Fuzzy linear regression model based on fuzzy scalar product

The new concept and method of imposing imprecise (fuzzy) input and output data upon the conventional linear regression model is proposed in this paper. We introduce the fuzzy scalar (inner) product to formulate the fuzzy linear regression model. In order to invoke the conventional approach of linear regression analysis for real-valued data, we transact the α-level linear regression models of the fuzzy linear regression model. We construct the membership functions of fuzzy least squares estimators via the form of “Resolution Identity” which is a well-known formula in fuzzy sets theory. In order to obtain the membership value of any given least squares estimate taken from the fuzzy least squares estimator, we transform the original problem into the optimization problems. We also provide two computational procedures to solve the optimization problems.     

Fuzzy least absolute linear regression

The distance between triangular fuzzy numbers is an important research topic in many fields. In this paper, we introduce a new distance between triangular fuzzy numbers, merge least absolute deviation method with the new distance and propose fuzzy regression model. We also investigate the properties and model algorithm of fuzzy least absolute linear regression model in detail by transforming this model into linear programming. Further, we use three numerical examples to illustrate our proposed model reasonable and make some comparisons with some existing fuzzy regression models. Finally, we investigate the robust property of our proposed model and apply our model in the missing data set to verify model data.     

Fuzzy least squares regression model based of weighted distance between fuzzy numbers

A new distance ND1 between fuzzy numbers based on an averaged representative of a fuzzy number (Weighting Average Based on Levels, WABL) is proposed. Based on this distance a new least squares regression model is proposed. The proposed model is studied for a broad class of fuzzy numbers and class of functions the membership of which is formed on the basis of the template μ_[.](x) = max(0.1 ? |x|^s).     

Monte Carlo methods in fuzzy linear regression

We apply our new fuzzy Monte Carlo method to a certain fuzzy linear regression problem to estimate the best solution. The best solution is a vector of triangular fuzzy numbers, for the fuzzy coefficients in the model, which minimizes one of two error measures. We use a quasi-random number generator to produce random sequences of these fuzzy vectors which uniformly fill the search space. We consider an example problem and show this Monte Carlo method obtains the best solution for one error measure and is approximately best for the other error measure.     

Error measures for fuzzy linear regression: Monte Carlo simulation approach

The focus of this study is to use Monte Carlo method in fuzzy linear regression. The purpose of the study is to figure out the appropriate error measures for the estimation of fuzzy linear regression model parameters with Monte Carlo method. Since model parameters are estimated without any mathematical programming or heavy fuzzy arithmetic operations in fuzzy linear regression with Monte Carlo method. In the literature, only two error measures (E₁ and E₂) are available for the estimation of fuzzy linear regression model parameters. Additionally, accuracy of available error measures under the Monte Carlo procedure has not been evaluated. In this article, mean square error, mean percentage error, mean absolute percentage error, and symmetric mean absolute percentage error are proposed for the estimation of fuzzy linear regression model parameters with Monte Carlo method. Moreover, estimation accuracies of existing and proposed error measures are explored. Error measures are compared to each other in terms of estimation accuracy; hence, this study demonstrates that the best error measures to estimate fuzzy linear regression model parameters with Monte Carlo method are proved to be E₁, E₂, and the mean square error. One the other hand, the worst one can be given as the mean percentage error. These results would be useful to enrich the studies that have already focused on fuzzy linear regression models.     

Fuzzy clusterwise linear regression analysis with symmetrical fuzzy output variable

The traditional regression analysis is usually applied to homogeneous observations. However, there are several real situations where the observations are not homogeneous. In these cases, by utilizing the traditional regression, we have a loss of performance in fitting terms. Then, for improving the goodness of fit, it is more suitable to apply the so-called clusterwise regression analysis. The aim of clusterwise linear regression analysis is to embed the techniques of clustering into regression analysis. In this way, the clustering methods are utilized for overcoming the heterogeneity problem in regression analysis. Furthermore, by integrating cluster analysis into the regression framework, the regression parameters (regression analysis) and membership degrees (cluster analysis) can be estimated simultaneously by optimizing one single objective function. In this paper the clusterwise linear regression has been analyzed in a fuzzy framework. In particular, a fuzzy clusterwise linear regression model (FCWLR model) with symmetrical fuzzy output and crisp input variables for performing fuzzy cluster analysis within a fuzzy linear regression framework is suggested. For measuring the goodness of fit of the suggested FCWLR model with fuzzy output, a fitting index is proposed. In order to illustrate the usefulness of FCWLR model in practice, several applications to artificial and real datasets are shown.     

基于模糊线性回归的脑卒中发病率预测模型

针对经典线性回归模型不能完全反映变量间的耦合关系而不适宜有模糊数的脑卒中发病率预测的问题,建立了一种模糊多元线性回归分析的脑卒中发病率预测模型。把历史数据分为建模数据样本和检测数据样本,采用线性规划法求出参数的中心值和模糊幅度值。实验结果表明,该模型具有较高的精确度和可操作性。     

Least squares estimation of a linear regression model with LR fuzzy response

The problem of regression analysis in a fuzzy setting is discussed. A general linear regression model for studying the dependence of a LR fuzzy response variable on a set of crisp explanatory variables, along with a suitable iterative least squares estimation procedure, is introduced. This model is then framed within a wider strategy of analysis, capable to manage various types of uncertainty. These include the imprecision of the regression coefficients and the choice of a specific parametric model within a given class of models. The first source of uncertainty is dealt with by exploiting the implicit fuzzy arithmetic relationships between the spreads of the regression coefficients and the spreads of the response variable. Concerning the second kind of uncertainty, a suitable selection procedure is illustrated. This consists in maximizing an appropriately introduced goodness of fit index, within the given class of parametric models. The above strategy is illustrated in detail, with reference to an application to real data collected in the framework of an environmental study. In the final remarks, some critical points are underlined, along with a few indications for future research in this field.     

基于灰色多元线性回归融合模型的故障率预测

为预测在设备使用年份期间的制氧系统故障率,提出灰色多元线性回归融合模型的新方法。该方法首先求出制氧系统各设备故障率的GM(1,1)模型;然后计算出制氧系统故障率、制氧系统各设备故障率与设备使用年份相关关系模型,并且将制氧系统各设备故障率的GM(1,1)模型代入该关系模型中;最后利用最小二乘法求出待定参数。通过对制氧系统故障率的预测分析表明,灰色多元线性回归融合模型在故障率预测精度上优于单一的灰色模型和线性回归模型,且不要求提供的历史数据具有典型的分布规律。该模型的预测结果可为制氧系统的维修工作提供决策依据。     

基于模糊子空间聚类的0 阶岭回归TSK 模糊系统

经典数据驱动型TSK模糊系统在利用高维数据训练模型时,由于规则前件采用的特征过多,导致规则的解释性和简洁性下降.对此,根据模糊子空间聚类算法的子空间特性,为TSK模型添加特征抽取机制,并进一步利用岭回归实现后件的学习,提出一种基于模糊子空间聚类的0阶岭回归TSK模型构建方法.该方法不仅能为规则抽取出重要子空间特征,而且可为不同规则抽取不同的特征.在模拟和真实数据集上的实验结果验证了所提出方法的优势.     

Semi-parametric partially logistic regression model with exact inputs and intuitionistic fuzzy outputs

Some accounting studies have focused on logistic regression relationships between exact/fuzzy inputs/outputs. However, intuitionistic fuzzy sets find application in many real studies instead of fuzzy sets. On the other hand, semi-parametric partially linear model also has attracted attentions in recent years. This study is an investigation of intuitionistic fuzzy semi-parametric partially logistic model for such cases with exact inputs, intuitionistic fuzzy outputs, intuitionistic fuzzy smooth function and intuitionistic fuzzy coefficients. For this purpose, a hybrid procedure is suggested based on curve fitting methods and least absolutes deviations to estimate the intuitionistic fuzzy smooth function and intuitionistic fuzzy coefficients. The proposed method is also compared with a common fuzzy logistic regression model as a real fuzzy data set. It is shown that the proposed intuitionistic fuzzy logistic regression model performs better and efficient results in regard to some goodness-of-fit criteria suggest that the proposed model could be successfully applied in many practical studies of intuitionistic fuzzy logistic regression model in expert systems.     

基于灰色GM(1,N)的模糊多元线性回归分析及应用

建立了基于对称三角模糊数的多元线性回归分析模型(简记为F L R模型),利用线性规划求出中心值和模糊度。以我国1995年到2008年粮食产量(来自《中国统计年鉴2009》)为原始数据,进行了多因素模糊拟合分析。利用GM(1,N)模型对2009年至2013年影响我国粮食产量的5个因素指标值进行了预测,将预测值代入FLR模型求出年度粮食产量,并与2009和2010年的实际产量比较,表明这种GM(1,N)模型和FLR模型有机结合形成的复合模型,预测精度高,可操作性强,且具有很高的可信度。     

T-S模糊随机系统的均方镇定

提出一类基于T-S模糊模型的非线性随机系统均方镇定的线性矩阵不等式(LMI)设计方法．利用非线性随机系统的Lyapunov稳定性理论，导出闭环系统均方稳定的若干LMI条件，并分析了这些条件之间的关系，最后通过数值例子说明了它们的应用．     

矿山产能分配的变量模糊线性规划模型

针对矿山资源开采过程中产能不确定的分配问题,引入了模糊结构元素理论。将产能用结构元表示,并利用结构元加权序将模糊数比较转化为单调函数比较,将含有模糊变量的线性规划问题等价转化为经典线性规划问题。以某矿山为例,建立矿山产能分配的变量模糊线性规划模型,并进行求解。结果表明：实现了将实际问题中的模糊事件进行精确表达,原问题的求解更简便。得到矿山产能取得最大可能利润时的可能分配。应用结构元加权序求解的线性规划模型优于结构元元序的。     

A novel hybrid classification model of artificial neural networks and multiple linear regression models

The classification problem of assigning several observations into different disjoint groups plays an important role in business decision making and many other areas. Developing more accurate and widely applicable classification models has significant implications in these areas. It is the reason that despite of the numerous classification models available, the research for improving the effectiveness of these models has never stopped. Combining several models or using hybrid models has become a common practice in order to overcome the deficiencies of single models and can be an effective way of improving upon their predictive performance, especially when the models in combination are quite different. In this paper, a novel hybridization of artificial neural networks (ANNs) is proposed using multiple linear regression models in order to yield more general and more accurate model than traditional artificial neural networks for solving classification problems. Empirical results indicate that the proposed hybrid model exhibits effectively improved classification accuracy in comparison with traditional artificial neural networks and also some other classification models such as linear discriminant analysis (LDA), quadratic discriminant analysis (QDA), K-nearest neighbor (KNN), and support vector machines (SVMs) using benchmark and real-world application data sets. These data sets vary in the number of classes (two versus multiple) and the source of the data (synthetic versus real-world). Therefore, it can be applied as an appropriate alternate approach for solving classification problems, specifically when higher forecasting accuracy is needed.     

Suboptimal filter for continuous‐time linear systems with unknown parameters

The filtering problem for continuous‐time linear systems with unknown parameters is considered. A new suboptimal filter is herein proposed. It is based on the optimal mean‐square linear combination of the local Kalman filters. In contrast to the optimal weights, the suboptimal weights do not depend on current observations; thus, the proposed filter can easily be implemented in real‐time. Examples demonstrate high accuracy and efficiency of the suboptimal filter. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

On properties of predictors derived with a two-step bootstrap model averaging approach—A simulation study in the linear regression model

In many applications of model selection there is a large number of explanatory variables and thus a large set of candidate models. Selecting one single model for further inference ignores model selection uncertainty. Often several models fit the data equally well. However, these models may differ in terms of the variables included and might lead to different predictions. To account for model selection uncertainty, model averaging procedures have been proposed. Recently, an extended two-step bootstrap model averaging approach has been proposed. The first step of this approach is a screening step. It aims to eliminate variables with negligible effect on the outcome. In the second step the remaining variables are considered in bootstrap model averaging. A large simulation study is performed to compare the MSE and coverage rate of models derived with bootstrap model averaging, the full model, backward elimination using Akaike and Bayes information criterion and the model with the highest selection probability in bootstrap samples. In a data example, these approaches are also compared with Bayesian model averaging. Finally, some recommendations for the development of predictive models are given.     

基于分散微粒群算法的二维模糊最大熵图像分割

该文研究了基于二维模糊信息熵的图像分割方法,针对二维模糊信息熵图像分割方法求取阈值时存在的计算复杂、时间长、实用性差等问题,提出了基于优化微粒群算法的二维最大熵图像分割方法。DPSO算法对图像的二维阈值空间进行全局搜索,并将搜索得到的二维熵最大值所对应的点灰度-区域灰度均值作为阈值进行图像分割。同时,为了避免该算法收敛到局部最优解的问题,在算法中引入了变异策略。通过实验显示了该算法在收敛性和计算效率上较QPSO在内其它优化算法具有更好的优越性。     

基于模糊模型的时滞不确定系统的模糊H∞鲁棒反馈控制

讨论了一类具有状态和控制时滞的不确定非线性系统的模糊H_∞ 状态反馈控制问题. 采用具有时滞的不确定Takagi-Sugeno(T-S)模糊模型对非线性系统进行建模, 提出了一套基于LMI的模糊鲁棒控制器的系统设计方法, 给出了模糊H_∞状态反馈控制器存在的充分条件, 以保证闭环模糊系统渐近稳定并满足从干扰输入到控制输出的H_∞范数界约束. 示例仿真表明了该方法的有效性.     

Stability analysis and controller synthesis for discrete uncertain singular fuzzy systems with distinct difference matrices in the rules

This paper mainly studies an extended discrete singular fuzzy model incorporating the multiple difference matrices in the rules and discusses its stability and design issues. By embracing additional algebraic constraint, traditional discrete Takagi-Sugeno (T-S) fuzzy model can be extended to a generalised discrete singular Takagi-Sugeno (GDST-S) model with individual difference matrices E_i in the locally singular models, where it can describe a larger class of physical or non-linear systems. Based on the linear matrix inequality (LMI) approach, we focus on deriving some explicit stability and design criteria expressed by the LMIs for the regarded system. Thus, the stability verification and controller synthesis can be performed by the current LMI tools. Finally, some illustrative examples are given to illustrate the effectiveness and validity of the proposed approach.