Form Of Probability Density Function Research Articles

Variability in the properties of the distribution of the relative humidity with respect to ice: implications for contrail formation

Abstract. Relative humidity with respect to ice (RHi) is a key variable in the formation of cirrus clouds and contrails. We document its probability density function (PDF) using long-term Measurements of Ozone, Water Vapour, Carbon Monoxide and Nitrogen Oxides by In-Service Airbus Aircraft (MOZAIC) and the In-service Aircraft for a Global Observing System (IAGOS) observations over the period 1995–2022 in the upper troposphere (UT) and the lower stratosphere (LS) between 325 and 175 hPa. The characteristics of the RHi PDF differ in the UT and in the LS of the high-latitude regions (HLs) and mid-latitude regions (MLs) of the Northern Hemisphere. In the LS, this PDF decreases exponentially with increasing RHi. In the UT, it first increases exponentially in subsaturated conditions and then decreases exponentially in supersaturated conditions. Because of these different behaviors, the PDF for the combined UT and LS is bimodal. In contrast to the HLs and the MLs, the RHi PDF in the tropical troposphere decreases exponentially with increasing RHi. The different forms of PDF, in the tropics and in the higher-latitude regions, lead to a global PDF of RHi in subsaturated tropospheric conditions that is almost uniform. These findings invite caution when using MOZAIC and IAGOS measurements to calibrate large-scale simulations of RHi. The variability in RHi properties associated with that of temperature also has implications for the formation of contrails. We examined the impact of switching fuel (from kerosene to bioethanol or liquid hydrogen) on the frequency of contrail formation using the Schmidt–Appleman criterion. We show that bioethanol and, to a larger extent, liquid hydrogen would produce more contrails. The impact of a potential change from kerosene to these alternative fuels decreases with decreasing pressure but increases when moving from the higher latitudes of the Northern Hemisphere to the tropics. Finally, we emphasize that investigations of the impact on contrail occurrence frequency as a result of switching from fossil kerosene to more sustainable fuels must be carried out in various meteorological conditions.

A statistical framework for a new Kavya-Manoharan Bilal distribution using ranked set sampling and simple random sampling

In survival and stochastic lifespan modeling, numerous families of distributions are sometimes considered unnatural, unjustifiable theoretically, and occasionally superfluous. Here, a novel parsimonious survival model is developed using the Bilal distribution (BD) and the Kavya-Manoharan (KM) parsimonious transformation family. In addition to other analytical properties, the forms of probability density function (PDF) and behavior of the distributions' hazard rates are analyzed. The insights are theoretical as well as practical. Theoretically, we offer explicit equations for the single and product moments of order statistics from Kavya-Manoharan Bilal Distribution. Practically, maximum likelihood (ML) technique, which is based on simple random sampling (SRS) and ranked set sampling (RSS) sample schemes, is employed to estimate the parameters. Numerical simulations are used as the primary methodology to compare the various sampling techniques.

Numerical modeling of ultrasonic shot peening with an accurate impact velocity

Ultrasonic shot peening (USP) uses the high-frequency motion of the shots with a random distribution of direction and velocity driven through ultrasonic resonance, thus obtaining a deeper gradient structural layer and a larger residual compressive stress. The variation rule of different parameters in the USP process is studied in detail through finite element simulation. To be closer to the experiment, the motion pattern of the shot is observed and repeatedly verified with the help of high-speed camera. After establishing the finite element model (FEM) of the impact of the random distribution of the shots, the motion characteristics are assigned to the shot in the form of probability density function (PDF). The evolution of shot peening coverage is calculated from the experimental and simulation results, and the patterns of residual stress, roughness, and plastic strain during the USP simulation are calculated. The results show that with the increase of shot peening coverage, the maximum residual stress (RS) and plastic strain increase gradually, and the variation rules of each roughness parameter with the coverage range are different, but the values are all within 10 μm. The modeling method used in this paper provides a new idea for the fine modeling of USP.

Enhanced estimation method and approximation method of the PDF of roll angular acceleration and jerk in beam seas

So far, the authors have established the theoretical estimation method of the Probability Density Function (PDF) of roll angular acceleration and jerk with the use of PLIM (PDF Line Integral Method), and estimations of the PDF of these properties were successfully achieved. PLIM, on the other hand, necessitates numerical integration along the complex shaped integral path. The integral path, in particular, becomes challenging in the calculation of the jerk property. However, if the restoring term is only represented by a linear component, then explicit expressions of PDF of roll angular acceleration can be obtained using the nonlinearization technique. It is noteworthy that the form of the nonlinear damping component does not constrain the applicability. This paper aims to show the derivation of the theoretical result and comparison with Monte Carlo Simulation (MCS) results. Furthermore, using the current form of the acceleration PDF, the PDF of roll angular jerk was obtained. The variance of roll angular rate determines the PDF form of the acceleration, which is the method's main advantage.

A novel current sensor indicator enabled WAFTR model for tool wear prediction under variable operating conditions

The health indicators (HIs) were extracted from the current sensor to represent the tool wear progression. The extracted HIs were found poorly correlated with the progression of tool wear as the raw current sensor signal was susceptible to the influence of other parts and structures in the machine tool. Hence, this paper proposed a novel current sensor-based HI that utilized the mean of inverse hyperbolic cosine function fitted to an envelope of the current signal to improve the correlation. Using the extracted HIs, many bespoke machine learning (ML) models have been developed by researchers. However, these models have many hyperparameters, difficult to interpret and especially poor prediction accuracy has been observed under variable operating conditions. This study overcame these issues by proposing a Weibull Accelerated Failure Time Regression (WAFTR) model, which combines process parameters data with HI for improving the prediction accuracy under variable operating conditions. This model mapped a functional relationship with tool wear in the form of probability density function to identify best HIs and acceleration/deacceleration factors which makes it interpretable. The acceleration/deacceleration factors are useful to deaccelerate the tool wear evolution by controlling the specific values of the machining parameters.

Distance distribution between two random points in arbitrary polygons

Distance distributions are a key building block in many subfields in mathematics, science and engineering. In this paper, we propose a novel framework for analytically computing the closed form probability density function (PDF) of the distance between two random points each uniformly randomly distributed in respective arbitrary polygon regions. The proposed framework is based on measure theory and uses polar decomposition for simplifying and calculating the integrals to obtain closed form results. We validate our proposed framework by comparison with simulations and published closed form results in the literature for simple cases. We illustrate the versatility and advantage of the proposed framework by deriving closed form results for a case not yet reported in the literature. Finally, we also develop a Mathematica implementation of the proposed framework which allows a user to define any two arbitrary (concave or convex) polygons, with or without holes, which may be disjoint or overlap or coincide and determine the distance distribution numerically.

Decentralized optimal large scale multi-player pursuit-evasion strategies: A mean field game approach with reinforcement learning

In this paper, the intelligent design for the pursuit-evasion game with large scale multi-pursuer and multi-evader has been investigated. Due to the vast number of agents, the notorious ”Curse of Dimensionality” can seriously challenge the traditional design in multi-player pursuit-evasion game, especially under harsh environment with limited communication resource to support information exchange among multi-players. To address this intractable challenge, the emerging Mean Field Games (MFG) theory has been utilized to solve the optimal pursuit-evasion strategies based on a new form of probability density function (PDF) instead of detailed information from all the other players/agents. As such, not only the information exchange is reduced, but also the computation dimension for the optimal strategy derivation is decreased. Specifically, the MFG has been integrated into the pursuit-evasion game to generate a hierarchical structure where the pursuers and the evaders form two mean field groups separately. To online solve the mean field equations, i.e., two coupled partial differential equations, the actor-critic reinforcement learning mechanism is adopted and further extended to a novel actor-critic-mass-opponent (ACMO) approach. In ACMO, the actor neural network estimates the optimal control, the critic neural network approximates the optimal cost function, the mass neural network learns the agent’s group PDF, and the opponent neural network predicts the opponents’ average states in the form of PDF that causes maximum cost for the agent’s group. The Lyapunov theory is utilized to provide the convergence analysis for all neural networks and the stability analysis for the closed-loop system. Eventually, a series of numerical simulations are conducted to demonstrate the effectiveness of the developed scheme.

Evaluating the Cost of Failure Risk: A Case Study of the Kang-Wei-Kou Stream Diversion Project

For water resource engineering projects, some potential risks that are unclear and difficult to investigate are the most crucial problem for engineers. This study presents a unique case of unknown risks, the Kang-Wei-Kou (KWK) Diversion project, which collapsed due to an unexpected hydraulic jump resulting from uneven regional rainfall. We used HEC-RAS incorporating Monte Carlo simulations for analyzing the potential risks to the original design of the KWK Diversion, concerning boundary conditions. An extreme value type I distribution was selected as the form of probability density function (PDF) to upstream discharge, whereas a normal distribution was selected as the form of PDF to downstream water depth, as boundary conditions. Applying the Monte Carlo method, the analysis revealed that the probability of failure was 7.5%, which was non-negligible and should have been considered in the design plan. This case study indicates that overlooking downstream conditions can lead a major facility to fail, and a stochastic analysis could be a way to find out potential risks. We also applied the cost–benefit analysis to review whether the design plans were appropriately estimated with careful risk based consideration.

Evasive behavior-based method for threat assessment in different scenarios: A novel framework for intelligent vehicle

The threat assessment process is a crucial part of intelligent vehicles (IVs) for evaluating the levels of criticality and taking possible measures to avoid the collision, especially for the collision avoidance systems (CAS). In this study, a novel threat assessment framework based on the driver’s evasive behavior, namely the CPIC, is proposed, which integrates the crash probability (CP) and inevitable crash (IC) state to be widely used by different CAS in different scenarios. In the first step of the CPIC framework, the detailed evasive driver behavior models (E-DBMs) in the form of probability density functions (PDFs) were introduced to generate more realistic collision-avoidance trajectories. Two techniques for sampling these trajectories, namely the Markov Chain Monte Carlo (MCMC) and adaptive Gaussian mixture framework (GMM) methods, were utilized to ensure the samples were from the area of high probability density in the E-DBMs. The CP value could be derived by considering multiple collision-avoidance trajectories. To confirm the IC state in step 2, the CPIC framework employed the driving limit-based approach for IC checking, which combined the CP value to double-check the unavoidable collision. A total of 82 critical events from the real-world naturalistic driving study, the Strategic Highway Research Program 2 (SHRP2), were extracted to verify the performance of the CPIC framework in different scenarios. Results show that the proposed method clearly revealed the risk levels when two vehicles were approaching, and 80 events were successfully identified as near crashes/crashes. Moreover, the real-time performance of the CPIC framework was also demonstrated. The findings indicate this CPIC framework could be used in practical applications of IVs in different scenarios.

Application of quadrature-based moment methods to the conditional moment closure

Two solution algorithms are developed for the conditional moment closure (CMC) using quadrature-based moment methods (QBMM). Their primary purpose is to eliminate the necessity of the additional grid for the conditioning variable (e.g., mixture fraction). As in almost every probability-density-function (PDF)-based description, the main technical hurdle is the solution of the molecular-mixing term. Here the focus is on solving this term coupled with chemistry, as opposed to spatial transport. In the first algorithm (referred to as SA-CMC), this is done by semi-analytically solving an equation for the deviation variable of the conditional scalar mean in terms of Jacobi polynomials. The mixture-fraction space is represented by the Gauss–Lobatto quadrature rule and a β-PDF. In the second algorithm (referred to as QBMM-CMC), a closed set of differential equations is written for the joint moments of a scalar and the mixture fraction, thereby eliminating the need to assume a form for the mixture-fraction PDF. Both solution algorithms are tested for multi-step H2 combustion, and the expected results are achieved with at most six quadrature nodes. Since in the proposed algorithms the scalar mean is ensured to be constant during molecular mixing, it is further concluded that the proposed algorithms are accurate, computationally cost-effective, and straightforward alternatives for traditional CMC solution methods.

Performance-based economic analysis to find the sustainable aggregate option for a granular roadway

Approximately 110,000 km of granular roadways exist in the 183,500 km road network in the State of Iowa, and operation and maintenance of these roadways costs roughly US$270 million annually. The major maintenance costs of these roads are aggregate cost and hauling costs from the quarries to the sites. Accordingly, acquiring a cost-effective, high performance, and environmentally sustainable surface material to be utilized in granular roadways can be a challenge. In this study, three conventional granular roadway materials and four coarser aggregate materials from different quarries were used to construct seven test sections. The long-term performance and mechanistic behaviors of the different surface materials, including stiffness, surface roughness and dust production were evaluated for a period of two years. Using the resulting data, a mechanistic-based life cycle benefit-cost analysis approach was developed to evaluate the use of coarse aggregate materials on granular roadways. The stochastic benefit-cost analysis (BCA) results for different aggregate materials are presented in the form of probability density functions. Two different scenarios are presented and the benefits are evaluated in terms of dust production and surface ride quality.

QSSTA: A Statistical Static Timing Analysis Tool for Superconducting Single-Flux-Quantum Circuits

Superconducting single flux quantum (SFQ) technology is an ultra-high performance and low power technology. The technology, however, lacks many of the design automation tools and capabilities that are commonplace in CMOS technology. This article describes methods to efficiently find the conditional probability density function (PDF) of the minimum workable clock period of SFQ circuits in view of manufacturing-induced process variations and presents qSSTA, a statistical static timing analysis tool targeting SFQ circuits. Following a grid-based correlation model, qSSTA represents spatial correlation of SFQ gates at different positions with respect to process parameters. By approximating timing characteristics of SFQ gates in a linear model, qSSTA is able to estimate the clock period as a normal random variable. Furthermore, process variations that generally result in extra delays in CMOS circuits can result in functional errors in SFQ circuits. qSSTA derives the closed form of the conditional PDF of the clock period under the scenario where all SFQ gates in the circuit work correctly. Compared to Monte Carlo simulations on look-up tables, experimental results show that the average percentage errors are 0.89% for the mean values, 8.04% for the standard deviation, and 0.61% for the 98-percentile point, whereas the runtime of qSSTA is 83% faster on average.

Impact of mapping strategies on rockfall frequency-size distributions

Rockfall frequency size distributions are used in Austria for the definition of a design block for the planning of technical rockfall protection. Rockfall size datasets are often incomplete. Here, we study fifteen catalogues of rockfall size in Austria, Italy, and the USA to analyse the impact of the data collection and mapping methods on the representativeness of the catalogues and on the estimates of frequency-size statistics. To describe and compare the catalogues of rockfall size, we first use Empirical Cumulative Distribution Functions (ECDFs), followed by parametric distribution estimates in the form of Probability Density Functions (PDFs), and Cumulative Distribution Functions (CDFs). We discuss the output of Kolmogorov-Smirnov tests, the position of the frequency-size distribution rollover, and the p-value and the standard errors associated to the distribution parameters estimates to determine the reliability of our model results. In addition, we analyse the variations in the modelled CDFs for different percentiles of the frequency-size distributions to describe and discuss the representativeness of the rockfall catalogues. Our results show that different mapping strategies may affect the estimates of frequency-size distribution of rock fall volume, a relevant information when evaluating the possible impacts of rockfall processes. We conclude offering recommendations for rockfall mapping, and the use and of a non-parametric statistical method being capable to deal with small datasets, which is very typical when dealing with rockfall data. Such recommendations help for a correct dimensioning of designing rockfall mitigation measures.

Theoretical and Practical Limits in Position Estimation Based on Received Signal Strength Measurements

In this study, theoretical limits in position estimation in Wireless Sensor Network (WSN) using a trilateration method are derived in the form of probability density functions (PDFs) and compared with practical limits obtained via received signal strength (RSS) measurements in an anechoic chamber. In particular, the PDF of the radius of error of an unknown node’s position has been derived as having a Nakagami-m distribution. The theoretical PDFs is validated via Kolmogorov–Smirnov hypothesis tests at 95% confidence level against the empirical results of position estimation in a 4 m × 4 m area using the RSS measured in an ideal environment of an anechoic chamber. High-resolution equipment is used in the RSS measurements to ensure the inaccuracies due to limited-capability WSN equipment can be quantified. A lower bound of positioning accuracy to be expected in real environments via the received signal strength method has been established to be in the range of 11 cm from the true position with 95% confidence level. Better results cannot be expected in the real environments with bigger variations than the variations observed in the anechoic chamber unless a more sophisticated algorithm, equipment with better resolution and/or more precise measurement methods are employed.

State estimation for the electro-hydraulic actuator based on particle filter with an improved resampling technique

State estimation for the electro-hydraulic actuator of civil aircraft is one of the most valuable but intractable issues. Recently, the state estimation approach based on particle filters has widely attracted attention. We pursue the benefits of the data-driven approach when physical model is deficienct, and put forward some improvements that are triggered by the shortcomings of particle filters algorithm. In order to solve the particles’ degeneracy phenomenon in particle filters, a kernel function that integrates the information of probability distribution is constructed; then, the established probability kernel function is designed to represent the probability density function of resampling and the regularization form of probability density function in Hilbert space is defined. Consequently, the probability density function of resampling is obtained by solving the support vector regression model. The novel resampling method based on support vector regression-particle filters can keep the diversity of particles as well as relieve the degeneracy phenomenon and eventually make the estimated state more realistic. The approach is simulated and applied to an electro-hydraulic actuator model. The estimation results validate the effectiveness of the proposed algorithm.

An approximate technique for determining in closed form the response transition probability density function of diverse nonlinear/hysteretic oscillators

An approximate analytical technique is developed for determining, in closed form, the transition probability density function (PDF) of a general class of first-order stochastic differential equations (SDEs) with nonlinearities both in the drift and in the diffusion coefficients. Specifically, first, resorting to the Wiener path integral most probable path approximation and utilizing the Cauchy–Schwarz inequality yields a closed-form expression for the system response PDF, at practically zero computational cost. Next, the accuracy of this approximation is enhanced by proposing a more general PDF form with additional parameters to be determined. This is done by relying on the associated Fokker–Planck operator to formulate and solve an error minimization problem. Besides the mathematical merit of the derived closed-form approximate PDFs, an additional significant advantage of the technique relates to the fact that it can be readily coupled with a stochastic averaging treatment of second-order SDEs governing the dynamics of diverse stochastically excited nonlinear/hysteretic oscillators. In this regard, it is shown that the technique is capable of determining approximately the response amplitude transition PDF of a wide range of nonlinear oscillators, including hysteretic systems following the Preisach versatile modeling. Several numerical examples are considered for demonstrating the reliability and computational efficiency of the technique. Comparisons with pertinent Monte Carlo simulation data are provided as well.

Non-stationary response statistics of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitation

An approximate analytical technique is developed for determining the non-stationary response amplitude probability density function (PDF) of nonlinear/hysteretic oscillators endowed with fractional derivative elements and subjected to evolutionary stochastic excitation. Specifically, resorting to stochastic averaging/linearization leads to a dimension reduction of the governing equation of motion and to a first-order stochastic differential equation (SDE) for the oscillator response amplitude. Associated with this first-order SDE is a Fokker–Planck partial differential equation governing the evolution in time of the non-stationary response amplitude PDF. Next, assuming an appropriately chosen time-dependent PDF form of the Rayleigh kind for the response amplitude, and substituting into the Fokker–Planck equation, yields a deterministic first-order nonlinear ordinary differential equation for the time-dependent PDF coefficient. This can be readily solved numerically via standard deterministic integration schemes. Thus, the non-stationary response amplitude PDF is approximately determined in closed-form in a computationally efficient manner. The technique can account for arbitrary excitation evolutionary power spectrum forms, even of the non-separable kind. A hardening Duffing and a bilinear hysteretic nonlinear oscillators with fractional derivative elements are considered in the numerical examples section. To assess the accuracy of the developed technique, the analytical results are compared with pertinent Monte Carlo simulation data.

Bayesian consideration of unknown sensor characteristics in fatigue-related structural health monitoring

In structural health monitoring, Bayesian updating is widely utilised in the analysis of noisy sequential data of dynamic systems with the objective of determining the state of damage of a structure or identifying its unknown dynamic characteristics or both. In the present work, this approach is enhanced to encompass the simultaneous handling of insufficient knowledge of sensor features – i.e. a non-applicable relation between state of damage and observations due to high uncertainty introduced by unknown measuring parameters – while given the nature of damage propagation as in fatigue-driven applications. Thereby, the statistical inversion problem of inferring unknown states of damage as well as unknown measuring and dynamic model parameters is addressed solely on the basis of observations and parameter-dependent functional expressions linking these quantities. As Bayesian updating provides the posterior belief on the unknown quantities in form of probability density functions, the question of state observability and parameter identifiability can be approached simultaneously. The methodology is applied to potential-drop measuring in a fatigue-loading scenario and its effectiveness is successfully demonstrated.

Thermoacoustic Damping Rate Determination From Combustion Noise Using Bayesian Statistics

In this paper, we present a method to determine the quantitative stability level of a lean-premixed combustor from dynamic pressure data. Specifically, we make use of the autocorrelation function of the dynamic pressure signal acquired in a combustor where a turbulent flame acts as a thermoacoustic driver. In the proposed approach, the unfiltered pressure signal including several modes is analyzed by an algorithm based on Bayesian statistics. For this purpose, a Gibbs sampler is used to calculate parameters like damping rates and eigenfrequencies in the form of probability density functions (PDF) by a Markov-chain Monte Carlo (MCMC) method. The method provides a robust solution algorithm for fitting problems without requiring initial values. A further advantage lies in the nature of the statistical approach since the results can be assessed regarding its quality by means of the PDF and its standard deviation for each of the obtained parameters. First, a simulation of a stochastically forced van-der-Pol oscillator with preset input values is carried out to demonstrate accuracy and robustness of the method. In this context, it is shown that, despite a large amount of uncorrelated background noise, the identified damping rates are in a good agreement with the simulated parameters. Second, this technique is applied to measured pressure data. By doing so, the combustor is initially operated under stable conditions before the thermal power is gradually increased by adjusting the fuel mass flow rate until a limit-cycle oscillation is established. It is found that the obtained damping rates are qualitatively in line with the amplitude levels observed during operation of the combustor.

Time-dependent probability density functions and information geometry in stochastic logistic and Gompertz models

A probabilistic description is essential for understanding growth processes in non-stationary states. In this paper, we compute time-dependent probability density functions (PDFs) in order to investigate stochastic logistic and Gompertz models, which are two of the most popular growth models. We consider different types of short-correlated multiplicative and additive noise sources and compare the time-dependent PDFs in the two models, elucidating the effects of the additive and multiplicative noises on the form of PDFs. We demonstrate an interesting transition from a unimodal to a bimodal PDF as the multiplicative noise increases for a fixed value of the additive noise. A much weaker (leaky) attractor in the Gompertz model leads to a significant (singular) growth of the population of a very small size. We point out the limitation of using stationary PDFs, mean value and variance in understanding statistical properties of the growth in non-stationary states, highlighting the importance of time-dependent PDFs. We further compare these two models from the perspective of information change that occurs during the growth process. Specifically, we define an infinitesimal distance at any time by comparing two PDFs at times infinitesimally apart and sum these distances in time. The total distance along the trajectory quantifies the total number of different states that the system undergoes in time, and is called the information length. We show that the time-evolution of the two models become more similar when measured in units of the information length and point out the merit of using the information length in unifying and understanding the dynamic evolution of different growth processes.