Gamma Shape Research Articles

Interference length reveals regularity of crossover placement across species

Crossover interference is a phenomenon that affects the number and positioning of crossovers in meiosis and thus affects genetic diversity and chromosome segregation. Yet, the underlying mechanism is not fully understood, partly because quantification is difficult. To overcome this challenge, we introduce the interference length Lint that quantifies changes in crossover patterning due to interference. We show that it faithfully captures known aspects of crossover interference and provides superior statistical power over previous measures such as the interference distance and the gamma shape parameter. We apply our analysis to empirical data and unveil a similar behavior of Lint across species, which hints at a common mechanism. A recently proposed coarsening model generally captures these aspects, providing a unified view of crossover interference. Consequently, Lint facilitates model refinements and general comparisons between alternative models of crossover interference.

Sample size considerations for single-arm clinical trials with time-to-event endpoint using the gamma distribution

BackgroundTime-to-event (TTE) endpoints are evaluated as the primary endpoint in single-arm clinical trials; however, limited options are available in statistical software for sample size calculation. In single-arm trials with TTE endpoints, the non-parametric log-rank test is commonly used. Parametric options for single-arm design assume survival times follow exponential distribution or Weibull distribution. MethodsThe exponential- or Weibull-distributed survival time assumption does not always reflect hazard pattern of real-life diseases. We therefore propose gamma distribution as an alternative parametric option for designing single-arm studies with TTE endpoints. We outline a sample size calculation approach using gamma distribution with a known shape parameter and explain how to extract the gamma shape estimate from previously published resources. In addition, we conduct simulations to assess the accuracy of the extracted gamma shape parameter and to explore the impact on sample size calculation when survival time distribution is misspecified. ResultsOur simulations show that if a previously published study (sample sizes ≥ 60 and censoring proportions ≤ 20 %) reported median and inter-quartile range of survival time, we can obtain a reasonably accurate gamma shape estimate, and use it to design new studies. When true survival time is Weibull-distributed, sample size calculation could be underestimated or overestimated depending on the hazard shape. ConclusionsWe show how to use gamma distribution in designing a single-arm trial, thereby offering more options beyond the exponential and Weibull. We provide a simulation-based assessment to ensure an accurate estimation of the gamma shape and recommend caution to avoid misspecification of the underlying distribution.

Generalized Bayesian Additive Regression Trees Models: Beyond Conditional Conjugacy

Bayesian additive regression trees have seen increased interest in recent years due to their ability to combine machine learning techniques with principled uncertainty quantification. The Bayesian backfitting algorithm used to fit BART models, however, limits their application to a small class of models for which conditional conjugacy exists. In this article, we greatly expand the domain of applicability of BART to arbitrary generalized BART models by introducing a very simple, tuning-parameter-free, reversible jump Markov chain Monte Carlo algorithm. Our algorithm requires only that the user be able to compute the likelihood and (optionally) its gradient and Fisher information. The potential applications are very broad; we consider examples in survival analysis, structured heteroskedastic regression, and gamma shape regression.

On the Influence of Sampling Scale on the In Situ Block Size Distribution

The modelling of discontinuities in rock mass is undertaken with different measurement techniques and used to determine the in situ block size distribution (IBSD). Two monitoring techniques are employed: televiewer logging of boreholes and photogrammetry of highwall faces in a quarry bench; televiewer performs at the borehole diameter scale, while photogrammetry surveys at the entire bench scale. Ground sampling distances were, respectively, about 1 and 8.5 mm. The discontinuities are modelled as a stochastic discrete fracture network (DFN), with the number of discontinuities used in the simulation calibrated by the intensity per unit length (P10) on the televiewer data, or by the fracture density (P21) on the photogrammetry data, leading to different fracture networks. From the discontinuity network models, the IBSDs are calculated and discussed as function of the sampling scale (i.e. televiewer or photogrammetry data source) and of the fracture density. The goal is to compare the results from both techniques for rock mass structural characterization, to assess their limitations and shortcomings, and to show their potential complementarity at different sampling scales. The televiewer data provides smaller block sizes than the photogrammetry, following the higher number of fractures observed in the former. All volumetric distributions obtained are extremely well represented by Gamma with a power law tail distribution. Despite different location parameters, it is particularly remarkable that all distributions present very similar Gamma shape parameters. The constant log–log slopes of the tails provide evidence of multi-scale validity and a scaling invariant structure (more than two orders of magnitude) of discontinuities of the rock mass. The IBSDs and the scale effect are discussed in the light of the fragment size distributions from blasts carried out in the area characterized.

High-Precision Calibration of Phase-Only Spatial Light Modulators

In the fields of optics and photonics, phase-only spatial light modulators (SLMs) play an increasingly important role in wave-front engineering. However, the SLMs are subject to wavefront distortion arising from the imperfection in the birefringence effect and physical structure of modulators. This paper presents a simple self-interference phase calibration method applicable to liquid-crystal SLM. We build an interferometric imaging system based on the Pancharatnam phase-shifting to measure the phase distribution of a light beam coming from SLMs. Two types of phase modulation errors of SLMs can be characterized in the measurement process: the erroneous gamma curve and shape aberration. The former belongs to dynamic phase distortion and is measured through a four-step Pancharatnam phase-shifting interference, which allows a one-shot recording of interference pattern via a polarization camera; the latter represents static phase distortion and is extracted from the interference between light waves coming from different regions of the SLM panel by using Zernike polynomial fitting. Our method has the advantages of high-precision pixel-wise phase correction and robustness against environmental disturbance and thus can facilitate the applications of SLM in optical field manipulation.

Phylogenetic tree analysis for Bali Cattle based on partial sequence 16S rRNA Mitochondrial DNA

Mitochondria DNA (mtDNA) as a source of genetic information based on the maternal genome, can provide important information for phylogenetic analysis and evolutionary biology. The objective of this study was to analyze the phylogenetic tree of Bali cattle with seven gene bank references (Bos indicus, Bos taurus, Bos frontalis, andBos grunniens) based on partial sequence 16S rRNA mitochondria DNA. The Bayesian phylogenetic tree was constructed using BEAST 2.4. and visualization in Figtree 1.4.4 (tree.bio.ed.ac.uk/software/figtree/). The best model of evolution was carried out using jModelTest 2.1.7. The most optimal was the evolutionary models GTR + I + G with p-inv (I) 0,1990 and gamma shape 0.1960. The main result indicated that the Bali cattle were grouped intoBos javanicus. Phylogenetic analysis also successfully classifyingBos javanicus, Bos indicus, Bos taurus, Bos frontalisandBos grunniens. These results will complete information about Bali cattle and useful for the preservation and conservation strategies of Indonesian animal genetic resources.

Evaluation of the AROME model's ability to represent ice crystal icing using in situ observations from the HAIC 2015 field campaign

AbstractSince pilots generally avoid intense convective areas, ice crystals icing (ICI) is an aeronautical weather incident that mainly occurs in the anvil of tropical deep convective clouds. Samples of favorable conditions for the occurrence of ICI and data from the High Altitude Ice Crystals (HAIC) 2015 field campaign in French Guiana are investigated and compared with simulations of the French operational mesoscale forecast system Application of Research to Operations at Mesoscales (AROME). To this end, a contextualization of convective systems into convective, stratiform, and cirriform regions is employed for both observations and AROME. General features of the microphysics of deep tropical convective systems are identified. The number concentration of crystals larger than 125 μm and total water content (TWC) are strongly correlated at each temperature level, and both decrease with increasing distance from convective cores. AROME can reproduce the general behavior of the observed microphysics, especially TWC, but seems unable to simulate extreme ICI events. Reasons are sought in the assumptions performed in the microphysical scheme ICE3, and guidelines are proposed to enhance its skills in the context of ICI. In particular, the representation of the snow particle size distribution is adjusted across observations using a generalized gamma shape. This shape is found to outperform the usual Marshall–Palmer and gamma shapes. Additionally, a temperature and snow content dependence of generalized gamma parameters is found. These changes are found to significantly improve the snow concentration diagnostic of ICE3, and these modifications open the way for improvements in the ICE3 scheme.

Spatial distribution of fungi from the analysis of aerobiological data with a gamma function

Fungi are an important component of ecosystems. Some fungi are widely distributed, while others are limited to certain habitats. Studies based on airborne fungal spores can help to know the geographical distribution of fungi in the territory. Our aim was to show that a gamma probability density function (gpdf) based on a database of 20 airborne fungal spore taxa concentrations in eight localities of Catalonia (NE Spain) for a period of 20 years was a useful tool to map the distribution of these taxa in this region, as well as to establish a general classification on their sporulation through the alpha parameter of the validated model. This allows a more efficient study of the atmospheric dynamics of the different taxa, since the number of taxa is reduced to a representative taxon for each of the categories of the generic classification. In general, the results obtained confirmed that the scale parameter of the gamma distribution changes from year to year, depending on the meteorological conditions, while the shape parameter remains fairly stable. At the temporal scale, airborne fungal spores of Agrocybe sp. showed the highest stability; at the spatial scale, Cladosporium sp. showed the highest stability. Regarding localities, Girona was the station with greater interannual variation, while Barcelona and Vielha showed the lowest. In addition, the results obtained allowed a non-subjective classification of these taxa in five groups, based on the gamma (shape) parameter. The taxa Alternaria sp., Cladosporium sp., Ganoderma sp., Pleospora sp., Leptosphaeria sp., Aspergillus sp.-Penicillium sp. were cosmopolitan and showed a similar behavior across the whole study area, with any of them possible candidates for used in predictive models; airborne fungal spores of Agrocybe sp., Arthrinium sp., Epicoccum sp., Drechslera sp.–Helminthosporium sp., Pithomyces sp., Thelephoraceae, Stemphylium sp., Xylariaceae can be used as meteorological indicators and Agaricus sp., Coprinaceae sp., Torula sp. can be used as indicators of anthropogenic activities. The results obtained could be used to reduce the number of spore taxa analyzed and subsequently develop generic predictive models.

Characteristics of Raindrop Size Distribution on the Eastern Slope of the Tibetan Plateau in Summer

Precipitation microphysics over the Tibetan Plateau (TP) remain insufficiently understood, due to the lack of observations and studies. This paper presents a comprehensive investigation of the raindrop size distribution (DSD) for rainfall that happened on the eastern slope of TP in summer. DSD differences between different rain types and under different rain rates are investigated. Confidential empirical relationships between the gamma shape and slope parameters, and between reflectivity and rain rate are proposed. DSD properties in this area are also compared with those in other areas. The results indicate that the stratiform and convective rains contribute to different rain duration and amount, with diverse rainfall macro- and microphysical properties. The rain spectra of two rain types can become broader with higher concentrations as the rain rate increases. DSDs in this area are different to those in other areas. The stratiform DSD is narrower than that in the non-plateau area. The two rain types of this area both have higher number concentrations for 0.437–1.625 mm raindrops than those of the mid-TP. The relationships of shape–slope parameters and reflectivity–rain rate in this area are also different from those in other areas. The rain spectra in this area can produce a larger slope parameter under the same shape parameter than in the mid-TP. The convective rain can produce a smaller rain rate under the same reflectivity. The accuracy proposed reflectivity–rain rate relationship in application to quantitative rainfall estimation is also discussed. The results show that the relationship has an excellent performance when the rain rate exceeds 1 mm h−1.

Three-parameter bivariate gamma regression model for analyzing under-five mortality rate and maternal mortality rate

Gamma regression is often used to model continuous, right-skewed and strictly positive data. This paper discusses three-parameter gamma regression, namely scale, shape, and location parameter, with two correlated response variables. The purpose of this study is to determine the estimator parameters, perform the testing for the parameters using Maximum Likelihood Ratio Test (MLRT) and its application in the cases of the Under-five Mortality Rate (U5MR) and the Maternal Mortality Rate (MMR) in North Sulawesi, Gorontalo, Central Sulawesi Provinces in 2016. The parameter estimation in this global model obtained through MLE. Nevertheless, the results showed an unsolvable equation in closed-form, then we used numerical optimization. In this study, we were using the Berndt-Hall-Hall-Hausman optimization method. After the estimation results are obtained, the estimation parameters need to be tested with a simultaneous test with the MLRT,while for test partially, we used the Z test. Moreover, the factors that influence U5MR and MMR are the poor population, the obstetric complications handled, the mothers received Fe3 during pregnancy, the teenage pregnancy,the children under-five received vitamin A and the households with clean and healthy life behavior.

On Use of the Standard Deviation of the Mass Distribution as a Parameter in Raindrop Size Distribution Functions

AbstractUse of the standard deviation σm of the drop mass distribution as one of the three parameters of raindrop size distribution (DSD) functions was introduced for application to disdrometer data supporting the Global Precipitation Measurement dual-frequency radar system. The other two parameters are a normalized drop number concentration Nw and the mass-weighted mean diameter Dm. This paper presents an evaluation of that formulation of the DSD functions, in two parts. First is a mathematical analysis showing that the procedure for estimating σm, along with the other DSD parameters, from disdrometer data is in essence another moment method. As such, it is subject to the biases and errors inherent in all moment methods. When the form of the DSD function is specified, it is inferior (like all moment methods) to the maximum likelihood technique for fitting parameters to sampled data. The second part is a series of sampling simulations illustrating the biases and errors involved in applying the formulation to the specific case of gamma DSDs. It leads to underestimates of σm and consequently to overestimates of the gamma shape parameter—with large root-mean-square errors. Comparison with maximum likelihood estimates shows the degree of improvement that could be obtained in the estimates of the shape parameter. The propensity to underestimate σm will be pervasive, and users of this DSD formulation should be cognizant of the biases and errors that can occur.

A clustering algorithm for scintillator signals applied to neutron and gamma patterns identification

In several nuclear applications, scintillators, coupled with a photomultiplier and pulse amplifier, are used in order to detect high energy particles, i.e. neutrons and gamma rays. The different particles incident on the scintillator produce electrical pulses having different shape; moreover, the amplitude of these signals is related to the particles energy. The electrical pulses of the scintillator chain are acquired by digital systems that, generally, perform a triggered acquisition consisting of a stream of pulse windows.The aim of this study is the development of a simplified clustering algorithm able to produce reference patterns in compliance with the pattern recognition algorithm based on the matched filter technique, starting from a stream of pulses generated by particles having different energy and type.This paper contains a general description of the clustering algorithm and of the main customizations performed for the scintillator signals. In order to test in real case the efficiency, the algorithm has been applied on the data acquired during a radiation test performed at Frascati Neutron Generator for Stilbene scintillator. The results show that this algorithm works properly, deriving the centroids of the clusters representing the neutron and gamma shapes, together with their occurrences in the analysed data stream.

Fast and Accurate Approximation of the Full Conditional for Gamma Shape Parameters

The gamma distribution arises frequently in Bayesian models, but there is not an easy-to-use conjugate prior for the shape parameter of a gamma. This inconvenience is usually dealt with by using either Metropolis–Hastings moves, rejection sampling methods, or numerical integration. However, in models with a large number of shape parameters, these existing methods are slower or more complicated than one would like, making them burdensome in practice. It turns out that the full conditional distribution of the gamma shape parameter is well approximated by a gamma distribution, even for small sample sizes, when the prior on the shape parameter is also a gamma distribution. This article introduces a quick and easy algorithm for finding a gamma distribution that approximates the full conditional distribution of the shape parameter. We empirically demonstrate the speed and accuracy of the approximation across a wide range of conditions. If exactness is required, the approximation can be used as a proposal distribution for Metropolis–Hastings. Supplementary material for this article is available online.

Association Between Muscle Strength and Modeling Estimates of Muscle Tissue Heterogeneity in Young and Old Adults.

Assessing aging muscle through estimates of muscle heterogeneity may overcome some of the limitations of grayscale analyses. The objectives of this study included determining statistical model parameters that characterize muscle echogenicity and are associated with strength in younger and older participants. Thirty-three community-dwelling participants were assigned to younger and older groups. Quantitative B-mode ultrasound scanning of the rectus femoris and isometric grip strength testing were completed. Shape or dispersion parameters from negative binomial distribution, Nakagami, gamma, and gamma mixture models were fitted to the grayscale histograms. The mean ages ± SDs of the younger and older groups were 24.0 ± 2.3 and 65.1 ± 6.5 years, respectively. Statistical model shape and dispersion parameters for the grayscale histograms significantly differed between the younger and older participants (P = .002-.006). Among all of the statistical models considered, the gamma mixture model showed the best fit with the grayscale histograms (χ2 goodness of fit = 62), whereas the Nakagami distribution displayed the poorest fit (χ2 goodness of fit = 2595). Grayscale values were significantly associated with peak grip strength force in younger adult participants (R2 = 0.36; P < .008). However, the negative binomial dispersion parameter k (adjusted R2 = 0.70; P < .001) and gamma shape parameter α (adjusted R2 = 0.68; P < .01) showed the highest associations with peak grip strength force in older adult participants. The negative binomial dispersion parameter k and the gamma shape parameter α have clinical relevance for the assessment of age-related muscle changes. Statistical models of muscle heterogeneity may characterize the association between muscle tissue composition estimates and strength better than grayscale measures in samples of community-dwelling older adults.

The gamma shape mixture model and influence of sample-unit size on estimation of tree diameter distributions: Forest modelling

The gamma shape mixture (GSM) model is based on a mixture of gamma distributions. A general Bayesian approach for estimating the unknown parameters of the GSM model was employed. Tree diameter at breast height (DBH) sets of complex and stratified forests are ideal data sources for testing the usefulness of theoretical distributions applied to modelling and simulating strongly differentiated data sets. The GSM model was useful in fitting these DBH structures. During the identification of homogenous forest patches of the similar DBH structures one should select sample plots of at least 0.15 ha in area, with a minimum of 30 trees. The use of GSM model has the potential to facilitate the presentation of not only DBH structures, but also other empirical distributions, describing e.g. various biological stages and processes. The GSM model can be employed especially for modelling multimodal, asymmetrical and heavy-tailed survey data.

Inference on the Gamma Distribution

ABSTRACTThis study develops inferential procedures for a gamma distribution. Based on the Cornish–Fisher expansion and pivoting the cumulative distribution function, an approximate confidence interval for the gamma shape parameter is derived. The generalized confidence intervals for the rate parameter and other quantities such as mean are explored. The proposed generalized inferential procedures are extended to construct prediction limits for a single future measurement and for at least p of m measurements at each of r locations. The performance of the proposed procedures is evaluated using Monte Carlo simulation. The simulation results show that the proposed procedures are very satisfactory. Finally, three real examples are used to illustrate the proposed procedures. Supplementary materials for this article are available online.

Forest modelling: the gamma shape mixture model and simulation of tree diameter distributions

• Key messageNew types of distribution functions are needed to model the dynamics of stands where important age classes are represented by few trees. In this study the gamma shape mixture model and two simulation methods were used for generating tree diameter data.• ContextTo analyse forest dynamics, it is necessary to know distribution of the characteristics (mainly tree diameters) of trees forming particular developmental phases. In many forest inventories, the measurement of large diameter at breast height (DBH) samples is practically impossible. In this case, DBH distributions can be generated using theoretical models.• AimsThe aim of this study was to assess the precision of the approximation of empirical DBH data using the gamma shape mixture (GSM) model and kernel density estimation. The strengths and weaknesses of the two simulation methods were presented and discussed.• MethodsThe GSM model was adopted to approximate empirical DBH data collected in 20 near-natural stands. Two simulation methods were used: (a) the procedure based on a multimodal distribution and gamma random numbers (MDGR procedure) and (b) MCMC techniques with Metropolis–Hastings sampling (MH method).• ResultsThe GSM model precisely fitted the investigated DBH distributions. The MDGR procedure was slightly more precise than the MH method, especially in the case of the samples of 250 DBHs. The level of homogeneity within the drawn DBH sets was similar for all samples.• ConclusionThe GSM model is very flexible. The DBH random variates, generated with the use of analysed procedures, represented all tree generations being significant from a biological point of view.

Estimation of Field Reliability Based on Aggregate Lifetime Data

Because of the exponential distribution assumption, many reliability databases recorded data in an aggregate way. Instead of individual failure times, each aggregate data point is a summation of a series of collective failures representing the cumulative operating time of one component position from system commencement to the last component replacement. The data format is different from traditional lifetime data and the statistical inference is challenging. We first model the individual component lifetime by a gamma distribution. Confidence intervals for the gamma shape parameter can be constructed using a scaled χ2 approximation to a modified ratio of the geometric mean to the arithmetic mean, while confidence intervals for the gamma rate and mean parameters, as well as quantiles, are obtained using the generalized pivotal quantity method. We then fit the data using the inverse Gaussian (IG) distribution, a useful lifetime model for failures caused by degradation. Procedures for point estimation and interval estimation of parameters are developed. We also propose an interval estimation method for the quantiles of an IG distribution based on the generalized pivotal quantity method. An illustrative example demonstrates the proposed inference methods. Supplementary materials for this article are available online.

Generic and realistic spatial deployment modeling on data transmission performance evaluation of multi‐hop smart grid metering system

In this research work, three types of deployment models, namely regular grid, uniform random, and actual town models are compared regarding communication performance prediction of multi‐hop wireless smart meter networks. Multi‐hop data transmission simulation and analysis are carried out to find the most suitable type of generic as well as realistic meter deployment model that would facilitate planning on access point location for each town area. We introduce a spatial analysis index that quantifies the randomness and uniformity of spatial deployment. In addition, our analytical investigation using gamma statistical distribution shows that such a random model with a larger gamma shape parameter is the most suitable generic and realistic meter deployment model for multi‐hop data transmission performance evaluation. © 2016 Institute of Electrical Engineers of Japan. Published by John Wiley &amp; Sons, Inc.

On second-order admissibilities of the estimators of gamma shape vector

ABSTRACTSimultaneous estimation problem of gamma shape vector is considered.First, it is shown that the maximum likelihood estimator (MLE), the bias corrected MLE, and the conditional MLE of shape vector are second-order inadmissible. Second, these estimators are improved up to the second order. Finally, we identify whether these improved estimators are second-order admissible or not. Simulation studies are also given.