Parameters Of Gamma Distribution Research Articles

Implementation of Fuzzy Sets in the Non-Isothermal Pyrolysis of Biomass

This paper proposes the study of the application of fuzzy logic on the relevant parameter of biomass pyrolysis. The frequency factor, the upper limit of ‘dE’, and the scale and shape parameters of the fuzzy Gamma distribution were fuzzified in order to estimate randomness and to accommodate realistic values. The distribution function, f (E), of the activation energies was assumed to follow the fuzzy Gamma distribution. Thermo-analytical data was found experimentally with the help of TGA/DTG analysis. The approximated solution of the DAEM was obtained using the asymptotic approach.

Entropy-Based Parameter Estimation for the Four-Parameter Exponential Gamma Distribution

Two methods based on the principle of maximum entropy (POME), the ordinary entropy method (ENT) and the parameter space expansion method (PSEM), are developed for estimating the parameters of a four-parameter exponential gamma distribution. Using six data sets for annual precipitation at the Weihe River basin in China, the PSEM was applied for estimating parameters for the four-parameter exponential gamma distribution and was compared to the methods of moments (MOM) and of maximum likelihood estimation (MLE). It is shown that PSEM enables the four-parameter exponential distribution to fit the data well, and can further improve the estimation.

The Effect of Nature and Parameters of Particle Size Distribution of Graphite on Physical and Mechanical Properties of Polimer Composites based on Aromatic Poliamide

The resultsof studies of the effect of nature and parameters of particle size distribution of graphite on physical and mechanical properties of polymer composites based on aromatic polyamide fenіlon C-2. The particle size of the filler and polymer for the theoretical gamma-distribution parameters (perimeter, thickness and diameter) have different values. Found the influence of fillers (natural graphite different bands), which differed ash content (5-15% and 0,05-2,5%), moisture and grinding fineness (dispersion) on the wear resistance of the samples of the polymer composite. There is non-linear connectionbetween the intensity and parameters graphite particles distribution.

SPARSE BAYESIAN LEARNING BOOSTED BY PARTIAL ERRONEOUS SUPPORT KNOWLEDGE.

Recovery of sparse signals with unknown clustering pattern in the case of having partial erroneous prior knowledge on the supports of the signal is considered. In this case, we provide a modified sparse Bayesian learning model to incorporate prior knowledge and simultaneously learn the unknown clustering pattern. For this purpose, we add one more layer to support-aided sparse Bayesian learning algorithm (SA-SBL). This layer adds a prior on the shape parameters of Gamma distributions, those modeled to account for the precision of the solution elements. We make the shape parameters depend on the total variations on the estimated supports of the solution. Based on the simulation results, we show that the proposed algorithm is able to modify its erroneous prior knowledge on the supports of the solution and learn the clustering pattern of the true signal by filtering out the incorrect supports from the estimated support set.

Increasing the regional availability of the Standardized Precipitation Index: an operational approach

The need to use a length of rainfall records of at least 30 years to calculate the Standardized Precipitation Index (SPI) limits its application in several Drought Early Warning Systems of developing countries. Therefore, in order to increase the number of weather stations in which the SPI may be applied, this study quantified the difference among SPI values derived from calibration periods (CP) smaller than 30 years in respect to those computed from the 30-year period of 1985 – 2014 in the State of Sao Paulo, Brazil (time scales ranging from 1 to 12 months were considered). The correlation, agreement and consistency of SPI values derived from CP ranging from the last 30 to 21 years have been evaluated. The Kolmogorov-Smirnov/Lilliefors test indicated, for all CP, that the 2-parameter gamma distribution may be used to calculate the SPI in the State of Sao Paulo. The normality test indicated that, even for the period of 1985 – 2014, the normally assumption of the SPI series is not always met. However, it was observed no remarkable difference in the rejection rates of the normality assumption obtained from the different CP. Finally, both absolute mean error and the modified index of agreement indicated a high consistence among SPI values derived from the calibration period of 1991 – 2014 (24 years) in respect to those derived from the 30-year period. Accordingly, it is possible to use weather stations with rainfall records starting in 1991 (or earlier) to calculate, in operational mode, the SPI in the State of Sao Paulo.

Multivariate EWMA control chart with highly asymmetric gamma distributions

In this paper, we investigated the performance of the multivariate exponentially weighted moving average (MEWMA) control chart under highly asymmetric gamma distributions. Simulation procedures were used to obtain probabilistic control limits of the chart using a specific underlying distribution with known parameters (Phase II). The performance of the chart was evaluated under three different mean vector shifts, and the best values for the smoothing constant were identified. Results were compared with the existing proposal of using normal limits. Our results suggested that for three or more variables, probabilistic control limits lead to a better performance of the chart. We extended the analysis to the T2 chart and compared the efficiency of both strategies. MEWMA chart performance was better than T2 chart in all the cases analysed. A real application with Phase I reference data is described. The parameters of the marginal gamma distributions were estimated and a specific algorithm was then employed to determine the appropriate values of the smoothing constant and upper control limit of the MEWMA chart (Phase II). The relevance of this method lies in the fact that it can be adapted to any practical situation in which gamma model is suitable.

Sampling variability in estimates of flow characteristics in coarse‐bed channels: Effects of sample size

AbstractAdequate description of hydraulic variables based on a sample of field measurements is challenging in coarse‐bed streams, a consequence of high spatial heterogeneity in flow properties that arises due to the complexity of channel boundary. By applying a resampling procedure based on bootstrapping to an extensive field data set, we have estimated sampling variability and its relationship with sample size in relation to two common methods of representing flow characteristics, spatially averaged velocity profiles and fitted probability distributions. The coefficient of variation in bed shear stress and roughness length estimated from spatially averaged velocity profiles and in shape and scale parameters of gamma distribution fitted to local values of bed shear stress, velocity, and depth was high, reaching 15–20% of the parameter value even at the sample size of 100 (sampling density 1 m−2). We illustrated implications of these findings with two examples. First, sensitivity analysis of a 2‐D hydrodynamic model to changes in roughness length parameter showed that the sampling variability range observed in our resampling procedure resulted in substantially different frequency distributions and spatial patterns of modeled hydraulic variables. Second, using a bedload formula, we showed that propagation of uncertainty in the parameters of a gamma distribution used to model bed shear stress led to the coefficient of variation in predicted transport rates exceeding 50%. Overall, our findings underscore the importance of reporting the precision of estimated hydraulic parameters. When such estimates serve as input into models, uncertainty propagation should be explicitly accounted for by running ensemble simulations.

Development and Evaluation of Stochastic Rainfall Models for Urban Drought Security Assessment

Development and Evaluation of Stochastic Rainfall Models for Urban Drought Security Assessment

A Comparative Study on the Derivation of Unit Hydrograph for Bharathapuzha 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 Bharathapuzha 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 2008. 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. The data related to these structures could not be obtained due to certain restriction in acquiring the data from authorized agencies.

New Approximate Bayesian Confidence Intervals for the Shape and the Scale Parameters of the Two Parameter Gamma Distribution

The aim of the present study is to construct confidence intervals for the shape and the scale parameters of the twoparameter Gamma Distribution. Using the square error loss function, closed form approximate Bayesian confidence intervals are derived. Numerical results show that the obtained Approximate Bayesian models rely only on the observations under study and have great coverage accuracy.

Impact of multipath effects on theoretical accuracy of TOA-based indoor VLC positioning system

This paper discusses the time-of-arrival (TOA) based indoor visible light communication (VLC) positioning system in a non-line-of-sight environment. The propagation delay is assumed to be gamma distributed. The generalized Cramer–Rao lower bound for multipath propagation is derived as the theoretical accuracy limitation. The performance of the positioning system is affected by the shape parameter and the scale parameter of gamma distribution. The influences on positioning accuracy of multipath effects are analyzed through discussing the physical meaning of the gamma distribution parameters. It is concluded that the lower bound of positioning accuracy is attained when variance of the non-line-of-sight propagation-induced path lengths is zero. The simulation result proves that the theoretical positioning accuracy is in the order of centimeters with the given scenario.

Flood Hydrograph with Synthetic Unit Hydrograph Routing

Synthetic unit hydrographs (SUH) based on geomorphology are used as a tool to produce flood hydrographs from rainfall records, especially in ungauged and partially gauged catchments. This study presents a flood hydrograph model formulated on SUH based approach using geomorphologic parameters derived from Survey of India maps and geographical information system (GIS) techniques to simulate basin runoff. It uses linear Muskingum routing model in which the routing parameters are determined from the kinematic approach rather than from runoff data. The model employs the unit hydrograph suggested by Central Water Commission (CWC) India, as the discharge data from the sub-catchments and routes the concurrent discharges generated from them to the watershed outlet, and compared with the unit hydrograph of the lumped catchment. Application of the model is demonstrated by using data of a small watershed in the Mahanadi basin, India. The study indicates the limitations of the CWC unit hydrograph approach both in the small hilly and large plane catchments. It is observed that the CWC unit hydrograph (UH) underestimates the peak discharges both for catchments smaller than 200 km2 with steeper slope and larger than 600 km2 with flatter slope. The study further emphasizes on using two parameter Gamma distribution for preparation of UH curve instead of adopting standard practice of drawing synthetic unit hydrograph using seven known points with approximate curve fitting.

유역단위의 토사발생 모의를 위한 식생뿌리깊이의 정량화

본 연구에서는 유역단위 토사발생 모의의 주요 인자인 식생뿌리의 깊이에 대한 정량화를 시도하였다. 본 연구에 사용된 자료는 2006년 7, 8월 집중호우로 발생한 강원도 산사태 피해지역의 사면파괴지역에서 측정한 것이다. 조사된 자료를 이용하여 사면경사와 뿌리깊이와의 관계를 확률밀도함수를 이용하여 정량화하였다. 본 연구에서는 뿌리깊이를 중력방향과 사면수직방향에 대해 각각 고려하였다. 마지막으로, 결정된 확률밀도함수를 이용하여 뿌리깊이를 모의하기 위한 방법을 제시하였다. 그 결과 뿌리깊이에 대한 적절한 확률밀도함수로 2변수 감마분포가 선정되었으며, 선정된 분포는 원자료의 특성을 잘 나타내고 있는 것으로 확인되었다. 그러나 뿌리깊이를 중력방향 또는 사면수직방향으로 정량화 하느냐는 특히 관측자료의 개수 및 기타 지형적 상황을 고려하여 적절히 선택할 수 있을 것으로 판단된다. 이러한 결과는 추후 유사지역의 유역단위 토사발생 모의에 있어 필요한 뿌리깊이의 추정에 적절히 이용될 수 있을 것이다. This study quantified the tree root depth, an important factor in the simulation of basin-scale sediment yield. The data used in this study were those measured in the area of landslide due to heavy rainfall around July and August, 2006. These field survey data was quantified the relationship between slope and tree root depth using a probability density function (pdf). The tree root depth in this study considers both the gravitational direction and perpendicular direction to the plane. Finally, a methodology to generate tree root depth data using the pdf decided. As results, the two parameter gamma distribution was selected and found to well represent the tree root depth data. However, one should consider the number of data available or other geomorphological conditions to decide if the quantification of tree root depth is with respect to the gravitational direction or perpendicular direction to the plane. The results in this study may be used effectively for the estimation of tree root depth in the basin-scale sediment yield.

Temporal and Spatial Interpolation of the Standardized Precipitation Index for Computational Efficiency in the Dynamic Drought Index Tool

AbstractThe feasibility of interpolating gamma-distribution parameters between different precipitation accumulation intervals (durations) is statistically evaluated. The interpolation of these parameters for a specific accumulation interval, but ending on different dates, is similarly assessed. Such interpolation increases the computational efficiency of drought-monitoring tools that require calculation of the standardized precipitation index (SPI) for any user-specified accumulation period on any given day. Spatial interpolation of the distribution parameters is also assessed. Given a 60-yr period of record, few statistically significant differences were found between gamma-distribution percentiles interpolated between fixed base durations and those computed directly. Shorter interpolation intervals (generally 30 days) were required for the shortest (e.g., 30 days) durations, whereas interpolation over periods of as long as 180 days could be used for the longest (between 360 and 720 days) durations. Interpolating the distribution parameters to different ending dates on the basis of those computed for the end of each month was also appropriate. The spatial interpolation of gamma-distribution parameters, although viable in practice for monitoring large-scale drought conditions, was associated with larger SPI differences than was the spatial interpolation of the SPI index itself or the interpolation of historical precipitation and the subsequent calculation of gamma-distribution parameters on the basis of these values.

Independent Gamma Variables of McKay Distribution for Rainfall Model

The application of mathematical models of rainfall is crucial in order to have a better understanding in terms of rainfall characteristics. The rainfall models has been used widely to improve water management, to construct hydrological structures and as an input in climatological studies. In this study, the gamma distribution is used to fit the marginals of monthly rainfall data. The parameters of gamma distributions, shape and scale, are estimated using the maximum likelihood estimation method. Then the sum of monthly rainfall amount is modelled using the Mckay ditribution for independent gamma variables.

Small Sample Tests for Shape Parameters of Gamma Distributions

The introduction of shape parameters into statistical distributions provided flexible models that produced better fit to experimental data. The Weibull and gamma families are prime examples wherein shape parameters produce more reliable statistical models than standard exponential models in lifetime studies. In the presence of many independent gamma populations, one may test equality (or homogeneity) of shape parameters. In this article, we develop two tests for testing shape parameters of gamma distributions using chi-square distributions, stochastic majorization, and Schur convexity. The first one tests hypotheses on the shape parameter of a single gamma distribution. We numerically examine the performance of this test and find that it controls Type I error rate for small samples. To compare shape parameters of a set of independent gamma populations, we develop a test that is unbiased in the sense of Schur convexity. These tests are motivated by the need to have simple, easy to use tests and accurate procedures in case of small samples. We illustrate the new tests using three real datasets taken from engineering and environmental science. In addition, we investigate the Bayes’ factor in this context and conclude that for small samples, the frequentist approach performs better than the Bayesian approach.

Statistical downscaling of probability density function of daily precipitation on the Polish coast

The aim of this study was to recognize the possibility of downscaling probability density function (PDF) of daily precipitation by means of canonical correlation analysis (CCA). Sea level pressure (SLP) over Europe and the North Atlantic was used as a predictor. A skilful statistical model could be used to generate projections of future changes of precipitation PDF driven by GCM (General Circulation Model) simulations. Daily precipitation totals from 8 stations located on the Polish coast of the Baltic Sea covering the period 1961-2010 were used to estimate the gamma distribution parameters, and only wet days (i.e. ≥0.1mm) were taken in the analysis. The results of the Kolmogorov-Smirnov test and comparison of empirical and theoretical (gamma-distributed) quan- tiles proved that gamma distribution gives a reliable description of daily precipitation totals. The validation of CCA models applied to gamma parameters revealed that the reliable reconstruction of precipita- tion PDF is possible only for average long-term conditions. In the case of individual months/seasons the agreement between empirical and reconstructed quantiles is poor. This study shows the potential of modelling of precipitation PDF, however efforts should be made to improve model performance by establishing more reliable links between regional forcing and the variability of the gamma parameters.

Superstatistical fluctuations in time series of leverage returns

We analyze to what extent the emergence of fat-tailed q-Gaussian distributions of daily leverage returns of North American industrial companies that survive default and de-listing between 2006 and 2012 can be described by superstatistics. To this end, we compare mean values of the Tsallis entropic parameter q obtained by two independent methods: (i) direct fitting of q-Gaussians to distributions of leverage returns; and (ii) derived from shape parameters of Gamma distributions fitted to histograms of inverted realized variances of these returns. For a vast majority of companies, we observe the striking consistency of average values of q obtained by both methods. This finding supports the applicability of superstatistical hypothesis, which assumes that q-Gaussians result from the superposition of locally normal distributions with Gamma-distributed precision (inverted variance).

Modified Moment Estimation for a Two Parameter Gamma Distribution

A study of a modification of moment estimation of a two parameter gamma distribution has been carried out. The scale parameter has been estimated using the Modified Moment Estimates (MMEs) and their performance is compared to the Maximum Likelihood Estimates (MLEs). Modified Moment Estimates are easy to compute relative to Maximum Likelihood Estimates. To adjust their bias, the Jackknife technique has been used. The efficiency of these estimators is compared by calculating their respective variances. The R-has been used to carry out the Monte Carlo simulation of data. The results obtained show that the Modified Moment Estimates perform as well as the Maximum Likelihood Estimates when bias as well as variation of data is taken into account.

The use of Gamma Distribution to Evaluate Water Pollutants in Asejire Reservior, Ibadan

This research work under-studies the application of probability functions to water pollution data with some related works. It figures out two parameter gamma distribution fitted to monitor the concentration of five water pollutants in Asejire reservoir; Turbidity, Colour, pH, Dissolved Oxygen and Alkalinity, by estimating the parameters and testing the goodness of fit. It was discovered that all the considered pollutants could be modeled with gamma distribution as confirmed by the Kolmogorov Smirnov (KS) test. It was also found that all the pollutants were significant except Dissolved Oxygen. The technique and results presented in this study provide a foundation for the use of gamma distribution to model water pollutants. The model is used as decision support tools for the management of the reservoir, as the attention of government and stake-holders must be drawn to keeping water safe for sustainable development.