Parameters Of Gamma Distribution Research Articles

Irregular Tessellation and Statistical Modeling Based Regionalized Segmentation for SAR Intensity Image

This paper presents a regionalized segmentation method for synthetic aperture radar (SAR) intensity images based on tessellation with irregular polygons. In the proposed method, the image domain is partitioned into a collection of irregular polygons, which are constructed using sets of nodes and are used to fit homogeneous regions with arbitrary shapes. Each partitioned polygon is taken as the basic processing unit. Assuming the intensities of the pixels in the polygon follow an independent and identical gamma distribution, the likelihood of the image intensities is modeled. After defining the prior distributions of the tessellation and the parameters for the likelihood model, a posterior probability model can be built based on the Bayes theorem as a segmentation model. To obtain optimal segmentation, a reversible-jump Markov chain Monte Carlo (RJMCMC) algorithm is designed to simulate from the segmentation model, where the move operations include updating the gamma distribution parameter, updating labels, moving nodes, merging polygons, splitting polygons, adding nodes, and deleting nodes. Experiments were carried out on synthetic and real SAR intensity images using the proposed method while the regular and Voronoi tessellation-based methods were also preformed for comparison. Our results show the proposed method overcomes some intrinsic limitations of current segmentation methods and is able to generate good results for homogeneous regions with different shapes.

Goodness of fit tests of the two-parameter gamma distribution against the three-parameter generalized gamma distribution

The generalized Gamma distribution is widely used in survival analysis, engineering and economics, and it includes the Gamma distribution as a special case. In real life applications, the Generalized Gamma distribution may be an unnecessarily complicated distribution to fit when one of the simpler models may be sufficient to analyze the data. The aim of this paper is to develop tests of goodness of fit of the two parameter gamma distribution against its three parameter counterpart. Specifically, we develop Wald, , score, likelihood ratio, and gradient tests. These tests are then compared in terms of empirical size and power using a simulation study.

The peak runoff model based on Existing Land Use and Masterplan in Sentul City area, Bogor

This research aimed to create a peak runoff mode based on existing land use (LU) and masterplan in Sentul City area. To determine the peak runoff by rational method, the study uses the formulation as follows: Q = 0.2778.C.I.A, in which Q is the peak runoff, C is the runoff coefficient of area, I is the average rain rate intensity, and A is the area of study. For recognizing the existing LU, the researcher used image analysis SPOT-6 (2017) by supervised classification. It estimated the gamma distribution parameter through the maximum likelihood method by using software QGIS 2.8, SAGA GIS, dan Arc-GIS 10.4.1. According to the analysis, the study result showed the existing LU peak runoff coefficient value and masterplan are 0.40 and 0.61, respectively, in which the difference is 0.21. The peak runoff increase is 25.32 m3/sec or 6,622,560 m3/year as the impact of land-use change.

Bivariate analysis of drought duration and severity for irrigation planning

In this study, the effectiveness of a bivariate analysis of agricultural drought characteristics for irrigation planning was evaluated. The case study was conducted in locations in central Italy and was based on olive crops which are widely grown in that area under rainfed or deficit irrigation regimes. For each locality, the available time series of daily precipitation and maximum and minimum temperatures were used to simulate the daily soil water dynamics (SWt) for olive crops. The simulation was performed assuming 10 irrigation strategies, different for both the volume and date of the interventions. By applying the Theory of Runs to SWt, with a threshold corresponding to the readily available water, the agricultural drought events in the time series were identified and characterised by their duration D (days) and severity S (i.e. the cumulative evapotranspiration deficit in mm) for each locality and strategy.A 2-parameter Gamma distribution was fitted to both D and S, whilst their dependence structure was modelled by a Gumbel copula. The evaluation of the best irrigation strategy in each locality was obtained by comparing the contour lines of the pairs (d,s) having a 10-year return period for the condition D ≥ d AND S ≥ s. Results showed a great influence of both climate and irrigation strategy on the joint distribution of D and S. Some strategies, despite a very similar behaviour in the severity, differed a lot in terms of the associated values of duration, thus denoting different dynamics of water stress (more or less intense). Among the single-irrigation strategies, the optimal one was usually represented by an application at the end of June. Among the double-irrigation strategies, the optimal one involved two interventions, one at the end of June and one at the end of July. These results were also discussed taking into account the corresponding probability of consecutive severe drought events.

Kappa Distributions and Isotropic Turbulence

In this work, the two-point probability density function (PDF) for the velocity field of isotropic turbulence is modeled using the kappa distribution and the concept of superstatistics. The PDF consists of a symmetric and an anti-symmetric part, whose symmetry properties follow from the reflection symmetry of isotropic turbulence, and the associated non-trivial conditions are established. The symmetric part is modeled by the kappa distribution. The anti-symmetric part, constructed in the context of superstatistics, is a novel function whose simplest form (called “the minimal model”) is solely dictated by the symmetry conditions. We obtain that the ensemble of eddies of size up to a given length r has a temperature parameter given by the second order structure function and a kappa-index related to the second and the third order structure functions. The latter relationship depends on the inverse temperature parameter (gamma) distribution of the superstatistics and it is not specific to the minimal model. Comparison with data from direct numerical simulations (DNS) of turbulence shows that our model is applicable within the dissipation subrange of scales. Also, the derived PDF of the velocity gradient shows excellent agreement with the DNS in six orders of magnitude. Future developments, in the context of superstatistics, are also discussed.

STUDY ON SUMMER RAINDROP SIZE DISTRIBUTION AND RETRIEVED POLARIMETRIC RADAR PARAMETER OVER LEIZHOU PENINSULA

Abstract. Leizhou peninsula is located in the south of Guangdong Province, near South China Sea, and has a tropical and subtropical monsoon climate. Based on observed drop size distribution (DSD) data from July 2007 to August 2007 with PARSIVEL disdrometers deployed at Zhanjiang and Suixi, the characterists of DSDs are studied. Non-linear least squares method is used to fit Gamma distribution. Convective and stratiform averaged DSDs are in good agreement with Gamma distribution, especially in stratiform case. Convective average DSDs have a wider spectrum and higher peak. Microphysical parameter differences between convective and stratiform are discussed, convective precipitation has a higher mass-weighted mean diameter (Dm) and generalized intercepts (Nw) in both areas. The constrained relations between Gamma distribution parameter (μ, Λ, N0) is derived. The retrieved polarimetric radar parameter (KDP, ZDR, Zh) have a good self-consistency, which can be used to improve the accuracy of KDP calculation. R-KDP-ZDR is superior to the R-KDP, R-ZDR-Zh in quantitative precipitation estimation (QPE), with a correlation coefficient higher than 0.98.

Gamma Distribution of FAST Feature for Statistical Image Modelling in Fingerprint Image Classification

A unique method for modelling Features from accelerated segment test (FAST) with the Gamma distribution for statistical image data is introduced. Instead of using an image's FAST feature instantly; we design the FAST function to decrease the other global extraction function. The method of moment is used to predict Gamma distribution parameters. FAST's mathematical depiction is the depiction of matrix and is too complicated to be implemented in the ranking of images. We are therefore proposing a fresh statistical function to display the picture in a few dimensional numbers. Our proposed feature utilizes FAST method of Gamma distribution that can be used directly in the identification of fingerprint images. We demonstrate that the Gamma distribution works with FAST and has been effectively implemented in the identification of fingerprint images. To show the benefits of the proposed feature, Shivang Patel Fingerprint Database is used on which classifier evaluation and state-of - the-art comparison are performed.

A Comparative Study on the Derivation of Unit Hydrograph for Dhadhar River Basin

Several techniques are available for the development of the unit hydrograph. But most of these traditional methods require manual fitting of the unit hydrograph through few points, which does not guarantee the area under the unit hydrograph to be unity. More over most of the stations are ungauged, due to which it becomes difficult to develop the unit hydrograph. So in order to overcome these problems, two methods have been considered in this study for the development of the unit hydrograph for Dhadhar river basin. They are the “two parameter Gamma distribution” and “three parameter Beta distribution”, both of which are based on Probability Distribution Functions (pdfs). The unit hydrograph developed by the two parameter Gamma distribution match well with the one developed by CWC method, but the unit hydrograph developed by the three parameter Beta distribution does not match well with the one developed by the CWC method. From the unit hydrograph, runoff hydrograph is convoluted for the year 2018. For this the hourly rainfall are generated from daily rainfall values by disaggregation. But on plotting, the simulated discharge hydrograph is found to be greater than the observed discharge. This may be due to non-incorporation of the inflow outflow processes of many hydraulic structures such as dams, irrigation schemes etc., existing in the basin in the model study.

Classifying heart conditions based on class probability output networks

This paper presents a novel method of classifying heart conditions from an electrocardiography (ECG) signal. For this purpose, the R-R intervals of ECG signal are analyzed by Gamma distribution parameters and classified into normal (NR) or abnormal (AN) ECG waves. For the normal ECG waves, the heart condition is further investigated by analyzing the dynamic behavior of heart activity based on the correlation between successive R-R intervals and long-term analysis. The classification of heart conditions is made by estimating the conditional class probabilities using class probability output networks (CPONs). The simulation for classifying heart conditions using the MIT-BIH data sets reveals that the proposed approach is effective for classifying heart conditions and allows more accurate classification than the existing classifiers such as the k-NN and SVM.

Bayesian Estimation for Two Parameters of Gamma Distribution under Generalized Weighted Loss Function

This paper deals with, Bayesian estimation of the parameters of Gamma distribution under Generalized Weighted loss function, based on Gamma and Exponential priors for the shape and scale parameters, respectively. Moment, Maximum likelihood estimators and Lindley’s approximation have been used effectively in Bayesian estimation. Based on Monte Carlo simulation method, those estimators are compared in terms of the mean squared errors (MSE’s).

Simulating Rainfall Seasonal Pattern in Kandangan Subregency Using Statistical Method

Rainfall is an important climate element which affects water resources availability in a certain area. Water resources availability itself are an important factor for human life. In this research, daily rainfall from 2014 to 2016 was statistically analyzed to obtain form and scale parameter for 2-parameter gamma distribution for each month. Then obtained parameter is clustered based on seasons resulting in four clusters. Then each obtained parameter within each cluster is modeled as dependent variable using simple linear regression with Sea Surface Temperature Anomaly over Nino 3,4 Zone as its independent variable. Those processes resulted in the regression model for each gamma distribution parameter and for each season. Then obtained model used and combined with random seed algorithm to simulate rainfall during 2017. Resulted rainfall showed that statistical model that had been built was capable to simulate the temporal pattern of rainfall although with underestimated rainfall value.

Using willingness to pay to forecast the adoption of solar photovoltaics: A “parameterization + calibration” approach

Distributed energy resources, such as rooftop solar photovoltaics (PV), are likely to comprise a substantial fraction of new generation capacity in the United States. However, forecasting technology adoption based on people's willingness to pay (WTP) faces two major challenges: the stated-intention and omitted-variable biases. Previous solar adoption literature has neglected to address these two biases altogether. Here, we adopt a “parameterization + calibration” approach to address both biases and estimate customers’ WTP for PV. After collecting survey data on respondents’ WTP for adopting PV, we characterize its empirical cumulative density function using a gamma distribution. We further calibrate the gamma distribution parameters using a national distributed PV adoption simulation model, finding the parameters that produce the best fit between simulated and historic solar adoption. We then show that the calibrated gamma distribution improves the raw WTP data after correcting for the two biases. Finally, we use our optimally-calibrated WTP to forecast market demand for residential PV at the county-level of the United States in 2020. Improving estimates of customer willingness to pay has significant implications for policy directly, e.g. estimating the effect of a proposed policy on technology adoption, and other regulatory processes that use forecasting, e.g. integrated resource planning.

Ground Observation of Tephra Particles: On the Use of Weather Radar for Estimating Volcanic Ash Distribution

This paper reports a preliminary attempt to determine volcanic ash particle size distribution using the video drop size detector (VDSD) for estimating volcanic ash amount with X-band radar. The VDSD records an image showing the size and number of particles falling into the aperture by a charge coupled device camera. Size distribution spectra of a range of particles from fine ash to small lapilli were derived in discrete form from the VDSD observation. The parameterization of the particle size distribution following Gamma function was done using volcanic ash of eruptions at the Sakurajima Volcano between December 13–21, 2014. Three Gamma distribution parameters were determined analytically. The analytical results revealed a continuous distribution of particles characterized by shape, intercept, and slope. The distribution was used to determine volcanic mass concentration, ground deposit weight, and reflectivity. Verification of these results with X-band radar observations showed that the reflectivity obtained from analytical results is similar to that from radar observation. However, the ground deposit weight from analysis was overestimated, compared with the real weight of ash deposit on the ground. The algorithm proposed in this study is expected to provide a practical method for estimating ash distribution in the aftermath of a volcanic eruption using radar-reflectivity for cases where direct measurement at the location is not possible. An overview of the algorithm for volcanic ash retrieval from X-band radar observations is also presented.

Ranking the Scientific Output of Researchers in Fractional Calculus

Abstract This paper analyses the citation profiles (CP) of 130 researchers in fractional calculus. In a first phase, the Canberra distance is used to measure the similarities between the researchers’ CP, and the multidimensional scaling technique (MDS) is adopted for processing and visualizing the information. In a second phase, the gamma probability distribution is used to fit the normalized CP and the gamma parameters are used to characterize the researchers. The MDS results and the gamma distribution parameters are represented graphically in 2- and 3-dimensional locus depicting the relative positions of the researchers.

Bayesian Estimation for Two Parameters of Gamma Distribution Under Precautionary Loss Function

In the current study, the researchers have been obtained Bayes estimators for the shape and scale parameters of Gamma distribution under the precautionary loss function, assuming the priors, represented by Gamma and Exponential priors for the shape and scale parameters respectively. Moment, Maximum likelihood estimators and Lindley’s approximation have been used effectively in Bayesian estimation.
 Based on Monte Carlo simulation method, those estimators are compared depending on the mean squared errors (MSE’s). The results show that, the performance of Bayes estimator under precautionary loss function with Gamma and Exponential priors is better than other estimates in all cases.

The Dependence of Sea SAR Image Distribution Parameters on Surface Wave Characteristics

Modeling the statistical distribution of synthetic aperture radar (SAR) images is essential for sea target detection, which is an important aspect of marine SAR applications. The main goal of this study is to determine the effects of sea states and surface wave texture characteristics on the statistical distributions of sea SAR images. A statistical analysis of the Envisat Advanced Synthetic Aperture Radar (ASAR) wave mode images (imagettes), covering a variety of sea states and wave conditions, was carried out to investigate the suitability of the statistical distributions often used in the literature for sea states parameters. The results revealed the variation in the distribution parameters in terms of their azimuthal cutoff wavelength (ACW) and the peak-to-background ratio (PBR) of the SAR image intensity spectra. The shape parameters of Gamma and Weibull distribution are sensitive and monotonously decreasing with respect to PBR, while the scale parameter is sensitive to ACW. The K distribution was shown to perform well, with both high and stable accuracy. The results of this paper provide a parameterized scheme for sea state classifications and can potentially be used for choosing the most suitable distribution model according to sea state when performing sea target detection.

Determination of Drainage Coefficient and Bund Height of Rice Fields at Gaya and Bhagalpur Districts in Bihar

Rainfall plays an important role in agricultural production, particularly in rainfed areas. Less occurrence of rainfall causes drought like situation and crops suffer due to deficit of water, whereas heavy rainfall occurring for longer duration lead to flood like situation resulting in more runoff, soil erosion and crop damage. Rice can sustain water for little longer period compared to other crops, but this crop also needs drainage. 30 years daily rainfall data was collected at Gaya and Bhagalpur districts and analyzed. Two parameter Gamma distribution model was found fitting well in 1 to 7 consecutive days maximum rainfall corresponding to return periods varying from 2 to 20 years. In order to determine drainage coefficient at Gaya and Bhagalpur districts, depthduration- return period curves were developed. Tangents were drawn on the curves from 100 mm, 150 mm and 200 mm points and slope of these tangents gave the drainage coefficients corresponding to these rain water storage levels. Since 10 years return period is generally considered in design of agricultural structures, so the poin ton Y axis, where tangent drawn on the curves of 10 years return period crosses, gives the bund height. For both Gaya and Bhagalpur bund heights were found as 24 cm and corresponding drainage coefficients as 12.5 and 25 mm/day.

A comparison of three parameter estimation methods of the gamma distribution of annual maximum low flow duration and deficit in the Upper Vistula catchment (Poland)

Based on 30-year 24-hour flow sequences at 69 water gauging stations in the Upper Vistula catchment, it was determined that the probability distributions of the low flow duration and its maximum annual deficit can be described by the gamma distribution with the estimated parameters by the methods: MOM, the method of moments, LMOM, the method of linear moments, and MLE, the method of maximum likelihood. The stationarity of the time series was tested by the Mann-Kendall correlation using the Hamed and Rao variance correction. The low flows were defined by the SPA method, with the limit flowQ70%. The quality of the match was tested by the Anderson-Darling goodness of fit test. This test allowed accepting the gamma distribution in all analysed cases, regardless of the method used to estimate the distribution parameters, since thepv(p-values) values were greater than 5% (over 18% forTmaxand 7.5% forVmax). The highestpvvalues for individual water gauging stations, as well as the highest 90%TmaxandVmaxquantiles were noted using LMOM to estimate the gamma distribution parameters. The highest 90%TmaxandVmaxquantiles were observed in the uppermost part of the studied area.

Asymptotic Approximations to the Non-Isothermal Distributed Activation Energy Model for Bio-Mass Pyrolysis

Abstract This paper describes the influence of some parameters significant to biomass pyrolysis on the numerical solutions of the non-isothermal nth order distributed activation energy model (DAEM) using the Gamma distribution and discusses the special case for the positive integer value of the scale parameter (λ), i.e. the Erlang distribution. Investigated parameters are the integral upper limit, the frequency factor, the heating rate, the reaction order, and the shape and rate parameters of the Gamma distribution. Influence of these parameters has been considered for the determination of the kinetic parameters of the non-isothermal nth order Gamma distribution from the experimentally derived thermoanalytical data of biomass pyrolysis. Mathematically, the effect of parameters on numerical solution is also used for predicting the behaviour of the unpyrolysized fraction of biomass with respect to temperature. Analysis of the mathematical model is based upon asymptotic expansions, which leads to the systematic methods for efficient way to determine the accurate approximations. The proposed method, therefore, provides a rapid and highly effective way for estimating the kinetic parameters and the distribution of activation energies.

Estimating freeway travel time and its reliability using radar sensor data

ABSTRACTTravel time and its reliability are intuitive system performance measures for freeway traffic operations. This paper proposes a method to estimate travel times based on data collected from roadside radar sensors, considering spatially correlated traffic conditions. Link-level and corridor-level travel time distributions are estimated using these travel time estimates and compared with the ones estimated based on probe vehicle data. The maximum likelihood estimation is used to estimate the parameters of Weibull, gamma, normal, and lognormal distributions. According to the log-likelihood values, lognormal distribution is the best fit among all the tested distributions. Corridor-level travel time reliability measures are extracted from the travel time distributions. The proposed travel time estimation model can well capture the temporal pattern of travel time and its distribution.