聚类有效性函数：熵公式

依据香农信息熵理论。本文引入了一个新的划分熵公式。结合Ｊ．Ｃ．Ｂｅｚｄｅｋ给出的划分熵，定义了一个新的聚类有效性函数。通过四组数据对该聚类有效性函数的判决功能和鲁棒性进行了研究。     

多指标面板数据融合聚类分析

针对多指标面板数据的样品分类和历史时期划分问题,从多元统计分析理论角度提出一个多指标面板数据的融合聚类分析方法。该方法改进了多指标面板数据的因子分析和系统聚类方法,依据Fisher有序聚类理论,构造了Frobenius范数形式的离差平方和函数,提出了多指标面板数据的有序聚类方法。实证结果表明,该方法能够满足系统分析的统一性要求,保证指标之间的不相关;能够克服时间维度上均值处理造成的偏误,信息损失较少;能够解决面板数据有序聚类的问题;弥补了单一分析的片面性和局限性。     

基于核函数的混合C均值聚类算法

提出了一种基于核函数的混合C均值聚类算法.首先利用模糊C均值聚类算法和另一种类型的可能性C均值聚类算法的优点,设计出一种混合C均值聚类算法.然而鉴于该算法存在的不足,本文将Mercer核函数引入到该算法中,仿真实验结果证实了该方法的可行性和有效性.     

灰色最优聚类理论模型及其应用

本文根据灰色关联度的定义，运用广义加权距离构造目标函数，建立了一种灰色最优聚类理论模型，并结合实例说明了该模型的应用     

FCM聚类算法中模糊加权指数m的优选方法

模糊c-均值(FCM)聚类算法是一种通过目标函数的极小化来获得数据集模糊划分的方法。其中,模糊加权指数m对FCM算法的分类性能有着重要的影响,而调用FCM算法进行模糊聚类分析时又必须给m赋值。因此,模糊加权指数m的优选研究就变得很有意义。基于模糊决策的方法本文给出了一种对m的优选方法,实验结果表明该方法是有效的。     

关于聚类有效性函数熵公式HP（u,c）

对文[1]提出的聚类有效性函数HP(u,c）作了一定的理论分析,并就文[1]使用的数据及其他数据进行了计算机模拟。模拟结果显示：HP（u,c) 作为FCM算法的聚类有效性函数是不合适的。     

函数型聚类分析:基于距离的一步法框架

基于距离的函数型聚类分析包含曲线拟合和聚类两个独立步骤,最优曲线拟合未必有利于类别信息的提取和保留。根据曲线拟合与聚类分析的计算过程,重新梳理了函数型聚类算法;基于距离度量,提出了同时考虑拟合和聚类效果的函数型聚类一步法;在交替方向乘子法(ADMM)框架下推导并给出了迭代求解算法。模拟试验结果显示,该函数型聚类算法有助于提高聚类精度;针对北京市空气质量监测站点二氧化氮(NO_2)污染物小时浓度数据的实例验证分析表明,该函数型聚类算法对不同类别空气质量监测点具有更好的区分度。     

聚类问题的人工神经网络方法

在不同的实际问题中，往往视需要使用不同的准则对模式进行聚类。本文给出了一个聚类准则，并使用该准则用人工神经网络方法在计算机上进行了模拟。结果表明本文使用的聚类准则更适合于用人工神经网络实现，可以取得极好的聚类效果。     

一种基于区间数多指标信息的FCM聚类算法

针对一类具有不确定性区间数多指标信息的聚类分析问题，依据传统的基于数值信息的FCM聚类算法的思路，提出了一种新的聚类分析算法。章首先描述了具有区间数多指标信息的聚类分析问题；其次给出了基于区间数多指标信息的关于最优划分和最优聚类中心确定的两个定理；然后给出了基于区间数多指标信息的FCM聚类算法的计算步骤。该算法的特点是聚类中心的表现形式为精确的数值，给出的两个定理说明了该聚类算法的收敛性。最后，通过给出一个算例说明了本给出的聚类算法。     

聚类分析在管理信息系统子系统划分中的应用

本文介绍了一种运用聚类分析方法对管理信息系统子系统进行划分的方法，它提高了大系统子系统划分的有效性和科学性。     

AD HOC网络中的区域划分和资源分配研究

建立了新的Ad Hoc无线网络的区域划分和资源分配模型,讨论了网络覆盖率和抗毁性.通过构造Voronoi图对平面单连通区域的Ad Hoc网络建立区域划分优化模型;定义了网络抗毁性的评价指标连通率,并通过构造Delaunay三角网的最小生成树和蒙特卡罗实验,取得了较好的抗毁仿真结果.最后结合K-均值分簇和罚函数法,得到了近似最优的平面复连通区域的Ad Hoc网络的区域划分和信道安排.     

Cluster validity for fuzzy clustering algorithms

The proportion exponent is introduced as a measure of the validity of the clustering obtained for a data set using a fuzzy clustering algorithm. It is assumed that the output of an algorithm includes a fuzzy nembership function for each data point. We show how to compute the proportion of possible memberships whose maximum entry exceeds the maximum entry of a given membership function, and use these proportions to define the proportion exponent. Its use as a validity functional is illustrated with four numerical examples and its effectiveness compared to other validity functionals, namely, classification entropy and partition coefficient.     

Classroom notes Two‐phase equilibrium: an example

A mathematically simple example of the partition coefficient computation from first (statistical mechanical) principles is given. The physical system represented approximates the partition of iodine between water and carbon tetrachloride phases. The example shows clearly how the chemical potential operates to determine concentrations in phase.     

Solving the knapsack problem with imprecise weight coefficients using genetic algorithms

This paper investigates solving the knapsack problem with imprecise weight coefficients using genetic algorithms. This work is based on the assumption that each weight coefficient is imprecise due to decimal truncation or coefficient rough estimation by the decision-maker. To deal with this kind of imprecise data, fuzzy sets provide a powerful tool to model and solve this problem. We investigate the possibility of using genetic algorithms in solving the fuzzy knapsack problem without defining membership functions for each imprecise weight coefficient. The proposed approach simulates a fuzzy number by distributing it into some partition points. We use genetic algorithms to evolve the values in each partition point so that the final values represent the membership grade of a fuzzy number. The empirical results show that the proposed approach can obtain very good solutions within the given bound of each imprecise weight coefficient than the fuzzy knapsack approach. The fuzzy genetic algorithm concept approach is different, but gives better results than the traditional fuzzy approach.     

Maximal Multiplicative Properties of Partitions

Extending the partition function multiplicatively to a function on partitions, we show that it has a unique maximum at an explicitly given partition for any n ≠ 7. The basis for this is an inequality for the partition function which seems not to have been noticed before.     

Completeness of the Description of an Equilibrium Canonical Ensemble by a Two-Particle Partition Function

We show that in the equilibrium classical canonical ensemble of particles with pair interaction, the full Gibbs partition function can be uniquely expressed in terms of the two-particle partition function. This implies that for a fixed number N of particles in the equilibrium system and a fixed volume V and temperature T, the two-particle partition function fully describes the Gibbs partition as well as the N-particle system in question. The Gibbs partition can be represented as a power series in the two-particle partition function. As an example, we give the linear term of this expansion. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 1, pp. 123–132, October, 2005.     

Reduction formulae of Littlewood–Richardson coefficients

There are two well-known reduction formulae by Griffiths–Harris for Littlewood–Richardson coefficients. Our observation is that some special cases of the factorization theorem of Littlewood–Richardson coefficients by King, Tollu and Toumazet give reduction formulae including the Griffiths–Harris formulae. We provide explicit statements of those reduction formulae in more general forms, and extend them to their conjugated forms also. Eight useful reduction formulae deleting one or two rows (columns) of each partition are listed up as results. As an application, we prove that if the Littlewood–Richardson coefficient is 1 and each partition has distinct parts, then one of two types of our reduction formulae is always applicable and hence we have an algorithm to test if the Littlewood–Richardson coefficient is 1. Furthermore, our conjecture is that one of four types of our reduction formulae is always applicable to all triples of partitions if the corresponding Littlewood–Richardson coefficient is 1.     

Convex decompositions of fuzzy partitions

In this paper we investigate some algebraic and geometric properties of fuzzy partition spaces (convex hulls of hard or conventional partition spaces). In particular, we obtain their dimensions, and describe a number of algorithms for effecting convex decompositions. Two of these are easily programmable, and each affords a different insight about data structures suggested by the fuzzy partition decomposed. We also show how the sequence of partitions in any convex decomposition leads to a matrix for which the norm of the corresponding coefficient vector equals a scalar measure of partition fuzziness used with certain fuzzy clustering algorithms.     

三次分拆数模5的幂

三次分拆由Hei-Chi Chan引入,并由Byungchan Kim命名,因为它和Ramanujan的三次连分数联系在一起.Hei-Chi Chan证明了三次分拆函数具有模3的幂的Ramanujan型同余.在最近的一篇文章中,William Y.C.Chen和Bernard L.S.Lin研究了三次分拆函数模5的同余...     

A polymatroid associated with convex games

A cooperative game with side payments is called convex if its characteristic function is supermodular. We define a polymatroid with the characteristic function of the convex game, and investigate some properties of a partition of the players equivalent to the principal partition of the polymatroid as well as a partial order among the elements of the partition. The partition and partial order reflects players' interdependences and one-way dependences respectively. The critical value in the polymatroid theory is related to the minimum amount of reasonable requirement between coalitions at the time of their amalgamation. Also shown is an application to the problem of oligopoly.