Species-occupancy distribution removes an excessive parameter from species-area relationship

Aim Although species-occupancy distributions (SODs) and species-area relationships (SARs) arise from the two marginal sums of the same presence/absence matrices, the two biodiversity patterns are usually explored independently. Here, we aim to unify the two patterns for isolate-based data by constraining the SAR to conserve information from the SOD. Location Widespread. Methods Focusing on the power-model SAR, we first developed a constrained form that conserved the total number of occupancies from the SOD. Next, we developed an additive-constrained SAR that conserves the entire shape of the SOD within the power-model SAR function, using a single parameter (the slope of the endemics-area relationship). We then relate this additive-constrained SAR to multiple-sites similarity measures, based on a probabilistic view of Sørensen similarity. We extend the constrained and additive-constrained SAR framework to 23 published SAR functions. We compare the fit of the original and constrained forms of 12 SAR functions using 154 published data sets, covering various spatial scales, taxa and systems. Main conclusions In all 23 SAR functions, the constrained form had one parameter less than the original form. In all 154 data sets the model with the highest weight based on the corrected Akaike's information criteria (wAICc) had a constrained form. The constrained form received higher wAICc than the original form in 98.79% of valid pairwise cases, approaching the wAICc expected under identical log-likelihood. Our work suggests, both theoretically and empirically, that all SAR functions may have one unnecessary parameter, which can be excluded from the function without reduction in goodness-of-fit. The more parsimonious constrained forms are also easier to interpret as they reflect the probability of a randomly chosen occupancy to be found in an isolate. The additive-constrained SARs accounts for two complimentary turn-over components of occupancies: turnover between species and turnover between sites

Statistical mechanics characterization of spatio-compositional inhomogeneity

On the basis of a model system of pillars built of unit cubes, a two-component entropic measure for the multiscale analysis of spatio-compositional inhomogeneity is proposed. It quantifies the statistical dissimilarity per cell of the actual configurational macrostate and the theoretical reference one that maximizes entropy. Two kinds of disorder compete: i) the spatial one connected with possible positions of pillars inside a cell (the first component of the measure), ii) the compositional one linked to compositions of each local sum of their integer heights into a number of pillars occupying the cell (the second component). As both the number of pillars and sum of their heights are conserved, the upper limit for a pillar height hmax occurs. If due to a further constraint there is the more demanding limit h <= h* < hmax, the exact number of restricted compositions can be then obtained only through the generating function. However, at least for systems with exclusively composition degrees of freedom, we show that the neglecting of the h* is not destructive yet for a nice correlation of the h*-constrained entropic measure and its less demanding counterpart, which is much easier to compute. Given examples illustrate a broad applicability of the measure and its ability to quantify some of the subtleties of a fractional Brownian motion, time evolution of a quasipattern [28,29] and reconstruction of a laser-speckle pattern [2], which are hardly to discern or even missed.Comment: 17 pages, 5 figure

Conic Optimization Theory: Convexification Techniques and Numerical Algorithms

Optimization is at the core of control theory and appears in several areas of this field, such as optimal control, distributed control, system identification, robust control, state estimation, model predictive control and dynamic programming. The recent advances in various topics of modern optimization have also been revamping the area of machine learning. Motivated by the crucial role of optimization theory in the design, analysis, control and operation of real-world systems, this tutorial paper offers a detailed overview of some major advances in this area, namely conic optimization and its emerging applications. First, we discuss the importance of conic optimization in different areas. Then, we explain seminal results on the design of hierarchies of convex relaxations for a wide range of nonconvex problems. Finally, we study different numerical algorithms for large-scale conic optimization problems.Comment: 18 page

A Sums-of-Squares Extension of Policy Iterations

In order to address the imprecision often introduced by widening operators in static analysis, policy iteration based on min-computations amounts to considering the characterization of reachable value set of a program as an iterative computation of policies, starting from a post-fixpoint. Computing each policy and the associated invariant relies on a sequence of numerical optimizations. While the early research efforts relied on linear programming (LP) to address linear properties of linear programs, the current state of the art is still limited to the analysis of linear programs with at most quadratic invariants, relying on semidefinite programming (SDP) solvers to compute policies, and LP solvers to refine invariants. We propose here to extend the class of programs considered through the use of Sums-of-Squares (SOS) based optimization. Our approach enables the precise analysis of switched systems with polynomial updates and guards. The analysis presented has been implemented in Matlab and applied on existing programs coming from the system control literature, improving both the range of analyzable systems and the precision of previously handled ones.Comment: 29 pages, 4 figure