Rayleigh Probability Density Function Research Articles

Control Charts for a Novel Exponent Power Rayleigh Distribution

Abstract: In this Paper, we presented the distribution as A Novel Exponent Power Rayleigh (NovEPRD) probability density function, Distribution, and Survival and Hazard functions introduced by Bharathi et al (2022) and A Novel Exponent Power Rayleigh (NovEPR) distribution Properties and its Applications studied by Bharathi et al (2022) and then here we derived the Weighted Variance (WV), Weighted Standard Deviation (WSD) and Correction Factor (CF) for existed A Novel Exponent Power Rayleigh (NovEPR) distribution , this paper obtains the parameters of the control lines X̅ and R drawn from the A Novel Exponent Power Rayleigh (NovEPR) distribution. These methods are compared with Newton Raphson’s simulations. For these control charts, Type I risk is compared with different groups of A Novel Exponent Power Rayleigh (NovEPR) distribution. Simulation results show that the Skewness Correction (SC) method of Type I carries less risk than other methods. When the distribution is approximately equal, the type I risk of the Shewhart, W V, WSD and SC X̅ plans is comparable, while the type I risk of the SC R plan is smaller

Contingency Assessment for Power System Integrated with Wind Power

Electric demand in the last few years has widely increased, especially in Iraq, where there is a significant difference between the generation and the load in almost all months of the year, particularly in the summer season. One of the important aspects where the operational engineers must take appropriate action in case of an unforeseen catastrophe is power system security. Consequently, the security of the power system depends on the contingency analysis. In order to investigate the impact of wind power on the contingency analysis, three wind farms (WFs) are selected based on the wind's speed availability in Iraq, which are Shaikh Saad, Al-Dujaili, and Al-Fajar. In addition, the wind speeds for these locations are analysed using the Weibull and Rayleigh probability density functions. In this regard, this paper studied and analysed the impact of integrating wind power on the operation of the 132 kV Iraqi grid systems (zone 18). The results show that when line outage contingencies occur (single- and double-line outages), the risk of power flow violations based on the MVA rating will be reduced when integrating wind energy with a 100% integration rate with the Iraqi grid system. Matlab, a programming language, and the Power System Simulator for Engineering (PSS/E) software (Version 32) are used to simulate the proposed approach of integrating wind power into the grid.

Braid feature extraction and representation of active sonar data

As an active sonar system pings an object of interest, the moving platform collects data descriptive of both the target and background clutter. We hypothesize that the relevant features of the target echo tend to persist and smoothly morph over time in some domain as the sonar platform continues its route. This work leverages the preceding and presents initial findings of a feature extraction algorithm resulting in representations of active sonar data that persistently appear and are physically and statistically informed. This is done by isolating the targets’ features from background clutter using a statistically meaningful threshold, extracting features onto a manifold representation, and mathematically describing and quantifying the extracted features. The background clutter is modeled using a Rayleigh probability density function, and the echo is isolated by keeping statistically significant responses. Feature extraction is performed on a ping-by-ping basis by minimizing the angle between the isolated echo samples and assigning a pseudo probability to sequential samples. These methods are used to extract features from ping sequences that are unique to targets and quantified using a correlation metric. Results reported are on simulated data and experimental field data. [Work supported by NREIP, NDSEG, and ONR, Grant No. N00014-21-1-2420.]

Estimation of Wind Speed Probability Distribution Parameters by Using Four Different Metaheuristic Algorithms

The inclusion of energy produced from renewable energy sources (RES) such as solar and wind energy into existing energy systems is important to reduce carbon emissions, air pollution and climate change, and to ensure sustainable development. However, the integration of RES into the energy system is quite difficult due to their highly uncertain and intermittent nature. In this study, considering three different probability density functions in total, the scale and shape parameters of the Weibull probability density function (PDF), the scale parameter of the Rayleigh PDF, and the scale and shape parameters of the Gamma PDF were estimated for the wind speed data obtained from urban stations located in Istanbul by using the four different metaheuristic algorithms, namely Genetic Algorithm (GA), Differential Evolution (DE), Particle Swarm Optimization (PSO) and Grey Wolf Optimization (GWO) algorithms. Calculating the mean absolute error (MAE), root mean squared error (RMSE), and R2 values for each PDF at each station, the PDF that characterizes the wind speed probability distribution the best was identified.

A new stochastic diffusion process based on the Rayleigh density function

In this work we present a model of a stochastic diffusion process based on the Rayleigh distribution function. (its trend is proportional to the rayleigh probability density function). In the first part we start by obtaining the probabilistic characteristics of this model as the explicit expression of the process by applying the Ito formula. The conditional mean and mean function is analysed to obtain fits and predictions using simulated data. Then we developed the statistical inference of the model by estimating these parameters through the maximum likelihood methodology. Finally, in order to highlight the usefulness of this methodology, we include the results obtained from several simulation examples.

Statistical dependence analysis of Erbium doped fiber ring lasers (EDFRL) chaos

Utilizing ergodic nature of chaotic lasers, the probability density function (PDF) estimation is carried out for peak amplitudes of pulsed chaos in Erbium Doped Fiber Ring Laser (EDFRL) in relation to varying the four key cavity parameters i.e. cavity gain, pump power, modulation index and modulation frequency. Kernel based non-parametric probability distributions are found more suitable to fit the chaos peaks data instead of parametric Gaussian, Rayleigh or Poisson Probability Density Functions. The resulting non-parametric PDFs’ shape, modes, skew and variance is found controllable by changing cavity parameters which is beneficial for random number generation with desired properties. The non-parametric distribution changed its shape between uni-modal and bimodal Gaussian like PDFs alternately and the variance also changes. The PDF shape corresponds to density of points in chaotic data bifurcation plots. This statistical in depth analysis is important while employing the EDFRL for random number generation and in chaotic secure communication applications. EDFRL chaos is found here to be a non-Poisson doubly stochastic point process implying it does depend on history of the chaos generation process, which is hidden in the slowly varying population inversion density of the laser. The bimodality skew keeps altering between left and right while becoming uni-modal every time between switching with modulation index and pump power changes. At high gain the variance of PDF decreases and it approaches uni-modal Gaussian. At high modulation frequency the PDF is left skewed with higher number of shorter pulses bursting.

Chaotic darwinian particle swarm optimization for real-time hierarchical congestion management of power system integrated with renewable energy sources

The increasing penetration of renewable energy sources that have intermittent outputs challenge the Independent System Operator while managing transmission congestion. A new Real-Time Hierarchical Congestion Management (RHCM) technique is proposed that reschedules generators in two stages based on Available Congestion Clearing Time (ACCT) of the transmission lines in presence of renewable energy sources. Chaotic Darwinian Particle Swarm Optimization (CDPSO) is used for determining the optimal schedules of demand response loads and reschedules of conventional generators to mitigate congestion. The solar and wind energy sources are modelled using Rayleigh and Beta probability density functions; Latin Hyper cube sampling is adopted for sampling. IEEE 39 bus system is simulated for cases of overloads and contingencies and RHCM using CDPSO is tested to ensure the security of the system. The benefit of RHCM incorporating demand response is presented in terms of reduced congestion relieving costs and decreased power loss.

Comparison based on the technical and economical analysis of wind energy potential at onshore, coastal, and offshore locations in Çanakkale, Turkey

Offshore wind technology in a global context is developing with extraordinary speed, and technical constraints are gradually decreasing. However, in Turkey, offshore applications are yet to be established despite the installation of a significant number of onshore wind power plants over the last ten years. The city of Çanakkale is situated on both the Marmara and the Aegean Sea. It is located on both sides of the Çanakkale Strait, which connects the continents of Asia and Europe and occupies an important geographical position in terms of wind-energy potential. In this study, the wind-energy potential was investigated using data from 18 different meteorological stations located at onshore, coastal, and offshore stations in Çanakkale. Weibull and Rayleigh probability density functions were used to obtain wind speed distribution curves. Four wind turbines with power ranging from 1500 kW to 4500 kW for onshore locations and three wind turbines with power ranging from 2000 kW to 5000 kW for coastal and offshore locations were compared in the scope of the methods used in economic analysis [energy cost estimation (C), net present value (NPV), benefit cost ratio, payback period, and internal rate of return]. In this context, the best performances were obtained in onshore locations of Bozcaada, Çanakkale Airport, and coastal and offshore locations of Bozcaada/Damlacık L. and Lapseki/Zincirbozan L. for Vestas V112 (119 m) turbine and Vestas V90 (105 m) turbine, respectively. This study aims to contribute scientifically to the development of coastal and offshore wind energy in the region.

Modeling of micro-grinding forces considering dressing parameters and tool deflection

The prediction of cutting forces is critical for the control and optimization of machining processes. This paper is concerned with developing prediction model for cutting forces in micro-grinding. The approach is based on the probabilistic distribution of undeformed chip thickness. This distribution is a function of the process kinematics, properties of the workpiece, and micro-topography of the grinding tool. A Rayleigh probability density function is used to determine the distribution of the maximum chip thickness as an independent parameter. The prediction model further includes the effect of dressing parameters. The integration of the dressing model enables the prediction of static grain density of the grinding tool at various radial dressing depths. The tool deflection is also considered in order to account for the actual depth of cut in the modeling process. The dynamic cutting-edge density as a function of the static grain density, the local tool deflection, elastic deformation, and process kinematics can hence be calculated. Once the chip thickness is calculated, the single-grain forces for individual abrasive grains are predicted and the specific tangential and normal grinding forces simulated. The simulation results are experimentally validated via cutting-force measurements in micro-grinding of Ti6Al4V. The results show that the model can predict the tangential and normal grinding forces with a mean accuracy of 10% and 30%, respectively. The observed cutting forces further imply that the flow stress of the material did not change with changing the cutting speed and the cutting strain rate. Moreover, it was observed that the depth of cut and grinding feed rate had the same neutral effect on the resultant grinding forces.

UHF Propagation Through a Trunk‐Dominated Forest

AbstractTransport theory is used to calculate the intensity of a wave propagating though a trunk‐dominated forest. The trunks are replaced by parallel infinitely long dielectric cylinders, and a vertically polarized plane wave is normally incident on the layer of these randomly distribution parallel cylinders. The problem can be reduced to a two‐dimensional transport problem that can be solved by the eigenvalue technique for the coherent, incoherent, and total intensity of the propagating wave. Using the assumption that the distribution of trunks is sparse and that the trunks locations are independent from one another, the probability density function (PDF) of the total and incoherent intensity as a function of depth into the layer can be found. To verify these results, a Monte Carlo (MC) simulation is performed in the low frequency limit which takes the effect of multiple scattering into account. The results show that there is a very close correspondence between the transport and MC simulation findings; the coherent, incoherent, and the total intensity for these two methods agree very closely as a function of distance into the slab. In addition, the PDFs for these three quantities also agree with the MC based histograms. One sees that as the wave propagates into the layer, the Rician distribution for the magnitude tends to a Rayleigh PDF as the coherent wave decays.

Evaluation of mathematical models for QRS feature extraction and QRS morphology classification in ECG signals

It is plausible to assume that the component waves in ECG signals constitute a unique human characteristic because morphology and amplitudes of recorded beats are governed by multiple individual factors. According to the best of our knowledge, the issue of automatically classifying different ’identities’ of QRS morphology has not been explored within the literature. This work proposes five alternative mathematical models for representing different QRS morphologies providing the extraction of a set of features related to QRS shape. The technique incorporates mechanisms of combining the mathematical functions Gaussian, Mexican-Hat and Rayleigh probability density function and also a mechanism for clipping the waveform of those functions. The searching for the optimal parameters which minimize the normalized RMS error between each mathematical model and a given QRS search window enables to find an optimal model. Such modeling behaves as a robust alternative for delineating heartbeats, classifying beat morphologies, detecting subtle and anomalous changes, compression of QRS complex windows among others. The validation process evaluates the ability of each model to represent different QRS morphology classes within 159 full ECG signal records from QT database and 584 QRS search windows from MIT-BIH Arrhythmia database. From the experimental results, we rank the winning rates for which each mathematical model best models and also discriminates the most predominant QRS morphologies Rs, rS, RS, qR, qRs, R, rR’s and QS. Furthermore, the average time errors computed for QRS onset and offset locations when using the corresponding winner mathematical models for delineation purposes were, respectively, 12.87±8.5 ms and 1.47±10.06 ms.

Probability density function based modeling of spatial feature variation in capsule endoscopy data for automatic bleeding detection

Wireless capsule endoscopy (WCE) is a video technology to inspect abnormalities, like bleeding in the gastrointestinal tract. In order to avoid a complex and long duration manual review process, automatic bleeding detection schemes are developed that mainly utilize features extracted from WCE images. In feature-based bleeding detection schemes, either global features are used which produce averaged characteristics ignoring the effect of smaller bleeding regions or local features are utilized that cause large feature dimension. In this paper, pixels of interest (POI) in a given WCE image are determined using a linear separation scheme, local spatial features are then extracted from the POI and finally, a suitable characteristic probability density function (PDF) is fitted over the resulting feature space. The proposed PDF model fitting based approach not only reduces the computational complexity but also offers more consistent representation of a class. Details analysis are carried out to find the best suitable PDF and it is found that fitting of Rayleigh PDF model to the local spatial features is best suited for bleeding detection. For the purpose of classification, the fitted PDF parameters are used as features in the supervised support vector machine classifier. Pixels residing in the close vicinity of the POI are further classified with the help of an unsupervised clustering-based scheme to extract more precise bleeding regions. A large number of WCE images obtained from 30 publicly available WCE videos are used for performance evaluation of the proposed scheme and the effects on classification performance due to the changes in PDF models, block statistics, color spaces, and classifiers are experimentally analyzed. The proposed scheme shows satisfactory performance in terms of sensitivity (97.55%), specificity (96.59%) and accuracy (96.77%) and the results obtained by the proposed method outperforms the results reported for some state-of-the-art methods.

Quantitative evaluation of liver diseases using multi-PDF models

We have proposed the triple Rayleigh probability density function (TR-PDF) model as the in vivo QUS method of liver fibrosis. Additionally, the double Nakagami probability density function (DN-PDF) model was proposed to evaluate the amount and distribution condition of fat droplets in the fatty-liver livers. The DN-PDF permitted to detect early and moderate fatty livers in previous ex vivo studies with over 15 MHz ultrasound. In our recent studies, we are examining clinical data to determine whether two multi-PDF models can evaluate each disease even at low frequencies. In this talk, the results of in vivo studies for liver fibrosis and fatty liver will be introduced.We have proposed the triple Rayleigh probability density function (TR-PDF) model as the in vivo QUS method of liver fibrosis. Additionally, the double Nakagami probability density function (DN-PDF) model was proposed to evaluate the amount and distribution condition of fat droplets in the fatty-liver livers. The DN-PDF permitted to detect early and moderate fatty livers in previous ex vivo studies with over 15 MHz ultrasound. In our recent studies, we are examining clinical data to determine whether two multi-PDF models can evaluate each disease even at low frequencies. In this talk, the results of in vivo studies for liver fibrosis and fatty liver will be introduced.

Predictive modeling of grinding force in the inner thread grinding considering the effect of grains overlapping

Grinding force is an important factor to consider in the field of precision manufacturing. In this study, a model of the thread grinding force in inner thread grinding that takes into account the thread helix angle and effect of grains overlapping is presented. The average undeformed chip thickness of a single abrasive grain can be obtained based on the Rayleigh probability density function. The contact length is assumed to be the arc contact length and is calculated based on the movement of an active grain. In addition, the effective grain overlap coefficient can be defined as the ratio of the effective area removed by a single grain to the cutting area without overlapping grains. The normal force is generated by material deformation and friction. Similarly, the tangential force is produced by material deformation and tangential friction. The normal and tangential forces can be calculated by considering a micro abrasive grain. Experiments were conducted to calculate the coefficients of the grinding force model in order to validate the model. Scanning electron microscopy was performed to determine the relationship between the normal force and the largest pit on the inner thread surface. Normal and tangential forces were analyzed under different spindle speeds, workpiece speeds, grinding depths, and thread helix angles. The numerical results suggest that grinding force is affected by the thread helix angle and friction forces; moreover, grain overlap cannot be ignored. Finally, a linear relationship between the normal force and the largest pit diameter was derived and used to determine the optimal normal force for generating improved inner thread surfaces. The proposed model provides a theoretical basis for the optimization of high-speed high-precision inner thread grinding.

Investigation of wind power density distribution using Rayleigh probability density function

Abstract When modelling the wind speed in a given location, the probability distributions prove to be a useful tool. In this paper, the wind power density in an urban and a suburban location in Constanta county, Romania, have been analyzed during the period January 2017 - December 2017, based on the hourly measured mean wind speed data. In this study, the Rayleigh probability density function was used to calculate the wind power density for each location in order to classify them in terms of wind energy. The Rayleigh distribution function has been derived from the available data and is fitted to the measured probability distribution on yearly basis. The power densities reported for the two stations varied from 295.39 to 194.5 W / m2.

Ornstein-Uhlenbeck-Lévy Electricity Portfolios with Wind Energy Contracting

To leverage the potential of integrating renewable sources into electricity portfolios the risk and cost trade-off of intermittency needs to be assessed. From the perspective of a Load Serving Entity (LSE), this work present the theoretical implications of energy allocation from two type of markets: bilateral long-term contracts and real-time trading. The purchasing of energy on both markets and from two different sources: wind energy and conventional generation is formulated with a stochastic procurement model (SPM). The unexpected jumps of spot market prices are modeled with a mean-reverting Levy process. The wind energy availability is modeled with multiplicative Brownian motion transformed to a Rayleigh probability density function. The risk assessment is defined by the efficient frontier and a user defined risk level. The SPM is tested numerically. The contracted share of wind power is found to range between 8% and 16%. Moreover, the analysis shows the convergence of SPM to an optimal portfolio irrespectively of the wind farm autocorrelation decay rate.

Strongly interacting soliton gas and formation of rogue waves

We study numerically the properties of (statistically) homogeneous soliton gas depending on soliton density (proportional to number of solitons per unit length) and soliton velocities, in the framework of the focusing one-dimensional Nonlinear Schr{\"o}dinger (NLS) equation. In order to model such gas we use N-soliton solutions (N-SS) with $N\sim 100$, which we generate with specific implementation of the dressing method combined with 100-digits arithmetics. We examine the major statistical characteristics, in particular the kinetic and potential energies, the kurtosis, the wave-action spectrum and the probability density function (PDF) of wave intensity. We show that in the case of small soliton density the kinetic and potential energies, as well as the kurtosis, are very well described by the analytical relations derived without taking into account soliton interactions. With increasing soliton density and velocities, soliton interactions enhance, and we observe increasing deviations from these relations leading to increased absolute values for all of these three characteristics. The wave-action spectrum is smooth, decays close to exponentially at large wavenumbers and widens with increasing soliton density and velocities. The PDF of wave intensity deviates from the exponential (Rayleigh) PDF drastically for rarefied soliton gas, transforming much closer to it at densities corresponding to essential interaction between the solitons. Rogue waves emerging in soliton gas are multi-soliton collisions, and yet some of them have spatial profiles very similar to those of the Peregrine solutions of different orders. We present example of three-soliton collision, for which even the temporal behavior of the maximal amplitude is very well approximated by the Peregrine solution of the second order.

Modelling and experimental study of roughness in silicon wafer self-rotating grinding

Self-rotating grinding is the most widely used technology to thin silicon wafer. The roughness is an important indicator of thinning quality and processing accuracy. To get a better grinding quality, rigid control of roughness is required. Although the models of roughness in metal and ceramic machining were extensively studied, mechanism of roughness formation in silicon wafer self-rotating grinding was not well understood. In this article, starting from the mechanism of grinding grooves formation, Rayleigh probability density function was used to characterize the depth of grinding grooves. By establishing a relationship between the roughness and the depth of grooves, a theoretical model of roughness was developed. The overlapping effect of abrasive grains and wheel-workpiece deflection were also considered to improve the accuracy of the model. The model could identify the effects of processing parameters, abrasive grain size, material properties and grinding mark geometry on roughness quantitatively. Verification experiments under seven grinding conditions were performed to validate the theoretical model. The experimental values agreed with the predictive values with less than 20% deviation. Effects of wheel rotation speed, wafer rotation speed, feed rate and wafer radial distance on roughness were discussed in detail.

New Simplest Exact Forms for the 3-D and 4-D Multivariate Rayleigh PDFs With Applications to Antenna Array Geometries

Determining tractable analytical expressions for the probability density functions (PDFs) of correlated Rayleigh random variables (i.e., the multivariate Rayleigh joint PDF) is a longstanding problem in wireless communications. Recent results have reported simpler forms for the 3-D and 4-D Rayleigh joint PDFs. Yet, the new solutions are still mathematically complex. New representations for the general 3-D Rayleigh and the general 4-D Rayleigh fading PDFs are derived. An advantage of the new representations is that only a single-integral computation is needed to compute the 3-D Rayleigh PDF and only double integral computation is needed for the 4-D Rayleigh PDF, where the integrals are proper and definite on $[0,\pi )$ and ${[0,2\pi )}$ , respectively. The new expressions for the 3-D and 4-D Rayleigh PDFs are used to determine several special, “best” placements in the 2-D plane for three and four antennas. Closed-form solutions are found for three antennas with two or three independent antenna pairs, and four antennas with four or five independent antenna pairs.

Evaluating Gaussian and Rayleigh-Based Mathematical Models for T and P-waves in ECG

This paper presents a comparative study of modelling and segmentation of P and T waves in electrocardiograms, using three different mathematical models: Gaussian function, a composition of two Gaussian functions and Rayleigh probability density function (Rayleigh pdf). In order to evaluate the adaptability and the matching degree between each model and each characteristic wave, we compute the normalized root mean square (RMS) error, as well as the evolution of the fitting parameters related to each kernel throughout ECG records from the well-known QT database. Concerning the delineation of P and T-waves, we apply Wavelet Transform for estimating T-wave and P-wave peak locations and combine each developed model with an approach based on the computation of Trapezium's area to locate the end point of each T-wave and the beginning and end point of each P-wave. The composition of two Gaussian functions has produced the most accurate results concerning wave modelling, providing average values of normalized RMS errors equal to 9,15% and 18,70%, respectively for T-wave and P-wave. Rayleigh pdf provided the most stable fitting parameters. For T-wave end location, the most accurate results were computed when using the kernel composition of two Gaussian functions, for which the average time error was 4,49 ± 14,32 ms. For P-wave begin and P-wave end locations, the most accurate results were computed when using kernel Rayleigh pdf, for which the average time errors were, respectively, -4,23 ± 14,84 ms and 2,26 ± 13,14 ms.