A New Method for Low Density Distribution Modeling and Near Threatened Species: The Study Case of Plectrohyla Guatemalensis.

We introduce a model that can be used for the description of the distribution of species when there is scarcity of data, based on our previous work (Ballesteros et al. J Math Biol 85(4):31, 2022). We address challenges in modeling species that are seldom observed in nature, for example species included in The International Union for Conservation of Nature's Red List of Threatened Species (IUCN 2023). We introduce a general method and test it using a case study of a near threatened species of amphibians called Plectrohyla Guatemalensis (see IUCN 2023) in a region of the UNESCO natural reserve "Tacaná Volcano", in the border between Mexico and Guatemala. Since threatened species are difficult to find in nature, collected data can be extremely reduced. This produces a mathematical problem in the sense that the usual modeling in terms of Markov random fields representing individuals associated to locations in a grid generates artificial clusters around the observations, which are unreasonable. We propose a different approach in which our random variables describe yearly averages of expectation values of the number of individuals instead of individuals (and they take values on a compact interval). Our approach takes advantage of intuitive insights from environmental properties: in nature individuals are attracted or repulsed by specific features (Ballesteros et al. J Math Biol 85(4):31, 2022). Drawing inspiration from quantum mechanics, we incorporate quantum Hamiltonians into classical statistical mechanics (i.e. Gibbs measures or Markov random fields). The equilibrium between spreading and attractive/repulsive forces governs the behavior of the species, expressed through a global control problem involving an energy operator.

A model and a numerical scheme for the description of distribution and abundance of individuals.

We introduce a model in the context of ecology that can be used to describe the distribution and abundance of individuals when data from field work is extremely limited (for example, in the case of endangered species). Our procedure is based on an intuitive understanding of the physical properties of phenomena. The idea is that individuals have the tendency to be attracted (or repulsed) to certain properties of the environment. At the same time, they are spread in such a way that if there is no reason for them to be in some specific locations, then they are uniformly distributed throughout the region. Our model draws from quantum mechanics, by using quantum Hamiltonians in the context of classical statistical mechanics. The equilibrium between the spreading and the attractive (or repulsive) forces determines the behavior of the species that we model, and this is expressed in terms of a global control problem of an energy operator which is the sum of a kinetic term (spreading) and a potential (attraction or repulsion). We focus on the full probability measure and a global control of the model (instead of looking at conditional measures that generate a global measure). Furthermore, we propose a numerical solution to this global control problem that overcomes the well-known major difficulty of Gibbs sampling (annealing) which is the fact that a global control is hardly reachable when the number of variables is large (the algorithms get stuck in non-optimal states).

On a Class of Tensor Markov Fields.

Here, we introduce a class of Tensor Markov Fields intended as probabilistic graphical models from random variables spanned over multiplexed contexts. These fields are an extension of Markov Random Fields for tensor-valued random variables. By extending the results of Dobruschin, Hammersley and Clifford to such tensor valued fields, we proved that tensor Markov fields are indeed Gibbs fields, whenever strictly positive probability measures are considered. Hence, there is a direct relationship with many results from theoretical statistical mechanics. We showed how this class of Markov fields it can be built based on a statistical dependency structures inferred on information theoretical grounds over empirical data. Thus, aside from purely theoretical interest, the Tensor Markov Fields described here may be useful for mathematical modeling and data analysis due to their intrinsic simplicity and generality.

Tsallis-Entropy Segmentation through MRF and Alzheimer anatomic reference for Brain Magnetic Resonance Parcellation.

Quantifying the intracranial tissue volume changes in magnetic resonance imaging (MRI) assists specialists to analyze the effects of natural or pathological changes. Since these changes can be subtle, the accuracy of the automatic compartmentalization method is always criticized by specialists. We propose and then evaluate an automatic segmentation method based on modified q-entropy (Mqe) through a modified Markov Random Field (MMRF) enhanced by Alzheimer anatomic reference (AAR) to provide a high accuracy brain tissues parcellation approach (Mqe-MMRF). We underwent two strategies to evaluate Mqe-MMRF; a simulation of different levels of noise and non-uniformity effect on MRI data (7 subjects) and a set of twenty MRI data available from MRBrainS13 as patient brain tissue segmentation challenge. We accessed eleven quality metrics compared to reference tissues delineations to evaluate Mqe-MMRF. MRI segmentation scores decreased by only 4.6% on quality metrics after noise and non-uniformity simulations of 40% and 9%, respectively. We found significant mean improvements in the metrics of the five training subjects, for whole-brain 0.86%, White Matter 3.20%, Gray Matter 3.99%, and Cerebrospinal Fluid 4.16% (p-values < 0.02) when Mqe-MMRF compared to the other reference methods. We also processed the Mqe-MMRF on 15 evaluation subjects group from MRBrainS13 online challenge, and the results held a higher rank than the reference tools; FreeSurfer, SPM, and FSL. Since the proposed method improved the precision of brain segmentation, specifically, for GM, and thus one can use it in quantitative and morphological brain studies.

An Evaluation of Radarsat-1 and ASTER Data for Mapping Veredas (Palm Swamps).

Veredas (palm swamps) are wetland complexes associated with the Brazilian savanna (cerrado) that often represent the only available source of water for the ecosystem during the dry months. Their extent and condition are mainly unknown and their cartography is an essential issue for their protection. This research article evaluates some of the fine resolution satellite data both in the radar (Radarsat-1) and optical domain (ASTER) for the delineation and characterization of veredas. Two separate approaches are evaluated. First, given the known potential of Radarsat-1 images for wetland inventories, the automatic delineation of veredas is tested using only Radarsat-1 data and a Markov random fields region-based segmentation. In this case, to increase performance, processing is limited to a buffer zone around the river network. Then, characterization of their type is attempted using traditional classification methods of ASTER optical data combined with Radarsat-1 data. The automatic classification of Radarsat data yielded results with an overall accuracy between 62 and 69%, that proved reliable enough for delineating wide and very humid veredas. Scenes from the wet season and with a smaller angle of incidence systematically yielded better results. For the classification of the main vegetation types, better results (overall success of 78.8%) were obtained by using only the visible and near infrared (VNIR) bands of the ASTER image. Radarsat data did not bring any improvement to these classification results. In fact, when using solely the Radarsat data from two different angle of incidence and two different dates, the classification results were low (50.8%) but remained powerful for delineating the permanently moist riparian forest portion of the veredas with an accuracy better than 75% in most cases. These results are considered good given the width of some types often less than 50 m wide compared with the resolution of the images (12.5 - 15 m). Comparing the classification results with the Radarsat-generated delineation allows an understanding of the relation between synthetic aperture radar (SAR) backscattering and vegetation types of the veredas.