Kernel shapes of fuzzy sets in fuzzy systems for function approximation

The shapes of if-part fuzzy sets affect the approximating capability of fuzzy systems. In this paper, the fuzzy systems with the kernel-shaped if-part fuzzy sets are built directly from the training data. It is proved that these fuzzy systems are universal approximators and their uniform approximation rates can be estimated in the single-input-single-output (SISO) case. On the basis of these rates, the relationships between the approximating capability and the shapes of if-part fuzzy sets are developed for the fuzzy systems. Furthermore, the sinc functions that serve as input membership functions are proved to have the almost best approximation property in a particular class of membership functions. The theoretical results are confirmed from the simulation data. In addition, the estimations of the uniform approximation rates are extended to the multi-input-single-output (MISO) case.     

A parallel approach to calculate lower and upper approximations in dominance based rough set theory

Feature selection plays an important role in data mining and machine learning tasks. Rough set theory has been a prominent tool for this purpose. It characterizes a dataset by using two important measures called lower and upper approximation. Dominance based rough set approach (DSRA) is an extension to conventional rough set theory. It is based on persistence of preference order while extracting knowledge from datasets. Dominance principal states that objects belonging to a certain decision class should follow the preference order. Preference order states that an object having higher values of conditional attributes should belong to higher decision classes. However, some of the basic concepts like checking preference order consistency of a dataset, dominance based lower approximation and upper approximation are computationally too expensive to be used for large datasets. In this paper, we have proposed a parallel incremental approach called Parallel Incremental Approximation Calculation or PIAC for short, for calculating these measures of lower and upper approximations. The proposed approach incrementally calculates lower and upper approximations using parallel threads. We compare our method with the conventional approach using ten widely used datasets. Whilst achieving the same accuracy levels as the conventional approach, our approach significantly reduces the average computation time, i.e., 71% for the lower approximation and 70% for the upper approximation. Over all datasets, the decrease in memory usage achieved was 99%.     

动态粒度下的粗糙集双向近似

经典的粗糙集理论刻画目标概念运用静态的粒度分析,不便于刻画人们问题求解的动态认知过程。已有文献分别用正向近似和逆向近似对目标概念和目标决策进行刻画,并成功地应用于分层聚类算法和规则提取方面。基于动态粒度原理,提出双向近似的概念,获得双向近似的一些重要性质,并将其应用于决策表中决策规则的获取。     

Some experiments in waveform approximation

Two methods of waveform approximation were compared, one based on maximal homogeneous neighborhoods, the other on recursive subdivision of nonuniform subintervals. The compactness/error tradeoff was found to favor the former method.     

渐进迭代逼近方法在等距曲线逼近中的应用

渐进迭代逼近(PIA)方法在CAD领域有很好的自适应性和收敛稳定性,在曲线或曲面的逼近和拟合问题上具有很好的应用前景.文中将该方法应用于二维自由曲线的等距曲线(也称offset曲线)的逼近,提出基于PIA的等距曲线逼近算法.首先在等距曲线上采样数据点,采用Floater的方法对数据点进行参数化,并以这些采样点作为初始控制顶点,由这些初始控制顶点产生初始逼近曲线;然后考察相同参数值处采样点和逼近点的误差,并运用PIA方法逐步逼近等距曲线.该算法分别考虑了等距曲线的多项式逼近和有理逼近.数值实例结果表明,综合控制顶点数和算法误差这2项因素,文中算法具备较好的优势.     

一种改进的图像增强算法及其应用

为改进图像增强算法,使之更适合医学领域图片的处理,采用了粗糙集的上逼近和下逼近思想,将图像分为物体区和背景区,使用不同的函数进行增强,进而提出了一种改进的基于粗糙集的增强算法,并首次应用于医学图像处理领域.实验结果显示改进的基于粗糙集的增强效果优于直方图均衡化方法.     

Weierstrass-type approximation theorems with Pythagorean hodograph curves

We prove the Weierstrass-type approximation theorem that states every C¹ curve in the 2-dimensional or 3-dimensional Euclidean space or in the 3-dimensional Minkowski space can be uniformly approximated by Pythagorean hodograph curves in the corresponding space. This abundance of PH curves is another theoretical confirmation of the usefulness and the versatility of the PH curves. We also address some algorithmic aspects of proposed PH approximation schemes and their convergence rates.     

隐式曲面的光滑逼近

给出了隐式曲面的光滑逼近和保凸逼近曲面的构造,所构造的曲面是分段三次的,其光滑度高于Ｃ＾１。     

New topological approach of rough set generalizations

In this paper, we introduce a generalization of rough set theory using topological structures generated by any binary general relation. The class of all after-composed sets and the class of all fore-composed sets defined here are used to generate two topological spaces. These topologies are used to approximate sets topologically. Lower and upper approximations are defined topologically and some of their properties are studied using these topologies. Also, membership, equality, inclusion relations and power set notions in the generalized approximation space are studied. Many differences are shown between notions of ordinary set theory and notions of rough set theory. These notions introduced in this work are good future work to knowledge discovery and data mining.     

近似算法之测度视角

标准近似、微分近似和占优分析是三种不同的近似算法度量方法。标准近似比度量近似解偏离最优解的相对误差。微分近似关注近似解解值在最优解值和最差解值所形成的区间内所处的位置。占优分析考虑近似解在所有可行解中的排名。本文综述相关概念和主要成果,以及各测度方法的优缺点。尤其关注以PCP定理及唯一博弈猜想为代表的不可近似性成果。     

模糊系统逼近理论：现状与展望

近年来,各类模糊系统已被证明是万能函数逼 近器,它们能实现任意的非线性连续控制规律和动态模型．本文综述与分析了模糊系统逼近 理论研究的最新成果,并指出了进一步研究工作的方向．     

初始轨道的确定方法

卫星初轨确定在卫星轨道改进中起着重要作用.随着测量技术的发展,测量数据不断增多,为初轨确定提供了良好的基础.常见的方法有多项式逼近,切比雪夫多项式逼近.然而在实际初轨确定过程中,它们存在很大的缺点:主要是逼近精度不高.根据实际测量数据,提出了用样条函数逼近的方法来获取初始轨道,这种方法具有逼近精度高,实际容易操作的优点.并且通过计算结果进行了比较,指出了多项式逼近和切比雪夫多项式逼近存在的不足.     

On Constrained Facility Location Problems

Given m facilities each with an opening cost, n demands, and distance between every demand and facility, the Facility Location problem finds a solution which opens some facilities to connect every demand to an opened facility such that the total cost of the solution is minimized. The k-Facility Location problem further requires that the number of opened facilities is at most k, where k is a parameter given in the instance of the problem. We consider the Facility Location problems satisfying that for every demand the ratio of the longest distance to facilities and the shortest distance to facilities is at most ω, where ω is a predefined constant. Using the local search approach with scaling technique and error control technique, for any arbitrarily small constant > 0, we give a polynomial-time approximation algorithm for the ω-constrained Facility Location problem with approximation ratio 1 + ω + 1 + ε, which significantly improves the previous best known ratio (ω + 1)/α for some 1≤α≤2, and a polynomial-time approximation algorithm for the ω-constrained k- Facility Location problem with approximation ratio ω+1+ε. On the aspect of approximation hardness, we prove that unless NP■DTIME(nO(loglogn)), the ω-constrained Facility Location problem cannot be approximated within 1 + lnω - 1, which slightly improves the previous best known hardness result 1.243 + 0.316ln(ω - 1). The experimental results on the standard test instances of Facility Location problem show that our algorithm also has good performance in practice.     

闭环连续系统仿真若干问题的分析研究

分析研究了对闭环系统的纯滞后环节如何处理的问题,找到开环系统和闭环系统的理论输出值计算方法,并比较3种Pade近似和全极点近似的精度。将此算法应用到闭环系统中,进一步比较4种近似,以求得闭环系统中误差最小的近似方法。同时提出在计算过程中对纯滞后处理的改进方法,使得误差大大减小,精度提高了一个数量级。研究结果表明,在开环系统和闭环系统,全极点近似由于没有引入零点,使得误差最小;改进方法使误差精度提高了一个数量级。     

The hardness of approximation: Gap location

We refine the complexity analysis of approximation problems by relating it to a new parameter calledgap location. Many of the results obtained so far for approximations yield satisfactory analysis with respect to this refined parameter, but some known results (e.g.,max-k-colorability, max 3-dimensional matching andmax not-all-equal 3sat) fall short of doing so. As a second contribution, our work fills the gap in these cases by presenting new reductions.Next, we present definitions and hardness results of new approximation versions of some NP-complete optimization problems. The problems we treat arevertex cover (for which we define a different optimization problem from the one treated in Papadimitriou & Yannakakis 1991),k-edge coloring, andset splitting.     

Perturbed padé approximation

In the present work, we give a rational approximation of a function f(x) which has a convergent power series form. This approximation is called a perturbed Padé approximation of f(x).

The error of the introduced approximation vanishes at some points in the interval [0,1]. A comparison of the results obtained by our introduced approximation and the first type Padé approximation is given     

System reduction via truncated Hankel matrices

The problem of approximating Hankel operators of finite or infinite rank by lower-rank Hankel operators is considered. For efficiency, truncated Hankel matrices are used as the intermediate step before other existing algorithms such as theCF algorithms are applied to yield the desirable approximants. If the Hankel operator to be approximated is of finite rank, the order of approximation by truncated Hankel operators is obtained. It is also shown that when themths-number is simple, then rational symbols of the best rank-m Hankel approximants of thenth truncated Hankel matrices converge uniformly to the corresponding rational symbol of the best rank-m Hankel approximant of the original Hankel operator asn tends to infinity. Supported by SDIO/IST managed by the U.S. Army under Contract No. DAAL03-87-K-0025 and also supported by the National Science Foundation under Grant No. DMS 8602337. Supported by SDIO/IST managed by the U.S. Army under Contract No. DAAL03-87-K-0025. Supported by the National Science Foundation under Grant No. DMS 8602337.     

Application of weighting functions to the ranking of fuzzy numbers

This paper proposes a new ranking method for fuzzy numbers, which uses a defuzzification of fuzzy numbers and a weighting function. Following Saeidifar and Pasha (2008), first, we define a weighted distance measure on fuzzy numbers, and then, by minimizing this distance, the weighted interval and point approximations of fuzzy numbers are obtained. These indices are applied to rank the fuzzy numbers. This method is new and interesting for ranking fuzzy numbers, and it can be applied for solving and optimizing engineering and economics problems in a fuzzy environment.     

Wide‐band analysis of wire antenna using MoM combined with best uniform rational approximation and surface–surface configuration

In this article, an efficient method based on the method of moments (MoM) combined with the best uniform rational approximation and surface–surface configuration is presented for the wide‐band analysis of wire antenna. Compared with the asymptotic waveform evaluation technique, the major advantage of the Maehly approximation is that it can be easily implemented into an existing MoM computer code. Compared with the wire‐surface configuration, the surface–surface configuration is easy to ensure the current continuity at the junction and maintain the required good accuracy. Numerical results including the input admittance and return loss show that the CPU time took by Maehly approximation is about 1/10 of the CPU time used by the traditional MoM. Good agreement between the presented method and the exact solution is observed. © 2010 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2010.