Generalized Gamma Research Articles

THE COMPARISON SOME MODELS DAILY WIND SPEED DATA IN PEKANBARU CITY USING TWO, THREE AND FOUR PARAMETERS GAMMA DISTRIBUTIONS

Accurate wind speed modeling is critical in estimating wind energy potential for harnessing wind power effectively. One of the bases for assessment of wind energy potential for a specified region is the probability distribution of wind speed, therefore wind speed data is needed to produce statistical modeling, especially in determining the best probability distribution. For this purpose, several modified distributions will be used and tested to determine the best model to describe wind speed in Pekanbaru. Six continuous probability density function such as Two Parameters Gamma (GM), Three Parameters Slashed Quasi-Gamma (SQG), Three Parameters Quasi Gamma (QG), Three Parameters Generalized Gamma (GG), Four Parameters Poly Weighted Exponentiated Gamma (PWEG) and Four Parameters Exponentiated Gamma Exponential (EGE) distributions were fitted for these tasks using the maximum likelihood technique. To determine the model’s fit to the daily wind speed data, several goodness-of-fit tests were applied, including the graphical methos test (pdf plot) and numerical criteria method test (AIC). The QG and GG distributions are found to be the best-fitted probability distribution based on goodness-of-fit tests for the daily wind speed data at the Pekanbaru station.

Recursive generalized gamma kernel density estimation for nonnegative dependent data

In this article, we consider the recursive generalized gamma (GG) kernel density estimator (KDE) for nonnegative dependent data from stationary α-mixing process. Asymptotic bias, variance and mean integrated squared error are provided. Simulation experiments from an autoregressive conditional duration model and a stochastic volatility model are conducted to compare the recursive GG KDE with non-recursive GG KDE. Two applications are provided for nonnegative time series data.

Mobile network traffic analysis based on probability-informed machine learning approach

The paper proposes an approach to the joint use of statistical and machine learning (ML) models to solve the problems of the precise reconstruction of historical events, real-time detection of ongoing incidents, and the prediction of future quality of service-related occurrences for prospective development of the modern networks. For forecasting, a regression version of the deep Gaussian mixture model (DGMM) is introduced. First, the preliminary clustering based on the finite normal mixtures is performed. This information is then used as an input for some supervised ML algorithm. It is the basic concept of the probability-informed ML approach in the field of telecommunications networks. Using the real-world datasets from a Portuguese mobile operator as well as public cellular traffic data, the article compares this approach with methods such as random forests, support vector machine regression, gradient boosting and LSTM. Vector autoregression, informed by the parameters of the generalized gamma (GG) distribution, which has also been successfully used to reconstruct past traffic patterns, is also used as a benchmark. We demonstrate that DGMM-based regression is 6.82–22.8 times faster than LSTM for the dataset. Moreover, DGMM-based regression can achieve better results for the most important traffic characteristics (average and total traffic, the number of users). For metrics MAPE and RMSE, it surpasses the results of statistical methods up to 46.7% (RMSE) and 91.5% (MAPE) (median increases are 28.0% and 80.1%, respectively), as well as for ML methods up to 13.0% (RMSE) and 35.7% (MAPE) (median increases are 0.39% and 2.5%, respectively). Thus, the use of a probability-informed approach for telecommunication data seems optimal for the computational speed and accuracy trade-off. Also, we introduce a novel statistical hypothesis testing method based on GG distribution for detecting suspected anomalies in traffic.

A 39 Year mortality study of survivors exposed to sulfur mustard agent: A survival analysis

BackgroundThe primary objective of this study was to analyze the long-term survival of 48,067 chemical warfare survivors who suffered from pulmonary, cutaneous, and ocular lesions in the decades following the Iran-Iraq war. MethodsThe data for this study were obtained from the Veterans and Martyr Affair Foundation (VMAF) database. The survivors were divided into two groups based on whether they were evacuated/admitted (EA) to a hospital or not evacuated/admitted (NEA) to a hospital. The proportional hazard (PH) assumption for age categories, gender, exposure statuses, and eye severity was not satisfied. Therefore, we used a Generalized Gamma (GG) distribution with an Accelerated Failure Time (AFT) model for analysis. ResultsThe study included a total of 48,067 observations, and among them, 4342 (9.03 %) died during the study period. The mean (SD) age of the survivors was 55.99 (7.9) years. The mortality rate increased with age, and higher rates were observed in males. Survival probabilities differed significantly among age categories, provinces, lung severity, and eye severity based on log-rank tests (p-value<0.05 for all). The GG model results showed that higher age and being male were associated with a shorter time to death. The study also found that the mortality rate was significantly higher in the EA group compared to the NEA group. ConclusionThe present study showed no significant difference in survival time between the EA and NEA groups. The findings suggest that pulmonary lesions caused by mustard gas are more likely to be fatal compared to skin and eye lesions. The results also indicate a potential association between survival time and the severity of lung damage.

Error Performance Analysis of Serial-relayed Underwater Wireless Optical Communication Systems over Generalized Gamma Distribution Weak Turbulence Environment with Pointing Error

Error Performance Analysis of Serial-relayed Underwater Wireless Optical Communication Systems over Generalized Gamma Distribution Weak Turbulence Environment with Pointing Error

Bootstrap Intervals for the Mean of the Weighted Mixture Generalized Gamma Distribution

Bootstrap Intervals for the Mean of the Weighted Mixture Generalized Gamma Distribution

Longitudinal changes of health-related quality of life in childhood chronic kidney disease.

Few longitudinal studies have evaluated the impact of chronic kidney disease (CKD) duration on health-related quality of life (HRQOL). The study's aim was to determine how HRQOL changes over time in childhood CKD. Study participants were children in the chronic kidney disease in children (CKiD) cohort who completed the pediatric quality of life inventory (PedsQL) on three or more occasions over the course of two or more years. Generalized gamma (GG) mixed-effects models were applied to assess the effect of CKD duration on HRQOL while controlling for selected covariates. A total of 692 children (median age = 11.2) with a median of 8.3years duration of CKD were evaluated. All subjects had a GFR greater than 15ml/min/1.73 m2. GG models with child self-report PedsQL data indicated that longer CKD duration was associated with improved total HRQOL and the 4 domains of HRQOL. GG models with parent-proxy PedsQL data indicated that longer duration was associated with better emotional but worse school HRQOL. Increasing trajectories of child self-report HRQOL were observed in the majority of subjects, while parents less frequently reported increasing trajectories of HRQOL. There was no significant relationship between total HRQOL and time-varying GFR. Longer duration of the disease is associated with improved HRQOL on child self-report scales; however, parent-proxy results were less likely to demonstrate any significant change over time. This divergence could be due to greater optimism and accommodation of CKD in children. Clinicians can use these data to better understand the needs of pediatric CKD patients. A higher resolution version of the Graphical abstract is available as Supplementary information.

Performance analysis of UAV assisted triple-hop mixed FSO/RF/UWOC system with altitude optimization

Abstract In this paper, we propose a unmanned aerial vehicle (UAV) assisted triple hop mixed FSO/RF/UWOC system to establish a communication between distant users. In the proposed system, the source sends information signal towards a hovering UAV such as drone acting as a first intermediate node via FSO link modelled by Gamma-Gamma distribution. The drone further transmits the received signal to the second intermediate node located on a floating vehicle (FV) on the ocean surface via a RF link modelled by the Nakagami-m fading distribution and from where it is finally forwarded towards the deep ocean destination via visible light channel (VLC) link which is modelled by Exponential Generalized Gamma (EGG) fading distribution. Both the relay nodes act as decode and forward relays. For this asymmetric triple hop mixed communication system, we obtain a new exact closed form analytical expression for the outage probability and also study its behaviour at high signal to noise ratio (SNR) regime. Additionally, the UAV altitude optimization is also carried out for the proposed system to find the optimal altitude of the UAV and the optimal elevation angle to maximize the system’s performance. The numerical simulation results illustrate the accuracy of the proposed system model and present the influence of the various link parameters such as atmospheric path loss, turbulence, pointing errors, angle-of-arrival fluctuations, multipath fading parameter, water turbulence, and scintillation on the performance of the proposed system.

Performance analysis of a dual-hop parallel relayed mixed FSO-UWOC system

Abstract In this paper, we propose a dual-hop parallel relayed mixed free space optical-underwater optical communication (FSO-UWOC) system model to setup a high-speed communication link connecting a distant sender located at the lighthouse tower and the receiver inside the deep ocean to transmit and receive the data for the applications such as deep sea exploration, marine life study etc. The proposed model is the setup of multiple parallel cooperative dual hop communication link for the betterment of the overall system performance under the worst case scenarios. The receiver examines the end-to-end signal to noise ratio and a relay is selected which has the maximum SNR. The selected relay then decodes the incoming optical signal through FSO link into its corresponding visible light signal and further transmits it to the underwater exploration devices such as underwater drone via VLC link. The FSO link is modelled by the Gamma-Gamma fading distribution and the VLC link is modelled by Exponential Generalized Gamma distribution. The end-to-end SNR of the system is utilized to derived the outage probability of the FSO-UWOC system and the impact of the individual link parameters is studied and presented for the proposed system model.

Rethinking Generalized Beta family of distributions

We approach the Generalized Beta (GB) family of distributions using a mean-reverting stochastic differential equation (SDE) for a power of the variable, whose steady-state (stationary) probability density function (PDF) is a modified GB (mGB) distribution. The SDE approach allows for a lucid explanation of Generalized Beta Prime (GB2) and Generalized Beta (GB1) limits of GB distribution and, further down, of Generalized Inverse Gamma (GIGa) and Generalized Gamma (GGa) limits, as well as describe the transition between the latter two. We provide an alternative form to the "traditional" GB PDF to underscore that a great deal of usefulness of GB distribution lies in its allowing a long-range power-law behavior to be ultimately terminated at a finite value. We derive the cumulative distribution function (CDF) of the "traditional" GB, which belongs to the family generated by the regularized beta function and is crucial for analysis of the tails of the distribution. We analyze fifty years of historical data on realized market volatility, specifically for S\&P500, as a case study of the use of GB/mGB distributions and show that its behavior is consistent with that of negative Dragon Kings.

A New Formulation of Generalized Gamma: Some Results and Applications

We extend the 2-parameter Weibull to the generalized gamma distribution by adding a new partial parameter. The new shape parameter can be used to easily generate generalized gamma distributions with different shapes of the density function, hazard rate, and mean residual lifetimes that are useful in simulating various business processes such as manufacturing processes, and reliability systems. We derived some theoretical results and created visual presentations to show the influence (or effect) of this new shape parameter as well. A new Monte Carlo simulation based on the new parameter was proposed to assess the discrepancy between the generalized gamma and its subfamilies. The power analysis of the proposed test was evaluated via simulation studies. We also present some numerical examples.

AE-TPGG: a novel autoencoder-based approach for single-cell RNA-seq data imputation and dimensionality reduction.

Single-cell RNA sequencing (scRNA-seq) technology has become an effective tool for high-throughout transcriptomic study, which circumvents the averaging artifacts corresponding to bulk RNA-seq technology, yielding new perspectives on the cellular diversity of potential superficially homogeneous populations. Although various sequencing techniques have decreased the amplification bias and improved capture efficiency caused by the low amount of starting material, the technical noise and biological variation are inevitably introduced into experimental process, resulting in high dropout events, which greatly hinder the downstream analysis. Considering the bimodal expression pattern and the right-skewed characteristic existed in normalized scRNA-seq data, we propose a customized autoencoder based on a two-part-generalized-gamma distribution (AE-TPGG) for scRNA-seq data analysis, which takes mixed discrete-continuous random variables of scRNA-seq data into account using a two-part model and utilizes the generalized gamma (GG) distribution, for fitting the positive and right-skewed continuous data. The adopted autoencoder enables AE-TPGG to captures the inherent relationship between genes. In addition to the ability of achieving low-dimensional representation, the AE-TPGG model also provides a denoised imputation according to statistical characteristic of gene expression. Results on real datasets demonstrate that our proposed model is competitive to current imputation methods and ameliorates a diverse set of typical scRNA-seq data analyses.Electronic Supplementary MaterialSupplementary material is available in the online version of this article at 10.1007/s11704-022-2011-y.

Unified Performance Assessment of Optical Wireless Communication Over Multi-Layer Underwater Channels

In this paper, we model the multi-layer vertical underwater link as a cascaded channel and unify the performance analysis for the underwater optical communication (UWOC) system using generalized Gamma (GG), exponential GG (EGG), exponentiated Weibull (EW), and Gamma-Gamma ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Gamma \Gamma$</tex-math></inline-formula> ) oceanic turbulence models. We derive unified analytical expressions for probability density function (PDF) and cumulative distribution function (CDF) for the signal-to-noise ratios (SNR) considering independent and non-identical (i.ni.d.) turbulent models and zero bore-sight model for pointing errors. We develop performance metrics of the considered UWOC system using outage probability, average bit error rate (BER), and ergodic capacity with asymptotic expressions for outage probability and average BER. We develop the diversity order of the proposed system to provide a better insight into the system performance at a high SNR. We also integrate a terrestrial OWC (TOWC) subjected to the combined effect of generalized Malága atmospheric turbulence, fog-induced random path gain, and pointing errors to communicate with the UWOC link using the fixed-gain amplify-and-forward (AF) relaying. We analyze the performance of the mixed TWOC and multi-layer UWOC system by deriving PDF, CDF, outage probability, and average BER using the bivariate Fox H-function. We use Monte-Carlo simulation results to validate our exact and asymptotic expressions and demonstrate the performance of the considered underwater UWOC system using measurement-based parametric data available for turbulent oceanic channels.

Design of Multiple Deferred State Repetitive Group Sampling Plans for Life Tests based on Generalized Gamma Distribution

Reliability sampling plans is a procedure to determine the acceptance or non-acceptance of the lot(s) of finished products by performing tests on the sampled items. Lifetime of individual items is a quality characteristic that can be treated as a continuous random variable and can be modeled by an appropriate continuous probability distribution. In this article, Multiple Deferred State Repetitive Group Sampling (MDSRGS) Plans for life tests under hybrid censoring with a provision to draw a random sample and to admit a maximum of one failure in the sample is formulated assuming that the lifetime random variable follows a Generalized Gamma (GG) distribution. A methodical procedure for the selection of the plan parameters using mean life criterion with the desired discrimination protecting the interests of the producer and the consumer in terms of the acceptable mean life and unacceptable mean life is evolved.

Probabilistic Peak Demand Estimation Using Members of the Clayton Generalized Gamma Copula Family

Climate change impacts many aspects of life and requires innovative thinking on various issues. The electricity sector is affected in several ways, including changes in the production components and consumption patterns. One of the most important issues for Independent System Operators, a state-controlled organization responsible for ensuring the reliability, availability, and quality of electricity delivery in the country, is the response to climate change. This is reflected in the appropriate design of production units to cope with the increase in demand due to extreme heat and cold events and the development of models aimed at predicting the probability of such events. In our work, we address this challenge by proposing a novel probability model for peak demand as a function of wet temperature (henceforth simply temperature), which is a weighting of temperature and humidity. We study the relationship between peak demand and temperature using a new Archimedean copula family, shown to be effective for this purpose. This family, the Clayton generalized Gamma, is a multi-parameter copula function that comprises several members. Two new measures of fit, an economic measure and a conditional coverage measure, were introduced to select the most appropriate family member based on the empirical data of daily peak demand and minimum temperature in the winter. The Clayton Gamma copula showed the lowest cost measure and the best conditional coverage and was, therefore, proven to be the most appropriate member of the family.

The Generalized Gamma Distribution as a Useful RND under Heston’s Stochastic Volatility Model

We present the Generalized Gamma (GG) distribution as a possible risk neutral distribution (RND) for modeling European options prices under Heston’s stochastic volatility (SV) model. We demonstrate that under a particular reparametrization, this distribution, which is a member of the scale-parameter family of distributions with the mean being the forward spot price, satisfies Heston’s solution and hence could be used for the direct risk-neutral valuation of the option price under Heston’s SV model. Indeed, this distribution is especially useful in situations in which the spot’s price follows a negatively skewed distribution for which Black–Scholes-based (i.e., the log-normal distribution) modeling is largely inapt. We illustrate the applicability of the GG distribution as an RND by modeling market option data on three large market-index exchange-traded funds (ETF), namely the SPY, IWM and QQQ as well as on the TLT (an ETF that tracks an index of long-term US Treasury bonds). As of the writing of this paper (August 2021), the option chain of each of the three market-index ETFs shows a pronounced skew of their volatility ‘smile’, which indicates a likely distortion in the Black–Scholes modeling of such option data. Reflective of entirely different market expectations, this distortion in the volatility ‘smile’ appears not to exist in the TLT option data. We provide a thorough modeling of the option data we have on each ETF (with the 15 October 2021 expiration) based on the GG distribution and compare it to the option pricing and RND modeling obtained directly from a well-calibrated Heston’s SV model (both theoretically and also empirically, using Monte Carlo simulations of the spot’s price). All three market-index ETFs exhibited negatively skewed distributions, which are well-matched with those derived under the GG distribution as RND. The inadequacy of the Black–Scholes modeling in such instances, which involves negatively skewed distribution, is further illustrated by its impact on the hedging factor, delta, and the immediate implications to the retail trader. Similarly, the closely related Inverse Generalized Gamma distribution (IGG) is also proposed as a possible RND for Heston’s SV model in situations involving positively skewed distribution. In all, utilizing the Generalized Gamma distributions as possible RNDs for direct option valuations under the Heston’s SV is seen as particularly useful to the retail traders who do not have the numerical tools or the know-how to fine-calibrate this SV model.

The BER Performance of the LDPC-Coded MPPM over Turbulence UWOC Channels

Turbulence-induced fading is a critical performance degrading factor for underwater wireless optical communication (UWOC) systems. In this paper, we propose a quasi-cyclic (QC) low-density parity-check (LDPC) code with multiple-pulse-position modulation (MPPM) to overcome turbulence-induced fading. MPPM is adopted as a compromise between the low-power efficiency of on–off keying (OOK) and the low bandwidth efficiency of pulse position modulation (PPM). The bit error rate (BER) performance of LDPC-coded MPPM over turbulence UWOC channels is investigated. The log-likelihood ratio (LLR) of MPPM is derived, and a simplified approximation is used for iterative decoding. Subsequently, the closed-form expression of the BER, without forward error correction (FEC) code, is obtained for the generalized-gamma (GG) fading model. Finally, Monte-Carlo (MC) simulation results are provided to demonstrate the correctness of the derived closed-form expressions and the effectiveness of the LDPC code with simplified LLR to improve the BER performance for different MPPM formats over fading channels.

Stochastic Geometry Analysis of IRS-Assisted Downlink Cellular Networks

Using stochastic geometry tools, we develop a comprehensive framework to analyze the downlink performance of various types of users (e.g., users served by direct base station (BS) transmissions and indirect intelligent reflecting surface (IRS)-assisted transmissions) in a cellular network with multiple BSs and IRSs. For the proposed users, we provide the approximate expressions for the performance in terms of coverage probability, ergodic capacity, and energy efficiency (EE). The proposed stochastic geometry framework can capture the impact of channel fading, locations of BSs and IRSs, arbitrary phase-shifts and interference experienced by a typical user supported by direct transmission and/or IRS-assisted transmission. For IRS-assisted transmissions, we first model approximate the desired signal power from the nearest IRS as a sum of scaled generalized gamma (GG) random variables whose parameters are functions of the IRS phase shifts. Then, we derive the Laplace Transform (LT) of the received signal power in a closed form. Also, we approximate the aggregate interference from multiple IRSs as the sum of normal random variables. Then, we derive the LT of the aggregate interference from all IRSs and BSs. The derived LT expressions are used to calculate coverage probability, ergodic capacity, and EE for users served by direct BS transmissions as well as users served by IRS-assisted transmissions. Finally, we derive the overall network coverage probability, ergodic capacity, and EE based on the fraction of direct and IRS-assisted users, which is defined as a function of the deployment density of IRSs, as well as blockage probability of direct transmission links. Numerical results validate the derived analytical expressions and extract useful insights related to the number of IRS elements, large-scale deployment of IRSs and BSs, and the impact of IRS interference on direct transmissions.

Fragment size distributions in brittle deformed rocks

The production of breccias and cataclasites is commonly proposed to result in power-law or log-normal probability distributions for fragment (grain) size. We show that in both natural and experimental examples, the common best fit probability distributions for the complete distributions are members of the Generalised Gamma (GG), Extreme Value (GEV) and Pareto (GP) families; power-law and log-normal distributions are commonly, but not always, poor fits to the data. An hierarchical sequence, GG → GEV → GP, emerges as the sample mean of the fragment size decreases. The physical foundations (self-similar fragmentation, collisional fragmentation, shattering) for these distributions are discussed. Particularly important is the shattering continuous phase transition that results in the simultaneous development of both coarse fragments and ultra-fine particles (dust). This phase transition leads to Generalised Pareto fragment size distributions for the coarse fragments. Also included is a discussion of the relations between fragment size distribution, processes and deformation history in the context of monomineralic rocks. The overall reported size distributions are compatible with theoretical developments but the topic would benefit from observations and experiments conducted with the theories in mind.

Performance Analysis and Altitude Optimization of UAV-Enabled Dual-Hop Mixed RF-UWOC System

This paper presents and investigates a dual-hop mixed radio frequency (RF) and underwater optical communication (UWOC) system. In the proposed system, an unmanned aerial vehicle (UAV) such as drone (S) located at low altitude is transmitting data signal towards the destination (D) which is located under water such as a submarine through a decode and forward (DF) relay (R) mounted on a ship over the sea surface. The data signal is transmitted from the UAV towards the relay located on the ship via RF channel modelled by the Nakagami- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$ m$</tex-math></inline-formula> distributed fading statistics. The relay decodes the received RF data signal and re-transmits it towards the destination via visible light channel modelled by Exponential Generalized Gamma (EGG) distribution. Using the DF relaying protocol, we derive closed form analytical expressions for the outage probability and average bit error rate of the proposed mixed RF-UWOC system. Additionally, UAV optimal altitude analysis has been carried out to find the optimal elevation angle corresponding to the optimal altitude of the UAV where the system performance is maximized. The numerical simulation is carried out to support the performance analysis of the mixed RF-UWOC system and shows the influence of the various channel parameters such as air bubbles, UAV altitude, water salinity variations and scintillation on the end to end performance of the mixed RF-UWOC system.