Prediction of software faults using fuzzy nonlinear regression modeling

In this paper, a new approach of modeling for Artificial Neural Networks (ANNs) models based on the concepts of fuzzy regression is proposed. For this purpose, we reformulated ANN model as a fuzzy nonlinear regression model while it has advantages of both fuzzy regression and ANN models. Hence, it can be applied to uncertain, ambiguous, or complex environments due to its flexibility for forecasting purposes.

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 regression based on asymmetric support vector machines

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.

Modern high-assurance network telecommunications systems often must have high software reliability. Software quality models yield timely predictions of quality indicators on a module-by-module basis, enabling one to target enhancement techniques. This paper introduces fuzzy nonlinear regression (FNR) to practitioners in high-assurance systems engineering. FNR integrates fuzzy logic and neural networks techniques to generate good results. We present an innovative method that differs from other FNRs such that the statistics of the dependent variable are used to build the FNR instead of focusing on the cost function. We tested our model in a large network telecommunications system written in a Pascal-like proprietary language. Data splitting was applied in the case study. In order to avoid the unfair partition of the data set, we randomly split the data thirty times and performed the experiment for each iteration. We also conducted our experiment using multiple linear regression (MLR) and we found that the results of FNR were significantly better than those using MLR

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

A fuzzified radial basis function network (FRBFN) is a kind of fuzzy neural network that is obtained by direct fuzzification of the well known neural model RBFN. A FRBFN contains fuzzy weights and can handle fuzzy-in fuzzy-out data. This paper shows that a FRBFN can also be interpreted as a kind of fuzzy expert system. Hence it owns the advantages of simple structure and clear physical meaning. Some metrics for fuzzy numbers have been extended to the metrics for n-dimensional fuzzy vectors, which are applicable to computations in FRBFNs. The corresponding metric spaces for n-dimensional fuzzy vectors are proved to be complete. Further, FRBFNs are proved to be able to act as universal function approximators for any continuous fuzzy function defined on a compact set. This paper applies the proposed FRBFN to nonparametric fuzzy nonlinear regression problems for multidimensional LR-type fuzzy data. Fuzzy nonlinear regression with FRBFNs can be formulated as a nonlinear mathematical programming problem. Two training algorithms are proposed to quickly solve the two types of problems under different criteria and constraint conditions, namely, the two-stage and BP (Back-Propagation) training algorithms. Simulation studies are carried out to verify the feasibility and demonstrate the advantages of the proposed approaches.

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.

This article introduces a general fuzzy regression model, which separates the response function on a mode and spreads of an α-level set for an observed fuzzy number, to estimate a fuzzy relation between two fuzzy random variables. We construct the general fuzzy regression model using least squares estimation and best response functions on the mode and spread of an α-level set for the fuzzy number when the response variable is an LR-fuzzy number and independent variables are crisp numbers. Then we derive a crisp mean and variance of the predicted fuzzy number, and compare the accuracy of our proposed fuzzy regression model with other fuzzy regression models suggested by many authors.

An integrated fuzzy regression-data envelopment analysis algorithm for optimum oil consumption estimation with ambiguous data

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.

This study presents an integrated fuzzy regression and time-series technique to estimate and predict electricity demand. Furthermore, it is difficult to model uncertain behaviour of energy consumption with only conventional time-series and fuzzy regression, which could be an ideal substitute for such cases. After reviewing various fuzzy regression models and studying their advantages and shortcomings, the best model is selected. Also, the impact of data preprocessing and post-processing on the fuzzy regression performance is to study and to show that this method does not contribute to the efficiency of the model. In addition, another unique feature of this study is utilisation of autocorrelation function to define input variables versus trial and error method. At last, the comparison of actual data with fuzzy regression and ARIMA model, using Granger?Newbold test, is achieved. Monthly electricity consumption of Iran from 1995 to 2005 is considered as the case of this study.

Fuzzy linear regression analysis for fuzzy input-output data

This paper tests three formulations of linear programming problems used to derive a fuzzy linear regression model, which are based on three different indices representing the necessity and of sufficiency equivalence relations between fuzzy numbers. It is shown that those formulations are extensions of three traditional formulations for fuzzy data. The trade‐off between “fit” (between the model and the observed data) and the “fuzziness” (of the model) is considered to be a multiobjective decisionmaking problem. This leads to the formulation of a multiobjective programming problem for deriving the fuzzy linear regression model. Corresponding to this situation, an interactive decision‐making procedure is proposed, which is applied to the problem of estimating the pork demand.

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.