Rayleigh Distribution Research Articles

A versatile model for lifetime of a component under stress

In this study, a versatile model, called alpha-monotone inverse Weibull distribution (alphaIW), for lifetime of a component under stress is introduced by using the alpha-monotone concept. The alphaIW distribution is also expressed as a scale-mixture between the inverse Weibull distribution and uniform distribution on (0, 1). The alphaIW distribution includes alpha-monotone inverse exponential and alpha-monotone inverse Rayleigh distributions as submodels and converenges to the inverse Weibull, inverse exponential, and inverse Rayleigh distributions as limiting cases. Also, slash Weibull, slash Rayleigh, and slash exponential distribuitons can be obtained under certain variable transformation and parameter settings. The alphaIW distribution is characterized by its hazard rate function and characterizing conditions are provided as well. Maximum likelihood, maximum product of spacing, and least squares methods are used to estimate distribution parameters. A Monte-Carlo simulation study is conducted to compare the efficiencies of the considered estimation methods. In the application part, two practical data sets, Kevlar 49/epoxy and Kevlar 373/epoxy, are modeled via the alphaIW distribution. Modeling performance of the alphaIW distribution is compared with its rivals by means of some well-known goodness-of-fit statistics and results show that alphaIW distribution performs better modeling than them. Results of comparison also indicate that obtaining the alphaIW distribution by using the alpha-monotone concept is cost effective since the new shape parameter added to the distribution by using the alpha-monotone concept significantly increases the modeling capability of the IW distribution. As a result of this study, it is shown that the alphaIW distribution can be an alternative to the well-known and recently-introduced distributions for modeling purposes.

Generalized Fiducial Inference for the Generalized Rayleigh Distribution

This article focuses on the interval estimation of the generalized Rayleigh distribution with scale and shape parameters. The generalized fiducial method is used to construct the fiducial point estimators as well as the fiducial confidence intervals, and then their performance is compared with other methods such as the maximum likelihood estimation, Bayesian estimation and parametric bootstrap method. Monte Carlo simulation studies are carried out to examine the efficiency of the methods in terms of the mean square error, coverage probability and average length. Finally, two real data sets are presented to demonstrate the applicability of the proposed method.

A Heavy-Tailed Distribution Based on the Lomax–Rayleigh Distribution with Applications to Medical Data

In this paper, we extend the Lomax–Rayleigh distribution to increase its kurtosis. The construction of this distribution is based on the idea of the Slash distribution, that is, its representation is based on the quotient of two independent random variables, one being a random variable with a Lomax–Rayleigh distribution and the other a beta(q,1). Based on the representation of this family, we study its basic properties, such as moments, coefficients of skewness, and kurtosis. We perform statistical inference using the methods of moments and maximum likelihood. To illustrate this methodology, we apply it to two real data sets.

Length Biased Extended Rayleigh Distribution and its Applications

The generalization of the Rayleigh distribution has been introduced by several authors in recent years. In this article, generalization of Rayleigh distribution has been derived by length- biased transformer technique named length biased Rayleigh- Rayleigh distribution. (LB-RRD). The statistical properties of LB-RRD have been formulated. Estimation of parameter is carried out by method of maximum likelihood. Three data sets are taken to show the superiority of LB-RRD with other competing distributions.

A Review of the Rayleigh Distribution: Properties, Estimation & Application to COVID-19 Data

A Review of the Rayleigh Distribution: Properties, Estimation & Application to COVID-19 Data

Gaussian Mixture Model for Marine Reverberations

Ocean reverberations, a significant interference source in active sonar, arise as a response generated by random scattering at the receiving end, a consequence of randomly distributed clutter or irregular interfaces. Statistical analysis of reverberation data has revealed a predominant adherence to the Rayleigh distribution, signifying its departure from specific distribution forms like the Gaussian distribution. This study introduces the Gaussian mixture model, capable of simulating random variables conforming to a wide array of distributions through the integration of an adequate number of components. Leveraging the unique statistical attributes of reverberation, we initiate the Gaussian mixture model’s parameters via the frequency histogram of the reverberation data. Subsequently, model parameters are estimated using the expectation–maximization (EM) algorithm and the most suitable statistical model is selected based on robust model selection criteria. Through a comprehensive evaluation that encompasses both simulated and observed data, our results underscore the Gaussian mixture model’s effectiveness in accurately characterizing the distribution of reverberation data, yielding a mean squared error of less than 4‰.

Preventive Maintenance and Replacement Model for Mechanically Repairable Systems with Linearly Increasing Hazard Rate

Most machines fail due to lack of appropriate preventive maintenance (PM) and replacement schedule, and this failure leads to higher cost of repair maintenance, distortion of production schedule, elongated downtime period and reduced productivity. These could however be avoided by the utilization of optimal PM and replacement models suited for the specific kind of system. It is on this premise that this work develops an optimal PM and replacement model for mechanically repairable systems with linearly increasing hazard rate which failure distribution of the system is characterized by the Rayleigh distribution. The failure times of a Rolls Royce dredging machine was used as real-time data to obtain the PM and replacement schedule for the machine at respective cost ratios. The results showed that the model provided an effective maintenance schedule for the machine and ensures optimal performance.

Homogeneously Weighted Moving Average Control Chart for Rayleigh Distribution

In this paper, we have proposed Homogeneously Weighted Moving Average (HWMA) control chart for Rayleigh distribution. The Average Run Length (ARL1) is used to evaluate the performance of the proposed HWMA control charts. The ARL1 performance of HWMA control chart is compared to the Exponentially weighted moving average (EWMA) control charts with respect to the different shift size (i.e. 10%, 15%, 20%, 30%, 40% increase and decrease in shift). The results are calculated using sample size n=5. It is observed that with the increase in shift proposed HWMA chart shows more efficient results i.e. ARL1 values decrease with the increase in shifts. It is found that the proposed HWMA chart for Rayleigh distribution outperforms the existing EWMA control chart.

Truncated Inverse Generalized Rayleigh Distribution and Some Properties

Truncated distributions arise naturally in many practical situations. It’s a conditional distribution that develops when the parent distribution's domain is constrained to a smaller area. The distribution of a right truncated is one of the types of a single truncated that is restricted within a specific field and usually occurs when the specified period for the study is complete. Hence, this paper introduces Right Truncated Inverse Generalized Rayleigh Distribution (RTIGRD) with two parameters is introduced. Then, provided some properties such as; (probability density function, cumulative distribution function (CDF), survival function, hazard function, ‎rth moment, mean, variance, Moment Generating Function, Skewness, kurtosis, Median, and Mode for Right Truncated Inverse Generalized Rayleigh Distribution on [0,1].

The dependence of wave height probability distributions on physical parameters from measurements near Sakhalin Island

The data of long-term surface waves measurements with bottom sensors near Sakhalin Island were used to build instrumental probability distributions for exceedance of wave heights. Waves with heights exceeding the significant wave height by more than three times were recorded. Specific features of the observations conducted during the periods of open and ice-covered sea surfaces are discussed. The subsets of statistically homogeneous data are arranged through selection considering the natural physical dimensionless parameters of the task, that control the effects of finite depth and nonlinearity (namely, the wave steepness, the wave amplitude to water depth ratio, the Ursell parameter). The manifestation of these and composite parameters in theoretical probability distributions of wave heights is discussed. The effects of nonlinearity and the measurement point depth on probability distributions are estimated with a focus on abnormally high waves. In particular, it is shown that for the place of registration an increase in the wave amplitude to water depth ratio parameter leads to a decrease in the probability of abnormally high waves. This behavior is consistent with the Glukhovsky theoretical distribution. The waves characterized by relatively large dimensionless depth parameter exhibit a higher probability of substantial exceedance of the significant height and are better described by the Rayleigh distribution.

Inference and optimal design of accelerated life test using the geometric process for power rayleigh distribution under time-censored data

In this paper, a new extension of the standard Rayleigh distribution called the Power Rayleigh distribution (PRD) is investigated for the accelerated life test (ALT) using the geometric process (GP) under Type-I censored data. Point estimates of the formulated model parameters are obtained via the likelihood estimation approach. In addition, interval estimates are obtained based on the asymptotic normality of the derived estimators. To evaluate the performance of the obtained estimates, a simulation study of 4, 5 and 6 levels of stress is conducted for ALT in different combinations of sample sizes and censored times. Simulation results indicated that point estimates are very close to their initial true values, have small relative errors, are robust and are efficient for estimating the model parameters. Similarly, the interval estimates have small lengths and their coverage probabilities are almost converging to their 95% nominated significance level. The estimation procedure is also improved by the approach of finding optimum values of the acceleration factor to have optimum values for the reliability function at the specified design stress level. This work confirms that PRD has the superiority to model the lifetimes in ALT using GP under any censoring scheme and can be effectively used in reliability and survival analysis.

Assessing Flood Risk: LH-Moments Method and Univariate Probability Distributions in Flood Frequency Analysis

This study examines all of the equations necessary to derive the parameters for seven probability distributions of three parameters typically used in flood frequency research, namely the Pearson III (PE3), the generalized extreme value (GEV), the Weibull (W3), the log-normal (LN3), the generalized Pareto Type II (PG), the Rayleigh (RY) and the log-logistic (LL3) distributions, using the higher-order linear moments method (LH-moments). The analysis represents the expansion of previous research whose results were presented in previous materials, and is part of hydrological research aimed at developing a standard for calculating maximum flows based on L-moments and LH-moments. The given methods for calculating the parameters of the examined distributions are used to calculate the maximum flows on Romania’s Prigor River. For both methods, the criterion for selecting the most suitable distribution is represented by the diagram of the L-skewness–L-kurtosis and LH-skewness–LH-kurtosis. The results for Prigor River show that the PG distribution is the best model for the L-moments method, the theoretical values of the statistical indicators being 0.399 and 0.221. The RY distribution is the best model for the LH-moments technique, with values of 0.398 and 0.192 for the two statistical indicators.

Predicting the occurrence of rogue waves in short-crested mixed seas

ABSTRACT In this paper we have for the first time proposed a novel transformed Gaussian process method for the estimation of rogue wave occurrence in a short crested mixed sea. The saddle point approximation of a specific joint density is utilised to further improve the computational efficiency of the proposed method. The proposed method has been applied to forecast the rogue wave occurrence in four short-crested mixed sea states, two with directional wave spectra based on measured surface elevation data at the coast of Yura, and another two with typical directional bimodal wave spectra. It is shown in these four cases that the proposed method can offer more accurate forecasting results than those obtained from the linear simulation method, from the transformation based on Hermite polynomials and from using the Rayleigh distribution model. It is also demonstrated that the proposed method is more efficient than the nonlinear simulation method.

Comparison of deep learning algorithms for quantitative diagnosis of fatty liver using multidimensional-moments heatmaps

Non-alcoholic fatty liver disease (NAFLD) carries a high risk of progressing to non-alcoholic steatohepatitis (NASH), cirrhosis, and hepatocellular carcinoma. Therefore, the quantitative diagnosis of NAFLD, i.e., determining hepatic fat deposition, is desired during ultrasound inspection. It is known that the deposition of fat droplets within hepatocytes increases the attenuation in ultrasound images. Because the texture of the attenuated ultrasound image becomes a homogeneous speckle, the probability density function of the echo envelopes can be approximated by a Rayleigh distribution. The diagnosis methods of NAFLD using echo-envelope statistics of ultrasound images have been reported. In this study, we focus on the distribution and variance of echo-envelope statistics and propose a new diagnosis method to analysis the characteristics of echo-envelope statistics using a deep learning algorithm (DLA). In the proposed method, first- and third-order moments as echo-envelope statistics are calculated at each pixel in the ultrasound image. Then, the heatmap is formed from the distributions of both moments in each region of interest within the liver. The heatmaps are classified into the stages of fatty liver using DLA such as a convolutional neural network or a vision transformer. In this presentation, a comparative study on the accuracies by various DLA are reported.

Reliable Data Transmission Scheduling for UAV-Assisted Air-to-Ground Communications

Reliability is significant for unmanned aerial vehicle (UAV)-assisted air-to-ground (A2G) communications. In this letter, we address the reliability optimization problem by properly scheduling data transmissions under Non-Line-of-Sight (NLoS) channel fading. We capture the channel fading effect by the Rayleigh distribution and then formulate the A2G communication reliability from a probabilistic perspective. We propose a constrained optimization model that jointly takes into account the impacts of the aerial and ground nodes' mobility, the stochastic fading characteristics of the A2G channel, and the transmission constraints. From the model, we derive a closed-form reliability-optimal solution for data transmission scheduling and theoretically characterize the optimal reliability that is achievable. Simulation results verify our theoretical results and show the superior performance of the proposed scheduling solution over benchmark methods.

A new extension of linear failure rate distribution with estimation, simulation, and applications

In this article, we provide a new three-parameter model derived from the newly reduced Cauchy power-G family and linked to the linear failure rate model. The truncated Cauchy power linear failure rate (TCPLFR) is the name given to this distribution. The TCPLFR distribution also contains the truncated Cauchy power Rayleigh distribution and the truncated Cauchy power exponential distribution as sub-models. The TCPLFR distribution has rising, falling, and U-shaped hazard rate functions. The distribution characteristics of the TCPLFR are presented. To compute the population parameters’ point and estimated confidence intervals, the maximum likelihood approach is employed. We explore the behavior of the maximum likelihood estimates as well as the estimated confidence intervals for the model parameters using Monte Carlo simulation. To demonstrate the significance and flexibility of the TCPLFR distribution, the Akaike information criterion (D1), Bayesian information criterion (D2), consistent Akaike information criterion (D3), Hannan–Quinn information criterion (D4), and Kolmogorov–Smirnov (D5) statistic with its p-value (D6) were employed. According to a real-world data analysis, the truncated Cauchy power linear failure rate distribution outperforms alternative models with two, three, and four parameters.

A New Discrete Raleigh Distribution and its Application in Immunogold Assay Data

Recently, discretization of continuous distribution was enchanted by many researchers especially in the field of reliability engineering and life testing. This is because the existing models are unsuited for many practical situations. For discretizing continuous distributions, various methods are available in literature. Among them a newly developed method is the two-stage composite discretization method which includes three different methodologies. Using these methodologies, we propose some new discretized distributions obtained through the discretization of Rayleigh distribution. We compare the hazard rate functions of Rayleigh, Discrete Rayleigh with the newly proposed discrete distributions. The flexibility of the model is illustrated using immunogold assay data.

Effects and mechanism of focusing condition on machining quality and bubble behaviors in femtosecond laser micro-grooving of silicon wafer in liquid

In this paper, the effect and mechanism of how varying focusing conditions would affect the machining quality of microgrooves and bubble behaviors (cavitation bubbles (CB) and persistent bubbles (PB)) in liquid-assisted femtosecond laser micro-grooving of silicon wafer were investigated. Firstly, the width and depth of the microgroove machined under negative defocusing condition (NDC) were considerably uniform, which showed an increase by 100 % and 125 % compared with that machined under positive defocusing condition (PDC). Secondly, the temporal evolution images of CB were captured by a time-resolved shadowgraph imaging system. Compared with PDC, the radius and lifetime of CB induced under NDC increased by 71 % and 100 %, and the time evolution curve of the radius of CB exhibited a high consistence with the Rayleigh distribution, while that induced under PDC fluctuated irregularly. Thirdly, the PB induced under NDC showed a backward flush and contributed to the formation of tendentious liquid flow, while that induced under PDC occupied the laser transmission path and caused disturbances on subsequent laser pulses. Finally, the differences in laser transmission mechanism under both the negative and positive defocusing conditions were discussed and the NDC was confirmed as the stable and efficient processing method. Through the comprehensive study of the bubble behaviors, the correlation between focusing conditions and machining quality of liquid-assisted laser cutting was established.

Modeling of heterogeneous site energy distributions in precipitate nucleation

The simulation of heat changes resulting from phase transitions can help to interpret differential scanning calorimetry (DSC) measurements, e.g. of metallic alloy systems in which multiple reactions overlap during non-isothermal heat treatments. So far, simulated DSC curves mostly exhibit sharp reaction peaks as commonly just one mean energy value for a certain type of nucleation site is assumed. This work proposes an efficient model for treating heterogeneous nucleation site energy variations within the framework of classical nucleation theory (CNT). The site energies are assumed to vary according to a Rayleigh distribution and a scaling function. The effect on the nucleation behavior of precipitates is studied. A consideration of the distribution of heterogeneous site energies has the potential to significantly smoothen the numerical treatment of precipitation processes compared to the non-distributed case. The comparison to previously published simulations of DSC curves during the cooling of an AA6005 aluminum alloy demonstrates the advantages of this extension, especially for slow cooling rates.

Evolution and Statistical Analysis of Internal Random Wave Fields within the Benjamin–Ono Equation

This study investigates the numerical evolution of an initially internal random wave field characterized by a Gaussian spectrum shape using the Benjamin–Ono (BO) equation. The research focuses on analyzing various properties associated with the random wave field, including the transition to a steady state of the spectra, statistical moments, and the distribution functions of wave amplitudes. Numerical simulations are conducted across different Ursell parameters, revealing intriguing findings. Notably, it is observed that the spectra of the wave field converge to a stationary state in a statistical sense, while exhibiting statistical characteristics that deviate from a Gaussian distribution. Moreover, as the Ursell parameter increases, the positive skewness of the wave field intensifies, and the kurtosis increases. The investigation also involves the computation of the probability of rogue wave formation, revealing deviations from the Rayleigh distribution. Notably, the study uncovers distinct types of internal rogue waves, specifically referred to as the “two sisters” and “three sisters” phenomena.