Abstract:
Traditional change-points detection is based on exact data set which cannot reflect prior information of data. This paper introduced a regression-class mixture decomposition method for fuzzy point data. In the method, different regression classes were mined sequentially in fuzzy point data set, and then the regression change points can be determined. So the number of change points which can be gotten automatically, need not prespecifying. Experiments prove that the proposed method is very robust, and by using fuzzy point data we introduced the prior information of the analysis data into the process of mining regression classes, which made the change-point we got in fuzzy point data be more practical than we got in exact data set.

Abstract:
The uncertainties and its prediction normally tendto be complex phenomena. The randomness and fuzziness are twokinds of uncertainties possible in real time. The randomness dealswith the general uncertainties whereas; the fuzzy logic addressesthe linguistic uncertainties. The fuzzy logic and its allied field dealwith the every part of uncertainties in fuzzy way. For a situationwhere, complex predictions are to tackle then statistical regressionmethodology is used from many years. The next step in thisscenario for dealing with uncertainties is the ‘Fuzzy Regression’.This paper presents the elementary theory of fuzzy regression andthe philosophy behind its potential application.

Abstract:
The research presents epsilon hierarchical fuzzy twin support vector regression based on epsilon fuzzy twin support vector regression and epsilon twin support vector regression. Epsilon FTSVR is achieved by incorporating trapezoidal fuzzy numbers to epsilon TSVR which takes care of uncertainty existing in forecasting problems. Epsilon FTSVR determines a pair of epsilon insensitive proximal functions by solving two related quadratic programming problems. The structural risk minimization principle is implemented by introducing regularization term in primal problems of epsilon FTSVR. This yields dual stable positive definite problems which improves regression performance. Epsilon FTSVR is then reformulated as epsilon HFTSVR consisting of a set of hierarchical layers each containing epsilon FTSVR. Experimental results on both synthetic and real datasets reveal that epsilon HFTSVR has remarkable generalization performance with minimum training time.

Abstract:
A systematic approach is proposed to optimize value for fuzzy linear regression (FLR) analysis using minimum fuzziness criteria with symmetric triangular fuzzy numbers (TFNs). Firstly, a new concept of credibility is defined to evaluate the performance of FLR models with different values when a set of sample data pairs is given. Secondly, based on the defined concept of credibility, a programming model is formulated to optimize the value of . Finally, both the numerical study and the real application show that the approach proposed in this paper is effective and efficient; that is, optimal value for can be determined definitely with respect to a set of given sample data pairs. 1. Introduction Statistic regression analysis and fuzzy regression analysis are two types of methods underlying different philosophies to assess the functional relationship between the dependent and independent variables and determine the best-fit model for describing the relationship, by exploiting the knowledge from the given input-output data pairs. In statistical regression analysis, deviations between the observed values and the estimates are assumed to be random errors disturbed by a probabilistic distribution. Different from statistic regression analysis, in fuzzy regression analysis, the deviations are attributed to the imprecision of the observed values and/or the indefiniteness of model structure. Tanaka et al. [1] firstly proposed fuzzy linear regression (FLR) analysis using the fuzzy functions defined by Zadeh’s extension principle [2], in which the observed values can differ from the estimated values to a certain degree of belief [3]. Thus, the uncertainty in this type of regression model becomes fuzziness, not randomness, and the disturbance is incorporated into the fuzzy coefficients, and the final objective is to adjust the fuzzy coefficients from the available sample data pairs. According to [3], the existing FLR methods can be roughly classified into the following two categories based on criterion function, that is, FLR methods using minimum fuzziness criteria and FLR methods using fuzzy least-squares criteria. By using the first category of FLR methods, FLR model can be built by minimizing the system vagueness. The first FLR method in [1] was extended by using other types of fuzzy coefficients, including general LP-type fuzzy coefficients [4], exponential fuzzy coefficients [5], and triangular fuzzy coefficients [6, 7], Chen et al. depended on symmetric triangular fuzzy numbers to study determination Method for Parameters of Rock’s shear strength through least

Abstract:
Fuzzy linear regression has been used in predicting analysis as to handle uncertainty variables. Many methods of fuzzy linear regressions were introduced but most of the methods associated with substantial complex computation procedures. The model of matrix-driven fuzzy linear regression was proposed as to overcome the computational risk and was successfully tested in a civil engineering application. This study extends the application of the model to investigate the relationship between variables impacting car sales volume. The variables of petroleum prices, population, Gross Domestic Product (GDP) and Gross National Product (GNP) are predicted with the response variable of car sales volume. Thirty years time series data of the variables from various Malaysian agencies were fed into the models. It is found that the model successfully yield a fuzzy linear regression equation as to explain the relationship between predictors and response variable. It also notices that eighty eight percent variations in car sales volume attributed by price of petroleum, population, GNP and GDP. The model also successfully explained the contributions of left and right errors of fuzzy numbers of regression coefficients to the car sales volume. The fuzzy numbers that represent coefficients of regression certainly offer a new contribution to the relationships between the variable of car sales volume and the four predictors.

Abstract:
Some existed fuzzy regression methods have some special requirements for
the object of study, such as assuming the observed values as symmetric triangular
fuzzy numbers or imposing a non-negative constraint of regression
parameters. In this paper, we propose a left-right fuzzy regression method,
which is applicable to various forms of observed values. We present a fuzzy
distance and partial order between two left-right (LR) fuzzy numbers and we
let the mean fuzzy distance between the observed and estimated values as the
mean fuzzy error, then make the mean fuzzy error minimum to get the regression
parameter. We adopt two criteria involving mean fuzzy error (comparative
mean fuzzy error based on partial order) and SSE to compare the
performance of our proposed method with other methods. Finally four different
types of numerical examples are given to illustrate that our proposed
method has feasibility and wide applicability.

Abstract:
Product selection is always one of the troubles that decision makers are facing with it. Correct selection requires having suitable method for this important issue. In this article, we concern to introduce an approach of fuzzy decision making for selection to decision makers. The nature of decision making is usually complex and without structure. Totally, most of qualitative and quantitative factors such as quality, price, and flexibility should be concerned for determining a suitable product. In this study, it is attempted to use recent advances in ranking methods for product selection. The proposed study uses oral preferences language shown in terms of triangular and trapezoid fuzzy numbers. Then, a multi criteria hierarchical decision making is suggested on the basis of fuzzy collection theory for product selection where the proposed fuzzy VIKOR uses different qualitative and quantitative criteria.

Abstract:
A simple Bayesian approach to nonparametric regression is described using fuzzy sets and membership functions. Membership functions are interpreted as likelihood functions for the unknown regression function, so that with the help of a reference prior they can be transformed to prior density functions. The unknown regression function is decomposed into wavelets and a hierarchical Bayesian approach is employed for making inferences on the resulting wavelet coefficients.

Abstract:
The aim of this article is to promote the use of probabilistic methods in the study of problems in mathematical general relativity. Two new and simple singularity theorems, whose features are different from the classical singularity theorems, are proved using probabilistic methods. Under some energy conditions, and without any causal or initial/boundary assumption, simple conditions on the energy flow imply probabilistic incompleteness. Also we introduce a probabilistic notion of spacetime boundary which has none of the pathological defects that the classical boundaries may have.

Abstract:
We construct a fuzzy varying coefficient bilinear regression model to deal with the interval financial data and then adopt the least-squares method based on symmetric fuzzy number space. Firstly, we propose a varying coefficient model on the basis of the fuzzy bilinear regression model. Secondly, we develop the least-squares method according to the complete distance between fuzzy numbers to estimate the coefficients and test the adaptability of the proposed model by means of generalized likelihood ratio test with SSE composite index. Finally, mean square errors and mean absolutely errors are employed to evaluate and compare the fitting of fuzzy auto regression, fuzzy bilinear regression and fuzzy varying coefficient bilinear regression models, and also the forecasting of three models. Empirical analysis turns out that the proposed model has good fitting and forecasting accuracy with regard to other regression models for the capital market.