Analysis of Extended Fisher-Kolmogorov Equation in 2D Utilizing the Generalized Finite Difference Method with Supplementary Nodes

In this study,we propose an efficient numerical framework to attain the solution of the extended Fisher-Kolmogorov(EFK)problem.The temporal derivative in the EFK equation is approximated by utilizing the Crank-Nicolson scheme.Following temporal discretization,the generalized finite difference method(GFDM)with supplementary nodes is utilized to address the nonlinear boundary value problems at each time node.These supplementary nodes are distributed along the boundary to match the number of boundary nodes.By incorporating supplementary nodes,the resulting nonlinear algebraic equations can effectively satisfy the governing equation and boundary conditions of the EFK equation.To demonstrate the efficacy of our approach,we present three numerical examples showcasing its performance in solving this nonlinear problem.

Extended Generalized Riccati Equation Mapping for Thermal Traveling-Wave Distribution in Biological Tissues through a Bio-Heat Transfer Model with Linear/Quadratic Temperature-Dependent Blood Perfusion

Analytical thermal traveling-wave distribution in biological tissues through a bio-heat transfer (BHT) model with linear/quadratic temperature-dependent blood perfusion is discussed in this paper. Using the extended generalized Riccati equation mapping method, we find analytical traveling wave solutions of the considered BHT equation. All the travelling wave solutions obtained have been used to explicitly investigate the effect of linear and quadratic coefficients of temperature dependence on temperature distribution in tissues. We found that the parameter of the nonlinear superposition formula for Riccati can be used to control the temperature of living tissues. Our results prove that the extended generalized Riccati equation mapping method is a powerful tool for investigating thermal traveling-wave distribution in biological tissues.

On a Novel Extended Lomax Distribution with Asymmetric Properties and Its Statistical Applications

In this article,we highlight a new three-parameter heavy-tailed lifetime distribution that aims to extend the modeling possibilities of the Lomax distribution.It is called the extended Lomax distribution.The considered distribution naturally appears as the distribution of a transformation of a random variable following the logweighted power distribution recently introduced for percentage or proportion data analysis purposes.As a result,its cumulative distribution has the same functional basis as that of the Lomax distribution,but with a novel special logarithmic term depending on several parameters.The modulation of this logarithmic term reveals new types of asymetrical shapes,implying a modeling horizon beyond that of the Lomax distribution.In the first part,we examine several of its mathematical properties,such as the shapes of the related probability and hazard rate functions;stochastic comparisons;manageable expansions for various moments;and quantile properties.In particular,based on the quantile functions,various actuarial measures are discussed.In the second part,the distribution’s applicability is investigated with the use of themaximumlikelihood estimationmethod.The behavior of the obtained parameter estimates is validated by a simulation work.Insurance claim data are analyzed.We show that the proposed distribution outperforms eight well-known distributions,including the Lomax distribution and several extended Lomax distributions.In addition,we demonstrate that it gives preferable inferences from these competitor distributions in terms of risk measures.

The New Mixed Generalized Erlang Distribution

In probability theory, the mixture distribution M has a density function for the collection of random variables and weighted by wi ≥ 0 and . These mixed distributions are used in various disciplines and aim to enrich the collection distribution to more parameters. A more general mixture is derived by Kadri and Halat, by proving the existence of such mixture by wi ∈ R, and maintaining . Kadri and Halat provided many examples and applications for such new mixed distributions. In this paper, we introduce a new mixed distribution of the Generalized Erlang distribution, which is derived from the Hypoexponential distribution. We characterize this new distribution by deriving simply closed expressions for the related functions of the probability density function, cumulative distribution function, moment generating function, reliability function, hazard function, and moments.

Analysis of College Students’ Test Scores Based on Two-Component Mixed Generalized Normal Distribution

In order to improve the fitting accuracy of college students’ test scores, this paper proposes two-component mixed generalized normal distribution, uses maximum likelihood estimation method and Expectation Conditional Maxinnization (ECM) algorithm to estimate parameters and conduct numerical simulation, and performs fitting analysis on the test scores of Linear Algebra and Advanced Mathematics of F University. The empirical results show that the two-component mixed generalized normal distribution is better than the commonly used two-component mixed normal distribution in fitting college students’ test data, and has good application value.

Comparative study on isolation forest, extended isolation forest and generalized isolation forest in detection of multivariate geochemical anomalies

It is not easy to construct a model to describe the geochemical background in geochemical anomaly detection due to the complexity of the geological setting.Isolation forest and its improved algorithms can detect geochemical anomalies without modeling the complex geochemical background.These methods can effec-tively extract multivariate anomalies from large volume of high-dimensional geochemical data with unknown population distribution.To test the performance of these algorithms in the detection of mineralization-related geochemical anomalies,the isolation forest,extended isolation forest and generalized isolation forest models were established to detect multivariate anomalies from the stream sediment survey data collected in the Wu-laga area in Heilongjiang Province.The geochemical anomalies detected by the generalized isolation forest model account for 40%of the study area,and contain 100%of the known gold deposits.The geochemical anomalies detected by the isolation forest model account for 20%of the study area,and contain 71%of the known gold deposits.The geochemical anomalies detected by the extended isolation forest algorithm account for 34%of the study area,and contain 100%of the known gold deposits.Therefore,the isolation forest mo-del,extended isolation fo-rest model and generalized isolation forest model are comparable in geochemical anomaly detection.

Federated Learning Model for Auto Insurance Rate Setting Based on Tweedie Distribution

In the assessment of car insurance claims,the claim rate for car insurance presents a highly skewed probability distribution,which is typically modeled using Tweedie distribution.The traditional approach to obtaining the Tweedie regression model involves training on a centralized dataset,when the data is provided by multiple parties,training a privacy-preserving Tweedie regression model without exchanging raw data becomes a challenge.To address this issue,this study introduces a novel vertical federated learning-based Tweedie regression algorithm for multi-party auto insurance rate setting in data silos.The algorithm can keep sensitive data locally and uses privacy-preserving techniques to achieve intersection operations between the two parties holding the data.After determining which entities are shared,the participants train the model locally using the shared entity data to obtain the local generalized linear model intermediate parameters.The homomorphic encryption algorithms are introduced to interact with and update the model intermediate parameters to collaboratively complete the joint training of the car insurance rate-setting model.Performance tests on two publicly available datasets show that the proposed federated Tweedie regression algorithm can effectively generate Tweedie regression models that leverage the value of data fromboth partieswithout exchanging data.The assessment results of the scheme approach those of the Tweedie regressionmodel learned fromcentralized data,and outperformthe Tweedie regressionmodel learned independently by a single party.

A Novel Defender-Attacker-Defender Model for Resilient Distributed Generator Planning with Network Reconfiguration and Demand Response

To improve the resilience of a distribution system against extreme weather,a fuel-based distributed generator(DG)allocation model is proposed in this study.In this model,the DGs are placed at the planning stage.When an extreme event occurs,the controllable generators form temporary microgrids(MGs)to restore the load maximally.Simultaneously,a demand response program(DRP)mitigates the imbalance between the power supply and demand during extreme events.To cope with the fault uncertainty,a robust optimization(RO)method is applied to reduce the long-term investment and short-term operation costs.The optimization is formulated as a tri-level defenderattacker-defender(DAD)framework.At the first level,decision-makers work out the DG allocation scheme;at the second level,the attacker finds the optimal attack strategy with maximum damage;and at the third level,restoration measures,namely distribution network reconfiguration(DNR)and demand response are performed.The problem is solved by the nested column and constraint generation(NC&CG)method and the model is validated using an IEEE 33-node system.Case studies validate the effectiveness and superiority of the proposed model according to the enhanced resilience and reduced cost.

Robust multi-task distributed estimation based on generalized maximum correntropy criterion

False data injection(FDI) attacks are common in the distributed estimation of multi-task network environments, so an attack detection strategy is designed by combining the generalized maximum correntropy criterion. Based on this, we propose a diffusion least-mean-square algorithm based on the generalized maximum correntropy criterion(GMCC-DLMS)for multi-task networks. The algorithm achieves gratifying estimation results. Even more, compared to the related work,it has better robustness when the number of attacked nodes increases. Moreover, the assumption about the number of attacked nodes is relaxed, which is applicable to multi-task environments. In addition, the performance of the proposed GMCC-DLMS algorithm is analyzed in the mean and mean-square senses. Finally, simulation experiments confirm the performance and effectiveness against FDI attacks of the algorithm.

Brain Functional Network Generation Using Distribution-Regularized Adversarial Graph Autoencoder with Transformer for Dementia Diagnosis

The topological connectivity information derived from the brain functional network can bring new insights for diagnosing and analyzing dementia disorders.The brain functional network is suitable to bridge the correlation between abnormal connectivities and dementia disorders.However,it is challenging to access considerable amounts of brain functional network data,which hinders the widespread application of data-driven models in dementia diagnosis.In this study,a novel distribution-regularized adversarial graph auto-Encoder(DAGAE)with transformer is proposed to generate new fake brain functional networks to augment the brain functional network dataset,improving the dementia diagnosis accuracy of data-driven models.Specifically,the label distribution is estimated to regularize the latent space learned by the graph encoder,which canmake the learning process stable and the learned representation robust.Also,the transformer generator is devised to map the node representations into node-to-node connections by exploring the long-term dependence of highly-correlated distant brain regions.The typical topological properties and discriminative features can be preserved entirely.Furthermore,the generated brain functional networks improve the prediction performance using different classifiers,which can be applied to analyze other cognitive diseases.Attempts on the Alzheimer’s Disease Neuroimaging Initiative(ADNI)dataset demonstrate that the proposed model can generate good brain functional networks.The classification results show adding generated data can achieve the best accuracy value of 85.33%,sensitivity value of 84.00%,specificity value of 86.67%.The proposed model also achieves superior performance compared with other related augmentedmodels.Overall,the proposedmodel effectively improves cognitive disease diagnosis by generating diverse brain functional networks.

Statistical analysis of generalized exponential distribution under progressive censoring with binomial removals

The estimation of generalized exponential distribution based on progressive censoring with binomial removals is presented, where the number of units removed at each failure time follows a binomial distribution. Maximum likelihood estimators of the parameters and their confidence intervals are derived. The expected time required to complete the life test under this censoring scheme is investigated. Finally, the numerical examples are given to illustrate some theoretical results by means of Monte-Carlo simulation.

SPECTRUM DISTRIBUTION OF THE SEDOND ORDER GENERALIZED DISTRIBUTED PARAMETER SYSTEMS

Spectrum distribution of the second order generalized distributed parameter system was discussed via the functional analysis and operator theory in Hilbert space. The solutions of the problem and the constructive expression of the solutions are given by the generalized inverse one of bounded linear operator. This is theoretically important for studying the stabilization and asymptotic stability of the second order generalized distributed parameter system.

GENERALIZED OPERATORS AND OPERATOR-VALUED DISTRIBUTIONS IN QUANTUM FIELD THEORY

The relation between generalized operators and operator-valued distributions is discussed so that these two viewpoints can be used alternatively to explain quantum fields.

GENERALIZED PARETO DISTRIBUTION FIT TO MEDICAL INSURANCE CLAIMS DATA

How to choose an optimal threshold is a key problem in the generalized Pareto distribution （GPD） model. This paper attains the exact threshold by testing for GPD,and shows that GPD model allows the actuary to easily estimate high quantiles and the probable maximum loss from the medical insurance claims data.

The exponentiated generalized power Lindley distribution: Properties and applications

In this paper, we introduce a new extension of the power Lindley distribution, called the exponentiated generalized power Lindley distribution. Several mathematical properties of the new model such as the shapes of the density and hazard rate functions, the quantile function, moments, mean deviations, Bonferroni and Lorenz curves and order statistics are derived. Moreover, we discuss the parameter estimation of the new distribution using the maximum likelihood and diagonally weighted least squares methods. A simulation study is performed to evaluate the estimators. We use two real data sets to illustrate the applicability of the new model. Empirical findings show that the proposed model provides better fits than some other well-known extensions of Lindley distributions.

Prediction Based on Generalized Order Statistics from a Mixture of Rayleigh Distributions Using MCMC Algorithm

This article considers the problem in obtaining the maximum likelihood prediction (point and interval) and Bayesian prediction (point and interval) for a future observation from mixture of two Rayleigh (MTR) distributions based on generalized order statistics (GOS). We consider one-sample and two-sample prediction schemes using the Markov chain Monte Carlo (MCMC) algorithm. The conjugate prior is used to carry out the Bayesian analysis. The results are specialized to upper record values. Numerical example is presented in the methods proposed in this paper.

Importance of Generalized Logistic Distribution in Extreme Value Modeling

We consider a problem from stock market modeling, precisely, choice of adequate distribution of modeling extremal behavior of stock market data. Generalized extreme value (GEV) distribution and generalized Pareto (GP) distribution are the classical distributions for this problem. However, from 2004, [1] and many other researchers have been empirically showing that generalized logistic (GL) distribution is a better model than GEV and GP distributions in modeling extreme movement of stock market data. In this paper, we show that these results are not accidental. We prove the theoretical importance of GL distribution in extreme value modeling. For proving this, we introduce a general multivariate limit theorem and deduce some important multivariate theorems in probability as special cases. By using the theorem, we derive a limit theorem in extreme value theory, where GL distribution plays central role instead of GEV distribution. The proof of this result is parallel to the proof of classical extremal types theorem, in the sense that, it possess important characteristic in classical extreme value theory, for e.g. distributional property, stability, convergence and multivariate extension etc.

Generalized Pareto Distribution Model and Its Application to Hydrocarbon Resource Structure Prediction of the Huanghua Depression

The generalized Pareto distribution model is a kind of hydrocarbon pool size probability statistical method for resource assessment. By introducing the time variable, resource conversion rate and the geological variable, resource density, such model can describe not only different types of basins, but also any exploration samples at different phases of exploration, up to the parent population. It is a dynamic distribution model with profound geological significance and wide applicability. Its basic principle and the process of resource assessment are described in this paper. The petroleum accumulation system is an appropriate assessment unit for such method. The hydrocarbon resource structure of the Huanghua Depression in Bohai Bay Basin was predicted by using this model. The prediction results accord with the knowledge of exploration in the Huanghua Depression, and point out the remaining resources potential and structure of different petroleum accumulation systems, which are of great significance for guiding future exploration in the Huanghua Depression.

Likelihood and Quadratic Distance Methods for the Generalized Asymmetric Laplace Distribution for Financial Data

Maximum likelihood (ML) estimation for the generalized asymmetric Laplace (GAL) distribution also known as Variance gamma using simplex direct search algorithms is investigated. In this paper, we use numerical direct search techniques for maximizing the log-likelihood to obtain ML estimators instead of using the traditional EM algorithm. The density function of the GAL is only continuous but not differentiable with respect to the parameters and the appearance of the Bessel function in the density make it difficult to obtain the asymptotic covariance matrix for the entire GAL family. Using M-estimation theory, the properties of the ML estimators are investigated in this paper. The ML estimators are shown to be consistent for the GAL family and their asymptotic normality can only be guaranteed for the asymmetric Laplace (AL) family. The asymptotic covariance matrix is obtained for the AL family and it completes the results obtained previously in the literature. For the general GAL model, alternative methods of inferences based on quadratic distances (QD) are proposed. The QD methods appear to be overall more efficient than likelihood methods infinite samples using sample sizes n ≤5000 and the range of parameters often encountered for financial data. The proposed methods only require that the moment generating function of the parametric model exists and has a closed form expression and can be used for other models.

Evaluating Traffic Congestion Using the Traffic Occupancy and Speed Distribution Relationship: An Application of Bayesian Dirichlet Process Mixtures of Generalized Linear Model

Accurate classification and prediction of future traffic conditions are essential for developing effective strategies for congestion mitigation on the highway systems. Speed distribution is one of the traffic stream parameters, which has been used to quantify the traffic conditions. Previous studies have shown that multi-modal probability distribution of speeds gives excellent results when simultaneously evaluating congested and free-flow traffic conditions. However, most of these previous analytical studies do not incorporate the influencing factors in characterizing these conditions. This study evaluates the impact of traffic occupancy on the multi-state speed distribution using the Bayesian Dirichlet Process Mixtures of Generalized Linear Models (DPM-GLM). Further, the study estimates the speed cut-point values of traffic states, which separate them into homogeneous groups using Bayesian change-point detection (BCD) technique. The study used 2015 archived one-year traffic data collected on Florida’s Interstate 295 freeway corridor. Information criteria results revealed three traffic states, which were identified as free-flow, transitional flow condition (congestion onset/offset), and the congested condition. The findings of the DPM-GLM indicated that in all estimated states, the traffic speed decreases when traffic occupancy increases. Comparison of the influence of traffic occupancy between traffic states showed that traffic occupancy has more impact on the free-flow and the congested state than on the transitional flow condition. With respect to estimating the threshold speed value, the results of the BCD model revealed promising findings in characterizing levels of traffic congestion.