a fuzzy extension of the silhouette width criterion for cluster analysis the present paper proposes a new cluster validity measure as an additional criterion to help the decision making process in fuzzy cluster analysis this measure named fuzzy silhouette is a generalization to the fuzzy case of the average silhouette width criterion originally conceived to assess crisp non fuzzy data partitions the fuzzy silhouette is more appealing than its crisp counterpart in the context of fuzzy cluster analysis since it makes explicit use of the fuzzy partition matrix provided by the clustering algorithm in addition it has been designed to improve performance of the original silhouette criterion in detecting regions with higher data density when the data set involves overlapping clusters the performance of the fuzzy silhouette is evaluated and compared to that of five well known cluster validity measures six data sets are used to illustrate different scenarios in which the proposed fuzzy silhouette performs similar to or better than these other criteria thus becoming eligible to join a pool of measures to be used all together in fuzzy cluster analysis

fast and robust fuzzy c means clustering algorithms incorporating local information for image segmentation fuzzy c means fcm algorithms with spatial constraints fcm s have been proven effective for image segmentation however they still have the following disadvantages 1 although the introduction of local spatial information to the corresponding objective functions enhances their insensitiveness to noise to some extent they still lack enough robustness to noise and outliers especially in absence of prior knowledge of the noise 2 in their objective functions there exists a crucial parameter a used to balance between robustness to noise and effectiveness of preserving the details of the image it is selected generally through experience and 3 the time of segmenting an image is dependent on the image size and hence the larger the size of the image the more the segmentation time in this paper by incorporating local spatial and gray information together a novel fast and robust fcm framework for image segmentation i e fast generalized fuzzy c means fgfcm clustering algorithms is proposed fgfcm can mitigate the disadvantages of fcm s and at the same time enhances the clustering performance furthermore fgfcm not only includes many existing algorithms such as fast fcm and enhanced fcm as its special cases but also can derive other new algorithms such as fgfcm s1 and fgfcm s2 proposed in the rest of this paper the major characteristics of fgfcm are 1 to use a new factor s i j as a local both spatial and gray similarity measure aiming to guarantee both noise immunity and detail preserving for image and meanwhile remove the empirically adjusted parameter a 2 fast clustering or segmenting image the segmenting time is only dependent on the number of the gray levels q rather than the size n q of the image and consequently its computational complexity is reduced from o nci 1 to o qci 2 where c is the number of the clusters i 1 and i 2

spatial clustering with density ordered tree abstract clustering has emerged as an active research direction for knowledge discovery in spatial databases most spatial clustering methods become ineffective when inappropriate parameters are given or when datasets of diverse shapes and densities are provided to address this issue we propose a novel clustering method called scdot spatial clustering with density ordered tree by projecting a dataset to a density ordered tree scdot partitions the data into several relatively small sub clusters with a box plot method a heuristic method is proposed to find the genuine clusters by repeatedly merging sub clusters and an iteration strategy is utilized to automatically determine input parameters moreover we also provide an innovative way to identify cluster center and noise extensive experiments on both synthetic and real world datasets demonstrate the superior performance of scdot over the baseline methods

finding the number of clusters in a dataset one of the most difficult problems in cluster analysis is identifying the number of groups in a dataset most previously suggested approaches to this problem are either somewhat ad hoc or require parametric assumptions and complicated calculations in this article we develop a simple yet powerful nonparametric method for choosing the number of clusters based on distortion a quantity that measures the average distance per dimension between each observation and its closest cluster center our technique is computationally efficient and straightforward to implement we demonstrate empirically its effectiveness not only for choosing the number of clusters but also for identifying underlying structure on a wide range of simulated and real world datasets in addition we give a rigorous theoretical justification for the method based on information theoretic ideas specifically results from the subfield of electrical engineering known as rate distortion theory allow us to describe the behavior of the dist

an iterative method for solving nonlinear integral equations of hammerstein type suppose h is a real hilbert space and f k h h are continuous bounded monotone maps with d k d f h assume that the hammerstein equation u kfu 0 has a solution an explicit iteration process is proved to converge strongly to a solution of this equation no invertibility assumption is imposed on k and the operator f is not restricted to be angle bounded our theorem complements the galerkin method of brezis and browder to provide methods for approximating solutions of nonlinear integral equations of hammerstein type

fcm based model selection algorithms for determining the number of clusters clustering is an important research topic that has practical applications in many fields it has been demonstrated that fuzzy clustering using algorithms such as the fuzzy c means fcm has clear advantages over crisp and probabilistic clustering methods like most clustering algorithms however fcm and its derivatives need the number of clusters in the given data set as one of their initializing parameters the main goal of this paper is to develop an effective fuzzy algorithm for automatically determining the number of clusters after a brief review of the relevant literature we present a new algorithm for determining the number of clusters in a given data set and a new validity index for measuring the goodness of clustering experimental results and comparisons are given to illustrate the performance of the new algorithm

recent developments in nonparametric density estimation advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where restrictions are imposed upon the class of densities or on the form of the likelihood function nonparametric density estimators that are data adaptive and lead to locally smoothed estimators are also discussed these include variable partition histograms estimators based on statistically equivalent blocks nearest neighbor estimators variable kernel estimators and adaptive kernel estimators for the multivariate case extensions of methods of univariate density estimation are usually straightforward but can be computationally expensive a method of multivariate density estimation that did not spring from a univariate generalization is described namely projection pursuit density estimation in which both dimensionality reduction and density estimation can be pursued at the same time finally some areas of related research are mentioned such as nonparametric estimation of functionals of a density robust parametric estimation semiparametric models and density estimation for censored and incomplete data directional and spherical data and density estimation for dependent sequences of observations

consistency of orthogonal series density estimators based on grouped observations in this correspondence we show that nonparametric orthogonal series estimators of probability densities retain the mean integrated square error mise consistency when observations are grouped to the points of a uniform grid prebinned this kind of grouping is typical for computer rounding errors and may also be useful in data compression before calculating estimates e g using the fft the main result shows that mise consistency holds for all square integrable densities under mild conditions on the grid step size

nonparametric regression analysis of growth curves in recent years nonparametric curve estimates have been extensively explored in theoretical work there has however been a certain lack of convincing applications in particular involving comparisons with parametric techniques the present investigation deals with the analysis of human height growth where longitudinal measurements were collected for a sample of boys and a sample of girls evidence is presented that kernel estimates of acceleration and velocity of height and of height itself might offer advantages over a parametric fitting via functional models recently introduced for the specific problem considered both approaches are biased but the parametric one shows qualitative and quantitative distortion which both are not easily predictable data analytic problems involved with kernel estimation concern the choice of kernels the choice of the smoothing parameter and also whether the smoothing parameter should be chosen uniformly for all subjects or individually

community detection in hypernetwork via density ordered tree partition hypernetwork as a useful representation of natural and social systems has received increasing interests from researchers community is crucial to understand the structural and functional properties of the hypernetworks here we propose a new method to uncover the communities of hypernetworks we construct a density ordered tree dot to represent original data by combining density and distance and the community detection in hypernetwork is converted to a dot partition problem then an anomaly detection strategy using box plot rule is applied to partition dot and judge whether there is a significant community structure in the hypernetwork moreover visual inspection as a complementary approach of box plot rule can effectively improve the effectiveness of community detection finally the method is compared with existing methods in both synthetic and real world networks

maximum entropy estimation of density and regression functions abstract this paper presents an information efficient technique to determine the functional forms of density and regression functions by maximizing entropy subject to given side conditions a flexible functional form approach is established as special cases of this approach exponential polynomial cobb douglas translog generalized cobb douglas generalized leontief fourier flexible form and minflex laurent series etc can be derived in a monte carlo experiment the maximum entropy regression method shows good performance in terms of goodness of fit boxplots and isoquants produced by the maximum entropy method are compared with those produced by other well known functions

review papers recent developments in nonparametric density estimation abstract advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where re

on weighting clustering recent papers and patents in iterative unsupervised learning have emphasized a new trend in clustering it basically consists of penalizing solutions via weights on the instance points somehow making clustering move toward the hardest points to cluster the motivations come principally from an analogy with powerful supervised classification methods known as boosting algorithms however interest in this analogy has so far been mainly borne out from experimental studies only this paper is to the best of our knowledge the first attempt at its formalization more precisely we handle clustering as a constrained minimization of a bregman divergence weight modifications rely on the local variations of the expected complete log likelihoods theoretical results show benefits resembling those of boosting algorithms and bring modified weighted versions of clustering algorithms such as k means fuzzy c means expectation maximization em and k harmonic means experiments are provided for all these algorithms with a readily available code they display the advantages that subtle data reweighting may bring to clustering

strong convergence theorems for a common zero of a finite family of m accretive mappings suppose k is a closed convex subset of a strictly convex real banach space e which has a uniformly g teaux differentiable norm suppose that every nonempty closed convex bounded subset of e has the fixed point property for nonexpansive mappings a strong convergence theorem is proved for a common zero of a family of m accretive mappings from k to e as a consequence an iterative method is constructed to converge to a common fixed point assuming existence of a family of pseudocontractive mappings from k to e under certain mild condition

minimax e stable cluster validity index for type 2 fuzziness in this paper we concentrate on the usage of uncertainty associated with the level of fuzziness in determination of the number of clusters in fcm for any data set we propose a minimax e stable cluster validity index based on the uncertainty associated with the level of fuzziness within the framework of interval valued type 2 fuzziness if the data have a clustered structure the optimum number of clusters may be assumed to have minimum uncertainty under upper and lower levels of fuzziness upper and lower values of the level of fuzziness for fuzzy c mean fcm clustering methodology have been found as m 2 6 and 1 4 respectively in our previous studies our investigation shows that the stability of cluster centers with respect to the level of fuzziness is sufficient for the determination of the number of clusters

kernels for nonparametric curve estimation summary the choice of kernels for the nonparametric estimation of regression functions and of their derivatives is investigated explicit expressions are obtained for kernels minimizing the asymptotic variance or the asymptotic integrated mean square error imse the present proof of the optimality of the latter kernels is restricted up to order k 5 these kernels are also of interest for the nonparametric estimation of probability densities and spectral densities a finite sample study indicates that higher order kernels asymptotically improving the rate of convergence may become attractive for realistic finite sample size suitably modified kernels are considered for estimating at the extremities of the data in a way which allows to retain the order of the bias found for interior points

a method for comparing two hierarchical clusterings abstract this article concerns the derivation and use of a measure of similarity between two hierarchical clusterings the measure bk is derived from the matching matrix mij formed by cutting the two hierarchical trees and counting the number of matching entries in the k clusters in each tree the mean and variance of bk are determined under the assumption that the margins of mij are fixed thus bk represents a collection of measures for k 2 n 1 k bk plots are found to be useful in portraying the similarity of two clusterings bk is compared to other measures of similarity proposed respectively by baker 1974 and rand 1971 the use of k bk plots for studying clustering methods is explored by a series of monte carlo sampling experiments an example of the use of k bk on real data is given

orthogonal bases approach for comparing nonnormal continuous distributions summary we present an orthonormal bases approach for detecting general differences among continuous distributions an unknown density function is represented by a finite vector of its estimated fourier coefficients with respect to a suitable orthonormal basis for a wide class of orthonormal bases we establish asymptotic normality of the vector of estimated fourier coefficients and propose an unbiased and consistent estimator of its asymptotic covariance matrix fourier coefficients are modelled as functions of fixed and possibly random effects this approach allows simultaneous detection of distributional differences attributable to various factors in clustered and correlated data with sufficiently large numbers of observations per each cluster with the same fixed and random effects realisations this work was motivated by multi level clustered non gaussian datasets from genetic studies

an examination of procedures for determining the number of clusters in a data set a monte carlo evaluation of 30 procedures for determining the number of clusters was conducted on artificial data sets which contained either 2 3 4 or 5 distinct nonoverlapping clusters to provide a variety of clustering solutions the data sets were analyzed by four hierarchical clustering methods external criterion measures indicated excellent recovery of the true cluster structure by the methods at the correct hierarchy level thus the clustering present in the data was quite strong the simulation results for the stopping rules revealed a wide range in their ability to determine the correct number of clusters in the data several procedures worked fairly well whereas others performed rather poorly thus the latter group of rules would appear to have little validity particularly for data sets containing distinct clusters applied researchers are urged to select one or more of the better criteria however users are cautioned that the performance of some of the criteria may be data dependent

robust path based spectral clustering spectral clustering and path based clustering are two recently developed clustering approaches that have delivered impressive results in a number of challenging clustering tasks however they are not robust enough against noise and outliers in the data in this paper based on m estimation from robust statistics we develop a robust path based spectral clustering method by defining a robust path based similarity measure for spectral clustering under both unsupervised and semi supervised settings our proposed method is significantly more robust than spectral clustering and path based clustering we have performed experiments based on both synthetic and real world data comparing our method with some other methods in particular color images from the berkeley segmentation data set and benchmark are used in the image segmentation experiments experimental results show that our method consistently outperforms other methods due to its higher robustness

characteristic inequalities of uniformly convex and uniformly smooth banach spaces abstract let x be a real banach space with dual x and moduli of convexity and smoothness x e and x respectively for 1 p denotes the duality mapping from x into 2 x with gauge function t p 1 and j p denotes an arbitrary selection for j p let a r 0 0 t is strictly increasing and there exists c 0 such that t c x t 2 and f r 0 0 t is convex nondecreasing and there exists k 0 such that k x it is proved that x is uniformly convex if and only if there is a a such that x y p x p p j p x y x y x y x and x is uniformly smooth if and only if there is a f such that x y p x p p j p x y x y x y x where for given function f f x y is defined by f x y p 0 1 x ty x p t f t y x ty x dt these inequalities which have various applications can be regarded as general banach space versions of the well known polarization identity occurring in hilbert spaces

a meta learning approach for determining the number of clusters with consideration of nearest neighbors an important and challenging problem in data clustering is the determination of the best number of clusters a variety of estimation methods has been proposed over the years to address this problem most of these methods depend on several nontrivial assumptions about the data structure and such methods may thus fail to discover the true clusters in a dataset that does not satisfy those assumptions we develop a new approach that takes as a starting point the simple and intuitive observation that close objects should fall within the same cluster whereas distant ones should not based on this simple notion we utilize a new measurement of good clustering called disconnectivity as well as existing goodness measurements and we embed these measures into a meta learning approach for estimating the number of clusters a simulation experiment based on 13 representative models and an application to real world datasets are conducted to show the effectiveness of the proposed method

global iterative schemes for accretive operators abstract let e be a real q uniformly smooth banach space and a e 2e be an m accretive operator which satisfies a linear growth condition of the form ax c 1 x for some constant c 0 and for all x e it is proved that if two real sequences n and n satisfy appropriate conditions the sequence xn generated from arbitrary x0 e by xn 1 xn n un n xn z un axn n 0 converges strongly to some x a 1 0 furthermore if e is a reflexive banach space with a uniformly g teaux differentiable norm and if every weakly compact convex subset of e has the fixed point property for nonexpansive mappings and a d a e 2e is m accretive then for arbitrary z x0 e the sequence xn defined by xn 1 n un 1 n xn 1 z xn en for un axn where en e is such that en

estimating the density of unemployment duration based on contaminated samples or small samples in estimating a density function for the duration of unemployment we consider two departures from what would be ideal conditions if the so called digit preference effect produces local distortion in observed samples we can apply a maximum entropy density estimation method to establish the functional form of the density we maximize entropy subject to moment restrictions the global shape of the density is determined by the lower ordered sample moments which are not affected much by the digit preference effect as a by product of this method we can establish the local transition structure of the digit preference effect as a second case of departure from an ideal condition we consider coarse sample observations where unemployment duration was observed only for 4 10 14 26 and 52 weeks once the unemployment duration density is derived quintile behavior over time the lorenz curve and the gini coefficient for the distribution of unemployment duration can be obtained finally we discuss the ramifications of only focusing on the headcount ratio of unemployment when other information is available

the selection of terms in an orthogonal series density estimator abstract we show that kronmal and tarter s well known rule for selecting the terms in an orthogonal series density estimator can lead to poor performance and even inconsistency in certain cases these difficulties arise when the underlying density has a nonmonotone sequence of fourier coefficients as is likely to be the case with sharply peaked or multimodal distributions we suggest a way of overcoming these shortcomings

minimum spanning tree based split and merge a hierarchical clustering method most clustering algorithms become ineffective when provided with unsuitable parameters or applied to datasets which are composed of clusters with diverse shapes sizes and densities to alleviate these deficiencies we propose a novel split and merge hierarchical clustering method in which a minimum spanning tree mst and an mst based graph are employed to guide the splitting and merging process in the splitting process vertices with high degrees in the mst based graph are selected as initial prototypes and k means is used to split the dataset in the merging process subgroup pairs are filtered and only neighboring pairs are considered for merge the proposed method requires no parameter except the number of clusters experimental results demonstrate its effectiveness both on synthetic and real datasets

data clustering by minimizing disconnectivity identifying clusters of arbitrary shapes remains a challenge in the field of data clustering we propose a new measure of cluster quality based on minimizing the penalty of disconnection between objects that would be ideally clustered together this disconnectivity is based on analysis of nearest neighbors and the principle that an object should be in the same cluster as its nearest neighbors an algorithm called mindisconnect is proposed that heuristically minimizes disconnectivity and numerical results are presented that indicate that the new algorithm can effectively identify clusters of complex shapes and is robust in finding clusters of arbitrary shapes

curve fitting by polynomial trigonometric regression summary we consider a method of estimating an unknown regression curve by regression on a combination of low order polynomial terms and trigonometric terms estimation based on trigonometric functions alone is known to suffer from bias problems at the boundaries due to the periodic nature of the fitted functions we show that these boundary problems are alleviated by adding low order polynomial terms the utility of the method is illustrated with examples and asymptotic results show that the estimators are competitive with other nonparametric procedures applications to estimation in semiparametric models are also considered with particular emphasis on the case of analysis of covariance when the relationship between the response and the covariate is difficult to model parametrically