Diversity from genes to ecosystems : a unifying framework to study variation across biological metrics and scales

This work was assisted through participation in “Next Generation Genetic Monitoring” Investigative Workshop at the National Institute for Mathematical and Biological Synthesis, sponsored by the National Science Foundation through NSF Award #DBI-1300426, with additional support from The University of Tennessee, Knoxville. Hawaiian fish community data were provided by the NOAA Pacific Islands Fisheries Science Center's Coral Reef Ecosystem Division (CRED) with funding from NOAA Coral Reef Conservation Program. O.E.G. was supported by the Marine Alliance for Science and Technology for Scotland (MASTS). A. C. and C. H. C. were supported by the Ministry of Science and Technology, Taiwan. P.P.-N. was supported by a Canada Research Chair in Spatial Modelling and Biodiversity. K.A.S. was supported by National Science Foundation (BioOCE Award Number 1260169) and the National Center for Ecological Analysis and Synthesis. All data used in this manuscript are available in DRYAD (https://doi.org/dx.doi.org/10.5061/dryad.qm288) and BCO-DMO (http://www.bco-dmo.org/project/552879).Biological diversity is a key concept in the life sciences and plays a fundamental role in many ecological and evolutionary processes. Although biodiversity is inherently a hierarchical concept covering different levels of organisation (genes, population, species, ecological communities and ecosystems), a diversity index that behaves consistently across these different levels has so far been lacking, hindering the development of truly integrative biodiversity studies. To fill this important knowledge gap we present a unifying framework for the measurement of biodiversity across hierarchical levels of organisation. Our weighted, information-based decomposition framework is based on a Hill number of order q = 1, which weights all elements in proportion to their frequency and leads to diversity measures based on Shannon’s entropy. We investigated the numerical behaviour of our approach with simulations and showed that it can accurately describe complex spatial hierarchical structures. To demonstrate the intuitive and straightforward interpretation of our diversity measures in terms of effective number of components (alleles, species, etc.) we applied the framework to a real dataset on coral reef biodiversity. We expect our framework will have multiple applications covering the fields of conservation biology, community genetics, and eco-evolutionary dynamics.Publisher PDFPeer reviewe

Analysis of correlation and ionization from pair distributions in many-electron systems

This work was supported in part by the Spanish MINECO project FIS2014-59311-P (cofinanced by FEDER). A.L.M., J.C.A. and J.A. belong to the Andalusian research group FQM-020, and S.L.R. to FQM-239.Jensen–Shannon divergence is used to quantify the discrepancy between the Hartree–Fock pair density and the product of its marginals for different N-electron systems, enclosing neutral atoms (with nuclear charge Z = N) and singly-charged ions (N = Z ±1). This divergence measure is applied to determine the interelectronic correlation in atomic systems. A thorough study was carried out, by considering (i) both position and momentum conjugated spaces, and (ii) systems with a nuclear charge as far as Z = 103. The correlation among electrons was measured by comparing, for an arbitrary system, the double-variable electron-pair density with the product of the respective one-particle densities. A detailed analysis throughout the Periodic Table highlights the relevance not only of weightiness for the systems considered, but also of their shell structure. Besides, comparative computations between two-electron densities of different atomic systems (neutrals, cations, anions) quantify their dissimilarities, patently governed by shell-filling patterns throughout the Periodic Table.Spanish MINECO (FEDER) FIS2014-59311-

Fast depth-based subgraph kernels for unattributed graphs

In this paper, we investigate two fast subgraph kernels based on a depth-based representation of graph-structure. Both methods gauge depth information through a family of K-layer expansion subgraphs rooted at a vertex [1]. The first method commences by computing a centroid-based complexity trace for each graph, using a depth-based representation rooted at the centroid vertex that has minimum shortest path length variance to the remaining vertices [2]. This subgraph kernel is computed by measuring the Jensen-Shannon divergence between centroid-based complexity entropy traces. The second method, on the other hand, computes a depth-based representation around each vertex in turn. The corresponding subgraph kernel is computed using isomorphisms tests to compare the depth-based representation rooted at each vertex in turn. For graphs with n vertices, the time complexities for the two new kernels are O(n 2) and O(n 3), in contrast to O(n 6) for the classic Gärtner graph kernel [3]. Key to achieving this efficiency is that we compute the required Shannon entropy of the random walk for our kernels with O(n 2) operations. This computational strategy enables our subgraph kernels to easily scale up to graphs of reasonably large sizes and thus overcome the size limits arising in state-of-the-art graph kernels. Experiments on standard bioinformatics and computer vision graph datasets demonstrate the effectiveness and efficiency of our new subgraph kernels

On R\'enyi and Tsallis entropies and divergences for exponential families

Many common probability distributions in statistics like the Gaussian, multinomial, Beta or Gamma distributions can be studied under the unified framework of exponential families. In this paper, we prove that both R\'enyi and Tsallis divergences of distributions belonging to the same exponential family admit a generic closed form expression. Furthermore, we show that R\'enyi and Tsallis entropies can also be calculated in closed-form for sub-families including the Gaussian or exponential distributions, among others.Comment: 7 page

A Novel Nonparametric Distance Estimator for Densities with Error Bounds

The use of a metric to assess distance between probability densities is an important practical problem. In this work, a particular metric induced by an a-divergence is studied. The Hellinger metric can be interpreted as a particular case within the framework of generalized Tsallis divergences and entropies. The nonparametric Parzen's density estimator emerges as a natural candidate to estimate the underlying probability density function, since it may account for data from different groups, or experiments with distinct instrumental precisions, i.e., non-independent and identically distributed (non-i.i.d.) data. However, the information theoretic derived metric of the nonparametric Parzen's density estimator displays infinite variance, limiting the direct use of resampling estimators. Based on measure theory, we present a change of measure to build a finite variance density allowing the use of resampling estimators. In order to counteract the poor scaling with dimension, we propose a new nonparametric two-stage robust resampling estimator of Hellinger's metric error bounds for heterocedastic data. The approach presents very promising results allowing the use of different covariances for different clusters with impact on the distance evaluation