; Banchi, E.; Velasco, R.; Stevanato, P.; Eric van de Weg, W.; Di Guardo, M.; Kumar, S.; Laurens, F.; Bink, M.C.A.M.Muranty, H.; Troggio, M.; Sadok, I. B.; Mehdi, A. R.; Auwerkerken, A.; Banchi, E.; Velasco, R.; Stevanato, Piergiorgio; Eric van de Weg, W.; Di Guardo, M.; Kumar, S.; Laurens, F.; Bink, M. C. A. M

Searching for network modules

When analyzing complex networks a key target is to uncover their modular structure, which means searching for a family of modules, namely node subsets spanning each a subnetwork more densely connected than the average. This work proposes a novel type of objective function for graph clustering, in the form of a multilinear polynomial whose coefficients are determined by network topology. It may be thought of as a potential function, to be maximized, taking its values on fuzzy clusterings or families of fuzzy subsets of nodes over which every node distributes a unit membership. When suitably parametrized, this potential is shown to attain its maximum when every node concentrates its all unit membership on some module. The output thus is a partition, while the original discrete optimization problem is turned into a continuous version allowing to conceive alternative search strategies. The instance of the problem being a pseudo-Boolean function assigning real-valued cluster scores to node subsets, modularity maximization is employed to exemplify a so-called quadratic form, in that the scores of singletons and pairs also fully determine the scores of larger clusters, while the resulting multilinear polynomial potential function has degree 2. After considering further quadratic instances, different from modularity and obtained by interpreting network topology in alternative manners, a greedy local-search strategy for the continuous framework is analytically compared with an existing greedy agglomerative procedure for the discrete case. Overlapping is finally discussed in terms of multiple runs, i.e. several local searches with different initializations.Comment: 10 page

Structure and Interpretation of Dual-Feasible Functions

We study two techniques to obtain new families of classical and general Dual-Feasible Functions: A conversion from minimal Gomory--Johnson functions; and computer-based search using polyhedral computation and an automatic maximality and extremality test.Comment: 6 pages extended abstract to appear in Proc. LAGOS 2017, with 21 pages of appendi