Probability Density Distribution Research Articles

DUS Topp–Leone-G Family of Distributions: Baseline Extension, Properties, Estimation, Simulation and Useful Applications

This study introduces the DUS Topp–Leone family of distributions, a novel extension of the Topp–Leone distribution enhanced by the DUS transformer. We derive the cumulative distribution function (CDF) and probability density function (PDF), demonstrating the distribution’s flexibility in modeling various lifetime phenomena. The DUS-TL exponential distribution was studied as a sub-model, with analytical and graphical evidence revealing that it exhibits a unique unimodal shape, along with fat-tail characteristics, making it suitable for time-to-event data analysis. We evaluate parameter estimation methods, revealing that non-Bayesian approaches, particularly Maximum Likelihood and Least Squares, outperform Bayesian techniques in terms of bias and root mean square error. Additionally, the distribution effectively models datasets with varying skewness and kurtosis values, as illustrated by its application to total factor productivity data across African countries and the mortality rate of people who injected drugs. Overall, the DUS Topp–Leone family represents a significant advancement in statistical modeling, offering robust tools for researchers in diverse fields.

A Comprehensive Approach to Fault Classification of Helicopter Engines with Adaboost Ensemble Model

This work is based on the PHM North America 2024 Conference Data Challenge’s datasets of Helicopter turbine engine performance measurements. These datasets were large and moderately imbalanced. For dealing with these challenges, we demonstrate significant tools covering feature engineering, augmentation and selection, model exploration, visualizations, model explainability and confidence margin estimation. This work was performed in its entirety using MATLAB. All these tools will be generally applicable to data-driven modeling and prediction of health to real life applications. Initially, we explored the 742k observations in the training set, noting a 60-40 split between healthy and faulty labels, and identified two major operational clusters within the data. We enhanced the dataset by removing duplicates and engineered new features based on domain knowledge, expanding the feature set to 242 dimensions. However for the torque margin estimation, we trained a regression model on a limited subset (18 features), which includes engineered features using domain knowledge, quadratic terms and linear interaction between all the terms. For the final submission, we utilized a stepwise linear regression model to optimize feature selection. This approach achieved a perfect regression score on test data, validated by a consistent torque margin residual range of +/- 0.5%. The model's RMSE and MAE metrics were optimal for employing a normal distribution probability density function. For , we reduced the feature set to 58 using dimensionality reduction techniques and balanced the data with upsampling and down-weighing the minority class. We employed ASHA (Asynchronous Successive Halving Algorithm) in conjunction with AutoML to efficiently determine the most suitable model family, significantly saving compute time. Subsequently, we trained ensemble models, including bagged tree and AdaBoost (Adaptive Boosting), which minimized false negatives and positives, achieving robust classification performance. This was particularly critical given the high penalty for false negatives in the data challenge. The MathWorks team score on Testing Data was 0.9686 at the close of competition. This was further improved to 0.9867. Our approach demonstrates the effectiveness of combining strategic data processing, feature engineering, and model selection to enhance predictive accuracy in complex operational datasets.

A short-term load forecasting method based on QRLSTM considering multiple factors analysis

Abstract Complex characteristics of power loads terribly affect the safety of the power grid. Thus, accurate short-term load forecasting is regarded as a solution to mitigate the effect of load complications. In this regard, this work is putting forward a Long Short-Term Memory Network Quantile Regression (LSTMQR) for power load, obtaining the probability density function (PDF). Firstly, the quantile of future power load was obtained using the LSTMQR model. The conditional quantile obtained by the LSTMQR model is used as the input of the Kernel Density Estimation model (KDE) to predict the future probability density distribution of short-term power load at different prediction times. Furthermore, an assessment method is proposed to quantify the effect of different factors on power load to improve prediction accuracy. Finally, a numerical simulation based on actual data in China is established to validate the effectiveness of the proposed forecasting algorithm, indicating the proposed forecasting algorithm can effectively evaluate the relationship between different factors and power load, and probability density prediction can accurately quantify the uncertainty of load information. The proposed algorithm results in a reduction of 37.6% in forecast error compared to the traditional forecasting algorithm.

Robust optimization bidding strategy for user-side resource-side participation in the market distribution of electrical energy and peaking ancillary services considering risk expectations and opportunity constraints

Compared to traditional resources, user-side resources are of various types and have more significant uncertainty about their regulatory capacity, leading to difficulties in coordinating decisions about their simultaneous participation in the electric energy and peaking ancillary services markets. This paper proposes a joint bidding decision-making method for the day-ahead electricity energy and peak shaving auxiliary service market based on distributed robust opportunity constraints, which addresses the problem of difficulty in using an accurate probability density distribution to represent the uncertainty process of user-side resources. Firstly, a data-driven method for characterizing the uncertainty of load regulation capacity is investigated, and fuzzy sets are constructed without assuming specific probability distributions of random variables. Then, to minimize the risk expectation of the joint bidding cost on the customer side, a bidding strategy that considers the uncertainty is proposed. Finally, an example simulation verifies the reasonableness and effectiveness of the proposed joint bidding method, and the results show that the constructed model overcomes the problem of over-conservatism of the robust model, and the computational adaptability is better than that of the stochastic model, which achieves a better balance between robustness and economy.

Variation Characteristics of Actual Evapotranspiration and Uncertainty Analysis of Its Response to Local Climate Change in Arid Inland Region of China

Exploring the variation characteristics of actual evapotranspiration (ETa) and its response to climate change in the arid inland region of China is of great significance for strengthening regional water resources management and maintaining ecological environment security and stability. Taking the Dulan River Basin as the research area, based on the meteorological data from the Wulan Station and hydrological data from the Shanggaba Station from 1981 to 2020, the variation characteristics of ETa at the annual scale were analyzed. The ETa estimation model and joint distribution model of P and potential evapotranspiration (ET0) was constructed based on climate factors, and the uncertainty of ETa response to climate change was explored with the water balance method, multiple linear regression, marginal distribution function, Copula function, and Monte Carlo algorithm. The results showed that the multi-year mean value of ETa in the study area was 261.6 mm, and the interannual process showed an insignificant upward trend, and had no abrupt change during the period. There were two obvious main cycles, which were 19-year periodic changes on the 30-year time scale and 6-year periodic changes on the 9-year time scale. The ETa estimation model based on precipitation (P) and ET0 had good simulation accuracy. The optimal marginal distributions of P and ET0 were Pearson-III (P-III) distribution and Generalized Extreme Value (GEV) distribution, respectively. The Copula joint distribution probability density of P and ET0 was a symmetric saddle-shaped distribution. ETa showed an inverted ‘S’ distribution with the change in joint guarantee rate of P and ET0, ranging from 116.9 mm to 498.6 mm. ETa was an interval range under a certain joint guarantee rate. The research results can provide support for the assessment of ETa, and help to further understand the driving mechanism of climate change on ETa in the arid inland region of China.

A Short-term PV Power Prediction and Uncertainty Analysis Model Based on CEEMDAN and AHA-BP

The stochastic and intermittent nature of photovoltaic (PV) generation brings a series of scheduling problems to the power system. An effective prediction of PV power is essential to minimize the impact of uncertainty. Therefore, this paper presents an integrated prediction model with complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN), the artificial hummingbird algorithm (AHA), and the BP neural network (BPNN) for predicting power generation from PV power plants, and a methodology for uncertainty analysis by using the nonparametric kernel density estimation (NPKDE). First, one month of PV power is decomposed into an array of components using CEEMDAN. Then, the weights and thresholds of the BPNN are optimized by using AHA. These components are trained using the BPNN. Finally, the final prediction results are obtained by superimposing the components, and NPKDE is employed to compute the probability density and confidence interval of the prediction error. The proposed prediction method demonstrates superior predictive performance in comparison with other models. Also, the NPKDE approach better describes the accuracy of the probability density distribution.

The First Standardised Sampling Plan Designed to Scout Dalbulus maidis (Hemiptera: Cicadellidae) Adults in Corn Crops Using Yellow Sticky Traps

ABSTRACTThe corn leafhopper Dalbulus maidis (Hemiptera: Cicadellidae) is one of the main pests in corn crops causing yield reductions of up to 70%. Dalbulus maidis primarily damages corn by transmitting pathogens. Direct plant damage is caused by sap suction and toxin injection. Sampling corn fields to determine the D. maidis density for decision‐making systems can be challenging due to field size, time and cost. Yellow sticky traps (YST) are commonly employed to determine the presence and density of D. maidis because they are easy to use and allow quick and low‐cost assessment. This study aimed to propose and evaluate a new sampling plan for D. maidis in corn crops using YST. This research was carried out in commercial corn crops in the Brazilian Atlantic Forest and Cerrado biomes for 3 years. The pest density in traps and on plants were moderately (r = 0.54–0.66) and significantly (p < 0.0001) correlated. The density evaluated in the traps showed relative variance of less than 25% and sampling time of up to 2 min per sample unit. The negative binomial distribution was adequate to represent the D. maidis density probability distribution for the YST sampling method with K common (2.8979) among all the fields evaluated. The sampling plan consisted of installing one trap every two hectares. The plan's total cost ranged from US$/ha 0.41 to 0.50, with a total sampling time of up to 2.5 min/ha. We propose this new sampling plan for D. maidis, which is suitable for incorporation into management programmes in corn crops, as it is representative, accurate, fast and low cost.

STOCHASTIC DISTURBANCE BY “WHITE” NOISE WAVE FUNCTIONS IN THE PARABOLIC POTENTIAL AND STATISTICAL WAVE PACKET LIFETIME DISTRIBUTION

The work examines the parabolic potential, which as the whole is subject to the stochastic influence of the random process of “white” noise, and the dynamics of the wave function of a particle in it. Based on the found solu- tions to the nonstationary Schrödinger equation, the time evolution of the wave function is considered. The problem of destroying the wave packet of a particle is formulated and analyzed, which is realized when the condition of achieving the dispersion of the packet of a given size is met. For the random variable – time to destruction – the probability distribution density is given.

Research on bridge safety early warning method based on strain energy theory and health monitoring data

Bridges are technology-intensive and heavily invested permanent infrastructure. After completion and opening to traffic, bridge structures are easy to be affected by factors such as traffic load and atmospheric environment. Therefore, it is necessary to do safety warning and evaluation of bridges, especially the abnormal behavior in the early stages of bridge operation. In this research paper, a large-span continuous rigid frame bridge installed with the health monitoring system (HMS), of which a large amount of health monitoring data are collected by the HMS, is used as an example, and then a bridge safety early warning method is proposed when the bridge is during early operating period. First of all, the research finding that the internal stress of the prototype bridge obeys normal distribution through statistical analysis is used; next, we deduce that the strain energy inside the prototype bridge is subject to the Non-central Chi-square distribution combined with the strain energy and statistical theories; in the end, the key probability density distribution function of strain energy and its parameters are derived by using the key stress distribution function of the high performance concrete C50 strength grade used in the prototype bridge. The method recommended in this paper is conducive to the formulation of bridge preventive maintenance strategies.

Modeling of Tensile Stress Distribution Considering Anisotropy of Mechanical Properties of Thin-Walled AlSi10Mg Samples Obtained by Selective Laser Melting

The work that is being presented demonstrates that there is a critical point at which the engineering stress–strain diagram’s elastic–plastic region transitions to yield and fracture stresses. This transition is demonstrated using thin-walled specimens made using selective laser melting technology from high-strength aluminum alloys (AlSi10Mg) that have undergone preliminary heat treatment. It was discovered that the strain-hardening coefficient, which was determined in the section from yield strength to fracture strength, and the critical point have a highly statistically significant association (0.83 by Spearman and 0.93 by Pearson). It was possible to derive a regression equation that connected the strain-hardening coefficient with the crucial transition point. The type of stress distribution in the elastic–plastic region changes (the Weibull distribution changes to a normal distribution) as the plasticity of the thin-walled samples increases. Additionally, the contribution of the probability density of the stress distribution described by the Cauchy distribution increases in a mode near the point at which the probability density of the fracture increases.

Synthesis of a four-point model of an airplane

The modeling of echo signals from distributed radar objects taking into account the noise of their angular coordinates is considered. Relationships are presented that allow the transition from a multipoint model of an arbitrary object containing tens, hundreds and even thousands of emitting points to a model composed of four emitting points located at the vertices of a square. As an example, we synthesized an airplane model containing only four points. It is obtained on the basis of a multipoint model composed of 56 points. Using numerical modeling methods, it has been shown that angular noise generated by multi-point and four-point aircraft models have identical correlation functions and similar parameters of the probability density distribution of angular coordinate noise. The obtained result is also confirmed by semi-natural modeling methods using a matrix simulator.

Data-Driven State of Health Interval Prediction for Lithium-Ion Batteries

The accurate prediction of the state of health (SOH) for lithium-ion batteries is a key factor for improving the performance of battery management systems (BMS). However, traditional point prediction methods are difficult to effectively eliminate errors due to the uncertainty of variables and application environments. This paper presents a model for predicting the interval of lithium-ion batteries based on health indicators (HIs) during charging, which addresses the limitations of current point prediction in practical applications. First, twelve HIs are extracted from the current and voltage variables of the charging process. Secondly, feature selection is performed by random forest (RF) training, and the selected HIs are dimensioned using partial least squares (PLS). Finally, a long short-term memory network (LSTM) combined with quantile regression (QR) is used to derive the quantile values of the prediction points and each quantile is employed as input information for Gaussian kernel density estimation (KDE) to obtain the SOH probability density distribution. The experimental results based on the NASA PCOE Li-ion battery dataset and CALCE Li-ion battery dataset show that the SOH interval coverage is more than 90% and the average width of the interval is less than 0.294.

Effect of geometric defects on the mechanical properties of additive manufactured Ti6Al4V lattice structures

Lattice structures realized by additive manufacturing (AM) have the great potential for a broad range of engineering applications. However, the lattice structures are involved in geometric defects. This paper focuses the effect of geometric defects on the mechanical properties of Ti6Al4V lattice structures manufactured by laser powder bed fusion (L-PBF), three cell topologies, i.e., body centered cubic with vertical struts (BCCZ), face centered cubic with vertical struts (FCCZ), and face and body centered cubic with vertical struts (FBCCZ) were studied. X-ray computed tomography was used to extract the shape and the distribution of process-induced geometric defects of these three kinds of samples. Probability density distributions of geometric defects in each layer were also established to analyze the effect of the printing sequence on geometric defects. Then these distributions of geometric defects were inputted into Abaqus to build the modified statistical models to study the effect of geometric defects on mechanical properties. It is shown that the deviation of the cross-section radius exhibits normal distribution and the deviation of the center axis offset exhibits logarithmic distribution. And the middle layer of the sample has a better manufacturing precision. The modified statistical model can predict the mechanical properties within an error of 5 %. The strut thickness deviation had a more significant effect on the mechanical properties than the strut waviness.

Information theoretic measures on quantum droplets in ultracold atomic systems

We consider Shannon entropy, Fisher information, Rényi entropy, and Tsallis entropy to study the quantum droplet phase in Bose–Einstein condensates. In the beyond mean-field description, the Gross–Pitaevskii equation with Lee-Huang-Yang correction gives a family of quantum droplets with different chemical potentials. At a larger value of chemical potential, quantum droplet with sharp-top probability density distribution starts to form while it becomes flat top for a smaller value of chemical potential. We show that entropic measures can distinguish the shape change of the probability density distributions and thus can identify the onset of the droplet phase. During the onset of droplet phase, the Shannon entropy decreases gradually with the decrease of chemical potential and attains a minimum in the vicinity where a smooth transition from flat-top to sharp-top QDs occurs. At this stage, the Shannon entropy increases abruptly with the lowering of chemical potential. We observe an opposite trend in the case of Fisher information. These results are found to be consistent with the Rényi and Tsallis entropic measures.

Emergence of extreme events from randomly chirped condensate

We explore rogue wave emergence in generic wave systems governed by the nonlinear Schrödinger equation. We highlight the interplay between the stochastic focusing induced by a random chirp of a continuous wave condensate and modulation instability of the condensate to periodic and/or random perturbations. We show that regardless of its statistical properties, the random chirp of the condensate leads to a non-Gaussian, heavy-tailed probability density distribution of peak powers of the waves excited atop of the condensate, which is a statistical signature of rogue waves, or more broadly, extreme events in physics.

Characterization of Hydraulic Spool Clamping Triggered by Solid Particles Based on Mechanical Model and Experiment Research

Hydraulic spool valves may clamp under the action of sensitive particles when working in hydraulic oils that contain solid particles, which will then bring about a devastating detriment to the machines. According to the failure statistics of hydraulic systems organized by ISO, more than 80% of the operational failures of hydraulic systems are caused by fluid contamination, and particulate contamination is the most important factor causing spool valve stagnation. In this paper, we considered various factors, including the material, size, and concentration of particles and the spool postures, and built a systematic spool clamping mechanical model. A device was designed to measure the spool valve friction under the action of particles. The influence of particle material, concentration, and size on the friction force of spool valves was investigated. By experiments, we measured the spool clamping force under the action of each single factor and then fitted the datum quantity of spool clamping force and the empirical equation of pulsating quantity. The study results demonstrate three types of non-ideal postures of spools in a valve hole, which are off-center, tilting, and off-center with tilting. Those three postures can engender clamping risk zones with different ranges inside the clearance between spool valves, increasing the risk of spool clamping. The kind of particles is found to have a certain but limited impact on the spool clamping force. Usually, particles with a higher elastic modulus can trigger a larger spool clamping force, which is in line with the theoretical equation. Within a certain range, the probability density distribution of particle size tallies with the normal distribution function, where the “sensitive particles” take up 0.7–1 of the clearance between spool valves. A higher particle volume fraction in oils means a greater number of sensitive particles and a larger spool clamping force. For the particles of a similar size with the clearance between spool valves, when their volume concentration tops over the “sensitive concentration”, namely 5%, the risk of spool clamping rises in a drastic manner. This study provides a theoretical reference and an empirical equation for the mechanism of spool clamping under the action of particles, as well as a definite quantitative indicator for the prediction and estimation of spool clamping which is of positive significance for the study of the predictive maintenance of hydraulic equipment.

Inefficient star formation in high Mach number environments

Context. The star formation rate (SFR), the number of stars formed per unit of time, is a fundamental quantity in the evolution of the Universe. Aims. While turbulence is believed to play a crucial role in setting the SFR, the exact mechanism remains unclear. Turbulence promotes star formation by compressing the gas, but also slows it down by stabilizing the gas against gravity. Most widely used analytical models rely on questionable assumptions, including: i) integrating over the density PDF, a one-point statistical description that ignores spatial correlation, ii) selecting self-gravitating gas based on a density threshold that often ignores turbulent dispersion, iii) assuming the freefall time as the timescale for estimating SFR without considering the need to rejuvenate the density PDF, iv) assuming the density probability distribution function (PDF) to be log-normal. This leads to the reliance on fudge factors for rough agreement with simulations. Even more seriously, when a more accurate density PDF is being used, the classical theory predicts a SFR that is essentially 0. Methods. Improving upon the only existing model that incorporates the spatial correlation of the density field, we present a new analytical model that, in a companion paper, is rigorously compared against a large series of numerical simulations. We calculate the time needed to rejuvenate density fluctuations of a given density and spatial scale, revealing that it is generally much longer than the freefall time, rendering the latter inappropriate for use. Results. We make specific predictions regarding the role of the Mach number, ℳ, and the driving scale of turbulence divided by the mean Jeans length. At low to moderate Mach numbers, turbulence does not reduce and may even slightly promote star formation by broadening the PDF. However, at higher Mach numbers, most density fluctuations are stabilized by turbulent dispersion, leading to a steep drop in the SFR as the Mach number increases. A fundamental parameter is the exponent of the power spectrum of the natural logarithm of the density, ln ρ, characterizing the spatial distribution of the density field. In the high Mach regime, the SFR strongly depends on it, as lower values imply a paucity of massive, gravitationally unstable clumps. Conclusions. We provide a revised analytical model to calculate the SFR of a system, considering not only the mean density and Mach number but also the spatial distribution of the gas through the power spectrum of ln ρ, as well as the injection scale of turbulence. At low Mach numbers, the model predicts a relatively high SFR nearly independent of ℳ, whereas for high Mach, the SFR is a steeply decreasing function of ℳ.

Minimum Risk Quantification Method for Error Threshold of Wind Farm Equivalent Model Based on Bayes Discriminant Criterion

The error threshold is the cornerstone to balance the mathematical complexity and simulation speed of wind farm (WF) equivalent models, and can promote the standardization process of equivalent methodology. Due to differences in power system conditions and model evaluation standards in different countries, the form and indexes of error thresholds of WF equivalent models have not been unified yet. This paper proposes a theoretical method for quantifying the minimum risk of error threshold of WF equivalent models based on the Bayes discriminant criterion. Firstly, the Euclidean errors of WF equivalent models in different periods are quantified, and the probability density distributions of the errors are fitted by kernel density estimation. Secondly, the real-time weighted prior probability algorithm is used to obtain the prior probability of a valid WF equivalent model, and the different losses caused by the missed judgment and misjudgment of the model validity to power systems are taken into account. Thirdly, the minimum risk quantification model of error threshold is established based on the Bayes discriminant criterion, and the feasibility of the proposed method is verified by an actual WF with numerical examples. Compared with the existing error thresholds, the proposed error threshold can more quickly and accurately determine the validity of WF equivalent models.

A tail probability density distribution model of stress amplitude under band-limited stochastic vibration loadings

Resonance of engineering structures under stochastic vibration loadings is a primary cause of fatigue failure. Considering that cyclic loadings below the fatigue limit of the material contribute little to fatigue damage, the portion of the stress amplitude distribution that exceeds the fatigue limit is in focus in this paper, and a tail probability density distribution model of stress amplitude under band-limited stochastic vibration loadings is proposed. First, the response stress power spectral density functions under band-limited loadings are simulated by using the normal distribution and the Weibull distribution, respectively, and then the frequency domain signals are converted into time domain signals by adopting the inverse fast Fourier transform method. A two-parameter Weibull model of tail probability density distribution of stress amplitude is proposed, and the relationship between model parameters and power spectral moment parameters is given. Finally, the comparative analysis of the model accuracy and the vibration fatigue test of aluminum alloy notch specimen are carried out. It is demonstrated that the fatigue life predicted by the presented model is highly coincident with the results of tests and the time domain method.

Stabilization of nonhomogenous semi-Markov jump linear systems with synchronous switching delays via aperiodically intermittent control

A type of nonhomogenous semi-Markov jump linear systems with synchronous switching delays is considered in the present paper. Owing to the existence of time-varying delays and nonhomogenous semi-Markov switching process, the selection of Lyapunov functional and the transformation of some time-varying parameters become more complicated in comparison with many existing works. To overcome these difficulties, a sort of Lyapunov–Krasovskii functional related to switching delays is introduced. Simultaneously, the technique of convex combination is applied to tackle a variety of time-varying terms including transition probabilities (TPs), probability distribution function and probability density function (PDF). Then, a time-invariant stability criterion superior to many relevant works in conservativeness is obtained to ensure mean-square exponential stability (MES). Based on this result, time-invariant conditions for mean square exponential stability of the controlled system are also attained by the continuous feedback control. Subsequently, to stand on the point of cost reduction, an aperiodically intermittent control (AIC) is adopted in such a system and a stability criterion is attained as well under more relaxed restrictions on relative parameters. Finally, a numerical example is provided to verify the effectiveness and correctness of the results.