分数阶反向累加与累减生成算子及互逆性

采用反向累加与反向累减的信息处理方式,将一阶累加生成算子和一阶累减生成算子拓展至分数阶范围,给出了分数阶反向累加生成算子与分数阶反向累减生成算子的解析表达式,并证明了两者之间的互逆性,为建立基于分数阶反向累加的灰色预测模型和拓宽灰色预测模型的应用范围提供理论基础.     

分数阶反向累加DGM(2,1)模型在低渗透油井产能预测中的应用

针对常见的灰色模型不能很好的解决新信息优先的问题,将分数阶融入到DGM(2,1)模型中,给出改进的分数阶反向累加DGM(2,1)模型(简记为RDGM(2,1)模型).RDGM(2,1)模型能够充分应用最新获得的数据,从中获取最有价值的信息.由于分数阶的灵活性,RDGM(2,1)模型能更好的反映数据中包含的信息,体现灰色模型处理少数据时精度高的优点.将RDGM(2,1)模型运用到油井的产能预测中,并与DGM(2,1)模型等灰色模型做了对比,结果表明RDGM(2,1)模型的模拟精度更高.     

分数阶灰色累加生成算子与累减生成算子及互逆性

针对现有灰色预测模型主要以一阶累加生成序列作为建模序列,再累减还原为原始序列预测值,本文通过Gamma函数将累加生成算子和累减生成算子拓展到正实数领域,给出分数阶累加生成算子和分数阶累减生成算子的解析表达式,一阶和整数阶均是其特例,证明了两算子之间的互逆性.为建立分数阶灰色预测模型和拓宽灰色预测模型的应用范围提供理论基础.     

基于分数阶累加的灰色GM(1,1|sin)模型及其应用

本文研究了灰色模型对振荡序列的预测问题.在已有GM(1,1|sin)模型的基础上,利用分数阶算子对原始序列进行累加生成的方法,获得了分数阶累加GM(1,1|sin)模型的表达式;以平均相对误差最小化为目标,利用粒子群算法求解非线性优化问题,获得了模型的最优参数.最后以城市交通流的模拟预测为例,结果表明本文提出的模型比GM(1,1|sin)模型具有更高的模拟精度,推广了GM(1,1|sin)预测模型的结果.     

非线性灰色Bernoulli模型的改进及应用

非线性灰色Bernoulli模型是灰色预测模型的一类拓展,在捕捉序列非线性趋势性能上表现良好,但仍然存在许多改进的空间.在传统的非线性灰色Bernoulli模型的基础上提出一种改进的方法,结合优化初始值,采用Guass-Newton算法求解最优模型参数以及滚动建模机制三个方面对模型进行改进.数值结果表明,优化初始值能够提高模型的预测精度,Guass-Newton算法寻求最优参数以及滚动建模机制能进一步减少预测误差的产生.因此,改进的模型能够有效地提高非线性灰色Bernoulli模型的预测性能.     

基于分数阶GM(1,1)与BP神经网络的电力负荷预测

传统的灰色GM(1,1)和BP神经网络模型存在对原始序列依赖高,收敛速度慢等缺点.将分数阶累加的思想引入GM(1,1)模型,再用逐层训练算法改进传统的BP神经网络.基于我国2010-2014年的电力数据,构建分数阶GM(1,1)与BP神经网络组合模型,预测2015年和2016年的总发电量.实证结果表明,该组合模型比GM(1,1)模型,分数阶GM(1,1)模型以及GM(1,1)与BP神经网络组合模型具有更好的数据拟合效果,更高的预测精度.     

具有时滞和周期特征的短时交通流灰建模及其应用

针对短时交通流的延迟性、随机性和周期性特征,采用灰关联分析和分数阶累加生成方法建立了带时滞和周期特征的分数阶累加灰色新模型.针对短时交通流的延迟性,将短时交通流数据拆分成参考时间序列和对应的比较时间序列,进行关联度分析,得到计算时滞值的方法.针对短时交通流的随机性和周期性,利用分数阶累加生成方法,并引入tan(kp)为发展系数,sin(kp)为输入变量,建立了短时交通流的分数阶GM(1,1|tan(kp),sin(kp))模型,给出了模型参数的最小二乘估计和周期性参数与分数阶阶数的优化求解算法.最后将模型应用于长沙市芙蓉区某交叉路口的交通流建模及预测中,并与常规的五种模型进行了对比分析,结果表明,模型能较为准确地反映交通流的实际情况,且有较高的预测精度和较为稳定的结果.     

基于分数阶GM(1,1)的图像去噪

针对传统整数阶GM(1,1)模型无法调节阶次来改善图像去噪性能,采用分数阶GM(1,1)模型来弥补其不足.分数阶GM(1,1)模型可以精确调节累加数之间数量级来达到更好的去噪效果.先通过在经典图像中添加椒盐噪声的去噪对比实验,得出分数阶GM(1,1)模型较GM(1,1)模型有更好的视觉效果,更高的峰值信噪比和结构相似度...     

分数阶反向累加非等间距GM(1,1)模型及应用

针对非等间距递减序列的预测问题,首先构建了一阶反向累加非等间距GM(1,1)模型(简称为非等间距GOM(1,1)模型),并给出了模型参数的最小二乘解和可用于预测的离散时间响应式.为进一步提高模拟预测精度,利用分数阶累加思想,提出了分数阶非等间距GOM(1,1)模型.以平均模拟相对误差最小化为目标,建立非线性规划模型可求解得到最优阶数.最后,以数值模拟和钛合金疲劳强度随温度变化预测为例,证实了该文提出模型的有效性和实用性.     

基于灰关联的GOM((p,q))(1,2)-BP神经网络模型及AQI预测

基于分数阶反向累加生成构建一种新的GM(1,2)模型,为使所构建模型能更好贴近和反映两个累加生成序列指标之间的真实关联关系,提出了基于不同序列采用不同累加阶数的GOM⁽(p,q))(1,2)模型.首先通过灰关联模型识别并筛选与特征序列关联度最大的相关因素序列,然后建立不同累加阶数的灰色模型,通过带压缩因子的粒子群优化算法求解模型最优阶数p和q,最后运用BP神经网络修正GOM⁽(p,q))(1,2)的模型值,构建GOM⁽(p,q))(1,2)-BP神经网络组合模型.模型应用于武汉市空气质量指数的预测,结果表明与单一模型相比,组合模型具有更好的性能和建模精度.     

The novel fractional discrete multivariate grey system model and its applications

Fractional order accumulation is a novel and popular tool which is efficient to improve accuracy of the grey models. However, most existing grey models with fractional order accumulation are all developed on the conventional methodology of grey models, which may be inaccurate in the applications. In this paper an existing fractional multivariate grey model with convolution integral is proved to be a biased model, and then a novel fractional discrete multivariate grey model based on discrete modelling technique is proposed, which is proved to be an unbiased model with mathematical analysis and stochastic testing. An algorithm based on the Grey Wolf Optimizer is introduced to optimize the fractional order of the proposed model. Four real world case studies with updated data sets are executed to assess the effectiveness of the proposed model in comparison with nine existing multivariate grey models. The results show that the Grey Wolf Optimizer-based algorithm is very efficient to optimize the fractional order of the proposed model, and the proposed model outperforms other nine models in the all the real world case studies.     

An optimized NGBM(1,1) model for forecasting the qualified discharge rate of industrial wastewater in China

The Nonlinear Grey Bernoulli Model NGBM(1,1) performs well in the simulation and forecasting of series having non-linear variations. To improve the simulation and forecasting accuracy, the parameters optimization of an NGBM(1,1) model is formulated as a combinatorial optimization problem and is solved collectively using LINGO (an Operational Research software) in this paper. The optimized result has been verified by a numerical example of a fluctuating sequence and a case study of opto-electronics industry in Taiwan. Comparisons of the obtained simulation results from the optimized combinatorial NGBM(1,1) model with the traditional one demonstrates that the optimal algorithm is a good alternative for parameters optimization of the NGBM(1,1) model. The optimized NGBM(1,1) model is used to simulate and forecast the annual qualified discharge rate of industrial wastewater in 31 provinces of China for the period from 2001 to 2011. The modeling results can assist the government in developing future policies regarding environmental management.     

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.     

分数阶累加生成GM~λ(1,1)模型的江苏省农民收入倍增能力预测

江苏省提出居民收入7年倍增计划.那么其农民收入能否同步倍增?基于分数阶累加生成GM~λ(1,1)灰色模型,采用2007—2012年江苏省农民收入数据,对其收入能力进行了预测.经计算发现,在MAPE误差允许范围内,选择分数阶λ值,可使预测结果更合理、更准确;最终结果表明江苏农民只需6年时间其收入就可以倍增.     

灰色预测GM(1,1)模型的改进及应用

应用自动寻优定权的方法和最小二乘法,研究了灰色系统理论中灰色预测GM(1,1)模型的预测公式的形成过程,发现灰色预测GM(1,1)模型在形成预测公式时对背景值和初始值的规定是不尽合理的,且现有的改进方法对灰色预测GM(1,1)模型的改进还不尽完善.为了提高灰色预测GM(1,1)模型的预测精度,提出并使用自动寻优定权对背景值进行选择,基于最小二乘法原理对灰色预测GM(1,1)模型的初始值进行改进.实例结果表明,提出的改进方法是有效和完善的,对灰色预测GM(1,1)模型的预测精度也有较大的提高.     

Nonlinear dynamic analysis of fractional order rub-impact rotor system

Nonlinear dynamic characteristics of rub-impact rotor system with fractional order damping are investigated. The model of rub-impact comprises a radial elastic force and a tangential Coulomb friction force. The fractional order damped rotor system with rubbing malfunction is established. The four order Runge–Kutta method and ten order CFE-Euler method are introduced to simulate the fractional order rub-impact rotor system equations. The effects of the rotating speed ratio, derivative order of damping and mass eccentricity on the system dynamics are investigated using rotor trajectory diagrams, bifurcation diagrams and Poincare map. Various complicated dynamic behaviors and types of routes to chaos are found, including period doubling bifurcation, sudden transition and quasi-periodic from periodic motion to chaos. The analysis results show that the fractional order rub-impact rotor system exhibits rich dynamic behaviors, and that the significant effect of fractional order will contribute to comprehensive understanding of nonlinear dynamics of rub-impact rotor.     

Hyperchaos in fractional order nonlinear systems

We numerically investigate hyperchaotic behavior in an autonomous nonlinear system of fractional order. It is demonstrated that hyperchaotic behavior of the integer order nonlinear system is preserved when the order becomes fractional. The system under study has been reported in the literature [Murali K, Tamasevicius A, Mykolaitis G, Namajunas A, Lindberg E. Hyperchaotic system with unstable oscillators. Nonlinear Phenom Complex Syst 3(1);2000:7–10], and consists of two nonlinearly coupled unstable oscillators, each consisting of an amplifier and an LC resonance loop. The fractional order model of this system is obtained by replacing one or both of its capacitors by fractional order capacitors. Hyperchaos is then assessed by studying the Lyapunov spectrum. The presence of multiple positive Lyapunov exponents in the spectrum is indicative of hyperchaos. Using the appropriate system control parameters, it is demonstrated that hyperchaotic attractors are obtained for a system order less than 4. Consequently, we present a conjecture that fourth-order hyperchaotic nonlinear systems can still produce hyperchaotic behavior with a total system order of 3 + ε, where 1 > ε > 0.     

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.     

贫信息背景下朱兰质量成本灰色GM(1,1)测算模型

朱兰曲线将质量总成本表示为质量投入与直接质量损失的合成.针对不同质量水平下的质量成本变化趋势,建立保证成本和故障成本的正负指数模型来确定最佳质量成本.灰色均值GM(1,1)测算模型可以同时克服"少数据""贫信息"的缺陷,更加精确地拟合质量成本的变动趋势.与指数函数模型模拟效果进行对比,结果表明灰色系统理论系列模型具有较高的预测精度,将指数函数模型和灰色GM(1,1)模型结合起来可以为企业提高质量管理水平尤其是质量成本管理提供依据和指导.     

北京市水资源利用建模分析研究

以北京市为例,分别应用无偏灰色GM(1,1)模型和非线性模型对北京市2001年-2010年的用水量进行了建模,利用最优化方法,计算了上述两种模型的最优组合模型,通过三种模型分别计算了北京市2001年-2010年的水资源利用量,并与北京市2001年-2010年的实际用水量进行了对比,采用精度检验方法,分别对无偏灰色模型,非线性模型和组合模型进行了精度检验,计算结果表明,加权组合模型是三种模型中精度最高的模型,通过组合模型计算得出的用水量值与实际水资源利用量相比误差最小,由此得出,可以利用组合模型对北京市未来的水资源利用量进行预测,预测结果可为其他相关研究提供参考.