Probability Density Function Research Articles

Optimal solution of multiobjective stable environmental economic power dispatch problem considering probabilistic wind and solar PV generation

The Environmental Economic Power Dispatch (EEPD) problem, a widely studied bi-objective nonlinear optimization challenge in power systems, traditionally focuses on the economic dispatch of thermal generators without considering network security constraints. However, environmental sustainability necessitates reducing emissions and increasing the penetration of renewable energy sources (RES) into the electrical grid. The integration of high levels of RES, such as wind and solar PV, introduces stability issues due to their uncertain and intermittent nature. This paper addresses these concerns by formulating and solving the Stable Environmental Economic Power Dispatch (SEEPD) problem, which includes fixed zonal reserve capacity from conventional thermal generators and uncertain reserves from RES. Uncertainties in RES and load demand are modeled using random variable generation techniques, applying Gaussian, Weibull, and log-normal probability density functions (PDFs) for load demand, wind velocity, and solar irradiance, respectively. The stochastic SEEPD problem extends to multiple periods by replicating the single-period problem for each interval in the planning horizon, linking periods through intertemporal ramping costs, physical ramp rate, and fixed zonal reserve constraints on dispatch variables. Multi-Objective Evolutionary Algorithms (MOEAs) have gained importance for solving complex nonlinear problems involving multi-objective functions. This paper applies the latest MOEAs to tackle the proposed SEEPD problem, incorporating stochastic wind and solar PV power sources. Network security constraints, such as transmission line capacities and bus voltage limits, are considered along with constraints on generator capabilities and intertemporal spinning reserves, ramp-up and ramp-down constraints for thermal generators. A bidirectional coevolutionary-based multi-objective evolutionary algorithm is employed, integrating an advanced constraint-handling technique to ensure compliance with system constraints. The simulation results show that the proposed formulation achieves a better trade-off between various conflicting objective functions compared to other state-of-the-art MOEAs.

Filling in the blanks

Context. In the solar neighbourhood, only ∼2% of stars in the Gaia survey have a line-of-sight velocity (vlos) contained within the RVS catalogue. These limitations restrict conventional dynamical analysis, such as finding and studying substructures in the stellar halo. Aims. We aim to present and test a method to infer a probability density function (PDF) for the missing vlos of a star with 5D information within 2.5 kpc. This technique also allows us to infer the probability that a 5D star is associated with the Milky Way’s stellar Disc or the stellar Halo, which can be further decomposed into known stellar substructures. Methods. We use stars from the Gaia DR3 RVS catalogue to describe the local orbital structure in action space. The method is tested on a 6D Gaia DR3 RVS sample and a 6D Gaia sample crossmatched to ground-based spectroscopic surveys, stripped of their true vlos. The stars predicted vlos, membership probabilities, and inferred structure properties are then compared to the true 6D equivalents, allowing the method’s accuracy and limitations to be studied in detail. Results. Our predicted vlos PDFs are statistically consistent with the true vlos, with accurate uncertainties. We find that the vlos of Disc stars can be well-constrained, with a median uncertainty of 26 km s−1. Halo stars are typically less well-constrained with a median uncertainty of 72 km s−1, but those found likely to belong to Halo substructures can be better constrained. The dynamical properties of the total sample and subgroups, such as distributions of integrals of motion and velocities, are also accurately recovered. The group membership probabilities are statistically consistent with our initial labelling, allowing high-quality sets to be selected from 5D samples by choosing a trade-off between higher expected purity and decreasing expected completeness. Conclusions. We have developed a method to estimate 5D stars’ vlos and substructure membership. We have demonstrated that it is possible to find likely substructure members and statistically infer the group’s dynamical properties.

Nonlinear coupled asymmetric stochastic resonance for weak signal detection based on intelligent algorithm optimization

Stochastic resonance has been extensively studied for detecting weak signals. To improve the diagnostic ability of weak signals, a novel nonlinear coupled asymmetric stochastic resonance (NCASR) system is investigated in this paper. Firstly, the NCASR system is established by coupling the asymmetric bistable system with the monostable system. Next, the expressions for the steady-state probability density (SPD) function, the mean first passage time (MFPT) and the signal-to-noise ratio (SNR) of the proposed system are derived based on the adiabatic approximation theory. Furthermore, the impact of system parameters on the SPD, the MFPT and the SNR is analyzed. Then, by simulation experiments, we verify the effectiveness of detecting weak signals for the NCASR system with Lévy noise. Finally, the NCASR system optimized by Adaptive Weighted Particle Swarm Optimization (AWPSO) algorithm is applied to detect the bearing fault signal. Compared with the optimized classical bistable stochastic resonance (CBSR) system, it is found that the detection performance of the NCASR system is superior to the CBSR system in detecting bearing fault signals.

Approximate probability density function for nonlinear surging in irregular following seas

Approximate probability density function for nonlinear surging in irregular following seas

Dynamic uncertainty evaluation of cylindricity error based on Bayesian deep neural network propagation method

Most of the existing methods for measuring and evaluating product geometric tolerances are for static processes, which use lower dimensions, are more complex to calculate, and have poor traceability and dynamic manageability and control of the measurement process. In this paper, a dynamic evaluation model of cylindricity error uncertainty based on the Bayesian deep neural network propagation method is proposed. Firstly, the input random variable sampling sample containing error uncertainty is generated by proposing the improved Monte Carlo sampling method (IMCS), which has the characteristics of homogeneity and validity; then the principle and construction process of a Bayesian neural network model are introduced to generate the propagation model of multidimensional random error variables; further, the Gaussian mixture model is trained by the Monte Carlo method, and the probability density function of the target response is generated by using the Gaussian mixture model. Through numerical experiments, test analysis of bearing outer rings, and comparison with ISO standard methods, the results show that the model based on the dynamic Bayesian deep neural network propagation method can correctly and effectively assess the uncertainty dynamically, and the whole estimation process is stable.

Turbulence scale and strength analysis in the roughness and inertial sublayers over urban areas: A wind tunnel study

The dissimilarity of dynamics and turbulence structures between the roughness and inertial sublayers (RSL, ISL) over roughness elements show that ISL turbulence is much more homogeneous. Conventional logarithmic law of the wall is merely applicable for the RSL mean-wind-speed profiles. To characterize the turbulence scales and elucidate the mechanism, the flows in the atmospheric surface layer (ASL) over real urban morphology are measured in wind tunnel experiments fabricated by 3D-printing, reduced-scale model of downtown Kowloon Peninsula, Hong Kong. Elevated dispersive stress at urban canopy signifies the influence from individual buildings and inhomogeneous RSL flows. A positive momentum contribution appears in the low-frequency regime. The large-scale turbulence in RSL thus enhances the mixing and transport, resulting in inhomogeneous flows. The contribution from ejection Q2 and sweep Q4 increases and decreases, respectively, with increasing RSL motion strength. The conventional −5/3 law is observed by empirical mode decomposition (EMD) and the largest amplitude fluctuation occurs more often when the turbulence length scale is comparable to the turbulent boundary layer (TBL) thickness. The single-point amplitude modulation (AM) shows that the large- and small-scale turbulence correlates tightly at the bottom of RSL. Besides, the joint probability density function (JPDF) illustrates that accelerating large scales often occur with decelerating small scales, and they are intensified with increasing wall-normal distance. As a result, large-scale turbulence influences substantially the flow structures over urban areas and the small-scale turbulence (even) in RSLs in the vicinity of building obstacles.

A Framework for Incorporating Binding Energy Distribution in Gas-ice Astrochemical Models

Abstract One of the most serious limitations of current astrochemical models with the rate equation (RE) approach is that only a single type of binding site is considered in grain surface chemistry, although laboratory and quantum chemical studies have found that surfaces contain various binding sites with different potential energy depths. When various sites exist, adsorbed species can be trapped in deep potential sites, increasing the resident time on the surface. On the other hand, adsorbed species can be populated in shallow sites, activating thermal hopping and thus two-body reactions even at low temperatures, where the thermal hopping from deeper sites is not activated. Such behavior cannot be described by the conventional RE approach. In this work, I present a framework for incorporating various binding sites (i.e., binding energy distribution) in gas-ice astrochemical models as an extension of the conventional RE approach. I propose a simple method to estimate the probability density function (pdf) for the occupation of various sites by adsorbed species, assuming a quasi-steady state. By using thermal desorption and hopping rates weighted by the pdfs, the effect of binding energy distribution is incorporated into the RE approach without increasing the number of ordinary differential equations to be solved. This method is found to be accurate and computationally efficient, and enables us to consider binding energy distribution even for a large gas-ice chemical network which contains hundreds of icy species. The impact of the binding energy distribution on interstellar ice composition is discussed quantitatively for the first time.

Stochastic stability of random stacking of blocks

The paper explores the stability of a tower obtained by stacking identical rectangular blocks on top of each other with an inherent randomness due to slight positional offsets between successive blocks. With probabilistic modeling techniques, the diffusive behavior of the stacking process is studied and the collapse is seen as a first passage time problem. In the considered model, alignment errors are idealized as independent Gaussian random variables with zero mean and given standard deviation. We derive expressions for the joint probability density functions of block positions, and analyze their correlation. The study extends to the stochastic stability of a stack of given height, by exploring the statistical characteristics of the center of gravity of the part of tower above each block. Eventually, the probabilistic analysis of collapse is developed to quantify the statistics of the number of blocks that can be heaped up before the tower topples. Although this problem may initially appear playful, it offers an illustrated introduction to first passage problems on a non homogenous process. From a practical standpoint, this analysis offers a simple understanding of the influence of alignment errors on the overall stability of a stack, which may find several fields of application.

A perspective on conditional spectrum-based determination of response statistics of nonlinear systems

This work focuses on determining the stochastic response properties, in the frequency domain, of a general class of nonlinear systems with polynomial nonlinearities. Specifically, the results are presented in terms of the stationary power spectral densities of the system’s displacement and velocity. This is pursued by revisiting the conditional power spectrum concept, with the assumption that the response process is both ergodic and pseudo-harmonic and characterized by an amplitude, and a phase. A theoretical elucidation of an existing formula for the conditional spectrum is attempted. In particular, this concept is interpreted in conjunction with the time averaging approximation made in the definition of the stationary probability density function of a response amplitude quantity, associated with the original nonlinear system. It is shown that a proper definition of the stationary probability density of the response amplitude, along with a reasonable treatment of the distribution over the frequency domain of the amplitude contribution, lead to an improved approximation of the stationary response power spectral density. The treatment involves the averaging of a population of surrogate spectral densities of stationary random responses conforming with the system responses associated with individual values of the amplitudes of the responses. The semi-analytical results have been quite favourably juxtaposed with a large suite of à propos Monte Carlo simulations, both in terms of the shape and of the range of the involved germane frequencies, even for strongly nonlinear systems.

Fokker–Planck equation and Feynman–Kac formula for multidimensional stochastic dynamical systems with Lévy noises and time-dependent coefficients

The aim of this paper is to establish a version of the Feynman–Kac formula for the time-dependent multidimensional nonlocal Fokker–Planck equation corresponding to a class of time-dependent stochastic differential equations driven by multiplicative symmetric (or asymmetric) α-stable Lévy noise. First the forward nonlocal Fokker–Planck equation is derived by the adjoint operator method, overcoming the challenges posed by time-dependent multidimensional nonlinear symmetric α-stable Lévy noise. Subsequently, the Feynman–Kac formula for the forward multidimensional time-dependent nonlocal Fokker–Planck equation is established by applying techniques for the backward nonlocal Fokker–Planck equations, which is associated with the backward stochastic differential equation driven by the multiplicative symmetric α-stable Lévy noise. Notably, in the case of asymmetric α-stable Lévy noise case, the presence of the characteristic function in the nonlocal operator adds complexity to the analysis. Using the Feynman–Kac formula, it is demonstrated that the solution of the forward nonlocal Fokker–Planck equation can be readily simulated through Monte Carlo approximation, especially in scenarios involving long-time simulation settings with large steps. These concepts are illustrated with intriguing examples, and the dynamic evolution of the probability density function corresponding to the stochastic SIS model and the stochastic model for the MeKS network (reflecting the interactions among the MecA complex, ComK and ComS) are investigated over an extended period.

A Nonlinear Filter based on GSK for Relative Navigation Using Relative Orbital Elements

Due to the sensitivity of the relative orbital elements model to the measurement noise, the non-stationary heavy-tailed noise(NSHT) induced by the time-varying environment during the relative navigation usually leads to filter divergence. To address this problem, a new nonlinear filter based on Gaussian-Student's-Multivariate K(GSK) mixture distribution is proposed in this paper. A Dirichlet stochastic mixture vector fusing Gaussian, Student's t, and Multivariate K distributions is introduced, thus proposing a GSK mixture distribution modeling measurement likelihood; then the Kullback-Leibler Divergence (KLD) of the true posteriori probability density function(PDF) and the approximate posteriori PDF are minimized by a variational Bayesian(VB) technique to solve for the state and parameter approximate a posteriori estimations, and finally a new nonlinear filter based on the GSK mixture distribution is derived for angles-only relative navigation in time-varying environments. Simulation outcomes indicate that the filter can realize state estimation in non-stationary states effectively with 45.16% higher estimation accuracy than the existing advanced filters.

Experiments on kinematic characteristics and energy dissipation in rockfall movement on a slope

This paper presents an experimental methodology for tracking trajectories of rockfall-saltation and extracting kinematic parameters from collisions between rockfalls and a slope surface. We conducted a series of experiments, each featuring different initial impact angles. Rockfall trajectories and their three-dimensional angular velocities were measured by a high-speed camera and built-in Inertial Measurement Unit (IMU), respectively. Our experiments demonstrate that rockfall dissipates its total energy as it progresses along the slope, and the dissipation rates are largely determined by the initial impact angle. Following the classification of rockfall-bed collisions into two modes—Mode-1: saltation dominant and Mode-2: rolling and sliding dominant, we examined the correlations between impact angles and the probability density functions of kinetic, linear, and rotational kinetic energy, as well as the coefficients of kinetic friction and restitution in both modes. Our findings highlight the crucial role of three-dimensional angular velocities in rockfall kinematics, displaying a notable divergence of up to 60% when compared with their two-dimensional counterparts. This is particularly evident in Mode-2, where the increase in rotational energy following collisions exceeds that of Mode-1 × 25%. The experimental investigation contributes to a deeper understanding of the fundamental physical processes inherent in successive rockfall-slope collisions, thereby benefitting predictive capabilities for rockfall disasters in mountainous regions.

Statistical distribution of surface elevation by using quadratic forms of normal random variables

This paper is concerned with the statistical properties of surface elevation, which is crucial for the design and operation of marine and coastal structures and is helpful to the prediction of rogue waves. A semi-analytic quadratic model is proposed to describe the statistical distribution of surface elevation by using the theory of quadratic forms of normal random variables, in which the characteristic function and cumulants are given in analytical form, and the probability density is obtained by the numerical inversion of the characteristic function using the fast Fourier transform algorithm. Compared with Monte Carlo simulations, the quadratic model presents excellent performance in describing the probability density function of surface elevation, especially at the tails, for varying degrees of steepness, bandwidth, and directional spreading in both finite and infinite water depth. The effect of these parameters that characterize sea states on the statistical properties of surface elevation is also investigated. Through a comparative study, the quadratic model is superior to previously existing theoretical models.

Bivariate Tail Conditional Co-Expectation for elliptical distributions

In this paper, we consider a random vector X=(X1,X2) following a multivariate Elliptical distribution and we provide an explicit formula for E(X|X≤X˜), i.e., the expected value of the bivariate random variable X conditioned to the event X≤X˜, with X˜∈R2. Such a conditional expectation has an intuitive interpretation in the context of risk measures. Specifically, E(X|X≤X˜) can be interpreted as the Tail Conditional Co-Expectation of X (TCoES). Our main result analytically proves that for a large number of Elliptical distributions, the TCoES can be written as a function of the probability density function of the Skew Elliptical distributions introduced in the literature by the pioneering work of Azzalini (1985). Some numerical experiments based on empirical data show the usefulness of the obtained results for real-world applications.

Difference of Soil Thickness Depending on Climate Zones and Geological Classification: Based on a Survey of Mountain Slopes in Japan

There is little information which examines and compares climate zones or geology/topography in their relation to soil thickness along mountain slopes, where soil thickness is a possible contributing factor for the increase of sediment wash-off. In this study, the data obtained from handy penetration tests conducted throughout Japan are collected and attached to information on climate zones and geography/topography with the aim of clarifying the difference in soil thickness according to the climate zone and geological classification. The probability distribution of soil thickness was found to fit roughly with the probability density function of a log-normal distribution, regardless of the climate zone or geological age / lithology. A comparison of μ showed that the soil thickness at target sites in the Pacific climate zone, which have high rainfall, was large, at approximately 2.1 m. Meanwhile, the soil thickness was low, at approximately 1.5 m, at target sites in the Setouchi climate zone, which has low rainfall. A comparison of the geology showed that soil thicknesses of sites where the geology consisted of Tertiary and Quaternary sedimentary rocks were high, at approximately 2.1 m and 2.3 m, respectively. This suggests that, in high-precipitation regions, the weathering of rocks is promoted so that the soil layers tend to increase in thickness.

Structural and Dynamical Quality Assessment of Gap‐Filled Sea Surface Temperature Products

AbstractPrevious studies that intercompared global Level‐4 (L4) sea surface temperature (SST) analyses were centered on the assessment of the accuracy and bias of SST by comparing them with independent near‐surface Argo profile temperature data. This type of assessment is centered on the absolute value of SST rather than on SST spatial properties (gradients), which is more relevant to the study of oceanographic features (e.g., fronts, eddies, etc.) and ocean dynamics. Here, we use, for the first time, the spectrum of singularity exponents to assess the structural and dynamical quality of different L4 gap‐filled products based on the multifractal theory of turbulence. Singularity exponents represent the geometrical projection of the turbulence cascade, and its singular spectrum can be related to the probability density function of the singularity exponents normalized by the scale. Our results reveal that the different schemes used to produce the L4 SST products generate different singularity spectra, which are then used to identify a potential loss of dynamical information or structural coherence. This new diagnostic constitutes a valuable tool to assess the structural quality of SST products and can support data satellite SST producers efforts to improve the interpolation schemes used to generate gap‐filled SST products.

Effects of Socio-Economic and Demographic Factors on Meat Consumption Pattern in Iran: A Demand System Approach

Meat as one of the most important resources of protein has a special role in human nutrition. Understanding the meat consumption structure of households is essential for planning and policymaking in this regard. In this research, we studied consumption patterns of meat products including chicken, veal, lamb, and fish for households in Iran (Mashhad city) using demand system estimation. The hypothesis of this study is that chicken is a necessary goods and other types of meat are luxury goods. Given the cross-sectional nature of the data and presence of zero expenditure for some households, we used the censored demand model based on a consistent two-step approach. For this purpose, at first, four Probit models were estimated to determine the factors affecting the probability of purchasing each selected meat product. After that, the probability density function (PDF) and the cumulative distribution function (CDF) were calculated for each selected meat product, and the Almost Ideal Demand System (AIDS) considering PDF and CDF was estimated for all types of meat using a non-linear seemingly unrelated regression. Also, the effect of demographic variables on meat consumption pattern was considered in demand system. The results of expenditure elasticities confirmed the hypothesis. The highest own-price elasticity was related to veal. Based on compensated price elasticities, all types of meat were net substitutes for chicken and chicken was also a net complement for all types of meat. On the other hand, the only substitute for lamb and chicken was veal, but with compensating income effect fish also became a substitute for them. So, in the event of an increase of the price of lamb and chicken, we recommend subsidizing the consumers with low purchasing power in order to increase the diversity of consumption of protein products. This can increase the consumption of fish.

Goodness-of-fit testing for normality where alternative distributions have undefined or constant skewness and excess kurtosis

The aim of the paper is to examine goodness-of-fit testing for normality where alternative distributions have undefined or constant skewness and excess kurtosis. The first step involves collecting a set of normality-oriented goodness-of-fit tests (GoFTs) recommended for use in the source literature. The second step is to create a family of distributions with undefined or constant skewness and excess kurtosis. For greater clarity, the alternative distributions have been divided into three groups. Group I consisted of symmetric alternatives with undefined excess kurtosis, group II of symmetric alternatives with constant non-zero excess kurtosis, and group III comprised asymmetric alternatives with constants non-zero skewness and non-zero excess kurtosis. An appropriate similarity measure (SM) of a given alternative distribution to the normal distribution was applied. The third step involved a Monte Carlo simulation. The obtained results indicate that the GoFT power for the analysed alternatives and for a given sample size remain relatively unchanged despite the significant decrease in SM. Several GoFTs proved incapable of making a distinction between normal distribution and the alternatives. This situation occurs even when SM equals 0.15. If SM equals 1, then the probability density functions (PDFs) are identical. The best tests for group I are the Gel-Miao-Gastwirth (SJ) and Robust Jarque- Bera (RJB) tests (positive excess kurtosis) as well as the Coin and Chen-Shapiro (CS) tests (negative excess kurtosis), for group II the SJ, 1st Hosking and RJB tests, while the Zhang-Wu (ZA) and CS tests are best for group III.

Annual Variations of PM10, PM2.5 and PM1.0 Fraction at an Urban Site in Ansan, Korea

This study examines particulate matter (PM) concentrations, specifically PM10, PM2.5, and PM1.0, at an urban site in Ansan, South Korea. Three BAM-1020 continuous monitors measured PM10, PM2.5, and PM1.0 hourly from January to December 2021, with data acquisition rates of 97.1%, 97.3%, and 93.4%, respectively. Average annual concentrations were 45.7±48.4 µg/m³ (PM10), 26.4±22.7 µg/m³ (PM2.5), and 18.5±14.5 µg/m³ (PM1.0). PM10 was 91% of the annual air quality standard, while PM2.5 exceeded this standard by 176%, indicating severe pollution. Statistical analyses using probability density function (PDF) and cumulative distribution function (CDF) showed 98.5% of data converged at 150 µg/m³ (PM10), 75 µg/m³ (PM2.5), and 50 µg/m³ (PM1.0), with the remaining 1.5% corresponding to approximately 131 hours, annually. In 2021, all six observed yellow dust events occurred in spring (March to May). PM10, PM2.5, and PM1.0 concentrations increased from January to March, peaked in March, decreased until September, and rose again. The coarse fraction ratio increased stepwise from February to May, likely due to yellow dust and episodes of high PM concentration in the spring. The PM1.0 concentration was higher in summer and fall compared with that in spring and winter. The PM1.0/PM2.5 ratio was highest in summer and fall even though the annual PM1.0concentration was lowest, suggesting a higher secondary formation rate of fine particles during these periods. This underscores the need for further scientific investigation on PM1.0. The study provides insights into the simultaneous observation and distribution characteristics of PM10, PM2.5, and PM1.0, highlighting the mass concentration, distribution, and ratio of PM1.0. These findings are crucial for understanding PM distribution in Ansan, an industrial complex, and can serve as essential data for chemical composition comparisons, data validation, public health strategies, and cost-benefit analyses for regional PM improvement.

State-transition-aware anomaly detection under concept drifts

Detecting temporal abnormal patterns over streaming data is challenging due to volatile data properties and the lack of real-time labels. The abnormal patterns are usually hidden in the temporal context, which cannot be detected by evaluating single points. Furthermore, the normal state evolves over time due to concept drifts. A single model does not fit all data over time. Autoencoders have recently been applied for unsupervised anomaly detection. However, they are trained on a single normal state and usually become invalid after distributional drifts in the data stream. This paper uses an Autoencoder-based approach STAD for anomaly detection under concept drifts. In particular, we propose a state-transition-aware model to map different data distributions in each period of the data stream into states, thereby addressing the model adaptation problem in an interpretable way. In addition, we analyzed statistical tests to detect the drift by examining the sensitivity and powers. Furthermore, we present considerable ways to estimate the probability density function for comparing the distributional similarity for state transitions. Our experiments evaluate the proposed method on synthetic and real-world datasets. While delivering comparable anomaly detection performance as the state-of-the-art approaches, STAD works more efficiently and provides extra interpretability. We also provide insightful analysis of optimal hyperparameters for efficient model training and adaptation.