Multiobjective fuzzy linear regression analysis and its application

Applications of Soft Computing to Human-centered Information Systems

Fuzzy regression analysis using fuzzy linear models with symmetric triangular fuzzy number coefficients has been formulated earlier. The goal of this regression is to find the output for all data sets. In this research, we have formulated a newer fuzzy linear regression model that uses symmetric triangular coefficient. The uniqueness of our model lies in the fact that this can be extended to use nonsymmetric fuzzy triangular coefficient that eradicates the drawbacks of existing fuzzy linear regression models.

Tanaka suggested that the parameters of a linear regression model should be made fuzzy to bring about the swing of a system and created a fuzzy linear regression model. This model can be formulated in a linear programming problem that minimizes a span between the upper and lower limits under constraints that include all data. In recent years, all the attention has been focused on a fuzzy number that has an indifferent zone. A fuzzy number is defi ned by using a Type 2 membership function. This paper addresses the fact that a Type 2 membership function has the upper and lower limits and shows that a Type 2 membership function can be identifi ed by expanding a fuzzy linear regression model into a fuzzy linear polynomial regression model. Finally, after a proposed fuzzy polynomial model is identifi ed, a mathematical model will be created for a fuzzy decision-making method that has an indifferent zone.

Error measures for fuzzy linear regression: Monte Carlo simulation approach

Fuzzy regression analysis is an important regression analysis method to predict uncertain information in the real world. In this paper, the input data are crisp with randomness; the output data are trapezoid fuzzy number, and three different risk preferences and chaos optimization algorithm are introduced to establish fuzzy regression model. On the basis of the principle of the minimum total spread between the observed and the estimated values, risk-neutral, risk-averse, and risk-seeking fuzzy regression model are developed to obtain the parameters of fuzzy linear regression model. Chaos optimization algorithm is used to determine the digital characteristic of random variables. The mean absolute percentage error and variance of errors are adopted to compare the modeling results. A stock rating case is used to evaluate the fuzzy regression models. The comparisons with five existing methods show that our proposed method has satisfactory performance.

Using fuzzy linear regression models, an estimation method to detect the human reasoning state in machine operation is discussed. Simple experiments of human operation using computer display show that the change of the human internal state at the reasoning in the operations is reflected in the system parameters of the fuzzy linear regression models. Fuzzy linear regression models are constructed using some fundamental operation data, and fuzziness parameters are calculated by traditional linear programming approaches. The characteristics of the human reasoning state are represented by the fuzziness parameters, and advantages of the proposed approaches, in human reliability are demonstrated by the example of vehicle driving operations.

Linear regression has been widely used for many years to forecast various socio-economic variables in marketing, management, sales, energy and so on. Demand and price are commonly estimated by means of regression models. However, where there is high uncertainty in the model, fuzzy regressions are applied. In this paper, a fuzzy-based approach is applied for transport energy demand forecasting using socio-economic and transport related indicators. The model is based on Gross Domestic Product (GDP), population and the number of vehicles as three inputs to the main model. Energy data from 1993 to 2005 are used to estimate each of the three inputs. By three individual fuzzy linear regression (FLR) models, a multi-level FLR model is designed. The input variables are transport energy demand in the last year, the number of vehicles, population and ratio of GDP overpopulation. The output variable is energy demand of the transportation sector in Iran. The inputs to the ending level are obtained as outputs of the starting levels. The estimation fuzzy problem for the model is formulated as a linear optimization problem. Comparison of the model predictions with data of the testing period shows validity of the proposed model. Furthermore, having obtained the fuzzy parameters, the transport energy demand is predicted for 2006 to 2020. It is noticeable that, without any price shock or efficiency improvement in the transportation sector, the energy consumption may reach to a threatening level of about 592 MBOE by 2020. Key words: Forecasting, multi-level fuzzy linear regression, transport energy demand, case study.

IntroductionThis is the first of four chapters on fuzzy regression. This chapter and Chapter 14 are about fuzzy linear regression and Chapters 12 and 13 consider fuzzy nonlinear regression. In this chapter the independent (predictor, explanatory) variables are crisp but the dependent (response) variable is fuzzy. In Chapter 14 both the independent variables and the dependent variable are fuzzy. This chapter is based on [1].Fuzzy linear regression has become a very large area of research. Put “fuzzy regression” into your search engine and you can get too many web sites to visit.“Fuzzy linear regression” will eliminate a lot of web sites but the list is still quite long. We have selected a few recent and key references on fuzzy linear regression which are: (1) books (or articles in these books) ([4],[8],[14]); and (2) papers ([2],[3],[5],[9],[10],[12],[13],[15],[19]-[24],[27]-[32]).As far as the authors know our research is the only research on using Monte Carlo techniques in fuzzy linear regression. However, there have been other approaches employing random search (genetic algorithms) and others using neural nets. If we put “genetic algorithms” and “fuzzy linear regression” into the search engine there are less than 200 references. A recent reference is [11]. We feel that one problem with using a GA is that it can converge to a local minimum and to avoid this you need to start it with many different randomly generated initial populations. Also, we believe that our Monte Carlo method is easier to apply than a genetic algorithm, once you have a quasi-random number generator in your computer. Next we searched for “neural nets” and “fuzzy linear regression” getting less than 100 references. A key reference on this topic is [6].

Fuzzy linear regression and crisp linear regression are different models which have different assumptions. Crisp linear regression makes statistical assumptions about model or data due to randomness while fuzzy linear regression on the other hand is a method which considers the vagueness or imprecision in data or model specification of a linear regression model. The uncertainty in prediction of a crisp linear regression are due to randomness and is represented by a statistical prediction interval while that of a fuzzy linear regression (due to imprecision or vagueness) is represented by a fuzzy number. The output of a classic linear regression is crisp or just an interval which makes it difficult to fully represent all uncertainties in the model. In this paper, a new method is introduced to construct fuzzy numbers from statistical prediction intervals of crisp linear regression models. The method produces a fuzzy-statistical output which is in some way comparable to the fuzzy output of a fuzzy linear regression model without the need of optimisation. A real data set is used to show the applicability of the proposed method

Fuzzy linear regression analysis for fuzzy input-output data

Fuzzy regression based on asymmetric support vector machines

Many researches have been carried out in fuzzy linear regression the past three decades. However, almost all of them are limited to type-1 fuzzy data. A fuzzy linear regression model with type-2 fuzzy output data and type-2 fuzzy coefficients is studied in this paper. A linear programming model based on goal programming is proposed to calculate the regression coefficients. To evaluate the proposed model, we present some indices based on the standard deviation, the correlation coefficient, and the standard error of estimate. The proposed method is illustrated by some numerical examples.

The characteristic of the fuzzy regression model is to enwrap all the given samples. An interval of fuzzy regression model is created by considering how far a sample is from the central values. That means when samples are widely scattered the size of an interval of the fuzzy model is widened. That is, the fuzziness of the fuzzy regression model is decided by the range of sample distribution. Therefore, many research results on a fuzzy regression model in order to describe the possibility of the target system have been reported. We have proposed two fuzzy robust regression models which remove influences of improper data such as unusual data and outliers. In this paper, we describe the model building of our fuzzy robust regressions by removing influences of improper data.

The theoretical background for abstract formalization of vague phenomenon of complex systems is fuzzy set theory. In the paper are defined vague data as specialized fuzzy sets - fuzzy numbers and there is described a fuzzy linear regression model as a fuzzy function with fuzzy numbers as vague regression parameters. To identify the fuzzy coefficients of model the genetic algorithm is used. The linear approximation of vague function together with its possibility area are analytically and graphically expressed. The suitable numerical experiments are performed namely in the task of two-dimensional fuzzy function modelling and the time series fuzzy regression analysis as well.

When we attempt to model a complex system including a human as an important component, it may be difficult to represent the system by a deterministic mathematical model. The main reason of this difficulty is that the system itself inherently has some fuzziness concerning subjective judgement of a human. In this paper, we propose a fuzzy nonlinear regression method with RFLN (RCE-based fuzzy learning network), which is capable of extracting knowledge of the experts automatically. RFLN is an extended RCE (restricted Coulomb energy) model, hence it needs few iterations in learning and its additional learning is easy. The proposed method has higher flexibility than fuzzy linear regression models. We propose learning algorithms to identify a nonlinear interval model which approximately includes all the given input-output data. The proposed method has characteristics of faster learning and of easier additional learning. The effectiveness of the method is shown by numerical experiments.