Joint Probability Distribution Function Research Articles

On the Bivariate Extension of the Extended Standard U-quadratic Distribution

This paper derives the bivariate version of the extended standard U-quadratic (eSU) distribution using the method of compounding or pseudo family of distributions. The joint probability and cumulative distribution functions of the derived distribution are obtained and it is observed that the said distribution can generate bivariate shape with the following properties: $(i)$ $X$ and $Y$ have bathtub shapes; $(ii)$ $X$ has constant and $Y$ has bathtub shapes; and $(iii)$ $X$ has inverted bathtub and $Y$ has bathtub shapes. Moreover, some properties of the derived distribution such as the marginal distribution, conditional distributions, conditional moments, product and ratio moments are derived. Further, the maximum likelihood estimation is performed to estimate the parameters of the derived distribution. Also, a simulation study is carried out to evaluate the behavior of the estimates of the parameters. Moreover, we derive a new bivariate Kumaraswamy distribution and use it to simulate bivariate data with $X$ and $Y$ having bathtub shapes. Furthermore, A new bivariate version of the Cubic Transmuted Uniform (CTU) distribution is also derived. Finally, the proposed Bivariate eSU distribution is applied to simulated data and compared with the said Bivarite Cubic Transmuted Uniform distribution. The result shows that the proposed Bivariate eSU distribution provides a better fit for the said simulated dataset as compared with the Bivariate CTU distribution.

Rapid simulation of two-dimensional spectra with correlated anisotropic dimensions.

A new algorithm has been developed to simulate two-dimensional (2D) spectra with correlated anisotropic frequencies faster and more accurately than previous methods. The technique uses finite-element numerical integration on the sphere and an interpolation scheme based on the Alderman-Solum-Grant algorithm. This method is particularly useful for numerical calculations of joint probability distribution functions involving quantities with a parametric orientation dependence. The technique's efficiency also allows for practical least-squares fitting of experimental 2D solid-state nuclear magnetic resonance (NMR) datasets. The simulation method is illustrated for select 2D NMR methods, and a least-squares analysis is demonstrated in the extraction of paramagnetic shift and quadrupolar coupling tensors and their relative orientation from the experimental shifting-d echo 2H NMR spectrum of a NiCl2 · 2D2O salt.

Serviceability analysis of sea-crossing bridges under correlated wind and wave loads

With coupled wind and wave loads suffered by sea-crossing bridges, it is challenging to include uncertainties for assessing bridge serviceability. In this study, an efficient probabilistic peak response framework with surrogate models is developed for the serviceability analysis of sea-crossing bridges under coupled wind and wave loads. The joint probability distribution functions (JPDF) of the mean wind speed, significant wave height, and peak wave period are first derived based on long-term field measurement data and using a C-vine copula model. The JPDF serves as an input to the bridge to calculate the peak response of the bridge through finite-element analyses. To improve computational efficiency and make the problem manageable, the surrogate models are then generated by both the support vector machine and response surface method to predict the bridge peak responses. The exceeding probabilities for the serviceability assessment are finally obtained by the Monte-Carlo method using the pre-defined threshold values of bridge serviceability. A real long-span sea-crossing bridge is used as a case study to demonstrate the feasibility and effectiveness of the proposed framework. The results elucidate the effects of correlated wind and wave loads on bridge serviceability. Ignoring the correlation of wind and wave parameters will significantly overestimate the exceeding probability of bridge serviceability.

Brief Review of Recent Achievements in the Flamelet Manifold Selection and Probability Density Distribution for Flamelet Manifold Variables

Abstract The flamelet model is a commonly used tool for turbulent combustion simulations in the engineering field due to its computational efficiency and compatibility with complex chemical reaction mechanisms. Despite being widely used for decades, the flamelet model still faces challenges when applied to complex flame configurations, such as partially premixed flames, inhomogeneous inlets, supersonic combustion, or multiphase combustion. The principal challenges are posed by the uncertainty of the presumed shapes for probability density functions (PDFs) of the flamelet tabulation variables and the coupled process of turbulent diffusion and chemical reaction in turbulent combustion. Recent progress is reviewed from the viewpoint of the reaction manifold, with connections made to other combustion models, as well as the determination of joint (or conditional) PDFs for flamelet manifold parameters (e.g., progress variable, scalar dissipation rates, etc.). Promising improvements have been outlined in computational efficiency and the accuracy of predicted variable fields in simulating complex combustion systems (such as turbulent inhomogeneous combustion, combustion with multi-regime modes, and two-phase combustion). Advances in computational resources, direct numerical simulation data, artificial intelligence, stochastic simulation methods, and other dimension-reduction combustion models will contribute to the development of more accurate and efficient flamelet-like models for engineering applications.

Characterization and evolution of local streamline geometry in an incompressible turbulent channel flow

We investigated the statistical characterization and time evolution of local streamline geometry in typical regions of an incompressible turbulent channel flow at the friction Reynolds number Reτ∼1000. Local streamline structure is completely and uniquely determined by one magnitude factor—the magnitude of velocity gradient tensor (VGT) A—and four shape parameters—the second and third invariants of normalized VGT q and r, the intermediate eigenvalue of normalized strain-rate tensor a2, and the cosine of the angle between vorticity and the intermediate eigendirection of normalized strain-rate tensor | cos β|. As the distance to the wall decreases, the joint probability distribution function of q and r becomes more symmetrical and concentrated, while outside the viscous sublayer, the distribution of A in q–r plane gets dispersed. Interestingly, the inertial conditional mean trajectories (CMTs) exhibit a symmetrical picture only in the buffer layer, and outside the viscous sublayer, the pressure CMTs contributing to slow evolution from unstable focus compression geometry to stable focus stretching geometry tend to dominate the q–r plane as getting closer to the wall. Due to combined effects of inertia and pressure, the origin of the q–r plane (pure-shear geometry) acts as an attractor in the central region, the logarithmic region, and the upper part of the buffer layer while acts as a repeller in the lower part of the buffer layer and the viscous sublayer.

Modeling forecast errors for microgrid operation using Gaussian process regression

Microgrids, denoting small-scale and self-sustaining grids, constitute a pivotal component in future power systems with a high penetration of renewable generators. The inherent uncertainty tied to renewable power generation, typified by photovoltaic and wind turbine systems, necessitates counterbalancing mechanisms. These mechanisms encompass Energy storage systems or conventional thermal fossil-fuel generators imbued with heightened flexibility. Addressing the uncertainty stemming from renewable generators mandates a cost-effective assessment and operational strategy for said compensatory devices. To this end, myriad uncertainty factors warrant scrutiny, conceivably concretized into a unified probability distribution function (PDF) that takes into account their temporal inter-dependencies. Diverse uncertainty factors, characterized by varying marginal distributions and scales, can be assimilated into a multivariate probability distribution through a conversion to normal distributions via rank correlation. However, with the escalation in the number of uncertainty factors embraced within a microgrid context, the endeavour becomes notably intricate when aiming to define conditional probability distributions originating from joint PDFs. This paper presents a method proposing the modelling of net-load forecast error distribution, considering the interplay among uncertainty factors. The approach introduces a data-driven Gaussian process regression technique for training and validating conditional PDFs among these uncertainty factors. Notably, this approach facilitates the transformation of said factors into normal distributions while preserving their inherent marginal characteristics. The resultant conditional density function, as per the proposed methodology, exhibits enhanced suitability for estimating net-load error distribution. Consequently, the conditional density function stemming from this proposed approach demonstrates superior aptitude in approximating the distribution of net load error.

Electric Vehicle and Photovoltaic Power Scenario Generation under Extreme High-Temperature Weather

In recent years, with the intensification of global warming, extreme weather has become more frequent, intensifying the uncertainty of new energy output and load power, and seriously affecting the safe operation of power systems. Scene generation is an effective method to solve the uncertainty problem of stochastic planning of integrated systems of new energy generation. Therefore, this paper proposes a scenario generation and scenario reduction model of photovoltaic (PV) output and electric vehicle (EV) load power under extreme weather based on the copula function. Firstly, the non-parametric kernel density estimation method is used to fit a large number of sample data. The kernel density estimation expressions of PV and EV powers under extreme weather conditions are obtained and the corresponding goodness of fit tests are carried out. Then, a variety of joint distribution models based on the copula function are established to judge the goodness of fit of each model, and the optimal copula function is selected as the joint probability distribution function by combining the Kendall and Spearman correlation coefficients of each model. Finally, the optimal copula joint probability distribution is used to generate PV and EV power scenarios. The data of extremely hot weather in a certain province were selected for an example analysis. The results show that the output scenario obtained conforms to the correlation under this extreme weather, and has higher accuracy in reflecting the actual PV output and load power in this province under this extreme weather, which can provide a reference for reliability analyses of power systems and power grid planning.

Bayesian calibration and fitting of nuclear thermal–hydraulic models by Markov chain Monte Carlo methods using the Gibbs sampler

This paper applies Bayesian methods to obtain the joint probability distribution function (PDF) of the unknown parameters of some of the correlation types used in thermal hydraulics, which are conditioned to a set of results of separate effect experiments. The problem is solved by Markov chain Monte Carlo methods using the Gibbs sampler. In this paper it is shown how to obtain the conditional PDFs for each one of the parameters, which define two typical correlations that appear in thermal–hydraulic problems, i) the heat transfer in the walls of a pipe and ii) the penetration length of steam discharged in a subcooled pool. The equations for the conditional probabilities of each one of the parameters are deduced and then used to build the Gibbs sampler which is then programmed in R. The PDFs are obtained for each one of the parameters and from these the maximum a posteriori of each one of the parameters is obtained. The new correlations obtained in this way are compared with the experimental data and show satisfactory performance for the case of the penetration of steam in a subcooled pool. Also, we study two cases one with a small number of experimental data and another one with a large number, it is observed that if the number of parameters is small the Markov chain converges fast to the region of high probability of the parameters for both cases with a small number of data and with large number of data, but if the number of parameters increases then the convergence for the case with small number of data becomes poor, however the convergence continuous being very fast for the case with large number of experimental data.

Deep Learning of Joint Scalar PDFs in Turbulent Flames from Sparse Multiscalar Data

ABSTRACT We investigate the reconstruction of multi-dimensional joint scalar probability density functions starting with sparse multiscalar data in turbulent flames using deep operator networks (DeepONets). The formulation is based on 1) a dimensionality reduction to reduce the complexity of modeling the PDF using principal component analysis (PCA), and 2) the implementation of the deep operator neural network (or DeepONet) to construct a generalizable joint PDF of the principal components (PCs). We demonstrate the accuracy and generalizability of the DeepONet on the Sandia turbulent non-premixed flames D, E and F where the DeepONet is trained on data for flame E and is evaluated for the 3 flames. The DeepONet-based approach is able to construct multiple PDF shapes based on the PCs for all conditions. Combined with a recently proposed method for evaluating thermochemical scalar means conditioned on the PCs, the convolution of these means with the joint PDFs yields good predictions of thermochemical scalars’ unconditional means profiles in the flames.

Forecasting in Stock Markets Using the Formalism of Statistical Mechanics

The possibility and expediency of forecasting in the stock markets are analyzed analytically using the methods and approaches of statistical mechanics. The apparatus of statistical mechanics is used to analyze and forecast one of the most important indicators of the market – the distribution of its logarithmic profitability. The Lotka-Volterra model used in ecology to describe systems of the "predator-prey" type was used as the initial model. It approximates market dynamics adequately. In the article, its Hamiltonian property is used, which makes it possible to apply the apparatus of statistical mechanics. The apparatus of statistical mechanics (using the principle of maximum entropy) makes it possible to implement a probabilistic approach that is adapted to the conditions of stock market uncertainty. The canonical variables of the Hamiltonian are presented as logarithms of stock and bond prices, the joint probability distribution function of stock and bond prices is obtained as a Gibbs distribution. The Boltzmann factor, included in the Gibbs distribution, allows us to estimate the probability of the occurrence of certain stock and bond prices and obtain an analytical expression for calculating the logarithmic return, which gives more accurate results than the widely used normal (Gaussian) distribution. According to its characteristics, the resulting distribution resembles the Laplace distribution. The main characteristics of the resulting distribution are calculated – the mean value, variance, asymmetry, and kurtosis. Mathematical results are presented graphically. An explanation is given of the cause-and-effect mechanism that causes a change in the profitability of the market. For this, the idea of Theodore Modis about the competition between stocks and bonds for the attention and money of investors is developed (by analogy with the turnover of biomass in models of the "predator-prey" type in biology). The results of the study are of interest to investors, theorists, and practitioners of the stock market. They allow us to make thoughtful and balanced investment decisions due to a more realistic idea of the expected return and a more adequate assessment of investment risk.

Drought Hazard Analysis in the Jilin Province Based on a Three-Dimensional Copula Method

Based on the precipitation and potential evapotranspiration at 23 stations in the Jilin Province, the standardized precipitation evapotranspiration index (SPEI) was calculated. Based on run theory and combined with previous research in Jilin Province, we redefine the drought threshold, with three characteristics, that is, drought duration, intensity, and severity, were identified and extracted. Two- and three-dimensional copula functions were selected to fit and combine drought variables to calculate the joint recurrence period. In order to make the alternative copula model more complete, 25 representative copula functions are selected for the first time to construct the Joint probability distribution function. The Mann–Kendall test was used to examine abrupt changes in Meihekou County. SPEI trends and the joint cumulative probability of drought variables in the Changbai area were analyzed. The results show that: (1) the SPEI in the Meihekou area was unstable around 2000 and 2010, with abrupt change points; (2) the trend of the SPEI in the Changbai Mountains area indicates that the drought occurrence is similar at various timescales; the drought range and duration increase with increasing timescale; and (3) the correlation between the three drought variables is strong and the fit is good. The results of two- and three-dimensional joint recurrence period calculations are similar, with a high probability of recurrence within three years.

A drought index based on groundwater quantity and quality: Application of multivariate copula analysis

In this paper, a new Groundwater Quantity-Quality-based Drought Index (GQQI) is developed based on multivariate Copula analysis of groundwater quantity and quality indicators. For evaluating the developed index, its temporal and spatial distribution is studied and compared to those of some other indices, such as Standardized Salinity Index (SSI), Standardized Groundwater Index (SGI), and Standardized Water level Index (SWI). The proposed index is applied for the temporal and spatial evaluation of the quantity and quality of groundwater in theLake Urmia basin, having 1084 piezometric and 935 groundwater quality monitoring wells. For a more comprehensive analysis of drought, 24 marginal and 26 joint probability distribution functions are driven. Based on the results, the developed GQQI is correlated with drought indices of SSI, SGI, and SWI by 88%, 86%, and 61%, respectively. Moreover, the values of the developed drought index indicates the occurrence of more severe droughts, compared to those detected by other drought indices. This can be resulted from considering the combined effects of groundwater quantity and quality using the proposed index. In addition, the multivariate GQQI can spatially and temporally represent the drought severity over the basin, and identify severe and extreme drought conditions better than other univariate drought indices.

Heavy inertial particles in rotating turbulence: Distribution of particles in flow and evolution of Lagrangian trajectories.

We revisit the problem of heavy particles suspended in homogeneous box turbulence flow subjected to rotation along the vertical axis, which introduces anisotropy along the vertical and horizontal planes. We investigate the effects of the emergent structures due to rotation, on the spatial distribution and temporal statistics of the particles. The distribution of particles in the flow are studied using the joint probability distribution function (JPDFs) of the second and third principle invariants of the velocity gradient tensor, Q and R. At high rotation rates, the JPDFs of Lagrangian Q-R plots show remarkable deviations from the well-known teardrop shape. The cumulative probability distribution functions for times during which a particle remains in vortical or straining regions show exponentially decaying tails except for the deviations at the highest rotation rate. The average residence times of the particles in vortical and straining regions are also affected considerably due to the addition of rotation. Furthermore, we compute the temporal velocity autocorrelation and connect it to the Lagrangian anisotropy in presence of rotation. The spatial and temporal statistics of the particles are determined by a complex competition between the rotation rate and inertia of the particle.

Fatigue assessment of ship structures based on equivalent wave probability (EWP) concept (1st report): Proposal of EWP concept and its verification by 8600TEU container ship's onboard hull monitoring

A long-term fatigue assessment method based on EWP concept is proposed. ‘Equivalent wave probability (EWP)’ is the fictitious (HS, Tm)'s joint probability distribution function (JPDF), for which the frequency distribution of the stress variance R2, f(R2), calculated by spectral fatigue assessment agrees with the observed one. By choosing probability function p(R2) to fit f(R2), the R2's statistical model (R2SM) which represents the relation between the EWP parameters and R2's population parameters is developed, and the Bayesian inference, which can estimate the EWP parameters from the measured R2 data is developed. The EWP at the reference position (RP) can be determined by Bayesian inference from the measured R2 through the R2SM at RP. To accurately estimate the measured f(R2) at the target position (TP) from the EWP at RP, an R2SM correction factor at TP, denoted by αTP, is introduced in the process of assimilating R2SM. The resulting R2SM, which has been assimilated by Bayesian inference using measured data, is referred to as data-assimilated R2SM (DAR2SM). The fatigue assessment using EWP at RP as the input of DAR2SM at TP is called Bayes-EWP-DAR2SM analysis. The validity of Bayes-EWP-DAR2SM analysis is verified by using the long-term (about four years) multi(12)-position hull monitoring (HM) data of an 8,600TEU container ship. The fatigue damages estimated by Bayes-EWP-DAR2SM based solely on the stress history of a single sensor are in agreement with measurements with sufficient accuracy, independent of the chosen data assimilation period. This demonstrates that the multi-position fatigue assessment solely through HM at one RP based on EWP concept is realized.

Composite power system risk evaluation considering the accurate model of renewable power output

The fluctuation of renewable energy will threaten the reliable operation of the composite transmission and generation power system, so the risk level of the composite system with renewable energy attached must be studied. However, the accuracy of the renewable energy model has typically been overlooked in current risk assessment approaches. Therefore, this study provides an accurate hybrid wind/solar power output model that takes the wind farm’s seasonal and spatial characteristics, as well as the correlation between wind and solar PV power into account. For the first, build the seasonal Weibull model of the wind speed and correct the speed captured by each wind turbine considering different spatial characteristics of the wind farm. Secondly, build the probability distribution function of the solar power based on the kernel density estimation method. In the next, construct the joint probability distribution function considering the correlation between wind and solar PV power based on copula theory. Then, using the suggested accurate renewable energy model, the sequential Cross-entropy (SCE) technique is utilized to evaluate the risk indices of the composite system with renewable energy. To validate the models and technique, the modified IEEE-RTS79 is studied. The findings indicate that the accuracy of the renewable energy model influences the risk assessment outcomes.

Turbulence spectra in natural and forced convection

Fourier spectra have been traditionally used to get insight into the flow structures of turbulence. In wall-bounded flows these are not reported usually in Kolmogorov units. In this paper the spectra are evaluated in these units to compare with the case of isotropic turbulence, to investigate at which distance from the wall flow isotropy is achieved. We have found that improvement of universality of the spectra in the near-wall region is achieved by evaluating the dissipative length scale from the total turbulence kinetic energy dissipation, rather than through the rate of isotropic dissipation as customary. The database of Lee and Moser [1] for flow in plane channel and of Pirozzoli et al. [2] for flow in a circular pipe have been considered. We have found similarities between pipe and channel flow up to a non-dimensional distance 0.2. Farther from the wall the spectra tend to become universal and similar to isotropic turbulence in channels, whereas the geometry of the circular pipe prevents universality. Large part of the paper is also focused on turbulent natural convection in rectangular enclosures. In that flow, comparison between velocity and temperature spectra allows to answer about possible differences in the slope of the inertial ranges. In this respect, the importance of the spectral bottleneck in the cascade of kinetic and thermal energy is stressed. The large amplitude of the bottleneck in the temperature spectra might be the cause for the k−4/3 spectral range observed at wall distances greater than the thermal boundary layer thickness, defined in terms of the maximum turbulent thermal energy production. Also in this flow evaluating the Kolmogorov length scale through the total dissipation yields better collapse of the spectral range with exponential decay on that of isotropic turbulence in the near-wall region. Flow visualisations and joint PDFs are used for deeper understanding of the complex physics of turbulent convection.

Projections of offshore wind energy and wave climate in Guangdong’s nearshore area using CMIP6 simulations

A good knowledge of future coastal wind and wave resources in the context of climate change is crucial for the construction of offshore wind farms. In this study, the dataset of the coupled model intercomparison project phase 6 (CMIP6) was used to evaluate the future wind resources and wave conditions in the nearshore area of Guangdong of China. The long short-term memory (LSTM) algorithm was used to develop a statistical downscaling method to render high spatial resolution data. The Copula function was used to construct the joint probability distribution function. The key findings were as follows. First, over the whole Guangdong coastal area, the projection of wind speed (<i>H</i>_s) shows a generally increasing trend, the wave height (<i>W</i>_s) remains almost unchanged, and the growth is particularly pronounced in the western area. Second, it is clear that the joint probability distribution is less spread in the late 21^st century; therefore, the distribution of wind energy will also be more suitable for the optimal operating of wind turbines. Third, the joint probability distribution results show that large wind and wave conditions possibly occur at the same time, which must be considered when determining worst-case conditions. Future work is required to use more models and scenarios from the ongoing CMIP6.

Effect of Co- and Counterswirl Air on Swirl Airblast Atomization

For swirl-stabilized gas turbine combustor, liquid fuel distribution in the near field dictates local equivalence ratio, volumetric heat release, and heat transfer to the chamber wall, and hence its understanding is essential. The effect of inlet air aerodynamics on spray characteristics in a primary zone of a simulated gas turbine burner is studied using a phase Doppler particle analyzer, high-speed Mie scatter imaging, and an orthogonal decomposition method. By employing intense coswirl air, the luminous spray region shifts upstream to the burner exit, where lower mass flux can be observed in the central region and higher mass flux in the outer region of the spray due to the recirculation zone formation. Based on the size velocity joint probability distribution functions (JPDFs) and the individual droplet transport with acquisition time, we conclude that the recirculation zone entraps the smaller droplets and transports them from the downstream to the upstream spray region. Compared to coswirl, counterswirl air exhibits torsion instability, intensifies the concentration of drops in the central region, and improves secondary atomization. Finally, five distribution functions are curve-fitted to the experimental data to capture the atomization process accurately.

Risk Field of Rock Instability using Microseismic Monitoringdata in Deep Mining.

With the gradual depletion of surface resources, rock instability caused by deep high stressand mining disturbance seriously affects safe mining. To create effective risk management, a rockinstability risk field model using microseismic monitoring data is proposed in this study. Rockinstability risk was presented visually in 3D visualization. The in-situ microseismic monitoringdata was collected and analyzed to make calculation of peak ground velocity (PGV), peak groundacceleration (PGA), energy flux, energy and seismic moment. Indicator weights of PGV, PGA, energyflux are confirmed by using the analytic hierarchy process (AHP) to calculate risk severity. The Copulafunction is then used to solve the joint probability distribution function of energy and seismic moment.Then the spatial distribution characteristics of risk can be obtained by data fitting. Subsequently, thethree-dimensional (3D) risk field model was established. Meanwhile, the established risk field isverified by comparing monitoring data without disturbance and the blasting data with disturbance.It is suggested that the proposed risk field method could evaluate the regional risk of rock instabilityreasonably and accurately, which lays a theoretical foundation for the risk prediction and managementof rock instability in deep mining.

Topology of Magnetic and Velocity Fields at Kinetic Scales in Incompressible Plasma Turbulence

AbstractThe topology of the magnetic and velocity fields at the kinetic scales are investigated in the context of nearly incompressible magnetosheath plasma turbulence. Using the unprecedented high‐resolution data from the Magnetospheric MultiScale mission, the joint probability distribution functions of geometrical invariants characterizing the magnetic and velocity fields gradient tensor at the kinetic scales are computed. The topological features of the magnetic and velocity field gradient tensors and their symmetric component tensors present axisymmetric distribution patterns, indicating that the structure of the plasma turbulence at the kinetic scales are different from those in hydrodynamic and magnetohydrodynamic turbulence. Moreover, a strong correlation between the straining and rotational parts of the magnetic and velocity field gradient tensors was found, which manifests the dominance of sheet‐like structures at the kinetic‐scales dissipation in incompressible plasma turbulence.