Distributions on the simplex

The simplex, the sample space of compositional data, can be structured as a real Euclidean space. This fact allows to work with the coefficients with respect to an orthonormal basis. Over these coefficients we apply standard real analysis, inparticular, we define two different laws of probability trought the density function and we study their main properties

The Dirichlet distribution with respect to the Aitchison measure on the simplex - a first approach

The algebraic-geometric structure of the simplex, known as Aitchison geometry, is used to look at the Dirichlet family of distributions from a new perspective. A classical Dirichlet density function is expressed with respect to the Lebesgue measure on real space. We propose here to change this measure by the Aitchison measure on the simplex, and study some properties and characteristic measures of the resulting density

A new distribution on the simplex containing the Dirichlet family

The Dirichlet family owes its privileged status within simplex distributions to easyness of interpretation and good mathematical properties. In particular, we recall fundamental properties for the analysis of compositional data such as closure under amalgamation and subcomposition. From a probabilistic point of view, it is characterised (uniquely) by a variety of independence relationships which makes it indisputably the reference model for expressing the non trivial idea of substantial independence for compositions. Indeed, its well known inadequacy as a general model for compositional data stems from such an independence structure together with the poorness of its parametrisation. In this paper a new class of distributions (called Flexible Dirichlet) capable of handling various dependence structures and containing the Dirichlet as a special case is presented. The new model exhibits a considerably richer parametrisation which, for example, allows to model the means and (part of) the variance-covariance matrix separately. Moreover, such a model preserves some good mathematical properties of the Dirichlet, i.e. closure under amalgamation and subcomposition with new parameters simply related to the parent composition parameters. Furthermore, the joint and conditional distributions of subcompositions and relative totals can be expressed as simple mixtures of two Flexible Dirichlet distributions. The basis generating the Flexible Dirichlet, though keeping compositional invariance, shows a dependence structure which allows various forms of partitional dependence to be contemplated by the model (e.g. non-neutrality, subcompositional dependence and subcompositional non-invariance), independence cases being identified by suitable parameter configurations. In particular, within this model substantial independence among subsets of components of the composition naturally occurs when the subsets have a Dirichlet distribution

Vertebrates Limb Geometry in the Simplex space

A novel metric comparison of the appendicular skeleton (fore and hind limb) of different vertebrates using the Compositional Data Analysis (CDA) methodological approach it’s presented. 355 specimens belonging in various taxa of Dinosauria (Sauropodomorpha, Theropoda, Ornithischia and Aves) and Mammalia (Prothotheria, Metatheria and Eutheria) were analyzed with CDA. A special focus has been put on Sauropodomorpha dinosaurs and the Aitchinson distance has been used as a measure of disparity in limb elements proportions to infer some aspects of functional morphology

Experimental design on the simplex

Optimum experimental designs depend on the design criterion, the model and the design region. The talk will consider the design of experiments for regression models in which there is a single response with the explanatory variables lying in a simplex. One example is experiments on various compositions of glass such as those considered by Martin, Bursnall, and Stillman (2001). Because of the highly symmetric nature of the simplex, the class of models that are of interest, typically Scheff´e polynomials (Scheff´e 1958) are rather different from those of standard regression analysis. The optimum designs are also rather different, inheriting a high degree of symmetry from the models. In the talk I will hope to discuss a variety of modes for such experiments. Then I will discuss constrained mixture experiments, when not all the simplex is available for experimentation. Other important aspects include mixture experiments with extra non-mixture factors and the blocking of mixture experiments. Much of the material is in Chapter 16 of Atkinson, Donev, and Tobias (2007). If time and my research allows, I would hope to finish with a few comments on design when the responses, rather than the explanatory variables, lie in a simplex. References Atkinson, A. C., A. N. Donev, and R. D. Tobias (2007). Optimum Experimental Designs, with SAS. Oxford: Oxford University Press. Martin, R. J., M. C. Bursnall, and E. C. Stillman (2001). Further results on optimal and efficient designs for constrained mixture experiments. In A. C. Atkinson, B. Bogacka, and A. Zhigljavsky (Eds.), Optimal Design 2000, pp. 225–239. Dordrecht: Kluwer. Scheff´e, H. (1958). Experiments with mixtures. Journal of the Royal Statistical Society, Ser. B 20, 344–360. 1

Optimal convergence estimates for the trace of the polynomial L2-projection operator on a simplex

In this paper we study convergence of the L2-projection onto the space of polynomials up to degree p on a simplex in Rd, d >= 2. Optimal error estimates are established in the case of Sobolev regularity and illustrated on several numerical examples. The proof is based on the collapsed coordinate transform and the expansion into various polynomial bases involving Jacobi polynomials and their antiderivatives. The results of the present paper generalize corresponding estimates for cubes in Rd from [P. Houston, C. Schwab, E. Süli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39 (2002), no. 6, 2133-2163].

Occurrence of herpes simplex virus 1 and three periodontal bacteria in patients with chronic periodontitis and necrotic pulp

Viral and bacterial associations appear to be implicated in the development of periodontal infections. Little information is available describing the periodontopathic agents in root canals with necrotic pulp. In this study, the occurrence and the combinations among herpes simplex virus type 1 (HSV-1) and Dialister pneumosintes, Tannerella forsythia.. and Treponema denticola in patients with chronic periodontitis and necrotic pulp were evaluated. Clinical samples from healthy subjects and patients with periodontal or pulp infections were analyzed using a nested polymerase chain reaction PCR to detect HSV and PCR to detect the 3 periodontal bacteria. The presence of Tannerella forsythia and Treponema denticola was observed in healthy, periodontitis, and necrotic pulp patients. HSV was observed in periodontitis and necrotic pulp patients, and no healthy subject harbored D. pneumosintes or HSV. The occurrence of Tannerella forsythia was not statistically significant in patients with necrotic pulp (P = 0.704). Periodontal bacteria were observed varying from 10.3% to 20.7% in periodontitis and necrotic pulp patients. The presence of Treponema denticola - HSV association was predominant in patients showing necrotic pulp (24.1%); however, HSV alone was observed in one patient with periodontitis and in another patient with necrotic pulp. The presence of double association among bacteria or bacteria - HSV could indicate a role in both periodontitis and necrotic pulp, and Tannerella forsythia - Treponenta denticola - HSV and Tannerella forsythia - D. pneumosintes - Treponema denticola - HSV associations might be important in periodontitis.

Prevalência da infecção por papilomavírus humano, herpes simplex tipo 1 e 2 e Chlamydia trachomatis em um segmento da população feminina da grande Natal/RN

Sexually transmitted diseases (STDs) are among the largest public health problems, especially in developing countries. The acquisition of these infections during early sexual activity is common and many infections have a benign course. However, in some pathogens remain in the state of latency can be reactivated and cause productive infection that may progress to severe forms. In addition, some of them are transmitted vertically resulting in congenital infection, causing immediate damage or long-term child. The classic risk factors for sexually transmitted agents are: early onset of sexual and reproductive health, multiple sexual partners throughout life, use of oral contraceptives and co-infections with different pathogens. We present the results of a cross-sectional study aimed to estimate the prevalence of genital infection by human papillomavirus (HPV), Herpes simplex virus (HSV) and Chlamydia trachomatis (CT) in a segment of the female population of the metropolitan area Christmas, among those who enrolled voluntarily sought, Basic Health Units for the examination of cancer screening cervix in the period 2008 to 2010. All participants, a total of 261 women answered a standard questionnaire by which identified the socio-demographic characteristics, classical risk factors for STDs, reproductive and sexual activity and smoking. Of each patient were obtained two samples, one for the completion of the Pap test for detection of cellular changes and the other processed for DNA extraction and analyzed by PCR (polymerase chain reaction) to detect the three pathogens studied. The population of the study was composed of sexually active women aged between 13 and 79 years, mean 38.7 years, most of them being married, low education levels and low incomes. The majority (87%) had normal results on cytology and only 2.7% had low-grade cytological abnormalities. Prevalence rates were 37.9% for HPV, 4.6% for CT and 26% for HSV. HPV prevalence was higher in women under 25, unmarried and in those who had multiple sexual partners. Women with simultaneous infection by HSV-1 and 2 had higher prevalence of HPV infection. The prevalence of HSV infection showed no association whatsoever with the risk factors analyzed and HSV-1 was the predominant type among the cases of genital HSV infection. The overall prevalence of C. Trachomatis was relatively low, thus providing greater value in younger women aged less than or equal to 20 years

Implementação paralela escalável e eficiente do algoritmo simplex padrão em arquitetura multicore

This work presents a scalable and efficient parallel implementation of the Standard Simplex algorithm in the multicore architecture to solve large scale linear programming problems. We present a general scheme explaining how each step of the standard Simplex algorithm was parallelized, indicating some important points of the parallel implementation. Performance analysis were conducted by comparing the sequential time using the Simplex tableau and the Simplex of the CPLEXR IBM. The experiments were executed on a shared memory machine with 24 cores. The scalability analysis was performed with problems of different dimensions, finding evidence that our parallel standard Simplex algorithm has a better parallel efficiency for problems with more variables than constraints. In comparison with CPLEXR , the proposed parallel algorithm achieved a efficiency of up to 16 times better

Cellular immune responses in mice immunized with Anisakis simplex larval antigens

Cellular immune responses to Anisakis simplex L3 antigens were investigated in BALB/c mice injected subcutaneously with a homologous crude extract (CE). Popliteal lymph nodes (PLN) were found to be increased in size and weight after A. simplex CE footpad injection. The effects of A. simplex CE in vitro proliferation were assayed with non-fractionated PLN cells or nylon-wool purified T cells derived from pooled lymph node cells of mice subcutaneously injected with CE. Spleen cells from immunized animals (antigen alone, or larva alone, or antigen plus larva) were studied by flow cytometry. The immunization induced a high proportion of CD4 + and TCR alpha beta + T cells. The number of B cells (CD45 + and TCR alpha beta-) in pre-immunized and infected mice was lower than that observed in animals subjected to infection only. The number of CD4 + T cells increased in the infected and in the pre-immunized and infected mice. In the latter, a decrease of CD8a + T cells was noted. The greatest increase in CD8a+ and TCR alpha beta- T cells was found in mice that had been subjected to infection only. Histological analysis showed that the most prominent lesions were gastric and intestinal in animals infected orally with one larva.

Reformulation of variational inequalities on a simplex and compactification of complementarity problems

Many variational inequality problems (VIPs) can be reduced, by a compactification procedure, to a VIP on the canonical simplex. Reformulations of this problem are studied, including smooth reformulations with simple constraints and unconstrained reformulations based on the penalized Fischer-Burmeister function. It is proved that bounded level set results hold for these reformulations under quite general assumptions on the operator. Therefore, it can be guaranteed that minimization algorithms generate bounded sequences and, under monotonicity conditions, these algorithms necessarily nd solutions of the original problem. Some numerical experiments are presented.

Planejamento da expansão de sistemas de transmissão usando algoritmos tipo dual simplex especializados em uma estrutura branch and bound

Pós-graduação em Engenharia Elétrica - FEIS

OPUS: interactive software for solving linear programming models using the simplex algorithm

Includes bibliography

Regressão Simplex aplicada a delineamentos de mistura e utilização do Algoritmo Boosting

In the composition of this work are present two parts. The first part contains the theory used. The second part contains the two articles. The first article examines two models of the class of generalized linear models for analyzing a mixture experiment, which studied the effect of different diets consist of fat, carbohydrate, and fiber on tumor expression in mammary glands of female rats, given by the ratio mice that had tumor expression in a particular diet. Mixture experiments are characterized by having the effect of collinearity and smaller sample size. In this sense, assuming normality for the answer to be maximized or minimized may be inadequate. Given this fact, the main characteristics of logistic regression and simplex models are addressed. The models were compared by the criteria of selection of models AIC, BIC and ICOMP, simulated envelope charts for residuals of adjusted models, odds ratios graphics and their respective confidence intervals for each mixture component. It was concluded that first article that the simplex regression model showed better quality of fit and narrowest confidence intervals for odds ratio. The second article presents the model Boosted Simplex Regression, the boosting version of the simplex regression model, as an alternative to increase the precision of confidence intervals for the odds ratio for each mixture component. For this, we used the Monte Carlo method for the construction of confidence intervals. Moreover, it is presented in an innovative way the envelope simulated chart for residuals of the adjusted model via boosting algorithm. It was concluded that the Boosted Simplex Regression model was adjusted successfully and confidence intervals for the odds ratio were accurate and lightly more precise than the its maximum likelihood version.