Gaussian Distribution Research Articles

Entropy-Based Volatility Analysis of Financial Log-Returns Using Gaussian Mixture Models.

Volatility in financial markets refers to the variation in asset prices over time. High volatility indicates increased risk, making its evaluation essential for effective risk management. Various methods are used to assess volatility, with the standard deviation of log-returns being a common approach. However, this implicitly assumes that log-returns follow a Gaussian distribution, which is not always valid. In this paper, we explore the use of (differential) entropy to evaluate the volatility of financial log-returns. Estimation of entropy is obtained using a Gaussian mixture model to approximate the underlying density of log-returns. Following this modeling approach, popular risk measures such as Value at Risk and Expected Shortfall can also be computed. By integrating Gaussian mixture modeling and entropy into the analysis of log-returns, we aim to provide a more accurate and robust framework for assessing financial volatility and risk measures.

Crystal Structure Prediction Using Generative Adversarial Network with Data-Driven Latent Space Fusion Strategy.

Crystal structure prediction (CSP) is an important field of material design. Herein, we propose a novel generative adversarial network model, guided by a data-driven approach and incorporating the real physical structure of crystals, to address the complexity of high-dimensional data and improve prediction accuracy in materials science. The model, termed GAN-DDLSF, introduces a novel sampling method called data-driven latent space fusion (DDLSF), which aims to optimize the latent space of generative adversarial networks (GANs) by combining the statistical properties of real data with a standard Gaussian distribution, effectively mitigating the "mode collapse" problem prevalent in GANs. Our approach introduces a more refined generation mechanism specifically for binary crystal structures such as gallium nitride (GaN). By optimizing for the specific crystallographic features of GaN while maintaining structural rationality, we achieve higher precision and efficiency in predicting and designing structures for this particular material system. The model generates 9321 GaN binary crystal structures, with 16.59% reaching a stable state and 24.21% found to be metastable. These results can significantly enhance the accuracy of crystal structure predictions and provide valuable insights into the potential of the GAN-DDLSF approach for the discovery and design of binary, ternary, and multinary materials, offering new perspectives and methods for materials science research and applications.

Fokker-Planck equation for the crystal-size probability density in progressive nucleation and growth with application to lognormal, Gaussian and gamma distributions

The Fokker Planck (FP) equation for the probability density function (PDF) of crystal size in phase transformations ruled by progressive nucleation and growth, has been derived. Crystals are grouped in sub-sets, we refer to as τ-crystals, where τ is the birth time of the set. It is shown that the size PDF is the superposition of the PDF of the crystal sub-sets (τ-PDFs), with weight given by the nucleation rate. The growth and diffusion coefficients entering the FP equations are estimated as a function of both τ-PDFs and nucleation rate. The functional form of these coefficients is studied for solutions of the FP equation for τ-crystals given by the lognormal, Gaussian and gamma distributions. For the first two distributions, the effect of fluctuations, nucleation rate and growth rate, on the shape of the distribution has been investigated. It is shown that for an exponential decay of the fluctuation term, the shape of the PDF is mainly governed by both the time constant for nucleation and the strength of the fluctuation. It is found that τ-PDFs given by the one-parameter gamma distributions are suitable to deal with KJMA (Kolmogorov Johnson Mehl Avrami) compliant phase transformations, where the fluctuation term is proportional to crystal size. The connection between the FP equation for the size PDF and the evolution equation for the density of crystal populations is also discussed.

A Topological Investigation of the Cosmic Web Formation

N-body simulations of the evolution of the large-scale structure of the universe (the Cosmic Web) were run while altering attractive gravity laws and initial conditions, in order to infer which properties of the universe are revealed by its current large-scale topology. The Cosmic Web was found to develop regardless of the gravitational alteration made. Significantly altering the initial conditions from Gaussian distribution ones was found to eliminate the Cosmic Web. Thus, we require small random perturbations in otherwise uniform initial conditions in the early universe, under an attractive gravitational force, for the Cosmic Web to develop.

An advancement in AdaSyn for imbalanced learning: An application to fraud detection in digital transactions

Imbalanced Learning is a significant issue in machine learning, affecting the performance and accuracy of binary or multi-classification algorithms, especially in large-scale data handling and classification. There are some popular techniques to covert this imbalanced data into a balanced one such as undersampling, under-sampling with tomek links, randomized oversampling, synthetic minority oversampling technique (SMOTE), and adaptive synthetic generation (ADASYN). Generally, the ADASYN algorithm could be used to propagate minority sample points to rise the imbalanced ratio between majority and minority sample points, but in some cases, it may conflict with decision boundary points and noisy points. This paper proposed a Refitted AdaSyn Algorithm (RAA) with Gaussian Distribution (GD). So that new minority samples are distributed much closer to the center of the minority sample to spotlight the conflicts. The classification accuracy has improved with RAA over formal ADASYN. For examining the proposed work the imbalanced benchmark datasets like European, Banksim, Paymentcard, and UCI credit card are considered. Vanilla Generative Adversarial Network (GAN) is a deep learning model used to classify fraud and non-fraud transactions, demonstrating significant differences between balanced and imbalanced learning approaches and achieving an accuracy of 97.5% on dataset DS4.

GS‐Octree: Octree‐based 3D Gaussian Splatting for Robust Object‐level 3D Reconstruction Under Strong Lighting

AbstractThe 3D Gaussian Splatting technique has significantly advanced the construction of radiance fields from multi‐view images, enabling real‐time rendering. While point‐based rasterization effectively reduces computational demands for rendering, it often struggles to accurately reconstruct the geometry of the target object, especially under strong lighting conditions. Strong lighting can cause significant color variations on the object's surface when viewed from different directions, complicating the reconstruction process. To address this challenge, we introduce an approach that combines octree‐based implicit surface representations with Gaussian Splatting. Initially, it reconstructs a signed distance field (SDF) and a radiance field through volume rendering, encoding them in a low‐resolution octree. This initial SDF represents the coarse geometry of the target object. Subsequently, it introduces 3D Gaussians as additional degrees of freedom, which are guided by the initial SDF. In the third stage, the optimized Gaussians enhance the accuracy of the SDF, enabling the recovery of finer geometric details compared to the initial SDF. Finally, the refined SDF is used to further optimize the 3D Gaussians via splatting, eliminating those that contribute little to the visual appearance. Experimental results show that our method, which leverages the distribution of 3D Gaussians with SDFs, reconstructs more accurate geometry, particularly in images with specular highlights caused by strong lighting. The source code can be downloaded from https://github.com/LaoChui999/GS-Octree.

0.2-mm-Step Verification of the Dual Gaussian Distribution Model with Large Sample Size for Predicting Tap Success Rates

The Dual Gaussian Distribution Model can be utilized for predicting the success rates of tapping targets. However, previous studies have shown that the prediction error increases to as much as 10 points, where "points" represent the percentage difference between the observed and predicted values of the tap success rate, particularly for a small target width W such as 2 mm. We hypothesize that this could be due to the experimental designs with sparse W levels performed by few participants, rather than the model itself. Our experiment involving horizontal and vertical bar targets with W = 2-8 mm (step: 0.2 mm) performed by more than 180 participants showed that the maximum prediction errors were relatively small: 2.769 and 3.185 points, respectively. Furthermore, the correlation between W and the prediction error was statistically small (Pearson's |r| < 0.2), and W was not a significant contributor to changing prediction errors (p>0.05). As these results do not support the concerns that the Dual Gaussian Distribution Model has an issue when used with small targets, the development of applications and refined models is encouraged to continue.

A Multi-Scale Model for Predicting Physically Short Crack and Long Crack Behavior in Metals.

The fatigue behavior of metal specimens is influenced by defects, material properties, and loading. This study aims to establish a multi-scale fatigue crack growth model that describes physically short crack (PSC) and long crack (LC) behavior. The model allows the calculation of crack growth rates for uniaxial loading at different stress ratios based on the material properties and specimen geometry. Furthermore, the model integrates the Gaussian distribution theory to consider material heterogeneity and the experimental measurement errors that cause fatigue scatter. The crack growth rate and fatigue life of metal specimens with different notch geometry were predicted. The curves generated by the multi-scale model were mainly consistent with the test data from the published literature at the PSC and LC stages.

Mem‐Transistor‐Based Gaussian Error–Generating Hardware for Post‐Quantum Cryptography Applications

AbstractQuantum computing can potentially hack the information encrypted by traditional cryptographic systems, leading to the development of post‐quantum cryptography (PQC) to counteract this threat. The key principle behind PQC is the “learning with errors” problem, where intentional errors make encrypted information unpredictable. Intentional errors refer to Gaussian distributed data. However, implementing Gaussian distributed errors is challenging owing to computational and memory overhead. Therefore, this study proposes a Gaussian error sampler that employs the intrinsic Gaussian properties of nanometer‐scale semiconductor devices. The proposed Gaussian error sampler significantly reduces computational and memory overhead. This work comprehensively evaluates the effectiveness of the proposed device by conducting statistical normality tests and generating quantile–quantile plots. The optimal programming voltage is identified to be −5.25 V, and the experimental results confirmed the Gaussian distribution of error data generated by the proposed module, aligning closely with software‐generated Gaussian distributions and distinct from uniform random distributions.

Based on improved crayfish optimization algorithm cooperative optimal scheduling of multi-microgrid system

In order to solve the influence of the complex interaction relationships among subjects on the system solution accuracy and speed of the Multi-Microgrid system under the high penetration rate of new energy. Firstly, the paper establishes the bi-level optimal scheduling Stackelberg game model based on shared energy storage, considering the inter-subject interaction in MMG. Subsequently, based on the four improvement methods of Chaotic Map, Quantum Behavior, Gaussian Distribution, and Nonlinear Control Strategy, the Chaotic Gaussian Quantum Crayfish Optimization Algorithm is proposed to solve the optimization scheduling model. The improved algorithm exhibits superior initial solutions and enhanced search capability. In comparison to the original algorithm, the relative errors of the CGQCOA optimization outcomes are 98%, 20.96%, 98.74% and 16.55%, respectively, enhancing the model-solving accuracy and the speed of convergence to the optimal solution. Finally, the simulation demonstrates that the revenue of Microgrid 1, Microgrid 2, and Microgrid 3 have increased by 0.73%, 1.17%, and 1.04%, respectively. Concurrently, the penalty cost of pollutant emission has decreased by 5.9%, 11.5%, and 12.68%, respectively. Furthermore, the revenue of the shared storage have increased by 1.91%. These findings validate the efficacy of the methodology proposed in enhancing the revenue of the various subjects and reducing pollutant gas emission.

Exploring statistical complexity and enabling speckle-free imaging with Dye-Kaolinite colloidal random lasers

This article extends our previous investigation into random lasers (RLs), focusing on the incorporation of Rhodamine 6G (R6G) dye in methanol with Kaolinite nanoclay scatterers as a disordered active medium for colloidal RLs. Building upon the prior exploration, which specifically focused on understanding the performance dependence related to the concentration of gain medium and scatterers, this report takes a closer look. It conducts a thorough statistical analysis of the incoherent dye-Kaolinite random lasing spectra, particularly in the context of replica symmetry breaking (RSB) and Lévy statistics. This study highlights that RSB is observed around the threshold energy and can serve as an indicator of the threshold. Gaussian distribution prevailed below, near, and far above the lasing threshold estimated from the levy (α) exponent, suggesting the absence of Lévy flight behavior. Additionally, Pearson correlation coefficients (PCC) were used to quantify the correlation between intensity fluctuations at different wavelengths, with the results visually represented by heatmaps. Furthermore, the random lasing output is utilized to achieve speckle-free imaging.

Rhythmic states and first-order phase transitions in adaptive coupled three-dimensional limit-cycle oscillators.

This paper reports a phase transition in coupled three-dimensional limit-cycle oscillators with an adaptive coupling. We reveal that the multiple-cluster rhythmic states emerge when natural frequencies of oscillators follow uniform distribution (case I), but disappear for Gaussian distribution (case II). Furthermore, as the coupling strength K increases, two first-order phase transitions occur sequentially. When K→0^{+}, an inevitable and nonhysteretic discontinuous phase transition occurs. For K>0, another discontinuous phase transition with a hysteresis loop emerges, and its occurrence depends on the width of the natural frequency distribution. From the microscopic perspective of the system, there are double switches between suppression and revival of oscillations as K varies in case I, but only one switch occurs in case II. Theoretical analyses of the incoherent states and the fixed points are given, which can be accurately verified by numerical simulations. This paper provides insights into our understanding of high-dimensional oscillators and their variants.

An exactly solvable model for non-Fickian transport in dynamically heterogeneous media

Abstract Diffusion, observed in various condensed phases, finds its theoretical background in Einstein’s theory of Brownian motion, characterized by the linear time-dependence of mean square displacement (MSD) denoting Fickian behavior and the Gaussian distribution of particle displacement. Nevertheless, diverse systems exhibit either non-linear, non-Fickian time-dependence of the MSD or non-Gaussian displacement distribution. Montroll and Weiss’s continuous-time random walk (CTRW) model and the stochastic diffusivity (SD) model have provided insights into anomalous diffusion phenomena and Fickian-yet-non-Gaussian transport in dynamically heterogeneous environments, respectively. Building upon these approaches, Song et al developed a generalized transport equation with an environment-dependent diffusion kernel, providing a quantitative explanation for non-Fickian MSD and non-Gaussian displacement distribution. Based on the generalized transport equation, this study introduces an exactly solvable model for a non-Gaussian displacement distribution, accommodating arbitrary time profiles in its MSD, including both Fickian and non-Fickian behaviors. Our findings confirm the model’s capability in describing such transport processes. Furthermore, the proposed model unifies the CTRW model under fast environmental fluctuations and the SD model under Fickian time dependencies, making it suitable for understanding tracer particle motion within explicit solvent or complex media.

Overlapping microbubble localization based on multiscale statistical features for ultrasound super-resolution imaging

Ultrasound localization microscopy enables visualization of the microvasculature through the localization and tracking of isolated microbubbles (MBs). Currently, improving the precision for localizing overlapping MBs at high concentration to decrease the acquisition time remains a challenge. This study introduces multiscale statistical feature localization (MSFL) to address this challenge. First, the overlapping MBs are separated by local energy and blobness filters constructed from the Hessian matrix of texture features and its eigenvalues. Then, multiscale and orientation kernels are modeled to adapt to the appearance of various MBs; their amplitude and gradient convolution response results jointly determine the credibility of the MBs. Finally, precise MB localization is obtained by the weighted average of the local maxima of the response maps at each scale to achieve super-resolution (SR) imaging. The proposed method is evaluated using simulation and in-vivo datasets. Simulation results demonstrate that MSFL can adapt to asymmetric Gaussian distributions and improve overlapping MB localization, with a 9.28 % improvement in localization precision and a 5 % greater Jaccard index compared to the Gaussian fitting method. Additionally, the strong robustness of MSFL under noise interference maintains its ability to locate more MBs, and reconstruct the same saturation vessels with 36 % acquisition time reduction compared to Radial symmetry method. For SR images, MSFL can draw smoother interlaced vessels when more overlapping MBs are present, and quickly reconstructs more saturated in vivo microvasculature with a resolution is 10.5 μm(λ/10). These results validate the advantages of MSFL in achieving precise SR imaging with a short acquisition time.

Dynamics of Lanthanide(III) complexes from crystallographic data and computational studies

In this study we take advantage of the large body of X-ray data reported for lanthanide complexes, as well as their similar coordination chemistry properties, to gain insight on the dynamic processes involving λ ↔ δ configurational changes leading to isomer interconversion. The analysis of X-ray data reveals that different donor groups are characterized by Gaussian distributions of the dihedral N–C–X–Y angles ϕ. In the case of ethylenediamine groups (X = C; Y = N) Gaussian distributions are very narrow, while acetates (X = C; Y = O) provide very broad distributions. The analysis of the potential energy surfaces (PESs) using DFT reveals that ethylenediamine groups are characterised by well-defined deep energy minima, while those of acetate groups are rather swallow. Other donor groups like acetamide, pyridine or phosphinate show intermediate Gaussian distributions with respect to ethylenediamine and acetates. The PESs were fitted to a potential incorporating quadratic, cubic and quartic terms. The quadratic force constants k11 obtained for the different donor types follow the trend of the full width at half maxima (FWHM) of Gaussian distributions obtained from X-ray data. The k11 values are very similar for a given donor group, while the anharmonic cubic (k111) and quartic (k1111) force constants vary significantly depending on the environment of the donor group. Overall, this work demonstrates that the analysis of X-ray crystallographic data in lanthanide complexes can provide very useful information on their coordination chemistry preferences and dynamic processes.

An Evidential Reasoning Method of Comprehensive Evaluation of Water Quality Based on Gaussian Distribution

This study provides an evidential reasoning method for water quality evaluation based on Gaussian distribution to handle the problem of comprehensive water quality evaluation for a region across a period (multiple sections and multiple time points). The method turns the collection of observed water quality indicator values into a probability distribution of water quality grades by using the Gaussian distribution to compute the confidence assessment of water quality grades over one period. It eliminates the subjectivity involved in determining confidence levels and the problem of information loss during data fusion that arises with conventional approaches. The probability distribution of each assessment grade is then determined by repeatedly synthesizing evidence of the same water quality grade using the improved evidential reasoning synthesis rule. To avoid the subjectivity included in experience-based weight settings, principal component analysis (PCA) is utilized to calculate the weights of water quality indicators based on contribution rates and load coefficients. In the end, utility theory is presented to modify the discrete probability distribution of precise numerical expressions, offering thorough results for the evaluation of water quality and facilitating the comparison of various water quality grades. Using the Xiangjiang River Basin as a case study, the proposed evaluation method is contrasted with popular techniques for assessing water quality, including the Single-Factor Evaluation Method, the Fuzzy Comprehensive Evaluation Method, and the Evidential Reasoning Comprehensive Evaluation Method. The findings suggest that the evidence reasoning approach for evaluating water quality that is based on Gaussian distribution is more rational, accurate, and scientific. Additionally, empirical studies on the annual water quality trends in various regions, the upstream, midstream, and downstream trends, and the water quality trends during wet and dry periods are conducted using this method to assess and analyze changes in water quality in the Xiangjiang River Basin during the “11th Five-Year Plan” and “12th Five-Year Plan” periods. The analysis findings demonstrate that, even if the rate of progress has slowed, the Xiangjiang River Basin’s overall water quality has been steadily improving since management and protection measures were put in place. This shows that the preventive and control efforts implemented in the “11th Five-Year Plan” and “12th Five-Year Plan” periods were successful; nevertheless, carrying out the current tactics might only have a limited impact. As a result, more advanced and creative approaches are required to encourage the ongoing enhancement of the water quality in the Xiangjiang River Basin.

A comparative study of different deep learning methods for time-series probabilistic residential load power forecasting

The widespread adoption of nonlinear power electronic devices in residential settings has significantly increased the stochasticity and uncertainty of power systems. The original load power data, characterized by numerous irregular, random, and probabilistic components, adversely impacts the predictive performance of deep learning techniques, particularly neural networks. To address this challenge, this paper proposes a time-series probabilistic load power prediction technique based on the mature neural network point prediction technique, i.e., decomposing the load power data into deterministic and stochastic components. The deterministic component is predicted using deep learning neural network technology, the stochastic component is fitted with Gaussian mixture distribution model and the parameters are fitted using great expectation algorithm, after which the stochastic component prediction data is obtained using the stochastic component generation method. Using a mature neural network point prediction technique, the study evaluates six different deep learning methods to forecast residential load power. By comparing the prediction errors of these methods, the optimal model is identified, leading to a substantial improvement in prediction accuracy.

Remote Sensing Image Segmentation Based on Probabilistic Fuzzy Local Information Clustering

Abstract. In this paper, a remote sensing image segmentation based on probabilistic fuzzy local information clustering algorithm is proposed. First, assuming that the spectral measure within the same ground object follows the same probability distribution. The dissimilarity between pixel and object is modeled by the negative logarithm of the Gaussian probability density function. It can improve the noise sensitive problem caused by the Euclidean distance which can only describe the data with isotropic distribution. Then, in order to consider the effect of local spatial constraint, on the one hand, the probability measure is used to modify the local fuzzy factor to establish the dissimilarity measure with spatial constraints. On the other hand, the hidden Markov random field is used to model the prior probability model of pixel membership. Next, the entropy regularization term of the objective function is built by combining the Kullback-Leibler(KL) maximum entropy model to further improve the robustness and noise resistance. The qualitative and quantitative analysis of simulated image and different types of real remote sensing images show that the proposed algorithm can effectively overcome the above problems and further improve the accuracy of image segmentation to over 95%.

Reducing semantic ambiguity in domain adaptive semantic segmentation via probabilistic prototypical pixel contrast

Domain adaptation aims to reduce the model degradation on the target domain caused by the domain shift between the source and target domains. Although encouraging performance has been achieved by combining contrastive learning with the self-training paradigm, they suffer from ambiguous scenarios caused by scale, illumination, or overlapping when deploying deterministic embedding. To address these issues, we propose probabilistic prototypical pixel contrast (PPPC), a universal adaptation framework that models each pixel embedding as a probability via multivariate Gaussian distribution to fully exploit the uncertainty within them, eventually improving the representation quality of the model. In addition, we derive prototypes from probability estimation posterior probability estimation which helps to push the decision boundary away from the ambiguity points. Moreover, we employ an efficient method to compute similarity between distributions, eliminating the need for sampling and reparameterization, thereby significantly reducing computational overhead. Further, we dynamically select the ambiguous crops at the image level to enlarge the number of boundary points involved in contrastive learning, which benefits the establishment of precise distributions for each category. Extensive experimentation demonstrates that PPPC not only helps to address ambiguity at the pixel level, yielding discriminative representations but also achieves significant improvements in both synthetic-to-real and day-to-night adaptation tasks. It surpasses the previous state-of-the-art (SOTA) by +5.2% mIoU in the most challenging daytime-to-nighttime adaptation scenario, exhibiting stronger generalization on other unseen datasets. The code and models are available at https://github.com/DarlingInTheSV/Probabilistic-Prototypical-Pixel-Contrast.

Bayesian Ensemble Kalman Filter for Gaussian Mixture Models

AbstractInverse theory and data assimilation methods are commonly used in earth and environmental science studies to predict unknown variables, such as the physical properties of underground rocks, from a set of measured geophysical data, like geophysical seismic or electromagnetic data. A new Bayesian approach based on the ensemble Kalman filter using Gaussian mixture models is presented to overcome the assumption of Gaussian distribution of the unknown variables commonly used in the data assimilation literature and to generalize the algorithm to inverse problems with multimodal probability distributions. In applications of subsurface characterization, the multimodality of the unknown variables is generally due to the presence of different rock types, also known as geological facies. In the proposed method, the weights of the Gaussian mixture model represent the facies proportions, and they follow a Markov chain model. The proposed Bayesian model generates the unknown model parameters conditioned on measured data using a Markov chain Monte Carlo sampler. The validity of the method is demonstrated on a data assimilation problem where the goal is to estimate the posterior distribution of the unknown rock density from a set of repeated measurements of acoustic wave velocity measured at different times. The proposed method provides accurate estimates with efficient computational times.