An identification algorithm in fuzzy relational systems

The paper presents an effective identification method in fuzzy relational systems. We propose an algorithm for constructing models on the basis of fuzzy and nonfuzzy data with the aid of fuzzy discretization and clustering techniques. The usefulness of the method provided is demonstrated by means of two numerical examples. Also a possible way of generating a linguistic decision-making algorithm is discussed.     

Local optimization of black-box functions with high or infinite-dimensional inputs: application to nuclear safety

Black-box optimization problems when the input space is a high-dimensional space or a function space appear in more and more applications. In this context, the methods available for finite-dimensional data do not apply. The aim is then to propose a general method for optimization involving dimension reduction techniques. Different dimension reduction basis are considered (including data-driven basis). The methodology is illustrated on simulated functional data. The choice of the different parameters, in particular the dimension of the approximation space, is discussed. The method is finally applied to a problem of nuclear safety.     

Information and dynamical systems: a concrete measurement on sporadic dynamics

We present a method for the study of dynamical systems based on the notion of quantity of information. Measuring the quantity of information of a string by using data compression algorithms, it is possible to give a notion of orbit complexity of dynamical systems. In compact ergodic dynamical systems, entropy is almost everywhere equal to orbit complexity. We have introduced a new compression algorithm called CASToRe which allows a direct estimation of the information content of the orbits in the 0-entropy case. The method is applied to a sporadic dynamical system (Manneville map).     

Solving the optimal network problem

A heuristic method for solving the optimal network problem is proposed and shown to yield high quality results. Solution methods based on Branch-and-Bound techniques are also considered in some detail. The effects of making various approximations, when calculating lower bounds, is discussed and the concept of forced moves introduced.The various methods are applied to a series of problems which include networks with link construction cost not proportional to length and with trip demands tij not all equal.     

Least-squares spectral element method for non-linear hyperbolic differential equations

The least-squares spectral element method has been applied to the one-dimensional inviscid Burgers equation which allows for discontinuous solutions. In order to achieve high order accuracy both in space and in time a space–time formulation has been applied. The Burgers equation has been discretized in three different ways: a non-conservative formulation, a conservative system with two variables and two equations: one first order linear PDE and one linearized algebraic equation, and finally a variant on this conservative formulation applied to a direct minimization with a QR-decomposition at elemental level. For all three formulations an h/p-convergence study has been performed and the results are discussed in this paper.     

On depth measures and dual statistics. A methodology for dealing with general data

A general depth measure, based on the use of one-dimensional linear continuous projections, is proposed. The applicability of this idea in different statistical setups (including inference in functional data analysis, image analysis and classification) is discussed. A special emphasis is made on the possible usefulness of this method in some statistical problems where the data are elements of a Banach space.The asymptotic properties of the empirical approximation of the proposed depth measure are investigated. In particular, its asymptotic distribution is obtained through U-statistics techniques. The practical aspects of these ideas are discussed through a small simulation study and a real-data example.     

A new perturbation solution for systems with strong quadratic and cubic nonlinearities

The new perturbation algorithm combining the method of multiple scales (MS) and Lindstedt–Poincare techniques is applied to an equation with quadratic and cubic nonlinearities. Approximate analytical solutions are found using the classical MS method and the new method. Both solutions are contrasted with the direct numerical solutions of the original equation. For the case of strong nonlinearities, solutions of the new method are in good agreement with the numerical results, whereas the amplitude and frequency estimations of classical MS yield high errors. For strongly nonlinear systems, exact periods match well with the new technique while there are large discrepancies between the exact and classical MS periods. Copyright © 2009 John Wiley & Sons, Ltd.     

A Bootstrap Approach to Nonparametric Regression for Right Censored Data

In this paper a two-stage bootstrap method is proposed for nonparametric regression with right censored data. The method is applied to construct confidence intervals and bands for a conditional survival function. Its asymptotic validity is established using counting process techniques and martingale central limit theory. The performance of the bootstrap method is investigated in a Monte Carlo study. An illustration is given using a real data.     

External circular crack under arbitrary shear loading

An exact closed form solution in terms of elementary functions has been obtained to the governing integral equation of an external circular crack in a transversely isotropic elastic body. The crack is subjected to arbitrary tangential loading applied antisymmetrically to its faces. The recently discovered method of continuity solutions was used here. The solution to the governing integral equation gives the direct relationship between the tangential displacements of the crack faces and the applied loading. Now a complete solution to the problem, with formulae for the field of all stresses and displacements, is possible.     

A Comparison of Discriminant Analysis versus Artificial Neural Networks

Artificial Neural Network (ANN) techniques have recently been applied to many different fields and have demonstrated their capabilities in solving complex problems. In a business environment, the techniques have been applied to predict bond ratings and stock price performance. In these applications, ANN techniques outperformed widely-used multivariate statistical techniques. The purpose of this paper is to compare the ANN method with the Discriminant Analysis (DA) method in order to understand the merits of ANN that are responsible for the higher level of performance. The paper provides an overview of the basic concepts of ANN techniques in order to enhance the understanding of this emerging technique. The similarities and differences between ANN and DA techniques in representing their models are described. This study also proposes a method to overcome the limitations of the ANN approach, Finally, a case study using a data set in a business environment demonstrates the superiority of ANN over DA as a method of classification of observations.     

On the combination of kernel principal component analysis and neural networks for process indirect control

ABSTRACT

A new adaptive kernel principal component analysis (KPCA) for non-linear discrete system control is proposed. The proposed approach can be treated as a new proposition for data pre-processing techniques. Indeed, the input vector of neural network controller is pre-processed by the KPCA method. Then, the obtained reduced neural network controller is applied in the indirect adaptive control. The influence of the input data pre-processing on the accuracy of neural network controller results is discussed by using numerical examples of the cases of time-varying parameters of single-input single-output non-linear discrete system and multi-input multi-output system. It is concluded that, using the KPCA method, a significant reduction in the control error and the identification error is obtained. The lowest mean squared error and mean absolute error are shown that the KPCA neural network with the sigmoid kernel function is the best.     

Prestructuring sparse matrices with dense rows and columns via null space methods

Several applied problems may produce large sparse matrices with a small number of dense rows and/or columns, which can adversely affect the performance of commonly used direct solvers. By posing the problem as a saddle point system, an unconventional application of a null space method can be employed to eliminate dense rows and columns. The choice of null space basis is critical in retaining the overall sparse structure of the matrix. A new one‐sided application of the null space method is also presented to eliminate either dense rows or columns. These methods can be considered techniques that modify the nonzero structure of the matrix before employing a direct solver and may result in improved direct solver performance.     

Mathematics and Neural Networks –A Glance at some Basic Connections

In the following paper, we present a brief and easily accessible introduction to the theory of neural networks under special emphasis on the rôle of pure and applied mathematics in this interesting field of research. In order to allow a quick and direct approach even for nonspecialists, we only consider three-layer feedforward networks with sigmoidal transfer functions and do not cover general multi-layer, recursive or radial-basis-function networks. Moreover, we focus our attention on density and complexity results while construction problems based on operator techniques are not discussed in detail. Especially, in connection with complexity results, we show that neural networks in general have the power to approximate certain function spaces with a minimal number of free parameters. In other words, under this specific point of view neural networks represent one of the best possible approximation devices available. Besides pointing out this remarkable fact, the main motivation for presenting this paper is to give some more mathematicians an idea of what is going on in the theory of neural networks and, perhaps, to encourage, at least a few of them, to start working in this highly interdisciplinary and promising field, too.     

Fuzzy linear regression and its applications to forecasting in uncertain environment

Linear regression has been used for many years in developing mathematical models for application in marketing, management, and sales forecasting. In this paper, two different sales forecasting techniques are discussed. The first technique involves non-fuzzy abstract methods of linear regression and econometrics. A study of the international market sales of cameras, done in 1968 by John Scott Armstrong, utilized these non-fuzzy forecasting techniques. The second sales forecasting technique uses fuzzy linear regression introduced by H. Tanaka, S. Uejima, and K. Asai, in 1980. In this paper, a study of the computer and peripheral equipment sales in the United States is discussed using fuzzy linear regression. Moreover, fuzzy linear regression is applied to forecasting in an uncertain environment. Finally, some possible improvements and suggestions for further study are mentioned.     

Multigrid technique for biharmonic interpolation with application to dual and multiple reciprocity method

A biharmonic-type interpolation method is presented to solve 2D and 3D scattered data interpolation problems. Unlike the methods based on radial basis functions, which produce a large linear system of equations with fully populated and often non-selfadjoint and ill-conditioned matrix, the presented method converts the interpolation problem to the solution of the biharmonic equation supplied with some non-usual boundary conditions at the interpolation points. To solve the biharmonic equation, fast multigrid techniques can be applied which are based on a non-uniform, non-equidistant but Cartesian grid generated by the quadtree/octtree algorithm. The biharmonic interpolation technique is applied to the multiple and dual reciprocity method of the BEM to convert domain integrals to the boundary. This makes it possible to significantly reduce the computational cost of the evaluation of the appearing domain integrals as well as the memory requirement of the procedure. The resulting method can be considered as a special grid-free technique, since it requires no domain discretisation. This revised version was published online in June 2006 with corrections to the Cover Date.     

Dealing with linear dependence during the iterations of the restarted block Lanczos methods

The Lanczos method can be generalized to block form to compute multiple eigenvalues without the need of any deflation techniques. The block Lanczos method reduces a general sparse symmetric matrix to a block tridiagonal matrix via a Gram–Schmidt process. During the iterations of the block Lanczos method an off-diagonal block of the block tridiagonal matrix may become singular, implying that the new set of Lanczos vectors are linearly dependent on the previously generated vectors. Unlike the single vector Lanczos method, this occurrence of linearly dependent vectors may not imply an invariant subspace has been computed. This difficulty of a singular off-diagonal block is easily overcome in non-restarted block Lanczos methods, see [12,30]. The same schemes applied in non-restarted block Lanczos methods can also be applied in restarted block Lanczos methods. This allows the largest possible subspace to be built before restarting. However, in some cases a modification of the restart vectors is required or a singular block will continue to reoccur. In this paper we examine the different schemes mentioned in [12,30] for overcoming a singular block for the restarted block Lanczos methods, namely the restarted method reported in [12] and the Implicitly Restarted Block Lanczos (IRBL) method developed by Baglama et al. [3]. Numerical examples are presented to illustrate the different strategies discussed.     

Empirical Bayes analysis of unreplicated microarray data

Because of the high costs of microarray experiments and the availability of only limited biological materials, microarray experiments are often performed with a small number of replicates. Investigators, therefore, often have to perform their experiments with low replication or without replication. However, the heterogeneous error variability observed in microarray experiments increases the difficulty in analyzing microarray data without replication. No current analysis techniques are practically applicable to such microarray data analysis. We here introduce a statistical method, the so-called unreplicated heterogeneous error model (UHEM) for the microarray data analysis without replication. This method is possible by utilizing many adjacent-intensity genes for estimating local error variance after nonparametric elimination of differentially expressed genes between different biological conditions. We compared the performance of UHEM with three empirical Bayes prior specification methods: between-condition local pooled error, pseudo standard error, or adaptive standard error-based HEM. We found that our unreplicated HEM method is effective for the microarray data analysis when replication of an array experiment is impractical or prohibited.     

Optimal control of a vector borne disease with horizontal transmission

The paper presents the optimal control applied to a vector borne disease with direct transmission in host population. First, we show the existence of the control problem and then use both analytical and numerical techniques to investigate that there are cost effective control efforts for prevention of direct and indirect transmission of disease. In order to do this three control functions are used, one for vector-reduction strategies and the other two for personal (human) protection and blood screening, respectively. We completely characterize the optimal control and compute the numerical solution of the optimality system using an iterative method.     

Effective direct methods for aerodynamic shape optimization

A direct method for aerodynamic shape optimization based on the use of Bézier spline approximation is proposed. The method is tested as applied to the optimization of the supersonic part of an axisymmetric de Laval nozzle. The optimization results are compared with the exact solution obtained by the control contour method (variational nozzle) and with nozzles constructed using another direct method, namely, local linearization. It is shown that both direct optimization methods can be used on rather coarse grids without degrading the accuracy of the solution. The optimization procedure involves the isoperimetric condition that the surface area of the nozzle is given and fixed, which prevents the use of the control contour method. Optimization with allowance for viscosity is performed using the method. For fairly short maximum possible nozzle lengths in the range of Reynolds numbers under consideration, it is shown that allowance for viscosity does not improve the nozzle shape produced by optimization based on the Euler equations. The role of viscosity is reduced to the determination of an optimal length.