The Gudermannian Growth Model: Theory, Application and Statistical Analysis

Abstract: Processes producing sigmoid curves are common in many areas such as biology, agrarian sciences, demography and engineering. Several mathematical functions have been proposed for modeling sigmoid curves. Some models such as the logistic, Gompertz, Richards and Weibull are widely used. This work introduces the Gudermannian function as an option for modeling sigmoid growth curves. The original function was transformed and the resulting equation was called the "Gudermannian growth model." This model was applied to four sets of experimental growth data to illustrate its practical application. The results were compared with those obtained by the logistic and Gompertz models. Since all these models are nonlinear in the parameters, the statistical properties of the least squares estimators were evaluated using measures of nonlinearity. For each experimental data set, the Akaike's corrected information criterion was utilized to discriminate among the models. In general, the Gudermannian model fitted better to the experimental data than the logistic and Gompertz models. The results showed that the Gudermannian model can be a good alternative to the classical sigmoid models.

Closed-form expressions to Gompertz-Makeham life expectancies: a historical note / Expressões de forma fechada para as expectativas de vida de Gompertz-Makeham: uma nota histórica / Expresiones de forma cerrada a las esperanzas de vida de Gompertz-Makeham: una nota histórica

Abstract Results well known in the actuarial community about closed-form expressions to Gompertz and Gompertz-Makeham life expectancies for a person aged x are still being independently rediscovered to this day. This note seeks to acknowledge previous results about closed-form expressions to Gompertz-Makeham life expectancies, especially in the actuarial science field, hoping to stimulate interdisciplinarity and provide the background for further developments, especially since the derivation of closed-form expressions for life expectancy (and annuities) based on particular mortality laws are matters of interest for multiple fields such as actuarial science, biology, demography, statistics among others.

Resumo Resultados bem conhecidos pela comunidade atuarial sobre expressões de forma fechada para esperança de vida de Gompertz e Gompertz-Makeham para uma pessoa de idade x ainda estão sendo redescobertos de forma independente nos dias atuais. Esta nota visa fornecer algum reconhecimento aos resultados anteriores sobre expressões de forma fechada para expectativa de vida de Gompertz e Gompertz-Makeham, especialmente no campo das ciências atuariais, na esperança de estimular a interdisciplinaridade e fornecer o pano de fundo para novos desenvolvimentos, em especial porque a derivação de expressões de forma fechadas para expectativa de vida (e anuidades) com base em leis de mortalidade despertam o interesse de várias áreas, como ciências atuariais, biologia, demografia, estatística, entre outras.

Resumen Los resultados bien conocidos por la comunidad actuarial sobre las expresiones de forma cerrada de las esperanzas de vida de Gompertz y Gompertz-Makeham para una persona de edad x todavía se están redescubriendo de forma independiente en la actualidad. Esta nota pretende reconocer algunos resultados anteriores sobre expresiones cerradas para la esperanza de vida de Gompertz y Gompertz-Makeham, en especial en el campo de las ciencias actuariales, con la esperanza de fomentar la interdisciplinariedad y proporcionar el telón de fondo para futuros desarrollos, sobre todo desde que la derivación de expresiones cerradas para la esperanza de vida (y anualidades) basado en leyes de mortalidad despertó el interés de varias áreas, como las ciencias actuariales, biología, demografía, estadística, entre otras.

Prospective production of fructose and single cell protein from date palm waste

BACKGROUND: Fructose and single cell protein are important products for the food market. Abundant amounts of low-grade dates worldwide are annually wasted. In this study, highly concentrated fructose syrups and single cell protein were obtained through selective fermentation of date extracts by Saccharomyces cerevisiae. RESULTS: The effect of air flow (0.1, 0.5, 0.75, 1, 1.25 and 1.5 vvm) and pH (4.5, 4.8, 5, 5.3 and 5.6) was investigated. Higher air flow led to lower fructose yield. The optimum cell mass production of 10 g/L was achieved at air flow of 1.25 vvm with the fructose yield of 91%. Similar cell mass production was obtained in the range pH of 5.0­5.6, while less cell mass was obtained at pH less than 5. Controlling the pH at 4.5, 5.0 and 5.3 failed to improve the production of cell mass which were 5.6, 5.9 and 5.4 g/L respectively; however, better fructose yield was obtained. CONCLUSIONS: Extension of the modified Gompertz enabled excellent predictions of the cell mass, fructose production and fructose fraction. The proposed model was also successfully validated against data from literatures. Thus, the model will be useful for wide application of biological processes.

Adjusting the growth curve of sugarcane varieties using nonlinear models / Ajuste da curva de crescimento de variedades de cana-de-açúcar utilizando modelos não lineares

ABSTRACT: Assessing sugarcane (Saccharum spp.) stalk growth helps to adequately manage the phenological stages of the crop. The aim of this study was to describe the height-growth curve of four sugarcane varieties (RB92579, RB93509, RB931530 and SP79-1011), in irrigated plant-cane and ratoon cane plantations, using the Logistic and Gompertz nonlinear models, while considering all deviations from assumptions. The model parameters were estimated based on the least squares method using the Gauss-Newton algorithm. To select the most suitable model, nonlinear measures, adjusted coefficient of determination (R2 adj), residual standard deviation (RSD), and corrected Akaike information criterion (AICc) were used. Based on the best models, stalk height growth rates and crop phenological stages were determined using critical points. All tests were performed in the free software environment for statistical computing and graphics, R. In general, the Logistic and Gompertz models without AR(1) better described the plant-cane and ratoon cane stalk height, respectively. All varieties showed early growth, and the RB92579 variety presented higher rates in both cycles.

RESUMO: A avaliação do crescimento dos colmos de cana-de-açúcar (Saccharum spp.) ajuda a adequar o manejo com as fases fenológicas da cultura. Objetivou-se com este trabalho descrever a curva de crescimento da altura de quatro variedades de cana-de-açúcar (RB92579, RB93509, RB931530 e SP79-1011), nos cultivos de cana planta e cana soca irrigados, utilizando os modelos não lineares Logístico e Gompertz, e considerando eventuais desvios de pressupostos. A estimação dos parâmetros dos modelos foi feita com base no método dos mínimos quadrados utilizando o Algoritmo de Gauss-Newton. Para selecionar o modelo mais adequado foram utilizadas as medidas de não linearidade, o coeficiente de determinação ajustado (R2 aj), o desvio padrão residual (DPR) e o critério de informação de Akaike corrigido (AICc). Com base nos melhores modelos foram determinadas as taxas de crescimento da altura dos colmos e as fases fenológicas da cultura por meio dos pontos críticos. Todas as análises foram realizadas no software estatístico R. No geral, os modelos Logístico e Gompertz sem AR(1) descreveram melhor a altura dos colmos em cana planta e cana soca, respectivamente. Todas as variedades apresentaram crescimento precoce, a variedade RB92579 apresentou maiores taxas em ambos os ciclos.

Reproduction and growth in a Neotropical arboreal mouse: Oecomys rutilus (Sigmodontinae: Cricetidae) in French Guiana / Reprodução e crescimento de um rato arbóreo neotropical: Oecomys rutilus (Sigmodontinae: Cricetidae) na Guiana Francesa

Abstract: Postnatal growth and development of the small Neotropical mouse Oecomys rutilus (Sigmodontinae: Cricetidae) were investigated from birth to day 143, in the laboratory. Morphometric measurements at age of 3 days, of both sexes combined, revealed body weight to be 3.4 ± 0.3 g, mean tail length as 27.4 ± 1.1 mm, and mean hind foot length as 9.3 ± 0.7 mm. Body weight was found to increase steadily until at least 69 days, whereas the instantaneous growth rates of other measurements declined earlier: the daily growth of hind foot length declined to a minimum at age of 24 days, and the growth of tail and of ear declined by the age of 33 days. Average litter size for 12 captive births was 2.5, ranging from 2 to 3. The preserved eye crystalline lens was weighted in 23 captive-born animals of known age, allowing a rough estimate of the age of reproduction in wild-caught animals. Based on the inferred relation between eye-lens weight and age, the youngest reproductive (pregnant) wild-caught females had an estimated age of 90 and 95 days.

Resumo: O crescimento pós-natal e o desenvolvimento do pequeno roedor neotropical Oecomys rutilus (Sigmodontinae: Cricetidae) foram investigados desde o nascimento até o dia 143, em laboratório. Mensurações morfométricas na idade de 3 dias, de ambos os sexos combinados, revelaram a massa corporal de 3,4 ± 0,3 g, comprimento médio da cauda de 27,4 ± 1,1 mm, e comprimento médio do pé de 9,3 ± 0,7 mm. A massa do corpo aumentou rapidamente até o dia 69, enquanto as taxas de crescimento instantâneo de outras medidas diminuíram mais cedo: o crescimento diário do pé diminuiu para o mínimo na idade de 24 dias, e os crescimentos da cauda e da orelha diminuíram na idade de 33 dias. O tamanho de prole médio para 12 nascimentos em cativeiro foi de 2,5, variando de 2 a 3. As lentes preservadas do cristalino do olho foram pesadas em 23 animais nascidos em cativeiro de idades conhecidas, permitindo fazer uma estimativa da idade de reprodução em animais capturados na natureza. Baseado na relação inferida entre peso da lente do cristalino e idade, as mais jovens fêmeas (prenhes) capturadas na natureza tiveram uma estimativa de 90 a 95 dias de idade.

Ajuste dos modelos Gompertz e Logístico aos dados de crescimento de frutos de coqueiro anão verde / The fit Gompertz and Logístic models to the growth data of green dwarf coconut fruits

O objetivo do trabalho foi estudar o ajuste dos modelos não-lineares, gompertz e logístico, na descrição do desenvolvimento de frutos de coqueiro da variedade anão verde, com base nos dados de diâmetros externos, longitudinal e transversal. Os resultados indicaram que tanto para o diâmetro longitudinal externo (DEL) como para o diâmetro transversal externo (DET), o modelo logístico foi o mais adequado na descrição do crescimento do fruto. O ajuste dos modelos, para ambas as variáveis, apresentou autocorrelação residual positiva. Para o DEL o resíduo foi modelado segundo um processo autoregressivo de primeira ordem (AR1) e para a variável DET o processo autoregressivo considerado foi de segunda ordem (AR2). A consideração do processo autoregressivo (AR1 ou AR2) resultou em estimativas dos parâmetros mais precisas. De acordo com o modelo logístico, o fruto do coqueiro anão verde, totalmente desenvolvido, tem diâmetros externos longitudinal e transversal, respectivamente, de 21,56cm e 15,35cm.

The aim of this study was to evaluate the quality fit of nonlinear models, gompertz and logistic data outside diameter, longitudinal (DEL) and transverse (DET) fruits of green dwarf coconut trees, as well as verification of the existence of residual autocorrelation. The results showed that for both, DEL and DET, the logistic model is most appropriate for describing the fruits growth. The fit of the models, for both variables, showed positive residual autocorrelation. For DEL, the residual was modeled by a first order autoregressive process, and for DET, the autoregressive process was considered of second order. Consideration of the autoregressive process resulted in more accurate estimates of the parameters. According to the logistic model, the fruit of the dwarf green, fully developed, have external diameters, longitudinal and transverse, respectively, of 21.56cm and 15.35cm.

Population Forecasts for Bangladesh, Using a Bayesian Methodology.

Population projection for many developing countries could be quite a challenging task for the demographers mostly due to lack of availability of enough reliable data. The objective of this paper is to present an overview of the existing methods for population forecasting and to propose an alternative based on the Bayesian statistics, combining the formality of inference. The analysis has been made using Markov Chain Monte Carlo (MCMC) technique for Bayesian methodology available with the software WinBUGS. Convergence diagnostic techniques available with the WinBUGS software have been applied to ensure the convergence of the chains necessary for the implementation of MCMC. The Bayesian approach allows for the use of observed data and expert judgements by means of appropriate priors, and a more realistic population forecasts, along with associated uncertainty, has been possible.