Probability Density Analysis Research Articles

PDFEstimator: An R Package for Density Estimation and Analysis

This article presents PDFEstimator, an R package for nonparametric probability density estimation and analysis, as both a practical enhancement and alternative to kernel-based estimators. PDFEstimator creates fast, highly accurate, data-driven probability density estimates for continuous random data through an intuitive interface. Excellent results are obtained for a diverse set of data distributions ranging from 10 to $10^6$ samples when invoked with default parameter definitions in the absence of user directives. Additionally, the package contains methods for assessing the quality of any estimate, including robust plotting functions for detailed visualization and trouble-shooting. Usage of PDFEstimator is illustrated through a variety of examples, including comparisons to several kernel density methods.

A novel self-adaptive fractional multivariable grey model and its application in forecasting energy production and conversion of China

Energy production and conversion have a significant impact on the economic development of all countries in the world. China’s energy production and conversion are large. Therefore, accurate mid-to-long term China’s energy production and conversion forecasting is becoming more and more important for integrating energy systems and energy strategic planning. For this purpose, a novel fractional grey sequence is proposed based on Grunwald–Letnikov fractional calculus. Furthermore, a novel self-adaptive fractional multivariable grey model is proposed based on the novel sequence. In this article, we compare several classical optimization algorithms and finally choose Particle Swarm Optimization (PSO) to compute the parameters. In addition, Monte-Carlo simulation and probability density analysis (PDA) are presented in this article to verify the model’s performance. Monte-Carlo simulation reduces the randomness of the results of the model runs to a certain extent. Probability density analysis visualizes this randomness through kernel density estimation (KDE). This paper compares the new model with the existing seven grey models and predicts the total energy consumption per capita, energy conversion efficiency and total renewable energy in China, respectively. The experimental results show that the new model is superior to the other seven models in terms of stability and prediction accuracy.

Probability density analysis of nonlinear stochastic dynamics of horizontal axis wind turbine blades

Wind turbine blades with horizontal axis are frequently subjected to cyclic and random loads during operation, which may decrease blade stability and impair power generation efficiency. In this paper, the finite element method (FEM) is mainly used to investigate the nonlinear stochastic dynamics of turbine blades and assess blade operation stability. The harmonic excitation and Gaussian white noise are combined to represent the external stimulation of the Mathieu-Duffing equation describing blade vibration, the Fokker-Planck (FP) equation is formulated and numerically solved using the FEM and the Crank Nicholson method to obtain the transient joint probability density function (PDF) and marginal PDF. The findings show that the blade's PDF exhibits an unstable split phenomenon in one vibration cycle, meaning that the blade's motion may jump randomly and has some uncertainty, potentially affecting blade operating stability and power generation efficiency. Furthermore, the sensitivity of blade stability to excitation strength is examined. The form and position of the PDF vary when the intensity of the harmonic excitation or noise is strong, increasing the random jump probability of blade vibration and compounding the instability and uncertainty of blade motion. Under severe circumstances, the blades may lose stability or even damage, endangering the safety and efficacy of blade operation. The current study might be used to evaluate blade stability and the overall safety of turbines.

A novel fractional structural adaptive grey Chebyshev polynomial Bernoulli model and its application in forecasting renewable energy production of China

Accurate mid-to-long term China's renewable energy forecasting is becoming more and more important for integrating renewable energy systems with smart grid and energy strategic planning. For this purpose, this paper proposes a novel fractional structural adaptive grey Chebyshev polynomial Bernoulli model. The new model is based on NGBM(1,1) model to describe nonlinear phenomena. The proposed adjacent accumulation operator can balance old and new information accumulation with less fluctuation. The introduction of the Chebyshev polynomial enables the model to adaptively adjust the structure of the model, which reduces the difficulty of the modeler's operation from the point of view of the expert system. Meanwhile, an improved Grey Wolf optimization algorithm (IGWO) was used to optimize the parameters. Four real cases of renewable energy in China, including production and consumption are used for verification. In addition, Monte Carlo simulation and probability density analysis illustrate the robustness and accuracy of the proposed model. Finally, the development of the four cases in the next 5 years is predicted.

Forecasting the economic indices of the high-tech industries in China using the grey multivariable convolution model

Reliable output forecasts are critical for decision-making and planning in high-tech industries. Several endogenous and exogenous factors could influence the output dynamics, and thereby a univariate model that only incorporates historical data and single method improvement is typically insufficient to generate accurate projections. In this paper, a novel grey multivariable convolution model is designed from a collaborative optimization perspective. The core innovations of this study can be summarized as follows. Initially, a collaborative framework integrating background value optimization, data preprocessing, and model structure improvement is constructed, which overcomes the internal deficiencies of the conventional grey models. Secondly, the Particle Swarm Optimization algorithm is selected to determine the optimal values of the damping accumulation parameters to enhance the adaptability and flexibility of the proposed model. Further, for validation purposes, experiments on forecasting the output of high-tech industries considering R&D investments are conducted from the national and provincial levels, among which six competing models are involved. Moreover, Monte-Carlo Simulation, incorporated with probability density analysis and statistical analysis, is introduced to evaluate the robustness of the newly-designed model. It can be demonstrated from the empirical results that the novel model is a reliable and promising method for predicting the economic indices of the high-tech industries with enhanced forecasting capability.

A novel structure adaptive fractional discrete grey forecasting model and its application in China’s crude oil production prediction

Crude oil resources are related to all aspects of people's life, and play a vital role in the development of the national economy. Using nonlinear discrete data to reasonably predict crude oil production can help the government adjust energy structure and formulate energy development strategy, which has great practical significance. In this paper, a novel r-order accumulation operation with a parameter is proposed, and a novel structure adaptive fractional discrete grey forecasting model is established. Several classical optimization algorithms are compared, and the Grey Wolf Optimizer (GWO) is selected to calculate the parameters. For testifying the effectiveness of the model, a prediction model is constructed based on the total crude oil production in Qinghai, Liaoning and Shaanxi provinces of China, and a performance comparison experiment is designed with the existing six grey models. In addition, Monte Carlo simulation and probability density analysis provide a new perspective to further illustrate the robustness and accuracy of the proposed model. The experimental results show that this model is superior to the other six models in terms of fitting accuracy, prediction accuracy and model stability.

Stationary Probability Density Analysis for the Randomly Forced Phytoplankton–Zooplankton Model with Correlated Colored Noises

In this paper, we propose a stochastic phytoplankton–zooplankton model driven by correlated colored noises, which contains both anthropogenic and natural toxins. Using Khasminskii transformation and the stochastic averaging method, we first transform the original system into an Itô diffusion system. Afterwards, we derive the stationary probability density of the averaging amplitude equation by utilizing the corresponding Fokker–Planck–Kolmogorov equation. Then, the stability of the averaging amplitude is studied and the joint probability density of the original two-dimensional system is given. Finally, the theoretical results are verified by numerical simulations, and the effects of noise characteristics and toxins on system dynamics are further illustrated.

A supervised multisegment probability density analysis method for incipient fault detection of quality indicator

The quality indicator monitoring has received widely attention and research in recent years, however, the detection of indicator-related incipient fault is still a challenging topic. In this paper, a supervised probability density analysis algorithm is proposed to detect the incipient fault in quality indicator. Firstly, the core process variable filter is introduced, and the regression model is constructed to extract the indicator-related information from process variable. Secondly, the data distribution extension and subsegment division strategy are presented, and a probability density estimation method is put forward for the indicator-related latent variable. Through the proposed symmetric divergence index, the distribution discrepancy between the online sample and the reference sample set is evaluated, which can be used for the incipient fault detection. Finally, a numerical example and the Tennessee Eastman process are used to demonstrate the effectiveness of the proposed method.

Multi-center bonds as resonance hybrids: A real space perspective.

The concept of distinct bonds within molecules has proven to be successful in rationalizing chemical reactivity. However, bonds are not a well-defined physical concept, but rather vague entities, described by different and often contradicting models. With probability density analysis, which can-in principle-be applied to any wave function, bonds are recovered as spin-coupled positions within most likely electron arrangements in coordinate space. While the wave functions of many systems are dominated by a single electron arrangement that is built from two-center two-electron bonds, some systems require several different arrangements to be well described. In this work, a range of these multi-center bonded molecules are classified and investigated with probability density analysis. The results are compared with valence bond theory calculations and data from collision-induced dissociation experiments.

Stochastic stability and stationary probability density analysis for a nutrient–phytoplankton model with multiplicative and additive noises

In this paper, the stability and stationary probability density for a stochastic nutrient–phytoplankton model with multiplicative and additive noises are explored. We first transform the original system into an Itô diffusion system by using Khasminskii transformation and stochastic averaging method. After that, we employ singular boundary theory to analyze the stability of the system. We then derive the stationary probability density of the averaging amplitude equation by the corresponding Fokker–Planck–Kolmogorov equation. Finally, the theoretical results are verified by numerical simulations and the influence of noise on system dynamics is further illustrated.

Analysis of the Motion Sickness and the Lack of Comfort in Car Passengers

Advanced driving assistance systems (ADAS) are primarily designed to increase driving safety and reduce traffic congestion without paying too much attention to passenger comfort or motion sickness. However, in view of autonomous cars, and taking into account that the lack of comfort and motion sickness increase in passengers, analysis from a comfort perspective is essential in the future car investigation. The aim of this work is to study in detail how passenger’s comfort evaluation parameters vary depending on the driving style, car or road. The database used has been developed by compiling the accelerations suffered by passengers when three drivers cruise two different vehicles on different types of routes. In order to evaluate both comfort and motion sickness, first, the numerical values of the main comfort evaluation variables reported in the literature have been analyzed. Moreover, a complementary statistical analysis of probability density and a power spectral analysis are performed. Finally, quantitative results are compared with passenger qualitative feedback. The results show the high dependence of comfort evaluation variables’ value with the road type. In addition, it has been demonstrated that the driving style and vehicle dynamics amplify or attenuate those values. Additionally, it has been demonstrated that contributions from longitudinal and lateral accelerations have a much greater effect in the lack of comfort than vertical ones. Finally, based on the concrete results obtained, a new experimental campaign is proposed.

Hierarchical Latent Variable Extraction and Multisegment Probability Density Analysis Method for Incipient Fault Detection

The incipient fault is difficult to detect because of its small amplitude and insignificant impact, however, ignoring such fault may cause irreversible damage to the system. In this article, a hierarchical latent variable extraction and multisegment probability density analysis method is proposed to detect the incipient fault. First, three data subspaces are constructed, which are named dominant, intermediate, and residual spaces, and key latent variables which contain more offline variance or online variation information will be retained. Afterward, the expanded data distribution interval and multiple data segmentsare constructed for the probability density estimation. Based on the improved symmetric divergence index, the distribution distance between the online data and offline modeling data can be evaluated, which has achieved 95.3% and 86.8% average detection rates for the faults in numerical case and Tennessee Eastman process. Finally, a real multiphase flow facility is used to demonstrate the effectiveness of the proposed method.

Effects of magnetic, electric, and intense laser fields on the optical properties of AlGaAs/GaAs quantum wells for terahertz photodetectors

We investigate optical absorption coefficients and changes in the refractive index in single- and double-step GaAs/AlGaAs quantum wells under the influence of magnetic, electric, and intense laser fields. For single-step quantum wells, increasing the intensity of the external laser and electric fields induces a red-shift in the total optical absorption coefficient and change in the refractive index and increases their amplitudes; however, increasing the magnetic field generates a blueshift in the optical absorption and change in the refractive index. For double-step quantum wells, the transition energy between the ground and first excited state is strongly decreased. In addition, the total optical absorption coefficients and changes in the refractive index incur a blue shift when the electric and magnetic fields are increased and a red-shift with increasing laser fields. Our analysis of coupling matrix elements, probability densities, and energy levels show that intense laser, electric, and magnetic fields can be harnessed to adjust and tune total optical absorption coefficients and refractive indices for terahertz applications.

Probability Density Analysis of Wind Storage Output Based on Monte Carlo Simulation

Due to the relatively large fluctuation amplitude of the output power of wind farms, when using energy storage to stabilize the output characteristics of wind power, the impact of the addition of energy storage on the probability density of wind power output needs to be further analyzed. This paper uses Monte Carlo simulation algorithm to analyze the probability density of active power output in consideration of the operating state of wind power and compares the change of the output probability density curve after adding energy storage. According to the output probability density curve obtained by the algorithm, the abscissa of the curve is shifted to the right by one energy storage capacity unit after the addition of energy storage, so that the output probability density curve of the wind storage system can reach the required value by adding energy storage.

In-Depth Molecular Dynamics Study of All Possible Chondroitin Sulfate Disaccharides Reveals Key Insight into Structural Heterogeneity and Dynamism.

GAGs exhibit a high level of conformational and configurational diversity, which remains untapped in terms of the recognition and modulation of proteins. Although GAGs are suggested to bind to more than 800 biologically important proteins, very few therapeutics have been designed or discovered so far. A key challenge is the inability to identify, understand and predict distinct topologies accessed by GAGs, which may help design novel protein-binding GAG sequences. Recent studies on chondroitin sulfate (CS), a key member of the GAG family, pinpointing its role in multiple biological functions led us to study the conformational dynamism of CS building blocks using molecular dynamics (MD). In the present study, we used the all-atom GLYCAM06 force field for the first time to explore the conformational space of all possible CS building blocks. Each of the 16 disaccharides was solvated in a TIP3P water box with an appropriate number of counter ions followed by equilibration and a production run. We analyzed the MD trajectories for torsional space, inter- and intra-molecular H-bonding, bridging water, conformational spread and energy landscapes. An in-house phi and psi probability density analysis showed that 1→3-linked sequences were more flexible than 1→4-linked sequences. More specifically, phi and psi regions for 1→4-linked sequences were held within a narrower range because of intra-molecular H-bonding between the GalNAc O5 atom and GlcA O3 atom, irrespective of sulfation pattern. In contrast, no such intra-molecular interaction arose for 1→3-linked sequences. Further, the stability of 1→4-linked sequences also arose from inter-molecular interactions involving bridged water molecules. The energy landscape for both classes of CS disaccharides demonstrated increased ruggedness as the level of sulfation increased. The results show that CS building blocks present distinct conformational dynamism that offers the high possibility of unique electrostatic surfaces for protein recognition. The fundamental results presented here will support the development of algorithms that help to design longer CS chains for protein recognition.

Algorithm for assessing changes in the functional state of muscles of the maxillofacial area of athletes using individual relaxation occlusal splints or mouth guards

The paper proposes an algorithm for assessing changes in the functional state of bilaterally located muscles. The proposed algorithm is based on the analysis of two-dimensional distribution function and two-dimensional probability density of the amplitude of electromyograms of the muscles under study. The paper proposes criteria for assessing the symmetry and asymmetry of muscle activity, presents the results of the study of changes in the functional state of the temporalis muscles and the masseter muscles for a group of athletes. During sports competitions and preparing to them, athletes are exposed to high physical and emotional stress which leads to a premature decrease in the functional activity of the muscles causing changes in the biomechanics of movements. The reasons that lead to a decrease in the functional activity of the muscles are as follows: occlusal disharmony; changes in bite and / or interalveolar distance, hypertonicity of the muscles of the maxillofacial region. Ultimately, these reasons lead to a deterioration in sports performance. The group of the athletes under study involved 10 males aged 23 to 24 years, who are engaged in combat sports at a professional level. Each athlete underwent a clinical examination, palpation and isometric tests of the muscles of the maxillofacial region (MR) and temporomandibular joints (TMJ), auscultation of the TMJ, control of occlusion (using copy paper and T-scan apparatus), cone-beam computed tomography (CBCT) of the dentoalveolar system (DAS) and electromyography (EMG) of the muscles of the maxillofacial region. At this stage of the research, each athlete received individual relaxation occlusal splints or mouthguards of two types: 2.0 mm thick soft one and 3.0 mm thick semi-rigid one. It was recommended to the athletes that they should wear the splints for 1–2 months in the following manner: to wear 2.0 mm thick soft splint on an ongoing basis, excluding food intakes and the training process, whereas 3.0 mm thick semi-rigid splints should be used during the training process and other types of physical activity, as well as at competitions.

Forecasting nuclear energy consumption in China and America: An optimized structure-adaptative grey model

Forecasting nuclear energy accurately is vital to ensure reliable electricity supply and alleviate environmental degradation problems. However, it is difficult to model the nuclear time series due to its complexity, nonlinearity, and uncertainty. To this end, this paper put forward an optimized structure-adaptive grey model by theoretically providing the generalized time response function and accurately modifying the background value based on Simpson's rule. Moreover, Monte-Carlo Simulation and probability density analysis (PDA) are employed to analyze the proposed model's merits, revealing its robustness and reliability. For validation and verification purposes, the optimized model is implemented to predict nuclear energy consumption in China and America, compared to seven benchmark models involving other prevalent grey models, conventional econometric technology, and artificial intelligence. Experimental results from two cases consistently demonstrate that the novel technique significantly outperforms the competitors from two different perspectives of PDA and level accuracy. Besides, further discussion over different forecasting horizons reveals that this new model can still deliver accurate forecasts with solid robustness and high reliability. Consequently, the proposed model is validated as a practical and promising model for forecasting nuclear energy consumption.

Application of a novel structure-adaptative grey model with adjustable time power item for nuclear energy consumption forecasting

Accurate estimations of nuclear energy consumption are an essential process for formulating appropriate policies and plans in the energy sector and associated companies. This paper presents a novel structure-adaptive grey model with an adjustable time power based on the nonlinear and complicated characteristics of nuclear energy consumption, in which three core innovations are summarized below. Initially, the generalized time response function for projections is theoretically deduced, which overcomes the fundamental flaws in the conventional grey model. Subsequently, the Cultural Algorithm is employed to determine the optimum values of the time power item to improve the adaptability and flexibility to confront diverse forecasting issues. Further, Monte-Carlo Simulation and Probability Density Analysis (PDA) are originally introduced to enhance the robustness of the proposed model. For illustration and verification purposes, experiments on predicting nuclear energy consumption in China and America are conducted in comparison with a range of benchmark models, including other prevalent grey models, conventional econometric technology, and artificial intelligences. The performance of the novel technique is evaluated from two different perspectives of PDA and level accuracy, confirming that this model is a very promising and powerful tool for predicting nuclear energy demands in China and America from 2019 to 2023.

Piecewise asymmetric exponential potential under-damped bi-stable stochastic resonance and its application in bearing fault diagnosis

It is difficult to extract weak signals in strong noise background, therefore a piecewise asymmetric exponential potential under-damped bi-stable stochastic resonance (PAEUBSR) system is proposed. First, the theoretical analysis of the steady-state probability density (SPD), mean first passage time (MFPT) and output signal-to-noise ratio (SNR) are derived under the adiabatic approximation theory. At the same time, the influence of different system parameters on system performance is explored. Then the PAEUBSR system is applied to the fault signal diagnosis of different types of bearings, and the parameters are optimized through the adaptive genetic algorithm (AGA). The test results are compared with the exponential potential over-damped symmetric bi-stable stochastic resonance (EOSBSR) system and the exponential potential under-damped symmetric bi-stable stochastic resonance (EUSBSR) system. Finally, the detection results on two sets of bearing fault data show that the PAEUBSR system has better effects on the enhancement and detection of bearing fault signals. This provides good theoretical support and application value for this system in subsequent theoretical analysis and practical engineering applications.

Rate-dependent bifurcation dodging in a thermoacoustic system driven by colored noise

Tipping in multistable systems occurs usually by varying the input slightly, resulting in the output switching to an often unsatisfactory state. This phenomenon is manifested in thermoacoustic systems. The thermoacoustic instability may lead to the disintegration of rocket engines, gas turbines and aeroengines, so it is necessary to design control measures for its suppression. It was speculated that such unwanted instability states may be dodged by changing the bifurcation parameters quickly enough, and compared with the white noise discussed in [1], colored noise with nonzero correlation time is more practical and important to the system. Thus, in this work, based on a fundamental mathematical model of thermoacoustic systems driven by colored noise, the corresponding Fokker–Planck–Kolmogorov equation of the amplitude is derived by using a stochastic averaging method. A transient dynamical behavior is identified through a probability density analysis. We find that both a relatively higher rate of change of parameters and change in the correlation time of the noise are beneficial to dodge thermoacoustic instability, while a relatively large noise intensity is a disadvantageous factor. More interestingly and importantly, power-law relationships between the maximum amplitude and the noise parameters are uncovered, and the probability of successfully dodging a thermoacoustic instability is calculated. These results serve as a guidance for the design of engines and to propose an effective control strategy, which is of great significance to aerospace-related fields.