Fuzzy nonlinear regression with fuzzified radial basis function network

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].

The neural network (NN) training algorithm based on gradient optimization can not avoid falling into the local minimum because of the inappropriate initial weight value. The paper applies the genetic algorithm (GA) to training the linkage weights of NN. The training result can be used as the weights of an initial network for a back propagation (BP) training algorithm, then online optimization work can be done by BP training algorithm. It succeeds in avoiding the GA's defect of high calculating cost of every step, and giving full play to GA's advantage of greater probability of global convergence. Thus the online BP training algorithm can be lifted out of local minimum with greater probability and the better training property of the network is gained. The result of simulation shows that the robustness of the control system is improved.

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

A multilayer perceptron with the extended Kalman filter (EKF) training algorithm is investigated as a communication receiver when intersymbol interference and white Gaussian noise are present. Besides discussing the complexity of the EKF algorithm, it is shown that the EKF has better performance than the conventional backpropagation (BP) training algorithm in the sense that is requires less training steps and also results in proper training even when the BP did not. With 2500 training symbols, the EKF resulted in about 4-dB performance improvement over the conventional BP. >

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.

Software quality models can predict the risk of faults in modules early enough for cost-effective prevention of problems. This paper introduces the fuzzy nonlinear regression (FNR) modeling technique as a method for predicting fault ranges in software modules. FNR modeling differs from classical linear regression in that the output of an FNR model is a fuzzy number. Predicting the exact number of faults in each program module is often not necessary. The FNR model can predict the interval that the number of faults of each module falls into with a certain probability. A case study of a full-scale industrial software system was used to illustrate the usefulness of FNR modeling. This case study included four historical software releases. The first release's data were used to build the FNR model, while the remaining three releases' data were used to evaluate the model. We found that FNR modeling gives useful results.

This paper presented the approach of prediction of pressure coefficient using feed-forward multilayer artificial neural network (ANN) by the backpropagation (BP) training algorithm. The regular shape square model of 1:1:2 (length: breadth: height) and a setback model which have a 20% setback at half of the height are considered in this study. The datasets of network architecture considered by the computational fluid dynamics method (CFD). Maximum 1% error is noticed between the Cp comparison of CFD and ANNs method. The predicted mean pressure coefficient (Cp) and fluctuation wind pressure coefficient (Cp,rms) from the calculated CFD data and predicted ANN data are plotted in the contour form. The study suggested that the setback model could control the pressure on different surfaces compared to the regular shape square model. The study also decided the ANN is a powerful tool to predict the pressure coefficient for the intermediate tapping points on the building, which may reduce the time and cost of wind tunnel study.KeywordsArtificial neural network (ANN)Backpropagation (BP) neural networkComputational fluid dynamics (CFD)Pressure coefficientPressure contourSetback buildingSquare buildingTapping pointsWind load

This paper explores feasibility of employing the non-recurrent backpropagation training algorithm for a recurrent neural network, Simultaneous Recurrent Neural network, for static optimisation. A simplifying observation that maps the recurrent network dynamics, which is configured to operate in relaxation mode as a static optimizer, to feedforward network dynamics is leveraged to facilitate application of a non-recurrent training algorithm such as the standard backpropagation and its variants. A simulation study that aims to assess feasibility, optimizing potential, and computational efficiency of training the Simultaneous Recurrent Neural network with non-recurrent backpropagation is conducted. A comparative computational complexity analysis between the Simultaneous Recurrent Neural network trained with non-recurrent backpropagation algorithm and the same network trained with the recurrent backpropagation algorithm is performed. Simulation results demonstrate that it is feasible to apply the non-recurrent backpropagation to train the Simultaneous Recurrent Neural network. The optimality and computational complexity analysis fails to demonstrate any advantage on behalf of the non-recurrent backpropagation versus the recurrent backpropagation for the optimisation problem considered. However, considerable future potential that is yet to be explored exists given that computationally efficient versions of the backpropagation training algorithm, namely quasi-Newton and conjugate gradient descent among others, are also applicable for the neural network proposed for static optimisation in this paper.

Some algorithms for solving non-linear integer mathematical programming problems

An elastic perfectly plastic structure at shakedown to given cyclić loading is under consideration. The stress-strain field of dissipative system in general is related to the history of loading. And only in a particular case, i.e. at the moment prior to the failure of an elastic perfectly plastic structure the distribution of the actual residual forces is unique for each prescribed history of loading (the safety factor of shakedown approaches unity). Nevertheless, there exist some domains where the plastic strains are equal to zero. The residual forces in the statically indeterminate parts of the structure may be non-unique: the stress field is only determined by the equilibrium equations. The extremum energy principle of minimum complementary energy allows to derive the actual residual forces out of all statically admissible residual forces at the moment prior to cyclic plastic failure. Then the stress-strain field analysis problem at the moment prior to the cyclic plastic failure is formulated as a problem of non-linear mathematical programming. Formulating the dual pair of non-linear programming problem (statical and kinematic formulation of analysis problem) the differential constraints are neglected or replaced by algebraic conditions. When the safety factor is approching a unity, the degeneracy of the statical formulation of the analysis problem often can occur. In this case a mathematical model is proposed for obtaining an upper bounds for the displacement at shakedown. It is pointed out that the known Kuhn-Tucker conditions of mathematical programming theory (i.e. compatibility equations of residual strains) in concert with restriction, limiting the maximum value of total energy dissipation, make up the adaptation conditions of the structure to given cyclic loading. Kuhn-Tucker conditions used in above—mentioned problem allow to correctly interprete the physical aspect of the degeneracy problem at shakedown. When the safety factor is larger than unity an artificial degeneracy situation for the statical formulation of analysis problem can be created. Then the mathematical models presented can be applied to the analysis of unloading elastoplastic structures. With this aim in view a fictitious equiplastic structure the behaviour of which is holonomic is derived. The displacements of the fictitious structure enclose the displacements of the actual structure subject to cyclic loading.

The method of optimal geometric synthesis of four-bar cylindrical gear with related gears with one external and one internal gearing is proposed. The method is based on a new calculation system that increases the number of synthesis output parameters to six, which makes it possible to considerably expand the domain of existence of the synthesis problem solution with the simultaneous synthesis of two gearing. The software implementing the proposed technique is described. To solve the problem of nonlinear mathematical programming the library of Optimization Toolbox functions included in Matlab is used. Examples of calculations are given.

An approach to the analysis of self-excited oscillations by using the method of harmonic linearization for controlled systems not satisfying the harmonic balance conditions in the first approximation, as to a problem of non-linear mathematical programming, is proposed.

Method of fictitious system for evaluation of frame shakedown displacements

For reinforcement, the photochromic field and the cooperation between the theoretical and experimental branches of physics, the computational, theoretical artificial neural networks (CTANNs) and the resilient back propagation (R[Formula: see text]) training algorithm were used to model optical characterizations of casting (Admantan-Fulgide) thin films with different concentrations. The simulated values of ANN are in good agreement with the experimental data. The model was also used to predict values, which were not included in the training. The high precision of the model has been constructed. Moreover, the concentration dependence of both the energy gaps and Urbach’s tail were, also tested. The capability of the technique to simulate the experimental information with best accuracy and the foretelling of some concentrations which is not involved in the experimental data recommends it to dominate the modeling technique in casting (Admantan-Fulgide) thin films.

The prediction of water quality parameters plays an important role in water resources and environmental systems. The use of electrical conductivity (EC) as a water quality indicator is one of the important parameters for estimating the amount of mineralization. This study describes the application of artificial neural network (ANN) and wavelet-neural network hybrid (WANN) models to predict the monthly EC of the Asi River at the Demirköprü gauging station, Turkey. In the proposed hybrid WANN model, the discrete wavelet transform (DWT) was linked to the ANN model for EC prediction using a feed-forward back propagation (FFBP) training algorithm. For this purpose, the original time series of monthly EC and discharge (Q) values were decomposed to several sub-time series by DWT, and these sub-time series were then presented to the ANN model as an input dataset to predict the monthly EC. Comparing the values predicted by the models indicated that the performance of the proposed WANN model was better than the conventional ANN model. The correlation of determination (R (2)) were 0.949 and 0.381 for the WANN and ANN models, respectively. The results indicate that the peak EC values predicted by the WANN model are closer to the observed values, and this model simulates the hysteresis phenomena at an acceptable level as well.