Gamma Distribution Model Research Articles

Gamma, Gaussian and logistic distribution models for airborne pollen grains and fungal spore season dynamics.

The characteristics of a pollen season, such as timing and magnitude, depend on a number of factors such as the biology of the plant and environmental conditions. The main aim of this study was to develop mathematical models that explain dynamics in atmospheric concentrations of pollen and fungal spores recorded in Rzeszów (SE Poland) in 2000–2002. Plant taxa with different characteristics in the timing, duration and curve of their pollen seasons, as well as several fungal taxa were selected for this analysis. Gaussian, gamma and logistic distribution models were examined, and their effectiveness in describing the occurrence of airborne pollen and fungal spores was compared. The Gaussian and differential logistic models were very good at describing pollen seasons with just one peak. These are typically for pollen types with just one dominant species in the flora and when the weather, in particular temperature, is stable during the pollination period. Based on s parameter of the Gaussian function, the dates of the main pollen season can be defined. In spite of the fact that seasonal curves are often characterised by positive skewness, the model based on the gamma distribution proved not to be very effective.

Indoor Exposure to Particulate Matter and Age at First Acute Lower Respiratory Infection in a Low-Income Urban Community in Bangladesh

The timing of a child's first acute lower respiratory infection (ALRI) is important, because the younger a child is when he or she experiences ALRI, the greater the risk of death. Indoor exposure to particulate matter less than or equal to 2.5 µm in diameter (PM2.5) has been associated with increased frequency of ALRI, but little is known about how it may affect the timing of a child's first ALRI. In this study, we aimed to estimate the association between a child's age at first ALRI and indoor exposure to PM2.5 in a low-income community in Dhaka, Bangladesh. We followed 257 children from birth through age 2 years to record their age at first ALRI. Between May 2009 and April 2010, we also measured indoor concentrations of PM2.5 in children's homes. We used generalized gamma distribution models to estimate the relative age at first ALRI associated with the mean number of hours in which PM2.5 concentrations exceeded 100 µg/m(3). Each hour in which PM2.5 levels exceeded 100 µg/m(3) was independently associated with a 12% decrease (95% confidence interval: 2, 21; P = 0.021) in age at first ALRI. Interventions to reduce indoor exposure to PM2.5 could increase the ages at which children experience their first ALRI in this urban community.

Multi-Mode Failure Models for Attribute Test Data in Reliability Systems, a Bayesian Analysis Approach Using Multi-Nomial Distribution Model

This paper describes the extending model of Multi-mode Failure Models by using the Weibull and Gamma distribution models presented in a conference [1,2]. Different than the models in the previous papers which are for variable test data, in this paper we will describe the use of attribute test data for our model. In reliability theory, the most important problem is to determine the reliability of a complex system from the reliability of its components. The weakness of most reliability theories is that the systems are described and explained as simply functioning or failed. In many real situations, the failures may be from many causes depending upon the age and the environment of the system and its components. Another problem in reliability theory is one of estimating the parameters of the assumed failure models. The estimation may be based on data collected over censored or uncensored life tests. In many reliability problems, the failure data are simply quantitatively inadequate. The Bayesian analyses are more beneficial than classical analyses in such cases. The Bayesian estimation analyses allow us to combine past knowledge or experience in the form of an apriori distribution with life test data to make inferences of the parameter of interest. In this paper, we have investigated the application of the Bayesian estimation analyses to multi-mode failure systems for attribute test data. The cases are limited to the models with independent causes of failure. We select our investigation by using the Multi-nomial distribution as our model. This distribution is widely used in reliability analysis for attribute test data. This model describes the likelihood function and follows with the description of the posterior function. A Beta prior is used in our analysis for each model and it is followed by the estimation of the point, interval, and reliability estimations.

Application of a novel type impinging streams reactor in solid–liquid enzyme reactions and modeling of residence time distribution using GDB model

Solid–liquid enzyme reactions constitute important processes in biochemical industries. The isomerization of d-glucose to d-fructose, using the immobilized glucose isomerase (Sweetzyme T), as a typical example of solid–liquid catalyzed reactions has been carried out in one stage and multi-stage novel type of impinging streams reactors. Response surface methodology was applied to determine the effects of certain pertinent parameters of the process namely axial velocity (A), feed concentration (B), nozzles’ flow rates (C) and enzyme loading (D) on the performance of the apparatus. The results obtained from the conversion of glucose in this reactor were much higher than those expected in conventional reactors, while residence time was decreased dramatically. Residence time distribution (RTD) in a one-stage impinging streams reactor was investigated using colored solution as the tracer. The results showed that the flow pattern in the reactor was close to that in a continuous stirred tank reactor (CSTR). Based on the analysis of flow region in the reactor, gamma distribution model with bypass (GDB) was applied to study the RTD of the reactor. The results indicated that RTD in the impinging streams reactor could be described by the latter model.

Characterization of knock intensity distributions: Part 2: parametric models

Knock intensity behaves as a random process which may be characterized using simple scalar metrics such as the mean and variance, or (more commonly) by the probability of knock events. However, such measures discard much of the information present in the signal. Several researchers have therefore sought to obtain a more complete characterization of the process by fitting parametric log-normal or gamma distribution models to knock intensity distributions. The present study extends this work both in terms of the range of engine operating conditions considered and in terms of the evaluation of the goodness of fit between two different models and the experimental data. In particular, new and arguably more application-appropriate measures of the goodness of fit provide a clearer assessment of the performance of the models, and a like-for-like comparison of log-normal and gamma distribution model forms demonstrates that the log-normal model better characterizes the experimental data used in this study.

A Comparison of Models to Infer the Distribution of Fitness Effects of New Mutations

Knowing the distribution of fitness effects (DFE) of new mutations is important for several topics in evolutionary genetics. Existing computational methods with which to infer the DFE based on DNA polymorphism data have frequently assumed that the DFE can be approximated by a unimodal distribution, such as a lognormal or a gamma distribution. However, if the true DFE departs substantially from the assumed distribution (e.g., if the DFE is multimodal), this could lead to misleading inferences about its properties. We conducted simulations to test the performance of parametric and nonparametric discretized distribution models to infer the properties of the DFE for cases in which the true DFE is unimodal, bimodal, or multimodal. We found that lognormal and gamma distribution models can perform poorly in recovering the properties of the distribution if the true DFE is bimodal or multimodal, whereas discretized distribution models perform better. If there is a sufficient amount of data, the discretized models can detect a multimodal DFE and can accurately infer the mean effect and the average fixation probability of a new deleterious mutation. We fitted several models for the DFE of amino acid-changing mutations using whole-genome polymorphism data from Drosophila melanogaster and the house mouse subspecies Mus musculus castaneus. A lognormal DFE best explains the data for D. melanogaster, whereas we find evidence for a bimodal DFE in M. m. castaneus.

THE CRITICAL DIAMETERS FOR RAINFALL ATTENUATION IN SOUTHERN AFRICA

This paper investigates the in∞uence of critical raindrop diameters on the speciflc rain attenuation in Durban (29 - 52 0 S, 30 - 58 0 E), South Africa. The total rainfall attenuation is evaluated by integrating over all the raindrop sizes and the difierential change in the attenuation is observed over a given range of drop size diameters. The major contribution to the speciflc attenuation for the drop size distribution (DSD) models considered is created by the raindrop diameters not exceeding 2mm especially at higher frequencies. The three parameter lognormal and gamma distribution models are employed for the purpose of analysis. For the DSD models considered in this work, the total percentage fraction created by raindrops in the diameter range 0:5mmD • 2:5mm and 1mmD • 3mm to the total speciflc attenuation is found to be critical for the overall and seasonal rainfall attenuation at 2.5GHz{ 100GHz in Durban. It is observed that the total speciflc attenuation increases with increased frequencies and a higher rainfall rate produces high rain attenuation. In this paper, both the overall and seasonal values of R0:01 determined for Durban are used.

OBTAINING RAINDROP SIZE MODEL USING METHOD OF MOMENT AND ITS APPLICATIONS FOR SOUTH AFRICA RADIO SYSTEMS

In this paper, the Raindrop Size Distribution (DSD) modeling and analysis are presented. Drop sizes are classifled into difierent rain types, namely: drizzle, widespread, shower and thunderstorm. The gamma and Lognormal distribution models are employed using the method of moments estimator, considering the third, fourth and sixth order moments. The results are compared with the existing raindrop size distribution models such as the three parameter lognormal distribution proposed by Ajayi and his colleagues and Singapore's modifled gamma and Lognormal models. This is then followed by the implementation of the proposed raindrop size distribution models on the computation of the speciflc rain attenuation. Finally, the paper suggests a suitable raindrop size distribution model for the region with its expressions. The proposed models are very useful for the determination of rain attenuation for terrestrial and satellite systems.

Gafchromic EBT2 film dosimetry in reflection mode with a novel plan‐based calibration method

A dosimetric system formed by Gafchromic EBT2 radiochromic film and Epson Expression 10000XL flatbed scanner was commissioned for dosimetry. In this paper, several open questions concerning the commissioning of radiochromic films for dosimetry were addressed: (a) is it possible to employ this dosimetric system in reflection mode; (b) if so, can the methods used in transmission mode also be used in reflection mode; (c) is it possible to obtain accurate absolute dose measurements with Gafchromic EBT2 films; (d) which calibration method should be followed; (e) which calibration models should be used; and (f) does three-color channel dosimetry offer a significant improvement over single channel dosimetry. The purpose of this paper is to help clarify these questions. In this study, films were scanned in reflection mode, the effect of surrounding film was evaluated and the feasibility of EBT2 film dosimetry in reflection mode was studied. EBT2's response homogeneity has been reported to lead to excessive dose uncertainties. To overcome this problem, a new plan-based calibration method was implemented. Plan-based calibration can use every pixel and each of the three color channels of the scanned film to obtain the parameters of the calibration model. A model selection analysis was conducted to select lateral correction and sensitometric curve models. The commonly used calibration with fragments was compared with red-channel plan-based calibration and with three-channel plan-based calibration. No effect of surrounding film was found in this study. The film response inhomogeneity in EBT2 films was found to be important not only due to differences in the fog but also due to differences in sensitivity. The best results for lateral corrections were obtained using absolute corrections independent of the dose. With respect to the sensitometric curves, an empirical polynomial fit of order 4 was found to obtain results equivalent to a gamma-distributed single hit model based on physical assumptions. Three-channel dosimetry was found to be substantially superior to red-channel dosimetry. Reflection mode with Gafchromic EBT2 radiochromic film was found to be a viable alternative to transmission mode. The same methods that are used in transmission mode can be followed in reflection mode. A novel plan-based method was developed for calibration and multichannel dosimetry. This novel method offers increased robustness against film response inhomogeneities and reduces considerably the time required for calibration.

Automatic Segmentation of Antenatal 3-D Ultrasound Images

The development of 3-D ultrasonic probes and 3-D ultrasound (3DUS) imaging offers new functionalities that call for specific image processing developments. In this paper, we propose an original method for the segmentation of the utero-fetal unit (UFU) from 3DUS volumes, acquired during the first trimester of gestation. UFU segmentation is required for a number of tasks, such as precise organ delineation, 3-D modeling, quantitative measurements, and evaluation of the clinical impact of 3-D imaging. The segmentation problem is formulated as the optimization of a partition of the image into two classes of tissues: the amniotic fluid and the fetal tissues. A Bayesian formulation of the partition problem integrates statistical models of the intensity distributions in each tissue class and regularity constraints on the contours. An energy functional is minimized using a level set implementation of a deformable model to identify the optimal partition. We propose to combine Rayleigh, Normal, Exponential, and Gamma distribution models to compute the region homogeneity constraints. We tested the segmentation method on a database of 19 antenatal 3DUS images. Promising results were obtained, showing the flexibility of the level set formulation and the interest of learning the most appropriate statistical models according to the idiosyncrasies of the data and the tissues. The segmentation method was shown to be robust to different types of initialization and to provide accurate results, with an average overlap measure of 0.89 when comparing with manual segmentations.

Empirical Bayesian regression model for estimation of small rates

The efficiency of adding an auxiliary regression variable to the logit model in estimation of small probabilities in large populations is considered. Let us consider two models of distribution of unknown probabilities: the probabilities have gamma distribution (model (A)), or logits of the probabilities have Gaussian distribution (model (B)). In modification of model (B) we will use additional regression variable for Gaussian mean (model (BR)). We have selected real data from Database of Indicators of Statistics Lithuania – Working-age persons recognized as disabled for the first time by administrative territory, year 2010 (number of populations K = 60). Additionally, we have used average annual population data by administrative territory. The auxiliary regression variable was based on data – Number of hospital discharges by administrative territory, year 2010. We obtained initial parameters using simple iterative procedures for models (A), (B) and (BR). At the second stage we performed various tests using Monte-Carlo simulation (using models (A), (B) and (BR)). The main goal was to select an appropriate model and to propose some recommendations for using gamma and logit (with or without auxiliary regression variable) models for Bayesian estimation. The results show that a Monte Carlo simulation method enables us to determine which estimation model is preferable.

Stationary-Wavelet-Based Despeckling of SAR Images Using Two-Sided Generalized Gamma Models

In this letter, a stationary-wavelet-based despeckling algorithm based on the two-sided generalized gamma distribution (GΓD) model is proposed. We first introduce the two-sided GΓD as a flexible and efficient model for the wavelet coefficients of logarithmically transformed synthetic aperture radar intensity or amplitude. The strength of the model is highlighted in terms of its fit to the data, its low computational cost, and the ease of parameter estimation. By empirical results, we then motivate the GΓD as model for the wavelet coefficients of the noise-free signal. The GΓD model parameters are estimated with moment methods, using both absolute central moments for the wavelet coefficients of the noisy signal and the noise. Finally, we exploit the prior information contained in the model by designing a Bayesian maximum a posteriori estimator for estimating the noise-free wavelet coefficients. Experimental results demonstrate the superiority of our method in terms of simultaneously reducing speckle and preserving structural details.

The gamma distribution model for pulsed-field gradient NMR studies of molecular-weight distributions of polymers

Self-diffusion in polymer solutions studied with pulsed-field gradient nuclear magnetic resonance (PFG NMR) is typically based either on a single self-diffusion coefficient, or a log-normal distribution of self-diffusion coefficients, or in some cases mixtures of these. Experimental data on polyethylene glycol (PEG) solutions and simulations were used to compare a model based on a gamma distribution of self-diffusion coefficients to more established models such as the single exponential, the stretched exponential, and the log-normal distribution model with regard to performance and consistency. Even though the gamma distribution is very similar to the log-normal distribution, its NMR signal attenuation can be written in a closed form and therefore opens up for increased computational speed. Estimates of the mean self-diffusion coefficient, the spread, and the polydispersity index that were obtained using the gamma model were in excellent agreement with estimates obtained using the log-normal model. Furthermore, we demonstrate that the gamma distribution is by far superior to the log-normal, and comparable to the two other models, in terms of computational speed. This effect is particularly striking for multi-component signal attenuation. Additionally, the gamma distribution as well as the log-normal distribution incorporates explicitly a physically plausible model for polydispersity and spread, in contrast to the single exponential and the stretched exponential. Therefore, the gamma distribution model should be preferred in many experimental situations.

Hydrodynamics of a Laboratory Scale Rotating Biological Contactor and Its Application for Decolorization of Textile Dyes by White Rot Fungus Irpex Lacteus

Textile industries produce huge amounts of colored wastewaters contaminated with various textile dyes. These waters represent serious environmental problem as common wastewater treatment procedures are not efficient to degrade the dyes. Therefore, new ways of textile wastewater treatment are looked for in order to tackle this environmental problem. An application of the white rot fungus Irpex lacteus, that is known to degrade many recalcitrant pollutants, seems to be one of possible options for the waste water decolorization. A laboratory scale Rotating Biological Contactor was constructed to perform the decolorization tests. The reactor vas equipped with 12 flat circular discs (o = 0.13 m, ?? = 0.01 m) mounted on a common shaft. The discs were made of the Filtren TM30 reticulated polyether foam. Volume of the liquid in the reactor was 1.5 dm3. Residence time distributions of the liquid phase in the reactor were measured to obtain hydrodynamic characteristics. The measurements were carried out at various liquid volumetric flow rates and at different rotational speeds of the discs uncovered with the biomass. The measured data were used to determine the liquid mean residence time and parameters of the gamma distribution model and of the model of the stirred tanks in series with the back-flow. The residence time distributions were also measured after 36 days of the reactor operation, i.e., with the discs covered with the Irpex lacteus mycelium biofilm. The flow behavior of the reactor without the fungus was close to the ideally stirred reactor behavior. The presence of the biomass in the reactor brought about deviations from the ideal mixing. Continuous decolorization experiments with the textile dye Remazol Brilliant Blue R and with a sample of an industrial textile waste water were performed. In the course of the experiments, activities of the enzymes involved in the dye decolorization, i.e., laccase, manganese-dependent peroxidase and lignin peroxidase were determined in the outlet stream of the reactor. The inlet and outlet concentrations of glucose were also measured. The results of the decolorization experiments will be presented and mutually compared and effects of hydrodynamic parameters on decolorization will be discussed.

Distribuição e probabilidade de ocorrência de precipitação em Cáceres (MT)

A precipitação é um elemento meteorológico de grande importância, e seu conhecimento histórico torna-se relevante para o monitoramento de impactos causados pelo seu excesso ou falta prolongados. Objetivou-se, no presente trabalho, analisar a distribuição da precipitação mensal e anual e os níveis de probabilidade de ocorrência de chuvas, utilizando-se o modelo probabilístico de distribuição Gama incompleta, para os dados de precipitação pluviométrica de Cáceres (MT). Utilizou-se uma série de dados de 26 anos, disponibilizada pelo Instituto Nacional de Meteorologia, o qual possui uma estação em Cáceres. A precipitação anual teve grande variabilidade, com mínima de 972,9 mm, em 1985, e máxima de 1.624,1 mm, em 1998. Observou-se, por meio dos dados mensais, que existem duas estações bem definidas, a seca (maio a setembro) e a chuvosa (outubro a abril). As menores médias mensais ocorreram nos meses de junho, julho e agosto, com 16,59 mm, 17,90 mm e 20,09 mm, respectivamente. As estimativas do parâmetro α variaram de 0,9, em junho e agosto, a 13,4, em março. O parâmetro β variou de 13,2, em março, a 33,1, em janeiro. O período com maior probabilidade de precipitação vai de dezembro a março, enquanto junho, julho e agosto foram os meses em que a probabilidade de ocorrência de precipitação foi mais baixa.

Comparison of six particle size distribution models on the goodness-of-fit to particulate matter sampled from animal buildings

Lognormal distribution is often used as a default model for regression analysis of particle size distribution (PSD) data; however, its goodness-of-fit to particle matter (PM) sampled from animal buildings and its comparison to other PSD models have not been well examined. This study aimed to evaluate and to compare the goodness-of-fit of six PSD models to total suspended particulate matter (TSP) samples collected from 15 animal buildings. Four particle size analyzers were used for PSD measurement. The models' goodness-of-fit was evaluated based on adjusted R 2, Akaike's information criterion (AIC), and mean squared error (MSE) values. Results showed that the models' approximation of measured PSDs differed with particle size analyzer. The lognormal distribution model offered overall good approximations to measured PSD data, but was inferior to the gamma and Weibull distribution models when applied to PSD data derived from the Horiba and Malvern analyzers. Single-variable models including the exponential, Khrgian-Mazin, and Chen's empirical models provided relatively poor approximations and, thus, were not recommended for future investigations. A further examination on model-predicted PSD parameters revealed that even the best-fit model of the six could significantly misestimate mean diameter, median diameter, and variance. However, compared with other models, the best-fit model still offered the relatively best estimates of mean and median diameters, whereas the best predicted variances were given by the gamma distribution model. Implications Particulate matter from animal buildings contains a large portion of coarse particles and, thus, has different particle size distribution (PSD) than ambient aerosols. The findings suggest a default use of the prevalent lognormal distribution model can lead to significant misestimates of PSD parameters such as mean diameter, median diameter, and variance. Other models, such as the gamma and Weibull distribution models, may be considered as alternative options. Because of a complexity in the shape of measured PSD profiles, the goodness-of-fit for each of the different PSD models must be compared before selecting one for data regression, parameter estimation, and reporting.

Syndromic surveillance models using Web data: The case of scarlet fever in the UK

Recent research has shown the potential of Web queries as a source for syndromic surveillance, and existing studies show that these queries can be used as a basis for estimation and prediction of the development of a syndromic disease, such as influenza, using log linear (logit) statistical models. Two alternative models are applied to the relationship between cases and Web queries in this paper. We examine the applicability of using statistical methods to relate search engine queries with scarlet fever cases in the UK, taking advantage of tools to acquire the appropriate data from Google, and using an alternative statistical method based on gamma distributions. The results show that using logit models, the Pearson correlation factor between Web queries and the data obtained from the official agencies must be over 0.90, otherwise the prediction of the peak and the spread of the distributions gives significant deviations. In this paper, we describe the gamma distribution model and show that we can obtain better results in all cases using gamma transformations, and especially in those with a smaller correlation factor.

Mammalian protein expression noise: scaling principles and the implications for knockdown experiments

The abundance of a particular protein varies both over time within a single mammalian cell and between cells of a genetically identical population. Here, we investigate the properties of such noisy protein expression in mammalian cells by combining theoretical and experimental approaches. The gamma distribution model is well-known to describe cell-to-cell variability in protein expression in a variety of common scenarios. This model predicts, and experiments show, that when protein levels are manipulated by altering transcription rates or mRNA half-life, protein expression noise, defined as the squared coefficient of variation, is constant. In contrast, we also demonstrate that when protein levels are manipulated by changing protein half-life, as mean levels increase, noise decreases. Thus, in mammalian cells, the scaling relationship between mean protein levels and expression noise depends on how mean levels are perturbed. Therefore it may be important to consider how common experimental manipulations of protein expression affect not only mean levels, but also noise levels. In the context of knockdown experiments, natural cell-to-cell variability in protein expression implies that a particular cell from the knockdown population may have higher protein levels than a cell from the control population. Simulations and experimental data suggest that approximately three-fold knockdown in mean expression levels can reduce such so-called "overlap probability" to less than ~10%. This has implications for the interpretation of knockdown experiments when the readout is a single cell measure.

A novel approach to motion segmentation based on gamma distribution

We provide a new motion segmentation method in image sequences based on gamma distribution. Motion segmentation is very important because it can be employed for video surveillance, object tracking, and action recognition. The Gaussian mixture model (GMM) has been widely used as a distribution model for motion segmentation. However, we found that the gamma distribution model is more suitable than the GMM for the optical flow based motion segmentation. Experimental results show that the proposed method is very effective in producing accurate motion segmentation results in image sequences.

Hydraulic modelling of horizontal-subsurface flow constructed wetlands: Influence of operation time and plant species

Hydraulic behaviour is a very important aspect in the design of a constructed wetland (CW). Different hydraulic models have been widely applied to obtain a better understanding of CW flow properties and provide tools that optimise the design of constructed wetlands as wastewater treatments. This work studied the effects of the time of operation and the plant species used on the hydraulic characteristics of horizontal subsurface-flow (HSSF) CWs. The plug-flow with dispersion (PFD) model, the detention-time gamma-distribution (DTGD) model and a newly developed model, the multi-flow detention-time gamma-distribution (MFDTGD) model, were applied for modelling the water flow over different operating periods in five pilot-scale constructed wetlands planted with different plant species: CW1, unplanted; CW2, Phragmites australis; CW3, Lythrum salicaria; CW4, Cladium mariscus; and CW5, Iris pseudacorus. The PFD model did not provide a good fit to the experimental effluent tracer concentrations, while the DTGD model provided a satisfactory fit and the MFDTGD model, a very good fit, enabling the differentiation of several flow pathways. When using the MFDTGD model, three different types of pathways could be observed: a ‘nominal’ pathway, a ‘tail’ pathway and a ‘preferential’ pathway. However, the high number of parameters needed by the MFDTGD model reduces the validity of this model for use as a design tool. The values obtained for the parameters of each model were in accordance with those in previously reported studies. Regarding the effects of the operation time and the plant species, as time increased, the N-values tended to increase, i.e. the systems tended to behave increasingly like an ideal plug-flow reactor, especially in CWs with more developed plant species (Iris pseudacorus in CW5 and Lythrum salicaria in CW3). Globally, the CWs tended towards more homogeneous distributions of the flow, probably due to biofilm growth and plant root development.