ISEL - Matemática
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Browsing ISEL - Matemática by Author "Aleixo, Sandra"
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- An Extension of Gompertzian Growth Dynamics Weibull and Frechet ModelsPublication . Rocha, J. Leonel; Aleixo, SandraIn this work a new probabilistic and dynamical approach to an extension of the Gompertz law is proposed. A generalized family of probability density functions, designated by Beta* (p, q), which is proportional to the right hand side of the Tsoularis-Wallace model, is studied. In particular, for p = 2, the investigation is extended to the extreme value models of Weibull and Frechet type. These models, described by differential equations, are proportional to the hyper-Gompertz growth model. It is proved that the Beta* (2, q) densities are a power of betas mixture, and that its dynamics are determined by a non-linear coupling of probabilities. The dynamical analysis is performed using techniques of symbolic dynamics and the system complexity is measured using topological entropy. Generally, the natural history of a malignant tumour is reflected through bifurcation diagrams, in which are identified regions of regression, stability, bifurcation, chaos and terminus.
- Dynamical analysis in growth models: Blumberg’s equationPublication . Rocha, J. Leonel; Aleixo, SandraWe present a new dynamical approach to the Blumberg's equation, a family of unimodal maps. These maps are proportional to Beta(p, q) probability densities functions. Using the symmetry of the Beta(p, q) distribution and symbolic dynamics techniques, a new concept of mirror symmetry is defined for this family of maps. The kneading theory is used to analyze the effect of such symmetry in the presented models. The main result proves that two mirror symmetric unimodal maps have the same topological entropy. Different population dynamics regimes are identified, when the intrinsic growth rate is modified: extinctions, stabilities, bifurcations, chaos and Allee effect. To illustrate our results, we present a numerical analysis, where are demonstrated: monotonicity of the topological entropy with the variation of the intrinsic growth rate, existence of isentropic sets in the parameters space and mirror symmetry.
- Dynamical behaviour on the parameter space: new populational growth models proportional to beta densitiesPublication . Aleixo, Sandra; Rocha, J. Leonel; Pestana, Dinis D.We present new populational growth models, generalized logistic models which are proportional to beta densities with shape parameters p and 2, where p > 1, with Malthusian parameter r. The complex dynamical behaviour of these models is investigated in the parameter space (r, p), in terms of topological entropy, using explicit methods, when the Malthusian parameter r increases. This parameter space is split into different regions, according to the chaotic behaviour of the models.
- Generalized models from Beta(p,2) densities with strong allee effect: dynamical approachPublication . Aleixo, Sandra; Rocha, J. LeonelA dynamical approach to study the behaviour of generalized populational growth models from Bets(p, 2) densities, with strong Allee effect, is presented. The dynamical analysis of the respective unimodal maps is performed using symbolic dynamics techniques. The complexity of the correspondent discrete dynamical systems is measured in terms of topological entropy. Different populational dynamics regimes are obtained when the intrinsic growth rates are modified: extinction, bistability, chaotic semistability and essential extinction.
- Modeling Allee Effect from Beta(p, 2) DensitiesPublication . Rocha, J. Leonel; Aleixo, SandraIn this work we develop and investigate generalized populational growth models, adjusted from Beta(p, 2) densities, with Allee effect. The use of a positive parameter leads the presented generalization, which yields some more flexible models with variable extinction rates. An Allee limit is incorporated so that the models under study have strong Allee effect.
- A spatial econometric analysis of the calls to the Portuguese National Health LinePublication . Simões, Paula; Carvalho, M. Lucília; Aleixo, Sandra; Gomes, Sérgio; Natário, IsabelThe Portuguese National Health Line, LS24, is an initiative of the Portuguese Health Ministry which seeks to improve accessibility to health care and to rationalize the use of existing resources by directing users to the most appropriate institutions of the national public health services. This study aims to describe and evaluate the use of LS24. Since for LS24 data, the location attribute is an important source of information to describe its use, this study analyses the number of calls received, at a municipal level, under two different spatial econometric approaches. This analysis is important for future development of decision support indicators in a hospital context, based on the economic impact of the use of this health line. Considering the discrete nature of data, the number of calls to LS24 in each municipality is better modelled by a Poisson model, with some possible covariates: demographic, socio-economic information, characteristics of the Portuguese health system and development indicators. In order to explain model spatial variability, the data autocorrelation can be explained in a Bayesian setting through different hierarchical log-Poisson regression models. A different approach uses an autoregressive methodology, also for count data. A log-Poisson model with a spatial lag autocorrelation component is further considered, better framed under a Bayesian paradigm. With this empirical study we find strong evidence for a spatial structure in the data and obtain similar conclusions with both perspectives of the analysis. This supports the view that the addition of a spatial structure to the model improves estimation, even in the case where some relevant covariates have been included.
- A spatio-temporal auto-regressive model for generating savings calls to a health linePublication . Paula Simões; M. Lucília Carvalho; Aleixo, Sandra; Gomes, Sérgio; Natário, IsabelUrgency admission is one of the most important factors regarding hospital costs, which can possibly be mitigated by the use of national health lines such as the Portuguese Saúde24 line (S24). Aiming future development of decision support indicators in a hospital savings context, based on the economic impact of the use of S24 rather than hospital urgency services, this study investigates spatio-temporal dependencies of the number of S24 calls generating savings in each Portuguese municipality, over the period 2010–2016, under an autoregressive approach. An econometric analysis of the savings obtained by the use of S24 is also carried out considering a savings index. Combining insights from classical spatial econometrics and from the analysis of spatio-temporal data, novel Bayesian Poisson spatio-temporal lag models are presented and applied in this paper. This extends to time the ideas of a Bayesian Poisson spatial lag model, considering both a parametric and a non-parametic structure for time and space-time effects. The results obtained for the savings index reveal that, over the last seven years, there has been a more comprehensive spatial effectiveness of the S24 line in solving the non-urgent emergency situations, that could be handled by primary health care services or in a self care basis.
- Synchronization in Von Bertalanffy’s modelsPublication . Rocha, J. Leonel; Aleixo, Sandra; Caneco, AcilinaMany data have been useful to describe the growth of marine mammals, invertebrates and reptiles, seabirds, sea turtles and fishes, using the logistic, the Gom-pertz and von Bertalanffy's growth models. A generalized family of von Bertalanffy's maps, which is proportional to the right hand side of von Bertalanffy's growth equation, is studied and its dynamical approach is proposed. The system complexity is measured using Lyapunov exponents, which depend on two biological parameters: von Bertalanffy's growth rate constant and the asymptotic weight. Applications of synchronization in real world is of current interest. The behavior of birds ocks, schools of fish and other animals is an important phenomenon characterized by synchronized motion of individuals. In this work, we consider networks having in each node a von Bertalanffy's model and we study the synchronization interval of these networks, as a function of those two biological parameters. Numerical simulation are also presented to support our approaches.
- Weighted coefficients to measure agreement among several sets of ranks emphasizing top and bottom ranks at the same timePublication . Aleixo, Sandra; Teles, JúliaIn this paper, two new weighted coefficients of agreement to measure the concordance among several (more than two) sets of ranks, putting more weight in the lower and upper ranks simultaneously, are presented. These new coefficients, the signed Klotz and the signed Mood, generalize the correspondent rank-order coefficients to measure the agreement between two sets of ranks previously proposed by the authors [1]. Under the null hypothesis of no agreement or no association among the rankings, the asymptotic distribution of these new coefficients was derived. To illustrate the worth of these measures, an example is presented to compare them with the Kendall’s coefficient and the van der Waerden correlation coefficient.
- Weighting lower and upper ranks simultaneously through rank-order correlation coefficientsPublication . Aleixo, Sandra; Teles, JuliaTwo new weighted correlation coefficients, that allow to give more weight to the lower and upper ranks simultaneously, are proposed. These indexes were obtained computing the Pearson correlation coefficient with a modified Klotz and modified Mood scores. Under the null hypothesis of independence of the two sets of ranks, the asymptotic distribution of these new coefficients was derived. The exact and approximate quantiles were provided. To illustrate the value of these measures an example, that could mimic several biometrical concerns, is presented. A Monte Carlo simulation study was carried out to compare the performance of these new coefficients with other weighted coefficient, the van der Waerden correlation coefficient, and with two non-weighted indexes, the Spearman and Kendall correlation coefficients. The results show that, if the aim of the study is the detection of correlation or agreement between two sets of ranks, putting emphasis on both lower and upper ranks simultaneously, the use of van der Waerden, signed Klotz and signed Mood rank-order correlation coefficients should be privileged, since they have more power to detect this type of agreement, in particular when the concordance was focused on a lower proportion of extreme ranks. The preference for one of the coefficients should take into account the weight one wants to assign to the extreme ranks.