Least-squares estimates in fuzzy regression analysis

Regression is a very powerful methodology for forecasting, which is considered as an essential component of successful OR applications. In this paper an idea stemmed from the classical least squares is proposed to handle fuzzy observations in regression analysis. Based on the extension principle, the membership function of the sum of squared errors is constructed. The fuzzy sum of squared errors is a function of the regression coefficients to be determined, which can be minimized via a nonlinear program formulated under the structure of the Chen–Klein method for ranking fuzzy numbers. To illustrate how the proposed method is applied, three cases, one crisp input-fuzzy output, one fuzzy input-fuzzy output, and one non-triangular fuzzy observations, are exemplified. The results show that the least-squares method of this paper is able to determine the regression coefficients with better explanatory power. Most important, it works for all types of fuzzy observations, not restricted to the triangular one.     

Regularized fuzzy clusterwise ridge regression

Fuzzy clusterwise regression has been a useful method for investigating cluster-level heterogeneity of observations based on linear regression. This method integrates fuzzy clustering and ordinary least-squares regression, thereby enabling to estimate regression coefficients for each cluster and fuzzy cluster memberships of observations simultaneously. In practice, however, fuzzy clusterwise regression may suffer from multicollinearity as it builds on ordinary least-squares regression. To deal with this problem in fuzzy clusterwise regression, a new method, called regularized fuzzy clusterwise ridge regression, is proposed that combines ridge regression with regularized fuzzy clustering in a unified framework. In the proposed method, ridge regression is adopted to estimate clusterwise regression coefficients while handling potential multicollinearity among predictor variables. In addition, regularized fuzzy clustering based on maximizing entropy is utilized to systematically determine an optimal degree of fuzziness in memberships. A simulation study is conducted to evaluate parameter recovery of the proposed method as compared to the extant non-regularized counterpart. The usefulness of the proposed method is illustrated by an application concerning the relationship among the characteristics of used cars.     

Intuitionistic fuzzy linear regression analysis

Linear regression analysis in an intuitionistic fuzzy environment using intuitionistic fuzzy linear models with symmetric triangular intuitionistic fuzzy number (STriIFN) coefficients is introduced. The goal of this regression is to find the coefficients of a proposed model for all given input–output data sets. The coefficients of an intuitionistic fuzzy regression (IFR) model are found by solving a linear programming problem (LPP). The objective function of the LPP is to minimize the total fuzziness of the IFR model which is related to the width of IF coefficients. An illustrative example is also presented to depict the solution procedure of the IFR problem by using STriIFNs.     

Enumeration of All Possibly Optimal Vertices with Possible Optimality Degrees in Linear Programming Problems with a Possibilistic Objective Function

In this paper, we treat linear programming problems with fuzzy objective function coefficients. To such a problem, the possibly optimal solution set is defined as a fuzzy set. It is shown that any possibly optimal solution can be represented by a convex combination of possibly optimal vertices. A method to enumerate all possibly optimal vertices with their membership degrees is developed. It is shown that, given a possibly optimal extreme point with a higher membership degree, the membership degree of an adjacent extreme point is calculated by solving a linear programming problem and that all possibly optimal vertices are enumerated sequentially by tracing adjacent possibly optimal extreme points from a possibly optimal extreme point with the highest membership degree.     

系数为LR-型模糊数的模糊线性最小二乘回归

针对输入、输出以及系数为LR-型模糊数的情况,建立模糊线性回归模型,提出该模型的最小二乘估计以及模型性能评价方法。当输入、输出以及系数都退化为精确值时,该估计退化为经典的最小二乘估计。该方法不仅适用于三角模糊数,也适用于其它LR-型模糊数(如指数型模糊数)。数值模拟表明,该方法的拟合效果较好。     

SUR模型回归系数的估计

本文证明了一个关于SUR模型回归系数最小方差线性无偏估计(MVLUE)的充要条件,并利用此充要条件讨论了几类SUR模型回归系数的MVLUE估计及两步估计.在方法上避免了与分块矩阵求逆有关的运算,所得结论推广和完善了已有的一些结果.     

模糊线性回归的稳健方法

考虑两类模糊回归模型：一类是设计点为实数,参数为模糊数;另一类是观察值为模糊数、参数为实数。就这两类回归模型,从稳健统计的角度提出相应的稳健方法,并通过例子与现有的方法进行比较,说明所提方法的稳健性。     

On the calculation of a membership function for the solution of a fuzzy linear optimization problem

In the present paper the fuzzy linear optimization problem (with fuzzy coefficients in the objective function) is considered. Recent concepts of fuzzy solution to the fuzzy optimization problem based on the level-cut and the set of Pareto optimal solutions of a multiobjective optimization problem are applied. Chanas and Kuchta suggested one approach to determine the membership function values of fuzzy optimal solutions of the fuzzy optimization problem, which is based on calculating the sum of lengths of certain intervals. The purpose of this paper is to determine a method for realizing this idea. We derive explicit formulas for the bounds of these intervals in the case of triangular fuzzy numbers and show that only one interval needs to be considered.     

模糊数据的线性回归模型

研究观测数据为模糊数据的统计线性回归模型 ,由该模型所得回归系数非模糊 ,易于应用。对于对称三角模糊数据一元线性回归给出最优解的解析表达式 ;将对称三角模糊数多元线性回归问题给出转化为一类二次规划问题的方法 ;证明了最优解的存在性和估计量的无偏性。     

某四阶微分算子的一个正则性定理

本文给出了某带参数λ的四阶微分算子B_λ的一个正则性定理,由此我们分别获得了关于该算子在“非线性情形”的二个同胚类及在“线性情形”的三个线性同构类。这对描述某飞行器在其运行过程中某些类双向稳定性质是有用与方便的。     

基于多目标规划法的模糊线性回归分析

讨论输入、输出均为模糊数,回归系数为实数时的模糊线性回归分析。由于模糊最小二乘线性回归容易受异常值的影响,而最小一乘法能有效地降低回归模型的误差。为此,基于最小一乘法,建立多目标规划模型并将其转化为非线性规划问题进行求解,从而实现模糊线性回归模型的参数估计。最后,结合一个数值实例,验证和比较该方法的合理性和优越性。     

Systems of linear fuzzy constraints

Linear Fuzzy constraints are linear constraints where coefficients are fuzzy numbers. This paper demonstrates that two points of view can be considered to extend classical linear constraints: either tolerance constraints, or approximate (in)equality constraints can be obtained. Resolution of systems of linear fuzzy constraints is shown to be made easier by the use of fuzzy numbers analytically represented through a given type of membership function and three parameters. Solution methods are provided in the case of non fuzzy variables; as an illustration, some numerical examples are presented. The fuzzy variable case is also evoked.This paper is part of Purdue University Electrical Engineering technical report TR-EE 78-13.     

Balanced generalized handcuffed designs

The definition of balanced generalized handcuffed designs (BHD) is of course more specific than that of the generalized handcuffed designs that we introduced in 1987.

In the first part of this paper, we present a particular property of a BHD, which is not necessarily that of a generalized handcuffed design.

Then, we provide the reader with a general procedure that enables one to obtain such designs, and is called a ‘difference method’. We also show how this difference method can be made more useful in the case where the set V on which a BHD is constructed is the residue classes of integers mod V.

The third part of this paper deals with the problem of the existence of a BHD; and a solution is given for a particular case. We assume that the method applied for solving this problem will allow for the constructing of many more theorems analogous to Theorem 3.     

系数为梯形模糊数的模糊回归分析的最小二乘法

由于模糊数往往可以用梯形模糊数来逼近,因此对梯形模糊数的模糊回归模型的研究就有一定的实用价值.采用最小二乘的方法,针对输入为精确数、输出和回归系数都是梯形模糊数的模糊线性回归模型,讨论了该模型回归系数的最小二乘估计及误差项的估计,实例说明了提出的参数估计的拟合度比较好.     

Maximum likelihood estimation from fuzzy data using the EM algorithm

A method is proposed for estimating the parameters in a parametric statistical model when the observations are fuzzy and are assumed to be related to underlying crisp realizations of a random sample. This method is based on maximizing the observed-data likelihood defined as the probability of the fuzzy data. It is shown that the EM algorithm may be used for that purpose, which makes it possible to solve a wide range of statistical problems involving fuzzy data. This approach, called the fuzzy EM (FEM) method, is illustrated using three classical problems: normal mean and variance estimation from a fuzzy sample, multiple linear regression with crisp inputs and fuzzy outputs, and univariate finite normal mixture estimation from fuzzy data.     

Two-dimensional homogeneous cubic systems: Classification and normal forms–III

This article is the third in a series of works devoted to two-dimensional homogeneous cubic systems. It considers the case where the homogeneous polynomial vector on the right-hand side of the system has a quadratic common factor with real zeros. The set of such systems is divided into classes of linear equivalence, in each of which a simplest system being a third-order normal form is distinguished on the basis of appropriately introduced structural and normalization principles. In fact, this normal form is determined by the coefficient matrix of the right-hand side, which is called a canonical form (CF). Each CF is characterized by an arrangement of nonzero elements, their specific normalization, and a canonical set of admissible values of the unnormalized elements, which ensures that the given CF belongs to a certain equivalence class. In addition, for each CF, (a) conditions on the coefficients of the initial system are obtained, (b) nonsingular linear substitutions reducing the right-hand side of a system satisfying these conditions to a given CF are specified, and (c) the values of the unnormalized elements of the CF thus obtained are given.     

Fuzzy Optimization Problems Based on Ordering Cones

The solution concepts of the fuzzy optimization problems using ordering cone (convex cone) are proposed in this paper. We introduce an equivalence relation to partition the set of all fuzzy numbers into the equivalence classes. We then prove that this set of equivalence classes turns into a real vector space under the settings of vector addition and scalar multiplication. The notions of ordering cone and partial ordering on a vector space are essentially equivalent. Therefore, the optimality notions in the set of equivalence classes (in fact, a real vector space) can be naturally elicited by using the similar concept of Pareto optimal solution in vector optimization problems. Given an optimization problem with fuzzy coefficients, we introduce its corresponding (usual) optimization problem. Finally, we prove that the optimal solutions of its corresponding optimization problem are the Pareto optimal solutions of the original optimization problem with fuzzy coefficients.     

Influence diagnostics in the varying coefficient model with longitudinal data

One or few observations can be highly influential on estimates of regression coefficients in the linear regression model. In this paper we derive influence diagnostics for the varying coefficients model with longitudinal data. We note that diagnostics in this context is quite different from the classical regression model in the sense that regression coefficients vary as time varies. A version of Cook’s distance is suggested to reflect this specific aspect of varying coefficient model. An algorithm to present some guidelines to determine influential observations deserving special attention is developed. An illustrative example based on the AIDS data is also given.     

社区教育评价的模糊等级和灰色关联度方法研究

阐述社区教育评价中应用模糊集合论方法的必要性;提出L-R型模糊数以及7个模糊等级的梯形模糊数和三角形模糊数表示方法及其清晰化计算公式;提出对象评判经模糊等级评价、综合并清晰化得出评价矩阵的方法;在提出理想矩阵基础上建立评价的灰色关系系数矩阵,最终求出每一评判对象的关联度,以此确定排序.举出由5名评价人员,对8个评判对象社区教育教学内容按"教学内容体现时代特色"、"教学内容体现社会关注"、"教学内容体现实际应用"3个指标进行评判的示例,得出绩效排序为Q_8优于Q_4优于Q_3优于Q_1优于Q_7优于Q_2优于Q_5优于Q_6的结果.     

Fuzzy polynomial regression with fuzzy neural networks

Recently, fuzzy linear regression is considered by Mosleh et al. [1]. In this paper, a novel hybrid method based on fuzzy neural network for approximate fuzzy coefficients (parameters) of fuzzy polynomial regression models with fuzzy output and crisp inputs, is presented. Here a neural network is considered as a part of a large field called neural computing or soft computing. Moreover, in order to find the approximate parameters, a simple algorithm from the cost function of the fuzzy neural network is proposed. Finally, we illustrate our approach by some numerical examples.