Approximation theory of fuzzy systems-MIMO case

In this paper, the approximation properties of MIMO fuzzy systems generated by the product inference are discussed. We first give an analysis of fuzzy basic functions (FBF's) and present several properties of FBF's. Based on these properties of FBF's, we obtain several basic approximation properties of fuzzy systems: 1) basic approximation property which reveals the basic approximation mechanism of fuzzy systems; 2) uniform approximation bounds which give the uniform approximation bounds between the desired (control or decision) functions and fuzzy systems; 3) uniform convergent property which shows that fuzzy systems with defined approximation accuracy can always be obtained by dividing the input space into finer fuzzy regions; and 4) universal approximation property which shows that fuzzy systems are universal approximators and extends some previous results on this aspect. The similarity between fuzzy systems and mathematical approximation is discussed and an idea to improve approximation accuracy is suggested based on uniform approximation bounds     

改进的FCM自动化建模方法

提出一种新的有限圆族包络（FCM）自动化建模方法：变深度包络法。该方法针对现有自动化建模方法在不重叠约束满足方面的不足,协同考虑布局组件尺寸与包络深度二因素,在包络圆近似模型中实现不重叠约束的精确表达。实例对比表明,变深度包络法不但能够较不完全包络法有效减少组件包络圆数目,简化布局优化预处理结果,且与不完全包络法相比,能够杜绝组件之间的重叠现象,避免最终布局结果错误的发生。     

粗糙集上下近似的矩阵刻画及应用

主要对粗糙集中上下近似的矩阵刻画及应用进行了研究。给出等价关系、一般二元关系、基于邻域的覆盖粗糙集下一种上下近似的新的矩阵刻画;作为应用,提出关系矩阵方程,并对上下近似的逆问题进行了研究,即在已知关系矩阵[MR,]上（下）近似[R(X)][(R(X))]的情况下反解[X,]给出了求解[X]的方法。     

2D Vector field approximation using linear neighborhoods

We present a vector field approximation for two-dimensional vector fields that preserves their topology and significantly reduces the memory footprint. This approximation is based on a segmentation. The flow within each segmentation region is approximated by an affine linear function. The implementation is driven by four aims: (1) the approximation preserves the original topology; (2) the maximal approximation error is below a user-defined threshold in all regions; (3) the number of regions is as small as possible; and (4) each point has the minimal approximation error. The generation of an optimal solution is computationally infeasible. We discuss this problem and provide a greedy strategy to efficiently compute a sensible segmentation that considers the four aims. Finally, we use the region-wise affine linear approximation to compute a simplified grid for the vector field.     

Approximation by Stancu–Chlodowsky polynomials

In this paper, we investigate convergence and approximation properties of a Chlodowsky type generalization of Stancu polynomials (we called Stancu–Chlodowsky polynomials).The rates of convergence of this generalization are obtained by means of modulus of continuity and by using the $K$-functional of Peetre. We also present and prove theorems on weighted approximation and the order of approximation of continuous functions by these operators on all positive semi-axis.     

The variational gaussian approximation revisited

The variational approximation of posterior distributions by multivariate gaussians has been much less popular in the machine learning community compared to the corresponding approximation by factorizing distributions. This is for a good reason: the gaussian approximation is in general plagued by an Omicron(N)(2) number of variational parameters to be optimized, N being the number of random variables. In this letter, we discuss the relationship between the Laplace and the variational approximation, and we show that for models with gaussian priors and factorizing likelihoods, the number of variational parameters is actually Omicron(N). The approach is applied to gaussian process regression with nongaussian likelihoods.     

Model approximation of multiple delay transfer function models using multiple-point step response fitting

Compared with model approximation for general SISO transfer function models with or without a single constant delay, the approximation problem for SISO transfer function models with multiple delays has received much less attention. In this paper, we attack this problem and thus present a multiple-point step response fitting based approximation method to derive the reduced models. A simple frequency-domain weighted recursive least squares (RLS) algorithm is proposed to determine a set of parameters so that the reduced models can approximate the original models by minimizing the defined frequency-domain squared-error between the step responses of the original and the reduced model. Numerical examples have demonstrated that the proposed approximation approach can not only introduce less dynamic approximation error, but also yield zero steady-state error.     

An axiomatic characterization of a fuzzy generalization of rough sets

In rough set theory, the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties. Several authors have proposed various fuzzy generalizations of rough approximations. In this paper, we introduce the definitions for generalized fuzzy lower and upper approximation operators determined by a residual implication. Then we find the assumptions which permit a given fuzzy set-theoretic operator to represent a upper (or lower) approximation derived from a special fuzzy relation. Different classes of fuzzy rough set algebras are obtained from different types of fuzzy relations. And different sets of axioms of fuzzy set-theoretic operator guarantee the existence of different types of fuzzy relations which produce the same operator. Finally, we study the composition of two approximation spaces. It is proved that the approximation operators in the composition space are just the composition of the approximation operators in the two fuzzy approximation spaces.     

The essential order of approximation for neural networks

There have been various studies on approximation ability of feedforward neural networks (FNNs). Most of the existing studies are, however, only concerned with density or upper bound estimation on how a multivariate function can be approximated by an FNN, and consequently, the essential approximation ability of an FNN cannot be revealed. In this paper, by establishing both upper and lower bound estimations on approximation order, the essential approximation ability (namely, the essential approximation order) of a class of FNNs is clarified in terms of the modulus of smoothness of functions to be approximated. The involved FNNs can not only approximate any continuous or integrable functions defined on a compact set arbitrarily well, but also provide an explicit lower bound on the number of hidden units required. By making use of multivariate approximation tools, it is shown that when the functions to be approximated are Lipschitzian with order up to 2, the approximation speed of the FNNs is uniquely deter     

Arbitrary-Order Fractance Approximation Circuits With High Order-Stability Characteristic and Wider Approximation Frequency Bandwidth

This paper discusses a novel rational approximation algorithm of arbitrary-order fractances, which has high order-stability characteristic and wider approximation frequency bandwidth. The fractor has been exploited extensively in various scientific domains. The well-known shortcoming of the existing fractance approximation circuits, such as the oscillation phenomena, is still in great need of special research attention. Motivated by this need, a novel algorithm with high order-stability characteristic and wider approximation frequency bandwidth is introduced. In order to better understand the iterating process, the approximation principle of this algorithm is investigated at first. Next, features of the iterating function and frequency-domain characteristics of the impedance function calculated by this algorithm are researched, respectively. Furthermore, approximation performance comparisons have been made between the corresponding circuit and other types of fractance approximation circuits. Finally, a fractance approximation circuit with the impedance function of negative 2/3-order is designed. The high order-stability characteristic and wider approximation frequency bandwidth are fundamental important advantages, which make our proposed algorithm competitive in practical applications.      

B-Spline Curve Approximation by Utilizing Big Bang-Big Crunch Method

The location of knot points and estimation of the number of knots are undoubtedly known as one of the most difficult problems in B-Spline curve approximation. In the literature, different researchers have been seen to use more than one optimization algorithm in order to solve this problem. In this paper, Big Bang-Big Crunch method (BB-BC) which is one of the evolutionary based optimization algorithms was introduced and then the approximation of B-Spline curve knots was conducted by this method. The technique of reverse engineering was implemented for the curve knot approximation. The detection of knot locations and the number of knots were randomly selected in the curve approximation which was performed by using BB-BC method. The experimental results were carried out by utilizing seven different test functions for the curve approximation. The performance of BB-BC algorithm was examined on these functions and their results were compared with the earlier studies performed by the researchers. In comparison with the other studies, it was observed that though the number of the knot in BB-BC algorithm was high, this algorithm approximated the B-Spline curves at the rate of minor error.     

The unreasonable effectiveness of neural network approximation

Results concerning the approximation rates of neural networks are of particular interest to engineers. The results reported in the literature have “slow approximation rates” O(1/√m), where m is the number of parameters in the neural network. However, many empirical studies report that neural network approximation is quite effective in practice. We give an explanation of this unreasonable effectiveness by proving the existence of approximation schemes that converge at a rate of the order of 1/m² by using methods from number theory     

粗集理论中知识的粗糙性研究

粗集理论是处理知识不精确和不完善的一种归纳学习方法,其基本思想是在保持分类能力不变的前提下,通过知识约简,导出概念的分类规则。熵作为对不确定性的一种度量,可用于描述近似空间(U,R)中对象的分类情况。在文中,知识的粗糙性定义为近似空间中的粗糙熵,近似空间上基于等价关系的划分过程是其粗糙熵不断减小的过程。同时讨论了信息系统中的若干粗糙熵性质。     

多维函数的进化逼近

提出一种多维函数的通用进化逼近方法,通过构造一类结合采样九样条函数优点的基本函数优点的基本函数族,提出一种单调函数逼近方法,并借助采样函数的有界变差特点,将该方法推广到一般函数情形,这两种函数的逼近都可通过遗传算法完成,该方法的优点在于可以简单一致地推广到更高维函数的逼近,并使逼近复杂度与维数成线性关系,降低学习算法难度,试验表明,方法是有效的,基于文中单调函数逼近技术提出的一种新的决策策略学习方     

An introduction to schoenberg's approximation

For a given function B and a non-zero real number h, Schoenberg's approximation defines from some data the function . For people not used to this kind of approximation, this paper intends to do a summary of the main definitions, properties and utilizations of Schoenberg's approximation: we show that the main tool to handle Schoenberg's approximation is the Fourier transform of B and even more its modified version, the transfer function of B; we give conditions for convergence of when h tends to zero, and we give various ways to define various B as combinations of translates of some function (usually is either some radial function, or obtained by a tensor product of some radial function), depending on the properties we want for the associated Schoenberg's approximation. Last, we show how multi-resolution analysis, subdivision techniques, and wavelets techniques, are nicely connected to Schoenberg's approximation.     

Effects of residual motion compensation errors on the performance of airborne along-track interferometric SAR

Two approximations, center-beam approximation and reference digital elevation model (DEM) approximation, are used in synthetic aperture radar (SAR) motion compensation procedures. They usually introduce residual motion compensation errors for airborne single-antenna SAR imaging and SAR interferometry. In this paper, we investigate the effects of residual uncompensated motion errors, which are caused by the above two approximations, on the performance of airborne along-track interferometric SAR (ATI-SAR). The residual uncompensated errors caused by center-beam approximation in the absence and in the presence of elevation errors are derived, respectively. Airborne simulation parameters are used to verify the correctness of the analysis and to show the impacts of residual uncompensated errors on the interferometric phase errors for ATI-SAR. It is shown that the interferometric phase errors caused by the center-beam approximation with an accurate DEM could be neglected, while the interferometric phase errors caused by the center-beam approximation with an inaccurate DEM cannot be neglected when the elevation errors exceed a threshold. This research provides theoretical bases for the error source analysis and signal processing of airborne ATI-SAR.     

A family of homogeneous analysis models for the design of scalable structures

Research in optimum structural design has shown that mathematical programming techniques can be employed efficiently only in conjunction with explicit approximate constraints. In the course of time a well-established approximation for homogeneous functions (scalable structures) has emerged based on the linear Taylor expansion of the displacement functions in the compliance design space (Reciprocal approximation). It has been shown that the quality of this approximation is based on the property that homogeneity of the constraints is maintained and consequently the approximation is exact along the scaling line.The present paper presents a new family of approximations of homogenous functions which have the same properties as the Reciprocal approximation and which produce more accurate models in most of the tested cases. The approximations are obtained by mapping the direct linear Taylor expansion of the constraints unto a space spanned by intervening variables (original design variables to a powerm). Taking the envelope of these constraints along the scaling line yields a new family of approximations governed by the parameterm. It is shown that the Reciprocal approximation is a particular member of this family of approximations (m = –1).The new technique is illustrated with classical examples of truss optimization. An optimal plate design is also reported. A parametric study of the results for various values of the exponentm is presented. It is shown that for special values of the exponentm the new approximations usually yield better models than those based on the Reciprocal approximation.     

Improved Approximation Bounds for Planar Point Pattern Matching

We analyze the performance of simple algorithms for matching two planar point sets under rigid transformations so as to minimize the directed Hausdorff distance between the sets. This is a well studied problem in computational geometry. Goodrich, Mitchell, and Orletsky presented a very simple approximation algorithm for this problem, which computes transformations based on aligning pairs of points. They showed that their algorithm achieves an approximation ratio of 4. We introduce a modification to their algorithm, which is based on aligning midpoints rather than endpoints. This modification has the same simplicity and running time as theirs, and we show that it achieves a better approximation ratio of roughly 3.14. We also analyze the approximation ratio in terms of a instance-specific parameter that is based on the ratio between diameter of the pattern set to the optimum Hausdorff distance. We show that as this ratio increases (as is common in practical applications) the approximation ratio approaches 3 in the limit. We also investigate the performance of the algorithm by Goodrich et al. as a function of this ratio, and present nearly matching lower bounds on the approximation ratios of both algorithms. This work was supported by the National Science Foundation under grants CCR-0098151 and CCF-0635099.     

Numerical comparison of least square-based finite-difference (LSFD) and radial basis function-based finite-difference (RBFFD) methods

In this paper, two mesh-free methods, i.e., least square-based finite difference (LSFD) and radial basis function-based finite difference (RBFFD), are compared numerically in terms of their accuracy and efficiency. These two mesh-free methods are based on different approximation schemes, that is, the least square approximation and radial basis function (RBF) approximation. The two mesh-free methods exhibit very different behaviors in many ways. In this study, we examine the performance of the two methods by applying them to two example problems: Poisson equation and two-dimensional incompressible viscous lid-driven cavity flow, and some interesting findings are observed.     

集合多覆盖问题的乘性权重更新分析

集合多覆盖问题的简单贪心算法的近似比是lnn＋1。本文提出简单贪心算法的一个变形,宽度优先贪心算法,并且证明其有近似比（lnn）／r＋lnlnn＋O（1）,其中r是覆盖要求。这个结果比由随机取整方法得到的近似比O（（lnn）/r＋√（lnn）＋1为优。宽度优先贪心算法的设计可以归入Arora等最近提出的乘性权重更新方法的框架。关键词集合多覆盖,宽度优先贪心算法,乘性权重更新方法