Fuzzy basis functions, universal approximation, and orthogonalleast-squares learning

Fuzzy systems are represented as series expansions of fuzzy basis functions which are algebraic superpositions of fuzzy membership functions. Using the Stone-Weierstrass theorem, it is proved that linear combinations of the fuzzy basis functions are capable of uniformly approximating any real continuous function on a compact set to arbitrary accuracy. Based on the fuzzy basis function representations, an orthogonal least-squares (OLS) learning algorithm is developed for designing fuzzy systems based on given input-output pairs; then, the OLS algorithm is used to select significant fuzzy basis functions which are used to construct the final fuzzy system. The fuzzy basis function expansion is used to approximate a controller for the nonlinear ball and beam system, and the simulation results show that the control performance is improved by incorporating some common-sense fuzzy control rules.     

Fuzzy system and CMAC network with B-spline membership/basis functions are smooth approximators

Due to their universal approximation, fuzzy system with B-spline membership functions and CMAC neural network with B-spline basis functions have been extensively used in control. In many practical applications, they are desired to approximate not only the assigned smooth function as well as its derivatives. In this paper, by designing a fuzzy system and CMAC neural network with B-spline basis functions, we prove that such a fuzzy system and CMAC can universally approximate a smooth function and its derivatives, i.e, for a given accuracy, we can approximate an arbitrary smooth function by such a fuzzy system and CMAC that not only the function is approximate within this accuracy, but its derivatives are approximated as well. The conclusions here provide solid theoretical foundation for their extensive applications. The authors would like to thank the referees for their invaluable suggestions.     

Fuzzy functions with LSE

“Fuzzy Functions” are proposed to be determined by the least squares estimation (LSE) technique for the development of fuzzy system models. These functions, “Fuzzy Functions with LSE” are proposed as alternate representation and reasoning schemas to the fuzzy rule base approaches. These “Fuzzy Functions” can be more easily obtained and implemented by those who are not familiar with an in-depth knowledge of fuzzy theory. Working knowledge of a fuzzy clustering algorithm such as FCM or its variations would be sufficient to obtain membership values of input vectors. The membership values together with scalar input variables are then used by the LSE technique to determine “Fuzzy Functions” for each cluster identified by FCM. These functions are different from “Fuzzy Rule Base” approaches as well as “Fuzzy Regression” approaches. Various transformations of the membership values are included as new variables in addition to original selected scalar input variables; and at times, a logistic transformation of non-scalar original selected input variables may also be included as a new variable. A comparison of “Fuzzy Functions-LSE” with Ordinary Least Squares Estimation (OLSE)” approach show that “Fuzzy Function-LSE” provide better results in the order of 10% or better with respect to RMSE measure for both training and test cases of data sets.     

Vectorizing NURBS surface evaluation with basis functions in power basis

Several known methods for direct evaluation of NURBS curves and surfaces are described. Runtime performance and simplicity of vectorization are discussed. An evaluation method, which uses basis functions precomputed in shifted power basis, is shown to be promising. This method for surfaces is vectorized with SSE2 intrinsics, yielding 3 times performance improvement over the non-vectorized version. Branchless vectorized linear search is proposed for span search, being most efficient for small number of knots. Binary search ending with a small linear search is shown to be most efficient for large number of knots, and good for general case. Performance comparison of the evaluation method and its equivalents from available geometric kernels is included.     

Fuzzy functions with support vector machines

A new fuzzy system modeling (FSM) approach that identifies the fuzzy functions using support vector machines (SVM) is proposed. This new approach is structurally different from the fuzzy rule base approaches and fuzzy regression methods. It is a new alternate version of the earlier FSM with fuzzy functions approaches. SVM is applied to determine the support vectors for each fuzzy cluster obtained by fuzzy c-means (FCM) clustering algorithm. Original input variables, the membership values obtained from the FCM together with their transformations form a new augmented set of input variables. The performance of the proposed system modeling approach is compared to previous fuzzy functions approaches, standard SVM, LSE methods using an artificial sparse dataset and a real-life non-sparse dataset. The results indicate that the proposed fuzzy functions with support vector machines approach is a feasible and stable method for regression problems and results in higher performances than the classical statistical methods.     

System identification with generalized orthonormal basis functions

A least-squares identification method is studied that estimates a finite number of expansion coefficients in the series expansion of a transfer function, where the expansion is in terms of recently introduced generalized basis functions. The basis functions are orthogonal in ₂, and generalize the pulse, Laguerre and Kautz bases. One of their important properties is that, when chosen properly, they can substantially increase the speed of convergence of the series expansion. This leads to accurate approximate models with only a few coefficients to be estimated. Explicit bounds are derived for the bias and variance errors that occur in parameter estimates as well as in the resulting transfer function estimates.     

Speech-based cursor control using grids: modelling performance and comparisons with other solutions

Speech recognition can be a powerful tool for use in human?–?computer interaction, especially in situations where the user's hands are unavailable or otherwise engaged. Researchers have confirmed that existing mechanisms for speech-based cursor control are both slow and error prone. To address this, we evaluated two variations of a novel grid-based cursor controlled via speech recognition. One provides users with nine cursors that can be used to specify the desired location while the second, more traditional solution, provides a single cursor. Our results confirmed a speed/accuracy trade-off with a nine-cursor variant allowing for faster task completion times while the one-cursor version resulted in reduced error rates. Our solutions eliminated the effect of distance, and dramatically reduced the importance of target size as compared to previous speech-based cursor control mechanisms. The results are explored through a predictive model and comparisons with results from earlier studies.     

Shape-adaptive radial basis functions

Radial basis functions for discrimination and regression have been used with some success in a wide variety of applications. Here, we investigate the optimal choice for the form of the basis functions and present an iterative strategy for obtaining the function in a regression context using a conjugate gradient-based algorithm together with a nonparametric smoother. This is developed in a discrimination framework using the concept of optimal scaling. Results are presented for a range of simulated and real data sets.     

Generalised chebyshev basis functions

The construction of generalised Chebyshev basis functions in one dimension is carried out for both linear and quadratic cases. The optimal selection of the point of reflection of the required Chebyshev Polynomial (s) is identified.     

修正核函数模糊聚类算法

应用核函数度量的紧致性和分离性,给出了一种新的聚类有效性指标KKW,由KKW指标得到最优聚类数并用于修正核函数模糊聚类算法(MKFCM),由于经过了修正核函数的映射,使原来没有显现的特征突显出来。用MKFCM对Wine和glass数据集进行聚类,每一类的聚类正确度大于90%;对于缺失数据的Wisconsin Breast Cancer 数据,错分率为4.72%。该聚类方法在性能上比经典聚类算法有所改进,具有更快的收敛速度以及较高的准确度。仿真实验的结果证实了修正核聚类方法的可行性和有效性。     

Multiscale approximation with hierarchical radial basis functions networks

An approximating neural model, called hierarchical radial basis function (HRBF) network, is presented here. This is a self-organizing (by growing) multiscale version of a radial basis function (RBF) network. It is constituted of hierarchical layers, each containing a Gaussian grid at a decreasing scale. The grids are not completely filled, but units are inserted only where the local error is over threshold. This guarantees a uniform residual error and the allocation of more units with smaller scales where the data contain higher frequencies. Only local operations, which do not require any iteration on the data, are required; this allows to construct the network in quasi-real time. Through harmonic analysis, it is demonstrated that, although a HRBF cannot be reduced to a traditional wavelet-based multiresolution analysis (MRA), it does employ Riesz bases and enjoys asymptotic approximation properties for a very large class of functions. HRBF networks have been extensively applied to the reconstruction of three-dimensional (3-13) models from noisy range data. The results illustrate their power in denoising the original data, obtaining an effective multiscale reconstruction of better quality than that obtained by MRA.     

Fuzzy functions based ARX model and new fuzzy basis function models for nonlinear system identification

In this study, auto regressive with exogenous input (ARX) modeling is improved with fuzzy functions concept (FF-ARX). Fuzzy function with least squares estimation (FF-LSE) method has been recently developed and widely used with a small improvement with respect to least squares estimation method (LSE). FF-LSE is structured with only inputs and their membership values. This proposed model aims to increase the capability of the FF-LSE by widening the regression matrix with lagged input–output values. In addition, by using same idea, we proposed also two new fuzzy basis function models. In the first, basis of the fuzzy system and lagged input–output values are structured together in the regression matrix and named as “L-FBF”. Secondly, instead of using basis function, the membership values of the lagged input–output values are used in the regression matrix by using Gaussian membership functions, called “M-FBF”. Therefore, the power of the fuzzy basis function is also enhanced. For the corresponding models, antecedent part parameters for the input vectors are determined with fuzzy c-means (FCM) clustering algorithm. The consequent parameters of the all models are estimated with the LSE. The proposed models are utilized and compared for the identification of nonlinear benchmark problems.     

Fast evaluation of radial basis functions: I

This paper describes some new techniques for the rapid evaluation and fitting of radial basic functions. The techniques are based on the hierarchical and multipole expansions recently introduced by several authors for the calculation of many-body potentials. Consider in particular the N term thin-plate spline, s(x) = Σ_j=1^N d_jφ(x−x_j), where φ(u) = |u|²log|u|, in 2-dimensions. The direct evaluation of s at a single extra point requires an extra O(N) operations. This paper shows that, with judicious use of series expansions, the incremental cost of evaluating s(x) to within precision ε, can be cut to O(1+|log ε|) operations. In particular, if A is the interpolation matrix, a_i,j = φ(x_i−x_j, the technique allows computation of the matrix-vector product Ad in O(N), rather than the previously required O(N²) operations, and using only O(N) storage. Fast, storage-efficient, computation of this matrix-vector product makes pre-conditioned conjugate-gradient methods very attractive as solvers of the interpolation equations, Ad = y, when N is large.     

Fuzzy modeling with multivariate membership functions: gray-boxidentification and control design

A novel framework for fuzzy modeling and model-based control design is described. The fuzzy model is of the Takagi-Sugeno (TS) type with constant consequents. It uses multivariate antecedent membership functions obtained by Delaunay triangulation of their characteristic points. The number and position of these points are determined by an iterative insertion algorithm. Constrained optimization is used to estimate the consequent parameters, where the constraints are based on control-relevant a priori knowledge about the modeled process. Finally, methods for control design through linearization and inversion of this model are developed. The proposed techniques are demonstrated by means of two benchmark examples: identification of the well-known Box-Jenkins gas furnace and inverse model-based control of a pH process. The obtained results are compared with results from the literature.     

Fuzzy nonlinear regression with fuzzified radial basis function network

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.     

Efficient preconditioning for image reconstruction with radial basis functions

Radial basis functions are a popular basis for interpolating scattered data during the image reconstruction process in graphic analysis. In this context, the solution of a linear system of equations represents the most time-consuming operation. In this paper an efficient preconditioning technique is proposed to solve these linear systems of equations. This algorithm consists of an iterative method which enforces at each iteration a projection of the residual onto a suitable subspace called coarse space. This constraint is ensured by solving an auxiliary problem at each iteration without regularisation. As increasing the number of the coarse space basis functions increases the computational cost of the algorithm, correct selection of coarse space basis is addressed in the paper. Numerical results illustrate the convergence properties of the proposed method with wavelet-like basis functions and regular distributed radial basis functions for image reconstruction.     

The finite volume element method with quadratic basis functions

The paper presents a box scheme with quadratic basis functions for the discretisation of elliptic boundary value problems. The resulting discretisation matrix is non-symmetrical (and also not an M-matrix). The stability analysis is based on an elementwise estimation of the scalar product <A _h u _h ,u _h >. Sufficient conditions placed on the triangles of the triangulation lead to discrete ellipticity. Proof of anO(h ²) error estimate is given for these conditions.