Probability Density Function Research Articles

Comparative study of turbulent round jet flows through direct numerical simulation at medium–high Reynolds numbers

This study examines a spatially evolving turbulent round jet at a Reynolds number of Re = 7000 based on the bulk velocity Ub and the orifice diameter D and a Prandtl number of Pr = 0.71 using direct numerical simulation (DNS). Statistical data are collected over 30 000D/Ub time units, including mean quantities, Reynolds stresses, heat fluxes, mixed moments, Reynolds stress and turbulent kinetic energy budgets, and probability density functions. Comparative analysis with a previous DNS at Re = 3500 (Nguyen and Oberlack, 2024a; 2024c) shows that the higher Reynolds number leads to a shift of the self-similarity to a greater distance from the orifice. A larger Reynolds number leads to higher magnitudes of different statistical moments, dissipation and production. In addition, a maximum visible in the near-field of the turbulence intensity components is smoothed at the larger Reynolds number. The probability density functions of the axial velocity and the passive scalar show Gaussian behavior on the centerline, but with larger standard deviations compared to the lower Reynolds number. The study highlights the significant impact of using a fully developed turbulent pipe flow profile as the inlet condition, which significantly reduces the size of the potential core by introducing initial turbulence intensities at the orifice.

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

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

Design considerations for a continuous mussel farm in New England Offshore waters. Part I: Development of environmental conditions for engineering design

Sustainable aquaculture in nearshore waters faces challenges such as stakeholder conflicts, environmental pollution, and spatial constraints. Offshore aquaculture offers a promising solution but requires robust engineering design to withstand extreme weather conditions. This study develops the environmental conditions essential for engineering the design of a continuous mussel dropper system in New England offshore waters. A potential farm location was identified using criteria including water depth, federal boundaries, seafloor suitability, farm size, and proximity to ports, based on bathymetric and sedimentary maps. Historical data from five wave monitoring stations and two current velocity stations were analyzed to model extreme environmental conditions, as waves and currents pose primary threats. The extreme wave and current conditions are modeled using the Weibull distribution, on the annual maximum hourly significant wave height data and the largest 0.3% of current speeds. A newly proposed method combining Spalding’s wall function with a fourth-order polynomial is used to enhance the current profile analysis. Additionally, a joint probability density function was developed for wave height and current velocity at a specific depth, providing insights into wave height, period, and wavelength for various return periods such as 10, 25, 50, and 100 years. The results suggest a 10-yr wave of 8 m significant wave height and a current speed of 1.68 m/s, while a 50-yr values are 9.4 m and 1.96 m/s respectively. These findings offer critical data on extreme wave and current conditions in New England’s offshore waters, providing practical guidance for the engineering design of offshore mussel farms. This research advances offshore mussel farming and benefits the development of all types of offshore aquaculture systems.

Analysis of a stochastic two-species predator-prey system in two-patch environments with Ornstein-Uhlenbeck process

To characterize the effects of diffusion and environmental noise on population dynamics, in the paper, we first develop a stochastic two-species predator-prey model in two-patch environments, where the fluctuations in the environment are depicted by an Ornstein-Uhlenbeck process. Then we analyze the dynamical behavior of the stochastic model in detail, including the existence and uniqueness of the global solution, the pth moment boundedness, asymptotic pathwise estimation and the existence of a stationary distribution. It is worth noting that by solving the six-dimensional algebraic equations corresponding to the stochastic model, we get the approximate expression of the probability density function around the quasi-coexistence equilibrium of the stochastic system. Finally, a series of numerical simulations are presented to confirm the feasibility and correctness of our theoretical analysis results. Our findings show the importance of considering the effect of this type of stochastic process on the population dynamics.

Asymptotic approximations for the distribution of the product of correlated normal random variables

We obtain asymptotic approximations for the probability density function of the product of two correlated normal random variables with non-zero means and arbitrary variances. As a consequence, we deduce asymptotic approximations for the tail probabilities and quantile functions of this distribution, as well as an asymptotic approximation for the widely used risk measures value at risk and tail value at risk.

Dimensional analysis of diffusive association rate equations

Diffusive adsorption/association is a fundamental step in almost all chemical reactions in diluted solutions, such as organic synthesis, polymerization, self-assembly, biomolecular interactions, electrode dynamics, catalysis, chromatography, air and water environmental dynamics, and social and market dynamics. However, predicting the rate of such a reaction is challenging using the equations established over 100 years ago. Several orders of magnitude differences between the theoretical predictions and experimental measurements for various systems, from self-assembled monolayers to protein-protein aggregations, make such calculations meaningless in many situations. I believe the major problem is that the time-dependent evolution curve of Fick’s gradient is an ideal assumption in most cases, and its slope is significantly overestimated. This paper digs into Fick’s gradient problem for 3D cases and provides a solution using the single-molecule diffusion probability density function discretely.

Quantification of contaminant mass discharge and uncertainties: Method and challenges in application at contaminated sites

Contaminant mass discharge (CMD) estimation involves combining multilevel concentration and flow measurements to quantify the contaminant mass passing through a control plane downgradient of a point source. However, geological heterogeneities and limited data introduce uncertainties that complicate CMD estimation and risk assessment. Although CMD is increasingly used in groundwater management, methods for quantifying and handling these uncertainties are still needed. This study develops and tests a CMD estimation method based on Bayesian geostatistics to quantify CMD uncertainties using data from a control plane perpendicular to the contaminant plume.By combining geostatistical conditional simulations of the spatial concentration distribution with the flow, an ensemble of CMD realizations is generated, from which a cumulative distribution function is derived. A key element of this approach is the use of a macrodispersive transport model to simulate the spatial concentration trend. This ensures that the estimated concentration reflects the expected physical behavior of the contaminant plume while also allowing the integration of site-specific conceptual information.The method is applicable to plumes with dissolved contaminants, such as chlorinated solvents, petroleum hydrocarbons, Per- and polyfluoroalkyl substances (PFAS) and pesticides. Site-specific conceptual understanding is used to inform the prior probability density functions of the structural model parameters and to define acceptable simulated concentration limits. We applied the method at three sites contaminated with chlorinated ethenes, demonstrating its robustness across varying information levels and data availability.Our results shows that strong site-specific conceptual knowledge and high sampling density constrain the CMD uncertainty (CV = 21 %) and results in estimated model parameters and a spatial concentration distribution that agrees well with the conceptual model. For a site with less data and limited conceptual knowledge, CMD and concentration distribution estimates are still feasible, though with higher uncertainty (CV = 41 %). Extending the method to account for multiple source zones and complex plume migration improved parameter identification and reduced the 95 % CMD confidence interval by 11 % ([4950–8750] to [5090–8480] g yr−1), while also providing a spatial concentration distribution in better agreement with the plume conceptualization.This study highlights the importance of integrating site-specific conceptual knowledge in CMD estimation, particularly for less-sampled sites. The method can furthermore assist in identifying remediation targets, evaluating remedial effectiveness, and optimizing sampling strategies.

Stochastic dynamics analysis for unilateral vibro-impact systems under combined excitation

Vibro-impact system, as an important type of non-smooth system, exhibits intricately nonlinear characteristics. Inevitably, the vibro-impact system will encounter random excitations, but the conventional methods ar7e not eligible for simultaneous determination of its transient responses and reliabilities. Commonly, existing methods of applying non-smooth transformation tend to ignore the essential non-smooth characteristics of vibro-impact system. To this end, this paper proposes a unified framework based on direct probability integral method (DPIM) to simultaneously determine stochastic dynamic responses and reliabilities of unilateral vibro-impact systems under combined harmonic and random excitation without non-smooth transformation, and captures their complicated dynamical behaviors. Firstly, the impact velocity dependent coefficient of restitution is introduced to establish the motion equation of vibro-impact system. Secondly, the probability density integral equation (PDIE) for the unilateral vibro-impact system is derived from the perspective of probability conservation. Then, the PDIE and governing differential equation of the system is solved in a decoupled and efficient way. Moreover, the first-passage reliability is assessed by introducing extreme value mapping of the stochastic dynamic response. Numerical results of three typical examples using the proposed framework are compared with those using Monte Carlo simulation (MCS), quasi-MCS and from the reference, which highlights the advantages of DPIM in computing the stochastic responses and reliabilities of vibro-impact system under random excitations and random parameters. The stationary probability density functions exhibit periodic fluctuations under combined harmonic and stochastic excitation. Specially, the noise intensity and frequency of harmonic excitation pose the great influence on the reliabilities of systems.

Deep feature response discriminative calibration

Deep neural networks (DNNs) have numerous applications across various domains. Several optimization techniques, such as ResNet and SENet, have been proposed to improve model accuracy. These techniques improve the model performance by adjusting or calibrating feature responses according to a uniform standard. However, they lack the discriminative calibration for different features, thereby introducing limitations in the model output. Therefore, we propose a method that discriminatively calibrates feature responses. The preliminary experimental results indicate that the neural feature response follows a Gaussian distribution. Consequently, we compute confidence values by employing the Gaussian probability density function, and then integrate these values with the original response values. The objective of this integration is to improve the feature discriminability of the neural feature response. Based on the calibration values, we propose a plugin-based calibration module incorporated into a modified ResNet architecture, termed Response Calibration Networks (ResCNet). Extensive experiments on datasets like CIFAR-10, CIFAR-100, SVHN, and ImageNet demonstrate the effectiveness of the proposed approach.

Non-Gaussian tails without stochastic inflation

We show, both analytically andnumerically,thatnon-Gaussian tails in the probability density function of curvature perturbations arise in ultra-slow-roll inflation from the δN formalism, without invoking stochastic inflation. Previously reported discrepancies between both approaches are a consequence of not correctly accounting for momentum perturbations. Once they are taken into account, both approaches agree to an excellent degree. The shape of the tail depends strongly on the phase space of inflation.

Zero-carbon-oriented optimal power scheduling of electric vehicles considering V2G response

Abstract This paper presents an electric vehicle (EV) optimization scheduling method that incorporates vehicle-to-grid (V2G) responses within the framework of a carbon market. Initially, the method characterizes the grid-connected and off-grid operational states of EVs. These states are represented by three types of probability density functions derived from historical data. Subsequently, a V2G response model is formulated based on consumer psychology theory. The approach then integrates the impact of carbon trading on the charging and discharging schedules of EVs, aiming to minimize carbon trading costs. The resulting optimization scheduling model is solved using the CPLEX solver.

Study on the pore structure of eco-regenerated mortar using corn cob based on nuclear magnetic resonance

This study aims to investigate the influence of corn cob particles on pore structure of mortar. By using NMR technology, the pore parameters of fast-hardening sulfoaluminate cement mortar (SAC) and ordinary Portland cement mortar (PO) were measured and compared. The influence of corncob particles on the mechanical properties of eco-mortar was evaluated by uniaxial compression test. The results indicate that medium and large pores of both SAC and PO increases with corn cob particle, as well as the porosity. The fractal dimensions of harmful and multiple harmful pores are negatively correlated with the volume fraction of corn cob particles. The addition of corn cob particles reduced the overall fractal dimension of the mortar and decreased the complexity of internal pores. The probability density function of grayscale values transitions from being “thin and tall” to “short and wide”, with an increasing proportion of high grayscale values. This leads to an increase in the proportion of large pores within the mortar, resulting in a continuous deterioration of pore structure. The addition of corn cob particles greatly reduces the strength of SAC and PO, but the peak stress of SAC is higher than that of PO. When corn cob particle content is 30wt% and 50wt%, the peak stress of SAC is increased by 23.37% and 14.25%, respectively, compared with the peak stress of PO.

A conditional deep learning model for super-resolution reconstruction of small-scale turbulent structures in particle-Laden flows

Super-resolution reconstruction of turbulent flows using deep learning has gained significant attention, yet challenges remain in accurately capturing physical small-scale structures. This study introduces the Conditional Enhanced Super-Resolution Generative Adversarial Network (CESRGAN) for reconstructing high-resolution turbulent velocity fields from low-resolution inputs. CESRGAN consists of a conditional discriminator and a conditional generator, the latter being called CoGEN. CoGEN incorporates subgrid-scale (SGS) turbulence kinetic energy as conditional information, improving the recovery of small-scale turbulent structures with the desired level of energy. By being aware of SGS turbulence kinetic energy, CoGEN is relatively insensitive to the degree of detail in the input. As shown in the paper, its advantages become more pronounced when the model is applied to heavily filtered input. We evaluate the model using direct numerical simulation (DNS) data of forced homogeneous isotropic turbulence. The analysis of Q-criterion isosurfaces, energy spectra, and probability density functions shows that the proposed CoGEN reconstructs fine-scale vortical structures more precisely and captures turbulent intermittency better compared to the traditional generator. Particle-pair dispersion simulations validate the physical fidelity of CoGEN-reconstructed fields, closely matching DNS results across various Stokes numbers and filtering levels. This paper demonstrates how incorporating available physical information enhances super-resolution models for turbulent flows.

The Probability Density Function Based Neuro-Fuzzy Wind Power Prediction With Global Convergence

The Probability Density Function Based Neuro-Fuzzy Wind Power Prediction With Global Convergence

Computational Prediction of Co-firing with Various Biomass Waste Using Turbulent Non-Premixed Combustion

Co-firing in coal power plants has limitations because the existing combustion systems are designed to provide optimal performance only with coal. Therefore, investigating the combustion aspects of co-firing by mixing coal with biomass before applying it to existing coal power plants is necessary. To address this, a new numerical model was developed to predict the co-firing behavior of coal with various types of biomass waste, specifically focusing on temperature and pollutant behavior. This study developed a co-firing model in a Drop Tube Furnace (DTF) using a composition of 25% Wood Chips (WC), 25% Solid Recovered Fuel (SRF), 25% Empty Fruit Bunch Fibers (EFFR), and 25% Rice Husk (RH). A structured grid arrangement and the Probability Density Function (PDF) were utilized to depict the relationship between chemical combustion and turbulence. The distributions of temperature and mass fractions of pollutants along the furnace axis were predicted. The highest temperature was observed with 25% EFFR, attributed to its highest volatile matter content. The simulation predicted that 25% RH would be the lowest SO2 emitter. However, it also showed a slight increase in NO and CO levels due to the increased oxygen content when coal was mixed with biomass. The simulation with 25% EFFR predicted a decrease in CO2 emissions compared to other biomass types. The results of this parametric investigation could support the implementation of biomass co-firing technology in existing coal-fired power plants.

Research on the P‐S‐N Curve Fitting Method of Notched Specimens Considering Small Sample Properties

ABSTRACTAiming at the issue of fatigue test data for large‐scale mechanical components of building steel are very limited, a method for fitting P‐S‐N curves under small sample data of notched specimens is proposed to predict fatigue life. First, a fatigue life subsample augmented and its reliability assessment method are established, based on Bayesian hierarchical modeling and modified Monte Carlo method. Second, a clustering combination weighting method is proposed, to define weights of hidden variables of the binomial mixture Weibull distribution, and the expectation–maximization algorithm is used to determine probability density function of the distribution. Finally, the P‐S‐N curves under various failure probabilities are fitted with Weibull distributed life models, and the convergence and prediction accuracy of the different models are compared. The results show that the fatigue data of small samples can be predicted better by using mixed Weibull distribution, and the fitting P‐S‐N curve is more reliable and accurate.

Quasiperiodic γ-Ray Modulations in the Blazars PKS 2155-83 and PKS 2255-282

While there has been an increase in interest in the possibility of quasiperiodic oscillations (QPOs) in blazars, the search has hitherto been restricted to sources with well-sampled light curves. Objects with light curves that include gaps have been, to our knowledge, overlooked. Here, we study two such curves, which have the interesting feature of pertaining to relatively high-redshift blazars—FSRQs, PKS 2155-83, and PKS 2255-282—observed by the Fermi Large Area Telescope. Their redshifts border the “cosmic noon” era of galaxy formation and merging, and their light curves exhibit a distinctive pattern of repetitive high and low (gap dominant) states for 15.6 yr. To accommodate for the gaps in the curves, data are integrated over extended time intervals of 1 month and 2 months. The resulting curves were also examined using methods suitable for sparsely sampled data. This investigation of PKS 2155-83 and PKS 2255-282 suggests QPOs with periods of 4.69 ± 0.79 yr (3σ) and 6.82 ± 2.25 yr (2.8σ), respectively. The probability density functions of the blazars’ fluxes, along with the correlation between their flux and spectral index, were also analyzed. Given the epochs in which the objects are observed, the plausibility of a binary black hole scenario as an origin of the apparent periodicity was examined. We estimated the prospective parameters of such a system using a simple geometric model. The total masses were estimated and found to be consistent, in principle, with independent (dynamical) measurements of the central black hole masses in the two host galaxies.

Theoretical wavelet ℓ1-norm from one-point probability density function prediction

Context. Weak gravitational lensing, which results from the bending of light by matter along the line of sight, is a potent tool for exploring large-scale structures, particularly in quantifying non-Gaussianities. It is a pivotal objective for upcoming surveys. In the realm of current and forthcoming full-sky weak-lensing surveys, convergence maps, which represent a line-of-sight integration of the matter density field up to the source redshift, facilitate field-level inference. This provides an advantageous avenue for cosmological exploration. Traditional two-point statistics fall short of capturing non-Gaussianities, necessitating the use of higher-order statistics to extract this crucial information. Among the various available higher-order statistics, the wavelet ℓ1 -norm has proven its efficiency in inferring cosmology. However, the lack of a robust theoretical framework mandates reliance on simulations, which demand substantial resources and time. Aims. Our novel approach introduces a theoretical prediction of the wavelet ℓ1-norm for weak-lensing convergence maps that is grounded in the principles of large-deviation theory. This method builds upon recent work and offers a theoretical prescription for an aperture mass one-point probability density function. Methods. We present for the first time a theoretical prediction of the wavelet ℓ1-norm for convergence maps that is derived from the theoretical prediction of their one-point probability distribution. Additionally, we explored the cosmological dependence of this prediction and validated the results on simulations. Results. A comparison of our predicted wavelet ℓ1 -norm with simulations demonstrates a high level of accuracy in the weakly nonlinear regime. Moreover, we show its ability to capture cosmological dependence. This paves the way for a more robust and efficient parameter-inference process.

Probabilistic Prediction and Assessment of Train-Induced Vibrations Based on Mixture Density Model

This study presents a probabilistic prediction method for train-induced vibrations by combining a deep neural network (DNN) with the mixture density model in a cascade fashion, referred to as the DNN-RMDN model in this paper. A benchmark example is conducted to demonstrate and evaluate the prediction performance of the DNN-RMDN model. Subsequently, the model is applied to a case study to investigate and compare the uncertainties of train-induced vibrations in the throat area and testing line area of a metro depot. After training, the model is capable of accurately predicting the probability density function (PDF) of train-induced vibrations at different distances from the track and at different frequencies. Utilizing the predicted PDF, probabilistic assessments can be performed to ascertain the likelihood of surpassing predefined limits. By employing a mixture density model instead of a single Gaussian distribution, the DNN-RMDN model achieves more accurate prediction of the PDF for train-induced vibrations. The proposed probabilistic assessment framework can effectively assist in vibration screening during the planning phase and in selecting and designing vibration mitigation measures of appropriate levels.

A deconstruction of methods to derive one-point lensing statistics

Gravitational lensing is a crucial tool for exploring cosmic phenomena, providing insights into galaxy clustering, dark matter, and dark energy. Given the substantial computational demands of N-body simulations, approximate methods like 𝙿𝙸𝙽𝙾𝙲𝙲𝙷𝙸𝙾 and 𝚝𝚞𝚛𝚋𝚘𝙶𝙻 have been proposed as viable alternatives for simulating lensing probability density functions (PDFs). This paper evaluates these methods and their effectiveness across both weak and strong lensing regimes, with a focus in the context where baryonic effects are negligible. Our comparative analysis reveals that these methods are effective for applications where lensing is mild, such as the majority of sources of electromagnetic and gravitational waves. However, both 𝙿𝙸𝙽𝙾𝙲𝙲𝙷𝙸𝙾 and 𝚝𝚞𝚛𝚋𝚘𝙶𝙻 break down for large values of convergence and magnification due to their loss of accuracy in capturing small-scale nonlinear matter fields, owing to oversimplified assumptions about internal halo structures and reliance on perturbation theory. 𝙿𝙸𝙽𝙾𝙲𝙲𝙷𝙸𝙾 yields second-to-fourth moments of the lensing PDFs, which are 6–10% smaller than those resulting from N-body simulations in regimes where baryonic effects are minimal. These findings aim to inform future studies on gravitational lensing of point sources, which are increasingly relevant with upcoming supernova and gravitational wave datasets.