Theoretical Probability Density Function Research Articles

Identification of probability distributions of SCFs in three-planar KT-joints

KT-joints are crucial in jacket platforms, providing structural integrity by effectively transferring loads between the main chord and braces. Their robust design ensures stability and durability in harsh offshore environments, preventing structural failures and enhancing the platform's overall safety and performance. This study presents the findings of finite element (FE) analysis conducted on 324 tubular three-planar KT-joints subjected to in-plane bending (IPB) loadings. The purpose of the research was to generate a probability distribution (PD) for the stress concentration factors (SCFs) under bending moments. To the authors' knowledge, no previous research has analyzed the PD of the maximum SCFs in three-planar KT-joints. Consequently, data representing the maximum SCFs were collected from the three-planar KT connections. The maximum likelihood method was used to determine the distribution parameters. The goodness-of-fit of the Generalized Extreme Value distribution was assessed using the Kolmogorov-Smirnov (KS) and Chi-square (CS) tests. The results indicated that this distribution is the most suitable for modeling three-planar KT-joints under IPB loading. In the subsequent stage, well-defined theoretical probability density functions (PDFs) and cumulative distribution functions (CDFs) were developed. Furthermore, probability-probability diagrams demonstrated that the proposed PD closely aligns with the observed data.

Probabilistic Analysis of Strength in Retrofitted X-Joints under Tensile Loading and Fire Conditions

In the present study, a total of 360 FE analyses were carried out on tubular X-joints strengthened with collar plates under brace tension under laboratory testing conditions (20 °C) and various fire conditions. The generated FE models were validated based on 31 tests. The FE analyses produced a comprehensive dataset that encapsulated resistance metrics, with detailed simulations of welds, contacts, and the incorporation of non-linear geometrical and material attributes. Twelve theoretical probability density functions (PDFs) were matched to the constructed histograms, with the maximum likelihood (ML) technique utilized to assess the parameters of these fitted PDFs. The theoretical PDFs, rigorously evaluated against the Anderson–Darling, Kolmogorov–Smirnov, and Chi-squared tests, identified the Generalized Petrov distribution as the optimal model for capturing the resistance behaviors of X-joints under tensile load and varying fire conditions. The findings have led to the proposition of five detailed theoretical PDFs and cumulative distribution functions (CDFs), introducing a novel perspective for assessing and reinforcing the structural resilience of strengthened CHS X-joints in engineering practices.

Experimental investigations on an FSO-fiber converged communication system under fog and turbulence

This paper presents experimental results for a free-space optical-fiber converged (FSO-FC) communication system under varying turbulence and foggy conditions. Based on the experimental measurements, we statistically characterize the FSO channel under different levels of turbulence and fog in terms of their respective probability density functions (PDFs). Our experimental PDFs fit well with the theoretical PDFs proposed in the literature. We experimentally evaluate the average bit error rate (ABER) of the considered FSO-FC communication system under different levels of turbulence and fog. Further, to compensate for the effects of turbulence and fog, we use multimode fiber as the receiver and compare the results with single-mode fiber as the receiver. Finally, we show the improvement in ABER under varying turbulence and fog by adding cyclic redundancy check bits to data bits.

Probability distribution functions for the ultimate strength of X-joints with collar plates in compressive load at room and fire conditions

In this paper, the results of performing finite element (FE) analysis on 360 tubular X-joints reinforced with collar plates have been used to propose a probability distribution model for ultimate strength at ambient and fire conditions. As far as the authors know, no study has been conducted on the probability distribution of the ultimate strength in X-joints with collar plates. Consequently, according to the reinforced X-joints, a total of ultimate capacity data sets was generated. The FE models were validated and compared with 35 experimental tests and numerical models. Also, in the present FE models, the welds connecting the plates to the chord member were created. Moreover, the contacts between the outer surfaces of the chord member and the inner surfaces of the reinforcing plates were modeled. In addition, nonlinear material and geometry properties were applied. In the following step, twelve theoretical probability density functions (PDFs) were fitted to relative frequency histograms of five databases. The maximum likelihood method was applied to determine the parameters of each distribution. Investigating the goodness-of-fit according to the Kolmogorov-Smirnov, Anderson Darling, and Chi-square tests showed that the Generalized Petro distribution is the best for the ultimate capacity values in collar plate retrofitted X-joints at ambient and different fire conditions. In the following step, six fully defined theoretical PDFs and cumulative distribution functions (CDFs) were established. Also, probability-probability graphs indicated that the proposed theoretical probability distributions are near the given data sets.

Determination of mathematical expectations of two weighting statistics for ensemble time scales

The Bureau International des Poids et Mesures uses the ALGOS algorithm to generate the Echelle Atomique Libre. The original and new ALGOS weighting algorithms employ two different statistics to calculate the weights, resulting in different experimental phenomena. These two statistics are believed to be the keys to analyzing these experimental phenomena, because the statistical characteristics (e.g. the mathematical expectations and variances) of the statistics completely determine the statistical characteristics of the weights, and subsequently the experimental phenomena. The core work conducted in this study is to derive the analytical expressions of the mathematical expectations of these two statistics under ideal conditions and to validate them empirically. This paper provides a theoretical analysis of these expressions and discusses the relationships between them and the mathematical expectations of the Allan variances. These expressions theoretically reveal the reasons behind the experimental phenomena, especially why hydrogen masers are assigned lower weights in the original ALGOS weighting algorithm but higher weights in the new one. According to the derivation process, these analytical expressions are universally applicable and can also be applied to time-keeping laboratories. In the future, this work can be extended to determine the theoretical statistic and weight performances of different types of frequency standards quantitatively for time-keeping laboratories under both ideal and non-ideal conditions, providing a theoretical basis for improving the design of ensemble time scale algorithms. In the , a method of obtaining the experimental variances and probability density functions (PDFs) of these two statistics is additionally proposed. The two-sample Kolmogorov–Smirnov test is presented to compare the experimental and theoretical PDFs of an example.

Variability of Extreme Air Temperatures and Precipitation in Different Natural Zones in Late XX and Early XXI Centuries According to ERA5 Reanalysis Data

Global temperature rise on the planet is accompanied by an increase in the frequency of extreme weather events, which can lead to significant disturbances in plant metabolic processes and the functioning of plant communities. Depending on the type of vegetation and climatic conditions, the extent and nature of this impact can vary significantly. Therefore, the aim of this work was to identify areas with different types of plant communities that have been most affected by extreme temperatures and precipitation in recent decades. To identify regions with temperature and precipitation extremes, we estimated the frequency of exceeding a given threshold (less than 5% quantile or greater than 95% quantile) for selected theoretical probability density functions for the air temperature and precipitation series. The number of cases of exceeding the extreme threshold is calculated for the whole globe, their spatial and temporal variability in different seasons of the year and in the first decades of the XXI century compared to the end of the XX century is analysed. The regions with different types of plant communities with the most pronounced amplification of extreme temperature and precipitation regimes under changing climate conditions were identified.

Generation and application of avatars in pharmacometric modelling

Simulations from population models have critical applications in drug discovery and development. Avatars or digital twins, defined as individual simulations matching clinical criteria of interest compared to observations from a real subject within a predefined margin of accuracy, may be a better option for simulations performed to inform future drug development stages in cases where an adequate model is not achievable. The aim of this work was to (1) investigate methods for generating avatars with pharmacometric models, and (2) explore the properties of the generated avatars to assess the impact of the different selection settings on the number of avatars per subject, their closeness to the individual observations, and the properties of the selected samples subset from the theoretical model parameters probability density function. Avatars were generated using different combinations of nature and number of clinical criteria, accuracy of agreement, and/or number of simulations for two examples models previously published (hemato-toxicity and integrated glucose-insulin model). The avatar distribution could be used to assess the appropriateness of the models assumed parameter distribution. Similarly it could be used to assess the models ability to properly describe the trajectories of the observations. Avatars can give nuanced information regarding the ability of a model to simulate data similar to the observations both at the population and at the individual level. Further potential applications for avatars may be as a diagnostic tool, an alternative to simulations with insurance to replicate key clinical features, and as an individual measure of model fit.

Theoretical Analysis of Random Scattering Induced by Microlensing

Theoretical investigations into the deflection angle caused by microlenses offer a direct path to uncovering principles of the cosmological microlensing effect. This work specifically concentrates on the the probability density function (PDF) of the light deflection angle induced by microlenses. We have made several significant improvements to the widely used formula from Katz et al. First, we update the coefficient from 3.05 to 1.454, resulting in a better fit between the theoretical PDF and our simulation results. Second, we developed an elegant fitting formula for the PDF that can replace its integral representation within a certain accuracy, which is numerically divergent unless arbitrary upper limits are chosen. Third, to facilitate further theoretical work in this area, we have identified a more suitable Gaussian approximation for the fitting formula.

Eigenfilter‐based analysis of uncoded space‐time labeling diversity with three transmit antennas in correlated Nakagami‐mfading channels

AbstractIn this paper, three‐transmit antennas uncoded space‐time labeling diversity (USTLD) scheme in arbitrarily correlated Nakagami‐fading channels are presented. The authors apply eigenfilter decomposition technique to synthesize the average branch signal‐to‐noise ratio (SNR), while deriving the theoretical probability density function (PDF) for the channel. The analytical mean bit error rate (BER) for the correlated channel was formulated using the moment generating function based on a virtual branch approach. Monte Carlo simulations are used to validate the derived analytical expression, which converge tightly at high values of SNR. Based on the decorrelation transformation technique, a further investigation of the USTLD scheme was performed in identically distributed fading channels. The results in this paper indicate that the best error performance is observed when the antenna spacings are 0.4for the 16QAM and 64QAM constellations respectively, whereis the antenna spacing distance. Simulations for the USTLD scheme with different values ofindicate that as the value forincreases, the error performance improved. Additionally, it can be seen that in similar conditions, the USTLD scheme in the identically distributed fading channels had illustrated an improved error performance over the traditional multiple‐input multiple‐output (MIMO) scheme respectively.

Probability distribution models for the ultimate strength of tubular T/Y-joints reinforced with collar plates at room and different fire conditions

In this research work, the simulation results of 360 T/Y-joints reinforced with collar plates have been used to propose theoretical probability distribution models for the ultimate capacity at ambient and fire conditions. So far, no study has been carried out on the probability distribution of ultimate strength in joints with collar plates at room or elevated temperature. Therefore, according to the reinforced T/Y-joint models, which were verified with 18 tests, a total of ultimate capacity data sets was produced. In the FE models, the welds along the collar/brace intersection and between the chord and collar have been generated. Also, the Augmented Lagrange method was applied as the contact algorithm for solving the contact between the collar and chord. In the next step, 13 theoretical probability density functions (PDFs) were fitted to the relative frequency histograms of the databases. For obtaining the parameters of each distribution, the maximum likelihood (ML) method was utilized. Assessing the goodness-of-fit based on the Kolmogorov-Smirnov and Anderson Darling tests indicated that the Johnson SB distribution is the best for the ultimate capacity values in the T/Y-joints reinforced with collar plates at 20, 400, 600, and 800 °C. But, for the reinforced joints at 200 °C, the Inverse Gaussian (3P) distribution is the best. In the next step, five fully defined theoretical PDFs were proposed for the collar plate reinforced T/Y-joints at room and different high temperatures. Probability difference and probability-probability graphs indicated that the proposed theoretical probability distributions are well near the given data sets.

Identification of probabilistic distribution parameters for the mesoscopic stochastic fracture model

As a kind of multiphase composite material, the basic mechanical behaviors of concrete are randomness and nonlinearity. The mesoscopic stochastic fracture model (MSFM) which can reflect the coupling effect of randomness and nonlinearity, has been widely used for the nonlinear analysis of concrete structures. In this paper, we proposed a new stochastic modeling principle to identify the probabilistic distribution parameters of MSFM. In order to reduce the modeling works, a dimension-reduced algorithm is proposed as well. In this paper, an overview of MSFM is firstly presented to introduce the background of the research. Then the stochastic harmonic function (SHF) representation is introduced to express the random field mentioned in the MSFM, and the probability density evolution method (PDEM) is applied to obtain the theoretical probability density function (PDF) of the stress–strain relationships. Furthermore, a stochastic modeling principle is proposed, in which minimizing the Kullback–Leibler divergence (KLD) is taken as the optimization object. Based on the framework of genetic algorithm, a dimension-reduced algorithm is proposed to identify the parameters with reference to the data from tested complete curves of uniaxial compressive and uniaxial tensile stress–strain relationship of concrete. The results indicate that the proposed principle and algorithm can be used to identify the parameters of MSFM accurately and efficiently.

A Low-Return-Period Rainfall Intensity Formula for Estimating the Design Return Period of the Combined Interceptor Sewers

The design rainfall intensity and its return period of the combined interceptor sewer is an important factor affecting combined sewer overflow (CSO) occurrence. However, we often use the interceptor ratio (or interceptor multiple, n0) to design the interceptor sewer, and its equivalent design return period is often ignored. In this study, a low return period rainfall formula modeling method was proposed to estimate this return period. First, a new rainfall event separation approach was especially developed, and the minimum interevent time (MIET) was set to time of concentration of the tributary area corresponding to the most downstream interceptor well. Second, a new rainfall intensity sampling algorithm, annual multi—event—maxima (AMEM) sampling algorithm, was put forward. For this sampling algorithm, several maxima of rainfall intensity should be sampled annually, and only one maximum is sampled for each rainfall event. In addition, the empirical frequency values of the above sampled rainfall intensities can be obtained according to the mathematical expectation formula (Weibull formula). After comparison, the lognormal distribution was selected for the theoretical probability density function. Finally, parameters of the low return period rainfall intensity formula were estimated using three-parameter Horner formula and MCMC (Markov Chain Monte Carlo) algorithm. A case study was conducted to demonstrate the proposed method based on the recorded rainfall data from a meteorological station in southwestern China and a combined sewer system. Results revealed that: (1) A MIET determination method was proposed according to independence of CSO events. (2) An annual multi-event-maxima (AMEM) sampling was proposed for collecting samples of the low return period rainfall intensity. (3) For the case study, the best-fit distribution for low return period rainfall intensity was lognormal distribution. (4) Resulted low return period rainfall intensity formula was provided.

Response of a stochastic multiple attractors wind-induced vibration energy harvesting system with impacts

Focusing on multiple attractors wind-induced vibration energy harvesting system with unilateral impact under the excitation of Gaussian white noise, the stochastic response characteristics, and energy harvesting performance of the systems are presented and discussed. The stochastic averaging of the energy envelope and a conversion mechanism for decoupling the electromechanical equations are introduced to derive the theoretical probability density functions. Numerical simulation results are given to verify the correctness of such theoretical methods. Effects of the noise intensity, restitution coefficient, time constant ratio and coupling coefficient on the mean square voltage are analyzed. It is found that increments of the restitution coefficient and the piezoelectric coupling coefficient of the electrical part have a positive effect on the output voltage of the system, while other factors are opposite. Analysis can help explain the physical mechanisms of the system and provide ideas for optimizing energy harvesting performance by properly tuning the coefficients.

Probability density functions of the DoB in tubular KT-joints of jacket-type platforms under out-of-plane bending loads

The degree of bending (DoB) has a profound effect on the fatigue behavior of tubular joints commonly found in offshore jacket structures. Probability density functions (PDFs) of the involved random variables are necessary for the fatigue reliability analysis of jacket structures. The objective of present research was the derivation of PDF for the DoB in tubular KT-joints commonly found in jacket-type platforms. A total of 243 finite element (FE) analyses were carried out on 81 FE models of KT-joints subjected to three types of out-of-plane bending (OPB) moment loading. Generated FE models were validated using experimental data, previous FE results, and available parametric equations. Based on the results of parametric FE study, a sample database was prepared for the DoB values. Thirteen theoretical PDFs were fitted to the developed histograms and the maximum likelihood (ML) method was applied to evaluate the parameters of fitted PDFs. In each case, the Kolmogorov-Smirnov test was used to evaluate the goodness of fit. After substituting the values of estimated parameters, six fully defined PDFs were presented for the DoB at the saddle positions of the central and outer braces in tubular KT-joints subjected to three types of OPB moment loading.

Dynamics of stochastically excited energy harvesting systems with impact

Focusing on rigid impact (RI) and elastic impact (EI) vibratory energy harvesting systems with the excitation of Gaussian white noise, the stochastic response characteristics, and energy harvesting performance of the systems are presented and compared. The stochastic averaging of the energy envelope is introduced with different processing methods in these systems to derive the theoretical probability density functions. Numerical simulation results verify the effectiveness of such theoretical methods. Effects of the coefficients and positions of impact barriers in the systems on the mean square voltage are analyzed. The analysis can help explain the physical mechanisms of the systems and provide ideas for optimizing energy harvesting performance by properly tuning the coefficients. Effects of the existence of three kinds of impact on the output energy are explored and compared. It is found that the existence of EI can help to enhance the energy harvesting performance of the system in most cases, and setting the impact barrier at the origin is more conducive to harvesting energy compared with other impact positions.

Simulation with Fuzzy Durations

Traditional stochastic discrete event simulation method requires the users to fit the task duration data with an empirical probability function or a theoretical probability density function. This has limited the application of discrete event simulation techniques from research stage to construction practice due to the tedious data fitting tasks required for the construction operation planners lacking of statistical background. On the other hand, usually the activity duration estimated by practitioners contains some kind of vagueness caused by estimator's subjective judgment. This type of vagueness may be more appropriate to be modeled by fuzzy numbers that can be assessed without data fitting. This paper proposes a new mechanism where fuzzy duration modeled by fuzzy numbers can be used in discrete event simulation. Consequently, the tedious data fitting tasks are eliminated but the uncertainty of task duration is still taken into account.

Probability distribution selection using PROMETHEE GDSS method

Rising inequalities between countries are widely investigated. Since they involve different aspects, in this paper they are examined through the distribution of countries’ growth and development, inclusion and intergenerational equity and sustainability. These aspects involve variables that are not normally distributed so different probability density functions (PDFs) have been proposed, tested and compared through literature. This paper compares different PDFs that are considered successful in describing distribution of the variables involved, as Weibull, gamma, lognormal, normal, loglogistic, Pareto, Burr and exponential. They are usually compared using diff erent goodness-of-fit measures including the log-likelihood, sum of squared errors, sum of absolute errors and chi-square statistics which can sometimes lead to divergent conclusions. This paper adds up to these goodness-of-fit measures and includes the Kolmogorov-Smirnov, Cramer-von Mises and Anderson-Darling statistics, the mean absolute and squared deviation between theoretical and empirical PDF, the mean absolute and squared deviation between theoretical and empirical cumulative distribution function, AIC and BIC as well as the deviation in skewness and kurtosis. Since diff erent distributions can be considered as alternatives and goodness-of-fit measures as conflicting criteria, the problem of finding the appropriate PDF can be viewed as multiple criteria decision making problem which can be solved using PROMETHEE method. However, diff erent preference parameters, lead to different ranking, so the final decision is obtained using PROMETHEE Group Decision Support System (GDSS) method. This paper therefore contributes to the existing literature on inequalities between countries and statistics in general by proposing a new approach for distribution modelling and selection.

The effect of liquid co-flow on gas fractions, bubble velocities and chord lengths in bubbly flows. Part I: Uniform gas sparging and liquid co-flow

Unique experiments were performed in a homogeneously sparged rectangular 400×200×2630 mm (W×D×H) bubble column with and without liquid co-flow. Bubbles in the range 4–7 mm were produced by needle spargers, which resulted in a very uniform bubble size. Dual-tip optical fibre probes were used to measure horizontal profiles of gas fractions, bubble velocities and bubble chord lengths for superficial gas velocities Usg in the range 0.63–6.25 cm/s and superficial liquid velocities Usl up to 20 cm/s. Images of the bubble column were captured and a Bubble Image Velocimetry technique was adopted to calculate bubble (parcel) velocities. For low gas fractions, when a homogeneous flow regime occurred, both methods agreed very well and the optical fibre probes were found to be rather accurate for our bubbles. A liquid co-flow was found to have a calming effect and to stabilize a homogeneous bubbly flow regime, with less spatial variation in gas fractions and bubble velocities. Bubble chord lengths were almost normally distributed and do not exhibit the theoretical triangular probability density functions. The mean cord lengths were in the range 1.9–3.5 mm and found to increase with Usg and to decrease slightly with increasing Usl, while a liquid co-flow significantly reduced the standard deviation of the chord length distribution.

SINR and Rate Distributions for Downlink Cellular Networks

Signal-to-Interference-plus-Noise Ratio (SINR) measures the quality of the radio link between a base station and a User Equipment (UE). This quality indicator is important for the operator as it is used to measure and to optimize network coverage and the maximum user bitrate (which can be deduced from the SINR using the Shannon formula). In this paper, we study the SINR distribution for the downlink assuming that the received signals from serving and interfering cells undergo a path-loss and log-normal shadowing fading. We show that SINR can be approximated by a log-normal random variable and we give analytical expressions of characteristics parameters of its distribution. We then derive the average SINR expression. Also, we give a closed-expression of the rate probability density function and the outage probability and we obtain closed expressions of the average rate and lower and upper bounds of the average rate. For validation, we first compare theoretical results to Monte-Carlo network simulations and show that all the theoretical curves approximate well those obtained by simulations. Then using real measurements obtained by a measurement campaign, we validate the SINR distribution assumption. Based on a test of normality, we find that the recorded SINR samples are normally distributed in the logarithmic domain. We also show that the theoretical probability density function curves and the histogram of the measured samples of SINR are close.

Analysis of Weighted Fractional Fourier Transform Based Hybrid Carrier Signal Characteristics

Recently, a weighted fractional Fourier transform (WFRFT) based hybrid carrier (HC) system has been proposed, which can converge single-carrier (SC) and multi-carrier (MC) systems. The cost and power dissipation of analog components often dominate in practical HC systems. Therefore, in this paper, we analyze the baseband HC signal characteristics, including average signal power, power spectral density (PSD), probability density function (PDF), peak-to-average power ratio (PAPR), and complementary cumulative distribution function (CCDF), using a continuous-time HC signal approximation method. Through theoretical analysis and simulation validation, it is proved that the approximation method does not change the average signal power, bandwidth and out-of-band emission (OOBE). The theoretical PDF expression of HC signal with the approximation method is then proposed. Furthermore, the PDF can also explain why the PAPR of baseband continuous-time HC signal is higher than that of discrete-time HC signal. The PAPR performance and the power amplifier (PA) efficiency of HC systems in different conditions are analyzed. Through the PDF analysis of HC signal envelope and the numerical computations of PAPR, it is shown that the approximation method can offer a precise characteristic description for HC signal, which can help to improve the system PAPR performance and the PA efficiency.