分数阶反向累加非线性灰色Bernoulli模型及其应用

非线性灰色Bernoulli模型相对于普通的GM(1,1)模型,能更好的反映数据序列的非线性增长趋势.分数阶蕴含"in between"思想,分数阶累加灰色模型相对一般的累加灰色模型具有更好的预测效果和适应性.为了更好地符合新信息优先原理,实现最小信息的最大挖掘,构造了分数阶反向累加非线性灰色Bernoulli模型,即FAONGBM(1,1)模型,并给出了该模型的具体求解过程.在参数优化方面,本文通过粒子群优化(PSO)算法实现分数阶阶数和非线性指数的最优搜索.最后运用FAONGBM(1,1)模型对我国水力发电总量进行实证分析,结果证明所提出的模型具有良好的拟合精度和预测精度.     

基于统一灰色生成算子的三参数离散灰色预测模型及其应用

现有三参数离散灰色预测模型的累加阶数取值范围局限于正实数,导致模型建模能力和作用空间受限。为此,论文首先引入实数域统一灰色生成算子;然后,基于统一灰色生成算子构造了新型三参数离散灰色预测模型,实现了其阶数从正实数到全体实数的拓展与优化,从而使得新型模型具备挖掘时序数据积分特性与差异信息的双重功能;最后,将该新模型应用于某装甲装备维修经费的建模,结果显示其精度优于其它同类灰色模型。本研究成果对完善灰色算子基础理论及提高灰色预测模型建模能力具有重要价值。     

基于三角模糊数序列的灰色预测模型

以GM(1,1)模型为代表的灰色预测模型是以精确数序列为基础,难以满足实际需要.为了使灰色模型适应于模糊数序列,具体给出了一种基于三角模糊数序列的建模方法,这种方法也可以实现对二元区间模糊数和梯形模糊数序列的建模.首先由三角模糊数序列得出三个含有等量信息的精确数序列:重心序列、隶属函数的覆盖面积序列和中界点序列,对这三个序列分别建模后,再导出原始三角模糊数序列的三个界点的预测模型.这种建模方法既保持了模糊数的整体性又提高了建模序列的光滑度,提高了预测精度.最后进行了多组随机三角模糊数序列的数据模拟,验证了模型的有效性.     

A Hybrid grey wolf optimizer and genetic algorithm for minimizing potential energy function

In this paper, we propose a new hybrid algorithm between the grey wolf optimizer algorithm and the genetic algorithm in order to minimize a simplified model of the energy function of the molecule. We call the proposed algorithm by Hybrid Grey Wolf Optimizer and Genetic Algorithm (HGWOGA). We employ three procedures in the HGWOGA. In the first procedure, we apply the grey wolf optimizer algorithm to balance between the exploration and the exploitation process in the proposed algorithm. In the second procedure, we utilize the dimensionality reduction and the population partitioning processes by dividing the population into sub-populations and using the arithmetical crossover operator in each sub-population in order to increase the diversity of the search in the algorithm. In the last procedure, we apply the genetic mutation operator in the whole population in order to refrain from the premature convergence and trapping in local minima. We implement the proposed algorithm with various molecule size with up to 200 dimensions and compare the proposed algorithm with 8 benchmark algorithms in order to validate its efficiency for solving molecular potential energy function. The numerical experiment results show that the proposed algorithm is a promising, competent, and capable of finding the global minimum or near global minimum of the molecular energy function faster than the other comparative algorithms.     

A novel multivariate grey model for forecasting the sequence of ternary interval numbers

A novel multivariate grey model suitable for the sequence of ternary interval numbers is presented in the paper. New model takes into account the influencing factors on the system behavior characteristic. New parameter setting makes the model directly applicable to the sequence of ternary interval number without the need to convert the sequence into real sequence. A compensation coefficient taken as a ternary interval number is added to the model equation. The accumulation method based on the new information priority is proposed to estimate coefficients. A connotative prediction formula is derived to replace the white response equation of the classical multivariate grey model. The single variable grey model, which takes into account the development trend of system behavior itself, is combined with the novel multivariate grey model based on the degree of grey incidence. Interval forecasts for China's electricity generation and consumer price index show that the new model has good performance.     

融合改进Tent混沌和模拟退火的灰狼算法

针对标准灰狼算法种群多样性差、后期收敛速度慢、易陷入局部最优的缺陷,提出一种改进灰狼算法.利用改进Tent混沌映射初始化种群,增加种群多样性;引入螺旋函数,提高算法收敛速度;融合模拟退火思想,避免陷入局部最优;设置搜索阈值,平衡全局搜索与局部搜索;利用改进Tent混沌映射产生新个体,替换性能较差个体并进行高斯扰动,增加寻优精度;将当前解和新解进行算术杂交,以保留当前解优点并减小扰动差异.使用基准测试函数和共享单车停车点选址及期初配置模型测试算法性能.结果表明,改进灰狼算法较标准灰狼算法、遗传算法和粒子群算法,收敛速度更快,寻优精度更高,性能更优越,并将该算法应用到共享单车停车选址上,验证了算法的有效性.     

角度化灰色T型关联度模型及应用研究

通过对现有灰色关联度模型及算法的分析,首次提出了角度化灰色T型关联度模型。在分段线性表示的基础上,使用相邻线段间的夹角构成的角度序列近似表示时间序列,并给出了相关灰色关联系数和灰色关联度的计算方法。角度化灰色T型关联度模型不仅能够反映序列的正负相关关系,并且满足对称性、唯一性、可比性和规范性等性质。最后,通过实证分析证明了该模型的实用性和有效性。     

A new method to mitigate data fluctuations for time series prediction

Although the classic exponential-smoothing models and grey prediction models have been widely used in time series forecasting, this paper shows that they are susceptible to fluctuations in samples. A new fractional bidirectional weakening buffer operator for time series prediction is proposed in this paper. This new operator can effectively reduce the negative impact of unavoidable sample fluctuations. It overcomes limitations of existing weakening buffer operators, and permits better control of fluctuations from the entire sample period. Due to its good performance in improving stability of the series smoothness, the new operator can better capture the real developing trend in raw data and improve forecast accuracy. The paper then proposes a novel methodology that combines the new bidirectional weakening buffer operator and the classic grey prediction model. Through a number of case studies, this method is compared with several classic models, such as the exponential smoothing model and the autoregressive integrated moving average model, etc. Values of three error measures show that the new method outperforms other methods, especially when there are data fluctuations near the forecasting horizon. The relative advantages of the new method on small sample predictions are further investigated. Results demonstrate that model based on the proposed fractional bidirectional weakening buffer operator has higher forecasting accuracy.     

Grey system model with the fractional order accumulation

The perturbation theory of least squares method is applied to explain why the traditional accumulated generating operator violates the principle of new information priority of Grey system theory. A new Grey system model with the fractional order accumulation is put forward and the priority of new information can be better reflected when the accumulation order number becomes smaller in the in-sample model. But Grey system model cannot deal with the systems with memory when the accumulation order number is 0 in the in-sample model. The results of practical numerical examples demonstrate that the new Grey model provides very remarkable predication performance compared with the traditional Grey model for small sample.     

Discrete grey forecasting model and its optimization

Although the grey forecasting model has been successfully adopted in various fields and demonstrated promising results, the literatures show its performance could be further improved. For this purpose, this paper proposes a novel discrete grey forecasting model termed DGM model and a series of optimized models of DGM. This paper modifies the algorithm of GM(1, 1) model to enhance the tendency catching ability. The relationship between the two models and the forecasting precision of DGM model based on the pure index sequence is discussed. And further studies on three basic forms and three optimized forms of DGM model are also discussed. As shown in the results, the proposed model and its optimized models can increase the prediction accuracy. When the system is stable approximately, DGM model and the optimized models can effectively predict the developing system. This work contributes significantly to improve grey forecasting theory and proposes more novel grey forecasting models.     

A Linearized Spectral-Galerkin Method for Three-Dimensional Riesz-Like Space Fractional Nonlinear Coupled Reaction-Diffusion Equations

In this paper, we establish a novel fractional model arising in the chemical reaction and develop an efficient spectral method for the three-dimensional Riesz-like space fractional nonlinear coupled reaction-diffusion equations. Based on the backward difference method for time stepping and the Legendre-Galerkin spectral method for space discretization, we construct a fully discrete numerical scheme which leads to a linear algebraic system. Then a direct method based on the matrix diagonalization approach is proposed to solve the linear algebraic system, where the cost of the algorithm is of a small multiple of $N^4$ ($N$ is the polynomial degree in each spatial coordinate) flops for each time level. In addition, the stability and convergence analysis are rigorously established. We obtain the optimal error estimate in space, and the results also show that the fully discrete scheme is unconditionally stable and convergent of order one in time. Furthermore, numerical experiments are presented to confirm the theoretical claims. As the applications of the proposed method, the fractional Gray-Scott model is solved to capture the pattern formation with an analysis of the properties of the fractional powers.     

分数阶灰色累减生成算子及其性质研究

给出了分数阶灰色累减生成算子的详细推导过程,并证明了分数阶灰色累减生成算子的不动点定理、信息优先原理、交换律与指数律,为分数阶灰色预测模型提供了理论基础.算例验证了分数阶灰色累减生成算子的特征,在灰色预测模型GM(1,1)中的应用证明了分数阶灰色累减生成算子的有效性.     

Examination and prediction of fog and haze pollution using a Multi-variable Grey Model based on interval number sequences

In this study, a new Multivariable Grey Model (1,m) aimed at interval grey number sequences with known possibility functions is built using the kernel and degree of greyness under new definitions. Based on the new model, formulae are deduced to calculate and predict the upper and lower bounds of interval grey numbers. Since the grey system model and fog- and haze-prone weather have the same characteristics of uncertainty, this model was applied to simulate and predict the measurable indicators of fog and haze in Nanjing, China. We selected visibility data and particulate matter data with a diameter of 2.5 µm to build a new Multivariable Grey Model (1,2) with a new kernel and degree of greyness sequence. In addition, we established the traditional Multivariable Grey Model (1,2) with the original kernel and degree of greyness and the Auto-Regressive Integrated Moving Average Model (1,1,0). The results show that the new Multivariable Grey Model (1,2) has the best simulation and prediction effects among the three models, with average relative errors of simulation and prediction at 1.32% and 0.32%, respectively. To further verify the validity and feasibility of the proposed model, we added another real-world example to establish the three models mentioned above. The results prove that the proposed model has evidently superior performance to another two models.     

A gray model for increasing sequences with nonhomogeneous index trends based on fractional‐order accumulation

By comparing the class ratio deviation and restoring error of first‐order accumulation with that of fractional‐order accumulation, a gray model for monotonically increasing sequences can obtain optimal simulation accuracy via selecting a proper cumulative order. In this study, a gray model for increasing sequences with nonhomogeneous index trends based on fractional‐order accumulation is proposed. To reduce the modeling error caused by the background value and to improve the prediction accuracy of the model, an optimized model using the 3/8 Simpson formula is constructed. Finally, the 2 proposed models are used to predict the total energy consumption in China and the monthly sales of new products in an enterprise. Compared with the GM(1,1) model based on fractional‐order accumulation, the proposed model exhibits better simulation and prediction accuracy.     

Computationally efficient learning of multivariate <Emphasis Type="Italic">t</Emphasis> mixture models with missing information

A finite mixture model using the multivariate t distribution has been well recognized as a robust extension of Gaussian mixtures. This paper presents an efficient PX-EM algorithm for supervised learning of multivariate t mixture models in the presence of missing values. To simplify the development of new theoretic results and facilitate the implementation of the PX-EM algorithm, two auxiliary indicator matrices are incorporated into the model and shown to be effective. The proposed methodology is a flexible mixture analyzer that allows practitioners to handle real-world multivariate data sets with complex missing patterns in a more efficient manner. The performance of computational aspects is investigated through a simulation study and the procedure is also applied to the analysis of real data with varying proportions of synthetic missing values.     

模糊需求下订购批量问题的研究

库存管理模型在现实生活中有着广泛的运用,它为管理决策者有效地确定最佳订购批量提供帮助.然而,由于历史数据的缺乏,需求量在很多情况下往往被主观地确定,因而带有一定的模糊性.本文针对两种不同类型的模糊需求:离散型与连续型,运用模糊理论分别建立了相应的模糊库存模型.该模型不同于已有的模糊库存模型如下:在现有的模糊库存的文献中,大多采用的是利用模糊集的知识对确定EOQ模型加以研究,而本文从模糊理论的角度对报童问题进行研究.     

Post-processing of Markov chain Monte Carlo output in Bayesian latent variable models with application to multidimensional scaling

In Bayesian analysis of multidimensional scaling model with MCMC algorithm, we encounter the indeterminacy of rotation, reflection and translation of the parameter matrix of interest. This type of indeterminacy may be seen in other multivariate latent variable models as well. In this paper, we propose to address this indeterminacy problem with a novel, offline post-processing method that is easily implemented using easy-to-use Markov chain Monte Carlo (MCMC) software. Specifically, we propose a post-processing method based on the generalized extended Procrustes analysis to address this problem. The proposed method is compared with four existing methods to deal with indeterminacy thorough analyses of artificial as well as real datasets. The proposed method achieved at least as good a performance as the best existing method. The benefit of the offline processing approach in the era of easy-to-use MCMC software is discussed.     

Fast evaluation and high accuracy finite element approximation for the time fractional subdiffusion equation

In this article, an efficient algorithm for the evaluation of the Caputo fractional derivative and the superconvergence property of fully discrete finite element approximation for the time fractional subdiffusion equation are considered. First, the space semidiscrete finite element approximation scheme for the constant coefficient problem is derived and supercloseness result is proved. The time discretization is based on the L1‐type formula, whereas the space discretization is done using, the fully discrete scheme is developed. Under some regularity assumptions, the superconvergence estimate is proposed and analyzed. Then, extension to the case of variable coefficients is also discussed. To reduce the computational cost, the fast evaluation scheme of the Caputo fractional derivative to solve the fractional diffusion equations is designed. Finally, numerical experiments are presented to support the theoretical results.     

Learning Fuzzy Grey Cognitive Maps using Nonlinear Hebbian-based approach

Recently, Fuzzy Grey Cognitive Maps (FGCM) has been proposed as a FCM extension. It is based on Grey System Theory, that it has become a very effective theory for solving problems within environments with high uncertainty, under discrete small and incomplete data sets. The proposed approach of learning FGCMs applies the Nonlinear Hebbian based algorithm determine the success of radiation therapy process estimating the final dose delivered to the target volume. The scope of this research is to explore an alternative decision support method using the main aspects of fuzzy logic and grey systems to cope with the uncertainty inherent in medical domain and physicians uncertainty to describe numerically the influences among concepts in medical domain. The Supervisor-FGCM, trained by NHL algorithm adapted in FGCMs, determines the treatment variables of cancer therapy and the acceptance level of final radiation dose to the target volume. Three clinical case studies were used to test the proposed methodology with meaningful and promising results and prove the efficiency of the NHL algorithm for FGCM approach.     

A Fast Conservative Scheme for the Space Fractional Nonlinear Schrödinger Equation with Wave Operator

A new efficient compact difference scheme is proposed for solving a space fractional nonlinear Schrödinger equation with wave operator. The scheme is proved to conserve the total mass and total energy in a discrete sense. Using the energy method, the proposed scheme is proved to be unconditionally stable and its convergence order is shown to be of $ \mathcal{O}( h^6 + \tau^2) $ in the discrete $ L_2 $ norm with mesh size $ h $ and the time step $ \tau $. Moreover, a fast difference solver is developed to speed up the numerical computation of the scheme. Numerical experiments are given to support the theoretical analysis and to verify the efficiency, accuracy, and discrete conservation laws.