Probability Density Distribution 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

Phase tracking using a Kalman filter based on probability density distribution in frequency-scanning interferometry

Frequency-scanning interferometry (FSI) utilizing external cavity diode lasers (ECDL) stands out as a potent technique for absolute distance measurement. Nevertheless, the inherent scanning nonlinearity of ECDL and phase noise pose a challenge, as it can compromise the accuracy of phase extraction from interference signals, thereby reducing the measurement accuracy of FSI. In this study, we propose a composite algorithm aimed at mitigating non-orthogonal errors by integrating the least-squares and Heydemann correction technique. Furthermore, we employ Kalman filtering for precise phase tracking. We introduce a parameter selection strategy based on the statistical distribution of instantaneous frequency to achieve the fusion estimation of phase observation values and theoretical models, which starts a new perspective for the application of multi-dimensional data fusion in FSI measurement. Through simulation and experimental validation, the efficacy of this approach is confirmed. The experimental results show promising outcomes: with an average phase error of 0.12%, a standard deviation of less than 1.7 µm in absolute distance measurement, and an average positioning accuracy error of 0.29 µm.

Molecular cloud matching in CO and dust in M33. I. High-resolution hydrogen column density maps from Herschel

This study is aimed to contribute to a more comprehensive understanding of the molecular hydrogen distribution in the galaxy M33 by introducing novel methods for generating high angular resolution ($18.2''$, equivalent to $75\ pc $ for a distance of $847\ kpc $) column density maps of molecular hydrogen ($N_ H_2 $). M33 is a local group galaxy that has been observed with Herschel in the far-infrared (FIR) wavelength range from $70$ to $500\ Previous studies have presented total hydrogen column density maps ($N_ H $), using these FIR data (partly combined with mid-IR maps), employing various methods. We first performed a spectral energy distribution (SED) fit to the $160$, $250$, $350,$ and $500\ continuum data obtain $N_ H $, using a technique similar to one previously reported in the literature. We also use a second method which involves translating only the $250\ map into a $N_ H $ map at the same angular resolution of $18.2''$. An $N_ H_2 $ map via each method is then obtained by subtracting the component. Distinguishing our study from previous ones, we adopt a more versatile approach by considering a variable emissivity index, $ and dust absorption coefficient, $ This choice enables us to construct a $ map, thereby enhancing the depth and accuracy of our investigation of the hydrogen column density. We address the inherent biases and challenges within both methods (which give similar results) and compare them with existing maps available in the literature. Moreover, we calculate a map of the carbon monoxide $ CO(1-0) $-to-molecular hydrogen (htwo ) conversion factor ( factor), which shows a strong dispersion around an average value of $1.8 throughout the disk. We obtain column density probability distribution functions ($N$-PDFs) from the $N_ H $, $N_ H_2 $, and $N_ hi $ maps and discuss their shape, consisting of several log-normal and power-law tail components.

Trend to Equilibrium for Run and Tumble Equations with Non-uniform Tumbling Kernels

We study the long-time behaviour of a run and tumble model which is a kinetic-transport equation describing bacterial movement under the effect of a chemical stimulus. The experiments suggest that the non-uniform tumbling kernels are physically relevant ones as opposed to the uniform tumbling kernel which is widely considered in the literature to reduce the complexity of the mathematical analysis. We consider two cases: (i) the tumbling kernel depends on the angle between pre- and post-tumbling velocities, (ii) the velocity space is unbounded and the post-tumbling velocities follow the Maxwellian velocity distribution. We prove that the probability density distribution of bacteria converges to an equilibrium distribution with explicit (exponential for (i) and algebraic for (ii)) convergence rates, for any probability measure initial data. To the best of our knowledge, our results are the first results concerning the long-time behaviour of run and tumble equations with non-uniform tumbling kernels.

Deep network fault diagnosis for imbalanced small-sized samples via a coupled adversarial autoencoder based on the Bayesian method.

Deep network fault diagnosis methods heavily rely on abundant labeled data for effective model training. However, small-sized samples and imbalanced samples often lead to insufficient features, resulting in accuracy degradation and even instability in the diagnosis model. To address this challenge, this paper introduces a coupled adversarial autoencoder (CoAAE) based on the Bayesian method. This model aims to solve the issue of insufficient samples by generating fake samples and integrating them with the original ones. Within the CoAAE framework, the probability density distribution of the original data is captured using an encoder and fake samples are generated by random sampling from this distribution and decoding them. This process is the adversarial interaction between the encoder and a classifier to obtain the prior distribution of the encoder's parameters. The encoder's parameters are updated through the decoder's reconstruction process, leading to the posterior distribution. Concurrently, the decoder is trained to enhance its ability to reconstruct samples accurately. To address the imbalance in the original samples, a parallel coupled network is employed. This network shares the weights of the extraction layer in the encoder, enabling it to learn the joint distribution between fault-related and normal samples. To evaluate the effectiveness of the proposed data augmentation method, experiments were conducted on a bearing database from Case Western Reserve University using ResNet18 as the deep learning diagnosis model representative. The results demonstrate that CoAAE can effectively augment imbalanced datasets and outperform other advanced methods.

The single and double inverse Pareto–Burr XII distribution: Properties, estimation methods, and real-life data applications

Researchers consistently improve statistical tools to provide a flexible method to fit or predict. The solution needs to be more suitable since the approach is sometimes complex and complicated. Generally, statistical tools are generated from ordinary least squares and wavelet methods. However, the reliability tool provides a good fit comparable to state-of-the-art methods, including the complex posterior setting of hyperparameters. In this article, a flexible extension of the Burr XII model, which is the odd inverse Pareto–Burr XII (OIPBXII) distribution, is studied and investigated using statistical tools such as Markov chain Monte Carlo methods. The mathematical properties of the probability density distribution of OIPBXII are derived, and it shows a behavior shape, decreasing, increasing, J-shaped, reversed-J-shaped, bathtub, upside-down bathtub, decreasing–increasing–decreasing hazard rates, and right-skewed, symmetrical, and concave down densities. The OIPBXII parameters are investigated by using seven classical approaches to estimation. Extensive simulation results are presented to explore the performance of these methods for small and large samples. An application to two real-life sets of data from engineering and medicine is analyzed, showing the flexibility of the OIPBXII distribution as compared to existing Burr XII distributions.

U–Pb Age and Mineral Inclusions in Zircon from Diamondiferous Garnet–Pyroxene Rocks of the Kumdy-Kol Microdiamond Deposit in Northern Kazakhstan

Abstract ––This paper describes the results of studying inclusions and determining the U–Pb age in zircon from the diamondiferous garnet–pyroxene rock of the Kumdy-Kol metamorphogenic diamond deposit, located in the Kokchetav subduction-collision zone. The distribution of rare earth elements in garnet and clinopyroxene is used as a basis for estimating the equilibrium pressure (5.5 ± 0.3 GPa) and temperature (993 ± 24 °C), which correspond to the diamond stability field. The composition of mineral inclusions in zircon indicates its formation at both the progressive and the regressive stage of metamorphism. The concordia diagram shows that the figurative points of zircon lie on a discordia with an upper intersection at 1953 ± 139 Ma and a lower intersection at 512 ± 4 Ma. The main peak on the graph of the probability density distribution of zircon ages corresponds to an age of 519 Ma. The presence of grossular–almandine garnet inclusions in zircon confirms the previous assumption that the basement rocks of the Kokchetav massif act as protoliths of garnet–pyroxene rocks.

Association of uninsured status and rurality with risk of financial toxicity after pediatric trauma

BackgroundPediatric traumatic injury is associated with long-term morbidity as well as substantial economic burden. Prior work has labeled the catastrophic out-of-pocket medical expenses borne by patients as financial toxicity. We hypothesized uninsured rural patients to be vulnerable to exorbitant costs and thus at greatest risk of financial toxicity. MethodsPediatric patients (<18 years) experiencing traumatic injury were identified in the 2016–2019 National Inpatient Sample. Patients were considered to be at risk of financial toxicity if their hospitalization cost exceeded 40% of post-subsistence income. Individual family income was computed using a gamma distribution probability density function with parameters derived from publicly available US Census Bureau data, in accordance with prior work. A multivariable logistic regression was developed to assess factors associated with risk of financial toxicity. ResultsOf an estimated 225,265 children identified for study, 34,395 (15.3%) were Rural. Rural patients were more likely to experience risk of financial toxicity (29.1 vs 22.2%, P < .001) compared to Urban patients. After adjustment, rurality (reference: urban status; adjusted odds ratio 1.45, 95% confidence interval 1.36–1.55) and uninsured status (reference: private; adjusted odds ratio 1.85, 95% confidence interval 1.67–2.05) remained linked to increased odds of risk of financial toxicity. Specifically among those with private insurance, Rural patients experienced markedly higher predicted risk of financial toxicity, relative to Urban. ConclusionOur findings suggest a complex interplay between rural status and insurance type in the prediction of risk of financial toxicity after pediatric trauma. To target policy interventions, future studies should characterize the patients and communities at greatest risk of financial devastation among rural pediatric trauma patients.

Estimation of Stress - Strength Reliability for Parallel Redundant System Via Burr of type X Distribution

Estimation for the reliability of a parallel redundant system with independent stress and strength Burr distribution (type ) probability density functions is considered. Estimation of the reliability parameters is conducted according to the non-Bayesian estimation methods, namely the maximum likelihood and shrinkage methods. The Bayesian estimation is also considered using the Jeffery and Gamma priors with squared error, quadratic and weighted loss functions. Finally, the reliability estimate is calculated and the best method for estimation for each case is given using the mean squared error criteria.

AIseed Simulation: A seed simulation sorting software for rapidly determining seed processing procedures and parameters

Seed processing is a crucial measure for improving seed quality after harvest. However, traditional seed processing procedures and parameter settings have largely depended on technicians' operational experience. To enhance the digitization of the seed processing procedures, AIseed Simulation software was developed and incorporates ''Simulated sorting'' and ''One-click sorting'' based on RGB images of seeds and impurities. Based on the sorting principles of current main seed sorting equipment, length, width, thickness, specific gravity, and color features (R, G, B, R/G, R/B, G/B) are selected. Through Spearman correlation analysis and probability density distribution plots, the sorting features and parameters are determined. By repeatedly comparing precision, recall, and F1 values of different sorting combinations, simulation sorting on a computer is achieved. The ''One-click sorting'' function has been integrated into the above procedures, facilitating automation and making the determination of optimal seed processing procedures and parameters easier and more rapid. The AIseed Simulation was individually validated on five crop seeds—Astragalus membranaceus, Perilla frutescens, Scutellaria baicalensis, Bupleurum chinense, and Platycodon grandiflorus—and the processing procedures and parameters established for each seed type were validated with actual processing equipment. The results validated that AIseed Simulation is a stand-alone, automated RGB image processing platform suitable for seed processing and parameter setting. Furthermore, the AIseed Simulation software exhibits versatility in a range of seed processing scenarios, suitable for both large-scale seed processing facilities and the demands of scientific research and education. These findings contribute significantly to enhancing seed quality and facilitate the intelligent development of the seed processing industry.

Kink of the nuclear charge radius isotope shift and overlaps of the neutron and proton orbitals in lead

Within the relativistic mean field approximation, we analyse the kink effect (KE) in the evolution of the charge radius isotope shift of lead isotopes as a function of the neutron number N. We show that if the interactions between neutron and proton states responsible for the KE are assumed to be proportional either to the overlaps of their corresponding wave functions or to those of their corresponding probability density distributions, it is not possible, by themselves, to explain the KE. However, we find that the small component of the single-particle Dirac spinors plays a relevant role in the kink formation. By considering the contribution of the N−126 valence neutrons to the proton central potential, we can explain the generation of the KE and why neutrons in the 1i11/2 orbital are more kinky than when they are in the 2g9/2 orbital.

Refined offshore wind speed prediction: Leveraging a two-layer decomposition technique, gated recurrent unit, and kernel density estimation for precise point and interval forecasts

Precise forecasting of offshore wind speeds is paramount for various applications, including offshore wind energy production, disaster prevention and mitigation, and maritime navigation. This study introduces a novel model for point and interval prediction of offshore wind speed, incorporating an innovative two-layer decomposition technique, gated recurrent unit (GRU), and kernel density estimation (KDE). Initially, the improved complete ensemble empirical mode decomposition with adaptive noise (ICEEMDAN) is utilized to decompose the raw series into multiple subsequences, thus alleviating the inherent data nonstationarity. Subsequently, we employ the refined composite multiscale fuzzy entropy (RCMFE) to reconstruct the subsequences into multiple components, thereby reducing the computational complexity. Variational mode decomposition (VMD) is utilized to enhance the modeling accuracy to decompose the high-frequency component, thereby generating a set of submodels. The GRU, a specialized recurrent neural network architecture, is leveraged to forecast the remaining components and submodels, producing a series of subresults. The subresults are linearly summed to obtain the final point prediction result. Finally, based on point prediction, employing KDE to gauge the probability density distribution of prediction errors culminates in the derivation of the offshore wind speed prediction interval. The experimental results substantiate the superior predictive performance of the proposed prediction model.

Numerical Study on Permeability of Reconstructed Porous Concrete Based on Lattice Boltzmann Method

The reconstruction of the porous media model is crucial for researching the mesoscopic seepage characteristics of porous concrete. Based on a self-compiled MATLAB program, a porous concrete model was modeled by controlling four parameters (distribution probability, growth probability, probability density, and porosity) with clear physical meanings using a quartet structure generation set (QSGS) along with the lattice Boltzmann method (LBM) to investigate permeability. The rationality of the numerical model was verified through Poiseuille flow theory. The results showed that the QSGS model exhibited varied pore shapes and disordered distributions, resembling real porous concrete. Seepage velocity distribution showed higher values in larger pores, with flow rates reaching up to 0.012 lattice point velocity. The permeability–porosity relationship demonstrated high linearity (the Pearson correlation coefficient is 0.92), consistent with real porous concrete behavior. The integration of QSGS-LBM represents a novel approach, and the research results can provide new ideas and new means for subsequent research on the permeability of porous concrete or similar porous medium materials.

Effects of gas velocity on nonequilibrium characteristics of fluidization

The gas velocity is a key factor of the macroscale operating conditions, affecting the nonequilibrium behavior and regime transitions of fluidization, but the mechanism of its effects remains unclear, resulting in a deterioration of the accuracy of the two-fluid model (TFM) simulation with the increase of the gas velocity. To seek its underlying mechanism, we performed time-resolved, particle-tracking velocimetry (PTV) measurements of a fluidized bed operated under bubbling to turbulent states, thereon the analysis of the probability density distribution of particle velocity, and hydrodynamic parameters. It is found that a stronger nonequilibrium exists with the increase of the gas velocity, in the sense that the bimodal distribution of particle velocity is more significant at turbulent fluidization. The total fluctuating energy and stress show a scale separation, suggesting that a reasonable continuum modeling should take into account the whole range of fluctuation.

Design and implementation of English vocabulary orthographical revision application based on Matlab

Apropos of English learning, vocabulary learning is one of the most essential parts. Thanks to modern technology, recent years has witnessed scads of vocabulary learning apps springing in the app market which include greatly comprehensive functions. However, among those popular apps, some universal handicaps still exist in the design of the review section, for example, the absence of the freedom for the users to classify the cognitive or priority grade of vocabulary and the low effect of some reviewing methods that could not satisfy the need of the language learners to fully master the orthography of words. This paper aims at contriving an app serving as a supplementary part of English vocabulary learning in order to remedy those inadequacies of mainstream applications or other self-learning methods. In this paper, literature study, graphology, and mathematical modelling are involved. Based on Matlab GUI/App Designer, this paper presents a new method to review English words in which users could customize their lexicon, as well as define the priority of each word and practice spelling of vocabulary on an hourly-based system instead of a daily-based one that most of the existing online platforms adopted. Through creating a discrete probability density function for each word, this app could forward a random word in the glossary for users to practice orthography, obeying the probability density distribution that has a mathematical relationship with familiarity and priority of the words as well as the time factor (Ebbinghaus forgetting curve). Thus, users could thoroughly grasp the orthography of target words.

Probability density distributions of morphological parameters of barchan dunes and their responses to the climate environment

Barchan dunes are a common aeolian geomorphology in sand seas on Earth and Mars. They occur primarily as barchan dune groups. Obvious regional differences occur in the size distribution of dune swarms in different dune fields because of sediment grain size, regional airflow, sand flux, and surface conditions. To explore dune size distributions in different barchan dune fields and clarify the effects of climate on barchan dune morphometry, we used high-resolution remote sensing images and extended time-series meteorological data. Barchan dune fields were located in the southeast edge of the Taklimakan Desert, the western and southern parts of the Qaidam Basin Desert, the Gonghe Basin Sandy Land, and the sand belt connecting the Badain Jaran Desert and the Tengger Desert. Our results revealed the probability density distribution of morphological parameters of barchan dunes, correlations among parameters, and responses to wind regime. The main conclusions are: (1) Probability density distributions of the lengths of windward and lee slopes, the lengths of the two horns, and the dune width, height, and spacing in different deserts followed a log-normal distribution function. (2) Good correlations were found among the dune morphological parameters; width and height, windward slope length and width, and height and spacing showed positive proportionality, and the basal area and the product of width and height were linearly correlated in a double logarithmic coordinate system. (3) Good correspondence was found between dune asymmetry and regional wind conditions, which can be explained by Tsoar's model. Morphological parameters of barchan dunes were linearly and positively correlated with the drift potential. These results provide essential data for further study of the morphological thresholds for dune-dune interactions and elucidation of the mechanisms of interaction among barchan dunes.

Mathematical Prediction of Electrical Solar Energy Based on Solar Data for Two Main Cities of Chad: Mongo in the Centre and Pala in the South of Chad

The comparative study of the solar powers between two main cities of Chad is performed in the present work, the city of Mongo in the Centre and that of Pala in the South, with an aim of knowing which one of the two cities is more adequate for an installation of the solar power station, taking into account the regional climatic and environmental conditions of both cities. To do this, the graphical statistical analysis of long-term solar irradiance data and temperature is performed. The data used is that of the decade (2010-2020), based on solar radiation data handed by the National Aeronautics and Space Administration (NASA) and Photovoltaic Geographical Information System (PGIS) for Mongo in the centre and Pala in the south of Chad. The shape of the mean monthly irradiation has been plotted and has been approximated using the sinusoidal function through the mean square analysis. The temperature data has been also obtained by the same process and plotted versus irradiance in order to find the adequate mathematical relationship between them. For the statistical analysis, the maximum entropy principle has been used. As results, it is found that the maximum irradiance is obtained in March, which are 226.26kWh/m&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; for Pala and 219.355kWh/m&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; for Mongo, while the minimum irradiances are obtained in August, which are 151.67kWh/m&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; for Pala and 158.9kWh/m&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; for Mongo. The temperature data is also obtained and the mean monthly data plotted, showing that apart for the months of March and April, the the shapes of irradiation and temperatures are similar for both sites. Then it is found that the frequency and probability density distributions reach their maximum at the same dates.

A Novel SMFL-Based Assessment Method for Corrosion Nonuniformity of Rebar and Its Application in Reliability Analysis of Corroded RC Beam

The reliability of corroded reinforced concrete (RC) structures relies on the accurate minimum cross-sectional area of corroded rebar. In this study, the accurate morphologies and self-magnetic flux leakage (SMFL) field strengths of twenty-eight non-uniformly corroded rebars were obtained using 3D structural light scanning and micromagnetic detection technologies, based on which three indices of the SMFL field variation ratio dH, the corrosion non-uniformity degree dSn, and the cross-sectional area ratio K0.25 are proposed. The statistical results show that the probability densities of dSn and K0.25 obey the Weibull distribution and Gamma distribution at the 95% confidence level, respectively, and their distribution parameters are linearly or inversely proportional to dH. The probability density distribution of the minimum cross-sectional area of corroded rebar can be determined using indices dSn and K0.25, based on which a feasible SMFL-based reliability assessment method of corroded RC structures is proposed. The case study of a real specific corroded RC beam shows that the reliability assessment error of the SMFL-based method is only 1.2%, which is much lower than the 20.7% error of the existing method. This SMFL-based method provides a novel idea that can automatically and accurately assess the effect of rebars’ corrosion non-uniformity on the reliability of specific in-serviceRC structures.

Adaptive directed support vector machine method for the reliability evaluation of aeroengine structure

Machine learning methods have been widely applied to structural reliability analysis, due to the excellent performance in modeling precision and efficiency. In this paper, an adaptive directed support vector machine model (ADSVM) is proposed by integrating the adaptive sampling technology (AST) and the developed directed support vector machine (DSVM), to improve the efficiency and accuracy of turbine blade reliability analysis. In the developed method, the AST is adopted to extract high quality samples, in respect of distributed probability density and the distances between sample values and response values. The DSVM was developed by introducing the penalty coefficients for all the slack variables, to improve the modeling accuracy of SVM approach. One numerical example and the reliability analysis of aeroengine high-pressure turbine blade were performed to validate the developed methods. As illustrated in this numerical investigation, the modeling precision of DSVM is improved by alleviating the lag effect of support vector regression. From the comparison of methods, it is demonstrated that the ADSVM has higher accuracy, efficiency and stability in reliability simulations, which are improved by ∼20 % and ∼46.2 %, respectively. The main efforts of this study are to propose a promising approach, ADSVM, for the reliability analysis of complex structures besides turbine blades, which hold the academical significance in enriching and developing mechanical reliability theory.

Inversion Symmetry Breaking in the Probability Density by Surface‐Bulk Hybridization in Topological Insulators

AbstractThe probability density distribution is analyzed in a topological insulator (TI) slab of finite thickness, where the bulk and surface states are allowed to hybridize. By using an effective continuum Hamiltonian approach as a theoretical framework, the wave functions are analytically obtained for each state near the ‐point. The results reveal that, under particular combinations of the hybridized bulk and surface states, the spatial symmetry of the electronic probability density with respect to the center of the slab can be spontaneously broken. This symmetry breaking arises as a combination of the parity of the solutions, their spin projection, and the material constants.