Double Gamma Research Articles

Research on Simulating Method of Optical Intensity Scintillation Series with Double Generalized Gamma Distribution in Atmospheric Turbulence

Research on Simulating Method of Optical Intensity Scintillation Series with Double Generalized Gamma Distribution in Atmospheric Turbulence

Nielsen's beta function and some infinitely divisible distributions

AbstractWe show that a large collection of special functions, in particular Nielsen's beta function, are generalized Stieltjes functions of order 2, and therefore logarithmically completely monotonic. This includes the Laplace transform of functions of the form , where f is itself the Laplace transform of a sum of dilations and translations of periodic functions. Our methods are also applied to ratios of Gamma functions, and to the remainders in asymptotic expansions of the double Gamma function of Barnes.

A Quantized Riemann–Hilbert Problem in Donaldson–Thomas Theory

Abstract We introduce Riemann–Hilbert problems determined by the refined Donaldson–Thomas theory. They involve piecewise holomorphic maps from the complex plane to the group of automorphisms of a quantum torus algebra. We study the simplest case in detail and use the Barnes double gamma function to construct a solution.

Clinical Pharmacokinetics and Bayesian Estimators for the Individual Dose Adjustment of a Generic Formulation of Tacrolimus in Adult Kidney Transplant Recipients.

Tacrolimus has a narrow therapeutic range and requires dose adjustment, usually based on the trough blood concentration but preferably on the area under the concentration-time curve over 12 h post-dose (AUC0-12h). The single-arm, multicentre, clinical study IMPAKT aimed: (i) to develop, in de novo kidney transplant recipients, pharmacokinetic models and maximum a-posteriori Bayesian estimators for a generic, immediate-release, oral formulation of tacrolimus to estimate tacrolimus AUC0-12h at different post-transplant periods using a limited sampling strategy, and considering the CYP3A5*3 polymorphism as a covariate and (ii) to compare the performance of these Bayesian estimators to those previously developed for the original formulation. Thirty patients were enrolled and 29 provided nine blood samples over 9 h at day 7 and months 1 and 3 post-transplant. Tacrolimus blood profiles measured with liquid chromatography-tandem mass spectrometrywere modelled using one-compartment, double gamma absorption, linear elimination models developed in-house. Different limited sampling strategies of three time-points within 4 h post-dose were tested for the maximum a-posteriori Bayesian estimator of tacrolimus AUC0-12h. The models and estimators were validated internally and their performance compared to that of models previously developed for the original formulation. The concentration-time curves, AUC0-12h/dose and trough blood concentration/dose exhibited wide inter-individual variability. The covariate-free pharmacokinetic models developed for the three post-transplant periods closely fitted the individual profiles. Maximum a-posteriori Bayesian estimators based on three different limited sampling strategies and no covariate yielded accurate AUC0-12h estimates, including for the five cytochrome P450 3A5 expressers and for the four patients without corticosteroids. The 0-1 h-3 h strategy finally chosen had very low bias (-4.0 to -2.5%) and imprecision (root mean square error 5.5-9.2%). The maximum a-posteriori Bayesian estimators previously developed for the reference product fitted the generic profiles with similar performance. We developed original pharmacokinetic models and accurate maximum a-posteriori Bayesian estimators to estimate patient exposure and adjust the dose of generic tacrolimus, and confirmed that the robust tools previously developed for the original formulation can be applied to this generic.

Comparing landslide size probability distribution at the landscape scale (Loess Plateau and the Qinba Mountains, Central China) using double Pareto and inverse gamma

Landsliding is a prominent geomorphological process in both the Loess Plateau and the Qinba Mountains in Central China. The size distribution of landslides plays an important role in quantifying their occurrence and magnitude, estimating erosion and sediment yields, and determining hazards. We generated landslide inventories for six study sites within the two study regions based on results from field surveying and remote sensing analyses. Landside size distribution differs considerably in both regions and can be described by the double Pareto and inverse gamma distributions. The power law decays faster in the Loess Plateau than in the Qinba Mountains; the locations of the rollover occur at larger landslide size in the Loess Plateau. Moreover, α, an exponent of power law scaling related to slope in the double Pareto function, is strongly related to ρ + 1, a control of power law decay in the inverse gamma. This study provides an insight into the landslide size probability distribution in different landscape types.

An improvement to the Kalos’ formula for double layer gamma ray exposure buildup factors for shielding materials of nuclear and radiological facilities

Gamma ray exposure buildup factors (BUFs) are used in point kernel method based shielding codes to estimate the scattered radiation dose rate. BUFs computed by assuming single material for multi-layer shielding configuration either overestimate or underestimate the dose rates. Hence, BUFs computed based on stratified shielding configuration give better estimates of scattered dose rate and provide scope for further optimization of shielding thickness. In nuclear and radiological facilities, shielding materials such as concrete (Con), iron (Fe), lead (Pb), and tungsten (W) are normally used as single or multi-layer shields. For the first time, buildup factors estimated based on Monte Carlo simulations (MCS) for double layer shielding configurations (DBUFs) of W and Fe, W and Con, Pb and Fe, Pb and Con, and Fe and Con are reported. The estimated DBUFs are compared with the values obtained using the empirical formulae published by several authors. A significant variation up to 40% is observed between MCS and empirical formulae. Hence, the correction factor in the empirical formula of Kalos’ is modified and standardized by carrying out further computations for various double layer shielding configurations and energies. Radiation shielding experiments performed to validate the MCS results are presented.

Average bit error rate performance of free-space optical systems over double generalized gamma fading channels based on avalanche photodiode detector

The performance of free-space optical (FSO) communication systems based on avalanche photodiodes (APDs) is investigated over the aggregated double generalized gamma (double GG) fading channels. The approximate average bit error rate (ABER) expression is theoretically derived in terms of the double GG distributions under moderate and strong turbulent atmospheric conditions considering thermal noise and shot noise. The union bound and Hermite polynomials are then used to estimate the ABER performance of M-ary pulse-position modulation (PPM)-based systems and orthogonal frequency division multiplexing scheme (OFDM)-based systems. The ABER performance is studied with different receiver noise temperatures, modulation orders, turbulence strengths, and average photon counts assuming plane wave and spherical wave propagations. The results show that adopting an optimal average APD gain affected by the receiver temperature minimizes the ABER value for both wave propagations. In addition, the present APD-based system offers better ABER performance than that of a PIN-based FSO system over the double GG fading channels at two receiver noise temperatures of 200 and 400 K for both the PPM and OFDM schemes. We provide a reference for FSO system design.

Unconditional density vs conditional density functions in estimating value-at-risk

ABSTRACT This research compares the performance of 10 Value-at-Risk (VaR) models, including pure density models (student’s t, mixture of normal, double gamma, kernel, and GPD) and conditional density models (GARCH-student’s t, GARCH-mixture of normal, GARCH-double gamma, GARCH-kernel, and GARCH-GPD). We employ these models and test their performance in Asian markets, including over the 2008 financial crisis period. Findings show that the density functions used in conjunction with the GARCH models are capable of capturing the high peak, fat-tails, and volatility clustering of returns and can improve the accuracy of VaR forecasts. Furthermore, the proposed GARCH-double gamma provides the best fit for all tested Asian markets, both developed and emerging markets, and for foreign exchange rates markets.

Triple the Gamma—A Unifying Shrinkage Prior for Variance and Variable Selection in Sparse State Space and TVP Models

Time-varying parameter (TVP) models are very flexible in capturing gradual changes in the effect of explanatory variables on the outcome variable. However, in particular when the number of explanatory variables is large, there is a known risk of overfitting and poor predictive performance, since the effect of some explanatory variables is constant over time. We propose a new prior for variance shrinkage in TVP models, called triple gamma. The triple gamma prior encompasses a number of priors that have been suggested previously, such as the Bayesian Lasso, the double gamma prior and the Horseshoe prior. We present the desirable properties of such a prior and its relationship to Bayesian Model Averaging for variance selection. The features of the triple gamma prior are then illustrated in the context of time varying parameter vector autoregressive models, both for simulated dataset and for a series of macroeconomics variables in the Euro Area.

Frequency–magnitude of landslides affected by the 27–29 November 2017 Tropical Cyclone Cempaka in Pacitan, East Java

Tropical cyclone (TC) Cempaka which occurred on 27–29 November 2017 has caused floods, landslides, and strong winds in certain areas of Java Island. Pacitan Regency was the most severely affected by TC Cempaka. The landslide frequency–area distribution curve of event inventory i.e. TC Cempaka can help to understand landslide susceptibility, hazard, vulnerability, and risk. Landslides were identified by using a local government database and by comparing pre- and post-event high-resolution satellite imageries. Field investigation was carried out in March 2018 to November 2018 to verify the landslide location and update the information. Power law, inverse gamma, and double Pareto model were employed to describe the frequency–magnitude of landslide (mLS) triggered by TC Cempaka. The exponent β values of power law, inverse gamma, and double Pareto were 2.6±0.28 (fitted for 8.5% of dataset), 2.2±0.08 (fitted for 83% of dataset), and 2.3±0.09 (best fitted for dataset), respectively. The P-values were 0.51, 0.67, and 0.91 for power law, inverse gamma, and double Pareto, respectively. This study revealed that rollover occurred at 200 and 300 m2 for double Pareto and inverse gamma, respectively. The cutoff points totaled 1096.49 ± 236.44 and 7235.4 ± 1896.7 m2 for double Pareto and power law, respectively. Rollover phenomenon was real and existed in the dataset because it was far from the minimum resolvable size of the landslide that the authors can delineate from the satellite images. mLS for Pacitan was distributed at around 2 to 4. The magnitude of large landslides was 3.2, that of medium landslides was less than 3, and that of small landslides was almost 4. Numerical estimation calculated a fixed mLS=3.01. Comparison analysis of β values obtained from several landslide inventories triggered by heavy rainfall suggests that the variability of β is related to the intensity and duration of rainfall. Triggering events, such as intensity and duration of rainfall, affect the proportion of large landslides that occur in an area. More complete landslide inventories and rainfall data or other landslide triggering factors from other areas are required for further relationship analysis between the β value and landslide triggering factors.

Bilateral multiple gamma returns: Their risks and rewards

The bilateral gamma model for returns is naturally derived from the lognormal model. Maximizing entropy in a random time change delivers the symmetric variance gamma model. The asymmetric variance gamma follows on incorporating skewness. Differential speeds for the upward and downward motions lead to the bilateral gamma. A further generalizations results in the bilateral double gamma model when the speed parameter of the bilateral gamma model is itself taken to be gamma distributed on entropy maximization. A rich five to seven parameter specification of preferences renders possible the extraction of physical densities from option prices. The quality of such extraction is measured by examining the uniformity of the estimated distribution functions evaluated at realized forward returns. The economic value of risky returns is seen to embed three/five risk premia for the bilateral gamma/bilateral double gamma. For the bilateral gamma they are up and down side volatilities compensated in up and down side drifts, and the down side drift compensated in the up side drift. For the bilateral double gamma one adds in addition compensations for skewness. Results reveal a drop in the down side risk premium since the crisis with an increase in the recent period.

Evaluation of the performance of Euro-CORDEX Regional Climate Models for assessing hydrological climate change impacts in Great Britain: A comparison of different spatial resolutions and quantile mapping bias correction methods

Regional Climate Models (RCMs) are an essential tool for analysing regional climate change impacts, such as hydrological change, as they provide simulations with more small-scale details and expected smaller errors than global climate models. There has been much effort to increase the spatial resolution and simulation skill of RCMs (i.e. through bias correction), yet the extent to which this improves the projection of hydrological change is unclear. Here, we evaluate the skill of five reanalysis-driven Euro-CORDEX RCMs in simulating precipitation and temperature, and as drivers of a hydrological model to simulate river flow on four UK catchments covering different physical, climatic and hydrological characteristics. We use a comprehensive range of evaluation indices for aspects of the distribution such as means and extremes, as well as for the structure of time series. We test whether high-resolution RCMs provide added value, through analysis of two RCM resolutions, 0.44° (50 km) and 0.11° (12.5 km), which are also bias-corrected employing the parametric quantile-mapping (QM) method, using the normal distribution for temperature, and the Gamma (GQM) and Double Gamma (DGQM) distributions for precipitation. The performance of these is considered for a range of meteorological variables and for the skill in simulating hydrological impacts at the catchment scale.In a small catchment with complex topography, the 0.11° RCMs clearly outperform their 0.44° version for precipitation and temperature, but when used in combination with the hydrological model, fail to capture the observed river flow distribution. In the other (larger) catchments, only one high-resolution RCM consistently outperforms its low-resolution version, implying that in general there is no added value from using the high-resolution RCMs in those catchments. Both resolutions produce river flow simulations that cover the observed flow duration curve, but the ensemble spread is large and therefore the simulations are difficult to use in practice. GQM decreases most of the simulation biases, except for extreme precipitation and high flows, which are further decreased by DGQM, which also reduces the multi-model simulation spread. Bias correction does not improve the representation of daily temporal variability measured by the Nash-Sutcliffe Efficiency Index, but it does for monthly variability, in particular when applying DGQM, which reduces most of the simulation biases. Overall, an increase in RCM resolution does not imply a better simulation of hydrology and bias-correction represents an alternative to ease decision-making.

Probabilistic Modelling for Earthquake Forecasting in the Northwestern Part of Haryana State, India

This study attempts to estimate the probability of the occurrence of large earthquakes (Mw ≥ 5.5) in the northwestern part of Haryana state, India, where a new nuclear power plant (NPP) is going to be constructed in the near future. First, an earthquake catalogue is developed for the period 1803–1986, and five stochastic models, namely lognormal, Weibull, gamma, Rayleigh, and double exponential, are then applied to past earthquake data. The performance of these models is checked using three statistical tests, and the lognormal, Weibull, and Rayleigh models are found to produce good approximations for this region, whereas the double exponential and gamma models yield intermediate and poor results. Hence, cumulative and conditional probabilities and related hazard curves for future earthquakes are estimated using the most suitable models. The cumulative probability of the occurrence of an earthquake (Mw ≥ 5.5) since the last event (1986) reached 0.95–0.98 as of 2018. The conditional probability of the occurrence of such an earthquake reaches 0.90–0.95 about 9–12 years from now (2027–2030), when the elapsed time will be 32 years (i.e., since 2018). The probability of earthquakes with different threshold magnitudes is then estimated, and based on the outcome of this investigation, earthquake magnitudes are classified from occasional (Mw ≤ 6.1) to very rare events (Mw ≥ 7.6) that this region may experience in the future. The findings of this study will be considered in seismic hazard assessment, liquefaction hazard assessment, and earthquake-resistant design of NPP components.

Density functional theory calculations of the double gamma prime Ni3Ta phase under pressure: Structural, mechanical, and electronic properties

In this study, we investigated the structural, mechanical, and electronic properties of the γ′′-Ni3Ta (body-centered tetragonal structure) phase under various pressures using first-principles methods. The negative formation enthalpy indicated that γ′′-Ni3Ta is energetically stable when external pressure is applied. The elastic constant and polycrystalline elastic moduli were calculated using the stress–strain and Voigt–Reuss–Hill approximations methods. The results showed that γ′′-Ni3Ta is mechanically stable and ductile. The mechanical anisotropies of γ′′-Ni3Ta were measured using the elastic anisotropic indexes (AU, AB, and AG, and A1, A2, and A3) and the calculated results suggest that γ′′-Ni3Ta is anisotropic. In addition, the sound velocities and Debye temperatures were calculated. Finally, the electronic densities of states were investigated to determine the internal mechanisms responsible for the structural and mechanical properties of γ′′-Ni3Ta.

Impacts of climate change on characteristics of daily‐scale rainfall events based on nine selected GCMs under four CMIP5 RCP scenarios in Qu River basin, east China

AbstractClimate change significantly influences characteristics of rainfall events including rainfall depth, rainfall duration, inter‐event time and temporal patterns that directly affect water resources management, flood defence and hydraulic structure design. In this study, a framework is proposed to analyse daily‐scale rainfall event characteristics based on global climate model (GCM) simulations. This framework includes bias correction of raw GCM‐simulated rainfall series, selection of good‐performing bias‐corrected GCMs based on the mean absolute percentage error (MAPE) and evaluation of selected GCMs' skills in simulating rainfall event characteristics and finally assessment of changes in rainfall event characteristics in the future. In this study, 17 GCMs, four representative concentration pathways (i.e., RCP2.6, RCP4.5, RCP6.0 and RCP8.5) and two future periods (i.e., 2041–2070 and 2071–2100) are considered. After bias correction of the GCMs using the monthly‐scale double gamma distribution, 9 out of 17 GCMs with MAPE values smaller than 20% in the historical period 1971–2000 are selected. In general, these selected GCMs well capture the rainfall characteristics of different rainfall event classes. The multi‐model ensembles suggest that compared to the historical period, the frequency of rainfall events with an extreme depth, short duration and long inter‐event time will increase in the two future periods and the change in 2071–2100 is generally larger than that in 2041–2070, indicating that more extreme climate conditions may occur in Qu River basin in the future. Moreover, the temporal patterns of heavy rainfall events will become more non‐uniform with more concentrated peak rainfall. The frequency of the delayed rainfall type (i.e., peaks occurring at the end of the rainfall event) will increase in the future, which can probably cause more severe floods and is very detrimental to flood defence in this study area.

Size and spatial distribution of loess slides on the Chinese Loess Plateau

ABSTRACTThe size and spatial distribution of loess slides are important for estimating the yield of eroded materials and determining the landslide risk. While previous studies have investigated landslide size distributions, the spatial distribution pattern of landslides at different spatial scales is poorly understood. The results indicate that the loess slide distribution exhibits a power-law scaling across a range of the size distribution. The mean landslide size and size distribution in the different geomorphic types are different. The double Pareto and inverse gamma functions can coincide well with the empirical probability distribution of the loess slide areas and can quantitatively reveal the rollover location, maximum probability, and scaling exponents. The frequency of loess slides increases with mean monthly precipitation. Moreover, point distance analysis showed that > 80% of landslides are located < 3 km from other loess slides. We found that the loess slides at the two study sites (Zhidan and Luochuan County) in northern Shaanxi Province, China show a significant clustered distribution. Furthermore, analysis results of the correlated fractal dimension show that the landslides exhibit a dispersed distribution at smaller spatial scales and a clustered distribution at larger spatial scales.

Achieving shrinkage in a time-varying parameter model framework

Shrinkage for time-varying parameter (TVP) models is investigated within a Bayesian framework, with the aim to automatically reduce time-varying parameters to static ones, if the model is overfitting. This is achieved through placing the double gamma shrinkage prior on the process variances. An efficient Markov chain Monte Carlo scheme is developed, exploiting boosting based on the ancillarity-sufficiency interweaving strategy. The method is applicable both to TVP models for univariate as well as multivariate time series. Applications include a TVP generalized Phillips curve for EU area inflation modeling and a multivariate TVP Cholesky stochastic volatility model for joint modeling of the returns from the DAX-30 index.

A two-variable generalization of the Kummer–Malmstén formula for the logarithm of the double gamma and double sine functions

A classical formula due to Kummer and Malmstén expresses the logarithm of the gamma function as a Fourier series. We prove a generalization of this formula which expresses the logarithm of the double gamma function as a double Fourier series. We also obtain a double Fourier series expansion for the logarithm of the double sine function S2(x,(ω1,ω2)), and of the two derivations given for this expansion, the first requires that ω1 and ω2 are incommensurable, and the second removes this restriction and holds in general. This second derivation relies upon an integral representation of the function log⁡S2(x,(ω1,ω2)) which appears to be new. A generalization of Raabe's formula for the gamma function to a two-variable version for the double gamma function is obtained, and double Fourier series expansions of the double Bernoulli polynomials are derived as well.

Bilateral Multiple Gamma Returns: Their Risks and Rewards

The bilateral gamma model for returns is naturally derived from the lognormal model. Maximizing entropy in a random time change delivers the symmetric variance gamma model. The asymmetric variance gamma follows on incorporating skewness. Differential speeds for the upward and downward motions lead to the bilateral gamma. A further generalizations results in the bilateral double gamma model when the speed parameter of the bilateral gamma model is itself taken to be gamma distributed on entropy maximization. A rich five to seven parameter specification of preferences renders possible the extraction of physical densities from option prices. The quality of such extraction is measured by examining the uniformity of the estimated distribution functions evaluated at realized forward returns.The economic value of risky returns is seen to embed three/five risk premia for the bilateral gamma/bilateral double gamma. For the bilateral gamma they are up and down side volatilities compensated in up and down side drifts, and the down side drift compensated in the up side drift. For the bilateral double gamma one adds in addition compensations for skewness. Results reveal a drop in the down side risk premium since the crisis with an increase in the recent period.

ABERs of LDPC-Coded Multi-Hop FSO Over Double GG Fading Channels With Pointing Error and Path Loss

The average bit error rate (ABER) performances of low-density parity-check (LDPC)-coded multi-hop free-space optical (FSO) communication system with decode-and-forward strategy over the aggregated double generalized gamma (double GG) fading channels for plane and spherical waves are investigated. In particular, on the basis of the unified closed-from expressions of the composite probability density function and cumulative distribution function in terms of the Meijer’s G function, the analytical expression of the ABER for this multi-hop FSO system is derived by generalized Gauss-Laguerre quadrature rule and verified by Monte Carlo simulation. Furthermore, LDPC codes are adopted to improve the system performance. The results show that the ABER performance over double GG distribution is mainly limited by both pointing error and path loss for these two transmission beams, whereas the fading can be significantly improved by LDPC codes, and the coding gain improvement becomes more apparent by decreasing the normalized beam-width, increasing the normalized jitter and hop numbers. This work is beneficial for the FSO system design.