automatic kernel clustering with bee colony optimization algorithm cluster analysis is important in data mining especially if there is unsupervised data recently many clustering methods have been proposed unfortunately most of these require the definition of the number of clusters in advance this study addresses this weakness by proposing a new automatic clustering algorithm automatic kernel clustering with bee colony optimization akc bco akc bco determines the appropriate number of clusters and assigns data points to correct clusters this is accomplished by the kernel function which increases clustering capability this method is validated using several benchmark data sets the result is compared with several existing automatic clustering methods the experiment results demonstrate that the proposed akc bco is more stable and accurate than others furthermore the proposed method is also applied to a real world medical problem

improving feature space based image segmentation via density modification feature space based approaches have been popularly used to perform low level image analysis in this paper a density modification framework that enhances density map based discriminability of feature values in a feature space is proposed in order to aid feature space based segmentation in images the framework embeds a position dependent property associated with each sample in the feature space of an image into the corresponding density map and hence modifies it the property association and embedding operations in the framework is implemented using a fuzzy set theory based system devised with cues from beam theory of solid mechanics and the appropriateness of this approach is established qualitative and quantitative experimental results of segmentation in images are given to demonstrate the effectiveness of the proposed density modification framework and the usefulness of feature space based segmentation via density modification

nonlinear equations invariant under the poincare similitude and conformal groups in two dimensional space time all realizations of the lie algebras p 1 1 sim 1 1 and conf 1 1 are classified under the action of the group of local diffeomorphisms of r3 the result is used to obtain all second order scalar differential equations invariant under the corresponding poincare similitude and conformal groups the invariant equations are in general nonlinear and the requirement of linearity turns out to be very restrictive group invariant solutions of some of the conformally invariant equations are obtained either by quadratures or by a linearizing transformation

i k means an iterative clustering algorithm based on an enhanced version of the k means abstract the k means tries to minimize the sum of the squared euclidean distance from the mean ssedm of each cluster as its objective function although this algorithm is effective it is too sensitive to initial centers so many approaches in the literature have focused on determining suitable initial centers however selecting suitable initial centers is not always possible especially when the number of clusters is increased this paper proposes an iterative approach to improve quality of the solution produced by the k means this approach tries to iteratively improve the quality of solution of the k means by removing one cluster minus dividing another one plus and applying re clustering again in each iteration this method called iterative k means minus plus i k means the i k means is speeded up using some methods to determine which cluster should be removed which one should be divided and how to accelerate the re clustering process results of experiments show that i k means can outperform k means to be known one of the accurate version of the k means in terms of minimizing ssedm for some instances the accuracy of i k means is about 2 times higher than both the k means and k means while it is faster than k means and has the reasonable runtime in comparison with the k means

multiplicative lidskii s inequalities and optimal perturbations of frames abstract in this paper we study two design problems in frame theory on the one hand given a fixed finite frame f f j j i n for c d we compute those dual frames g of f that are optimal perturbations of the canonical dual frame for f under certain restrictions on the norms of the elements of g on the other hand we compute those v f v f j j i n for invertible operators v which are close to the identity that are optimal perturbations of f that is we compute the optimal perturbations of f among frames g g j j i n that have the same linear relations as f in both cases optimality is measured with respect to submajorization of the eigenvalues of the frame operators hence our optimal designs are minimizers of a family of convex potentials that include the frame potential and the mean squared error the key tool for these results is a multiplicative analogue of lidskii s inequality in terms of log majorization and a characterization of the case of equality

two fast nearest neighbor searching algorithms for image vector quantization in this paper two efficient codebook searching algorithms for vector quantization vq are presented the first fast search algorithm utilizes the compactness property of signal energy on transform domain and the geometrical relations between the input vector and every codevector to eliminate those codevectors that have no chance to be the closest codeword of the input vector it achieves a full search equivalent performance as compared with other fast methods of the same kind this algorithm requires the fewest multiplications and the least total times of distortion measurements then a suboptimal searching method which sacrifices the reconstructed signal quality to speed up the search of nearest neighbor is presented this algorithm performs the search process on predefined small subcodebooks instead of the whole codebook for the closest codevector experimental results show that this method not only needs less cpu time to encode an image but also encounters less loss of reconstructed signal quality than tree structured vq does

fast modified global k means algorithm for incremental cluster construction the k means algorithm and its variations are known to be fast clustering algorithms however they are sensitive to the choice of starting points and are inefficient for solving clustering problems in large datasets recently incremental approaches have been developed to resolve difficulties with the choice of starting points the global k means and the modified global k means algorithms are based on such an approach they iteratively add one cluster center at a time numerical experiments show that these algorithms considerably improve the k means algorithm however they require storing the whole affinity matrix or computing this matrix at each iteration this makes both algorithms time consuming and memory demanding for clustering even moderately large datasets in this paper a new version of the modified global k means algorithm is proposed we introduce an auxiliary cluster function to generate a set of starting points lying in different parts of the dataset we exploit information gathered in previous iterations of the incremental algorithm to eliminate the need of computing or storing the whole affinity matrix and thereby to reduce computational effort and memory usage results of numerical experiments on six standard datasets demonstrate that the new algorithm is more efficient than the global and the modified global k means algorithms

aliasing and oblique dual pair designs for consistent sampling abstract in this paper we study some aspects of oblique duality between finite sequences of vectors f and g lying in finite dimensional subspaces w and v respectively we compute the possible eigenvalue lists of the frame operators of oblique duals to f lying in v we compute the spectral and geometrical structure of minimizers of convex potentials among oblique duals for f with norm restrictions as an application we show that these optimal duals are the closest to being tight frames and therefore have their spectrum as concentrated as possible among oblique duals with norm restrictions we obtain a complete quantitative analysis of the impact that the relative geometry between the subspaces v and w has in oblique duality we apply this analysis to compute those rigid rotations u for w such that the canonical oblique dual of u f minimize every convex potential we also introduce a notion of aliasing for oblique dual pairs and compute those rigid rotations u for w such that the canonical oblique dual pair associated to u f minimize the aliasing we point out that these two last problems are intrinsic to oblique duality within the context of consistent sampling

the minmax k means clustering algorithm applying k means to minimize the sum of the intra cluster variances is the most popular clustering approach however after a bad initialization poor local optima can be easily obtained to tackle the initialization problem of k means we propose the minmax k means algorithm a method that assigns weights to the clusters relative to their variance and optimizes a weighted version of the k means objective weights are learned together with the cluster assignments through an iterative procedure the proposed weighting scheme limits the emergence of large variance clusters and allows high quality solutions to be systematically uncovered irrespective of the initialization experiments verify the effectiveness of our approach and its robustness over bad initializations as it compares favorably to both k means and other methods from the literature that consider the k means initialization problem 2014 elsevier ltd all rights reserved

a new nonsmooth optimization algorithm for minimum sum of squares clustering problems the minimum sum of squares clustering problem is formulated as a problem of nonsmooth nonconvex optimization and an algorithm for solving the former problem based on nonsmooth optimization techniques is developed the issue of applying this algorithm to large data sets is discussed results of numerical experiments have been presented which demonstrate the effectiveness of the proposed algorithm

how novel is the novel black hole optimization approach due to abundance of novel optimization algorithms in recent years the problem of large similarities among methods that are named differently is becoming troublesome and general the question arises if the novel source of inspiration is sufficient to breed an optimization algorithm with a novel name even if its search properties are almost the same as or are even a simplified variant of the search properties of an older and well known method in this paper it is rigidly shown that the recently proposed heuristic approach called the black hole optimization is in fact a simplified version of particle swarm optimization with inertia weight additionally because a large number of metaheuristics developed during the last decade is claimed to be nature inspired a short discussion on inspirations of optimization algorithms is presented

optimality of klt for high rate transform coding of gaussian vector scale mixtures application to reconstruction estimation and classification the karhunen loeacuteve transform klt is known to be optimal for high rate transform coding of gaussian vectors for both fixed rate and variable rate encoding the klt is also known to be suboptimal for some non gaussian models this paper proves high rate optimality of the klt for variable rate encoding of a broad class of non gaussian vectors gaussian vector scale mixtures gvsm which extend the gaussian scale mixture gsm model of natural signals a key concavity property of the scalar gsm same as the scalar gvsm is derived to complete the proof optimality holds under a broad class of quadratic criteria which include mean squared error mse as well as generalized f divergence loss in estimation and binary classification systems finally the theory is illustrated using two applications signal estimation in multiplicative noise and joint optimization of classification reconstruction systems

clustering by analytic functions data clustering is a combinatorial optimization problem this article shows that clustering is also an optimization problem for an analytic function the mean squared error or in this case the squared error can expressed as an analytic function with an analytic function we benefit from the existence of standard optimization methods the gradient of this function is calculated and the descent method is used to minimize the function

random walks on symmetric spaces and inequalities for matrix spectra abstract using harmonic analysis on symmetric spaces we reduce the singular spectral problem for products of matrices to the recently solved spectral problem for sums of hermitian matrices this proves r c thompson s conjecture matrix spectral inequalities johns hopkins university press baltimore md 1988

improvement of the k means clustering filtering algorithm in this paper we present a modified filtering algorithm mfa by making use of center variations to speed up clustering process our method first divides clusters into static and active groups we use the information of cluster displacements to reject unlikely cluster centers for all nodes in the kd tree we reduce the computational complexity of filtering algorithm fa through finding candidates for each node mainly from the set of active cluster centers two conditions for determining the set of candidate cluster centers for each node from active clusters are developed our approach is different from the major available algorithm which passes no information from one stage of iteration to the next theoretical analysis shows that our method can reduce the computational complexity in terms of the number of distance calculations of fa at each stage of iteration by a factor of fc ac where fc and ac are the numbers of total clusters and active clusters respectively compared with the fa our algorithm can effectively reduce the computing time and number of distance calculations it is noted that our proposed algorithm can generate the same clusters as that produced by hard k means clustering the superiority of our method is more remarkable when a larger data set with higher dimension is used

mercer kernel based clustering in feature space the article presents a method for both the unsupervised partitioning of a sample of data and the estimation of the possible number of inherent clusters which generate the data this work exploits the notion that performing a nonlinear data transformation into some high dimensional feature space increases the probability of the linear separability of the patterns within the transformed space and therefore simplifies the associated data structure it is shown that the eigenvectors of a kernel matrix which defines the implicit mapping provides a means to estimate the number of clusters inherent within the data and a computationally simple iterative procedure is presented for the subsequent feature space partitioning of the data

kernel induced fuzzy clustering of image pixels with an improved differential evolution algorithm a modified differential evolution de algorithm is presented for clustering the pixels of an image in the gray scale intensity space the algorithm requires no prior information about the number of naturally occurring clusters in the image it uses a kernel induced similarity measure instead of the conventional sum of squares distance use of the kernel function makes it possible to partition data that is linearly non separable and non hyper spherical in the original input space into homogeneous groups in a transformed high dimensional feature space a novel search variable representation scheme is adopted for selecting the optimal number of clusters from several possible choices extensive performance comparison over a test suite of 10 gray scale images and objective comparison with manually segmented ground truth indicates that the proposed algorithm has an edge over a few state of the art algorithms for automatic multi class image segmentation

a fast vq codebook generation algorithm using codeword displacement in this paper we present a fast codebook generation algorithm called cgaucd codebook generation algorithm using codeword displacement by making use of the codeword displacement between successive partition processes by implementing a fast search algorithm named mfaupi modified fast algorithm using projection and inequality for vq encoding in the partition step of cgaucd the codebook generation time can be further reduced significantly using mfaupi the computing time of cgaucd can be reduced by a factor of 4 7 7 6 compared to generalized lloyd algorithm gla our proposed method can reduce the codebook generation time by a factor of 35 9 121 2 compared to the best codebook generation algorithm to our knowledge our approach can further reduce the corresponding computing time by 26 0 32 8 it is noted that our proposed algorithm can generate the same codebook as that produced by the gla the superiority of our method is more remarkable when a larger codebook is generated

fast k nearest neighbor search based on projection and triangular inequality in this paper a novel algorithm for finding k points that are closest to a query point is presented some inequalities are used to delete impossible data points and reduce distance computations our algorithm makes use of a data point s feature to reject unlikely candidates for a query point and can eliminate many of the unlikely data points which cannot be rejected by other available algorithms experimental results show that our algorithm is superior to other methods in terms of computing time and the number of distance calculations in most cases and is more remarkable if a larger data set with higher dimension is used compared with available approaches our method can reduce the computing time and number of distance calculations significantly

local extrema for procrustes problems in the set of positive definite matrices abstract given two positive definite matrices a and b a well known result by gelfand naimark and lidskii establishes a relationship between the eigenvalues of a and b and those of ab by means of majorization inequalities in this work we make a local study focused on the spectrum of the matrices that achieve the equality in those inequalities as an application we complete some previous results concerning procrustes problems for unitarily invariant norms in the manifold of positive definite matrices

duality in reconstruction systems we consider the notion of finite dimensional reconstructions systems rs s which includes the fusion frames as projective rs s we study erasures some geometrical properties of these spaces the spectral picture of the set of all dual systems of a fixed rs the spectral picture of the set of rs operators for the projective systems with fixed weights and the structure of the minimizers of the joint potential in this setting we give several examples

frame representations and parseval duals with applications to gabor frames let x n be a frame for a hilbert space h we investigate the conditions under which there exists a dual frame for x n which is also a parseval or tight frame we show that the existence of a parseval dual is equivalent to the problem whether x n can be dilated to an orthonormal basis under an oblique projection a necessary and sufficient condition for the existence of parseval duals is obtained in terms of the frame excess for a frame g g g induced by a projective unitary representation of a group g it is possible that g g g can have a parseval dual but does not have a parseval dual of the same type the primary aim of this paper is to present a complete characterization for all the projective unitary representations such that every frame g g g with a necessary lower frame bound condition has a parseval dual of the same type as an application of this characterization together with a result about lattice tiling we prove that every gabor frame g g l k again with the same necessary lower frame bound condition has a parseval dual of the same type if and only if the volume of the fundamental domain of l k is less than or equal to 1 2

a survey of kernel and spectral methods for clustering clustering algorithms are a useful tool to explore data structures and have been employed in many disciplines the focus of this paper is the partitioning clustering problem with a special interest in two recent approaches kernel and spectral methods the aim of this paper is to present a survey of kernel and spectral clustering methods two approaches able to produce nonlinear separating hypersurfaces between clusters the presented kernel clustering methods are the kernel version of many classical clustering algorithms e g k means som and neural gas spectral clustering arise from concepts in spectral graph theory and the clustering problem is configured as a graph cut problem where an appropriate objective function has to be optimized an explicit proof of the fact that these two paradigms have the same objective is reported since it has been proven that these two seemingly different approaches have the same mathematical foundation besides fuzzy kernel clustering methods are presented as extensions of kernel k means clustering algorithm

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

the hyperbolic smoothing clustering method the minimum sum of squares clustering problem is considered the mathematical modeling of this problem leads to a min sum min formulation which in addition to its intrinsic bi level nature has the significant characteristic of being strongly nondifferentiable to overcome these difficulties the resolution method proposed adopts a smoothing strategy using a special c differentiable class function the final solution is obtained by solving a sequence of low dimension differentiable unconstrained optimization subproblems which gradually approach the original problem the use of this technique called hyperbolic smoothing allows the main difficulties presented by the original problem to be overcome a simplified algorithm containing only the essentials of the method is presented for the purpose of illustrating both the reliability and the efficiency of the method a set of computational experiments was performed making use of traditional test problems described in the literature

j means a new local search heuristic for minimum sum of squares clustering abstract a new local search heuristic called j m eans is proposed for solving the minimum sum of squares clustering problem the neighborhood of the current solution is defined by all possible centroid to entity relocations followed by corresponding changes of assignments moves are made in such neighborhoods until a local optimum is reached the new heuristic is compared with two other well known local search heuristics k and h m eans as well as with h m eans an improved version of the latter in which degeneracy is removed moreover another heuristic which fits into the variable neighborhood search metaheuristic framework and uses j m eans in its local search step is proposed too results on standard test problems from the literature are reported it appears that j m eans outperforms the other local search methods quite substantially when many entities and clusters are considered

modified global k means algorithm for minimum sum of squares clustering problems k means algorithm and its variations are known to be fast clustering algorithms however they are sensitive to the choice of starting points and inefficient for solving clustering problems in large data sets recently a new version of the k means algorithm the global k means algorithm has been developed it is an incremental algorithm that dynamically adds one cluster center at a time and uses each data point as a candidate for the k th cluster center results of numerical experiments show that the global k means algorithm considerably outperforms the k means algorithms in this paper a new version of the global k means algorithm is proposed a starting point for the k th cluster center in this algorithm is computed by minimizing an auxiliary cluster function results of numerical experiments on 14 data sets demonstrate the superiority of the new algorithm however it requires more computational time than the global k means algorithm

image restoration of compressed image using classified vector quantization abstract to reduce communication bandwidth or storage space image compression is needed however the subjective quality of compressed images may be unacceptable and the improvement of quality for compressed images may be desirable this paper extends and modifies classified vector quantization cvq to improve the quality of compressed images the process consists of two phases the encoding phase and the decoding phase the encoding procedure needs a codebook for the encoder which transforms a compressed image to a set of codeword indices the decoding phase also requires a different codebook for the decoder which enhances a compressed image from a set of codeword indices using cvq to improve a compressed image s quality is different from the existing algorithm which cannot reconstruct the high frequency components for compressed images the experimental results show that the image quality is improved dramatically for images in the training set the improvement of psnr is about 3 db for images which are outside the training set the improvement of psnr is about 0 57 db which is comparable to the existing method

a simulated annealing algorithm for the clustering problem abstract in this paper we discuss the solution of the clustering problem usually solved by the k means algorithm the problem is known to have local minimum solutions which are usually what the k means algorithm obtains the simulated annealing approach for solving optimization problems is described and is proposed for solving the clustering problem the parameters of the algorithm are discussed in detail and it is shown that the algorithm converges to a global solution of the clustering problem we also find optimal parameters values for a specific class of data sets and give recommendations on the choice of parameters for general data sets finally advantages and disadvantages of the approach are presented

a genetic algorithm using hyper quadtrees for low dimensional k means clustering the k means algorithm is widely used for clustering because of its computational efficiency given n points in d dimensional space and the number of desired clusters k k means seeks a set of k cluster centers so as to minimize the sum of the squared euclidean distance between each point and its nearest cluster center however the algorithm is very sensitive to the initial selection of centers and is likely to converge to partitions that are significantly inferior to the global optimum we present a genetic algorithm ga for evolving centers in the k means algorithm that simultaneously identifies good partitions for a range of values around a specified k the set of centers is represented using a hyper quadtree constructed on the data this representation is exploited in our ga to generate an initial population of good centers and to support a novel crossover operation that selectively passes good subsets of neighboring centers from parents to offspring by swapping subtrees experimental results indicate that our ga finds the global optimum for data sets with known optima and finds good solutions for large simulated data sets

nonsmooth dc programming approach to the minimum sum of squares clustering problems abstract this paper introduces an algorithm for solving the minimum sum of squares clustering problems using their difference of convex representations a non smooth non convex optimization formulation of the clustering problem is used to design the algorithm characterizations of critical points stationary points in the sense of generalized gradients and inf stationary points of the clustering problem are given the proposed algorithm is tested and compared with other clustering algorithms using large real world data sets

an interior point algorithm for minimum sum of squares clustering an exact algorithm is proposed for minimum sum of squares nonhierarchical clustering i e for partitioning a given set of points from a euclidean m space into a given number of clusters in order to minimize the sum of squared distances from all points to the centroid of the cluster to which they belong this problem is expressed as a constrained hyperbolic program in 0 1 variables the resolution method combines an interior point algorithm i e a weighted analytic center column generation method with branch and bound the auxiliary problem of determining the entering column i e the oracle is an unconstrained hyperbolic program in 0 1 variables with a quadratic numerator and linear denominator it is solved through a sequence of unconstrained quadratic programs in 0 1 variables to accelerate resolution variable neighborhood search heuristics are used both to get a good initial solution and to solve quickly the auxiliary problem as long as global optimality is not reached estimated bounds for the dual variables are deduced from the heuristic solution and used in the resolution process as a trust region proved minimum sum of squares partitions are determined for the first time for several fairly large data sets from the literature including fisher s 150 iris

realizations of galilei algebras all inequivalent realizations of the galilei algebras of dimensions not greater than five are constructed using the algebraic approach proposed by shirokov the varieties of the deformed galilei algebras are discussed and families of one parametric deformations are presented in explicit form it is also shown that a number of well known and physically interesting equations and systems are invariant with respect to the considered galilei algebras or their deformations

initialization of k modes clustering using outlier detection techniques we considered the initialization of k modes clustering from the view of outlier detection we proposed an initialization algorithm for k modes clustering via the distance based outlier detection technique we presented a partition entropy based outlier detection technique and designed an initialization algorithm via it we proposed a new distance metric weighted matching distance metric the effectiveness of our initialization algorithms was shown on several uci data sets the k modes clustering has received much attention since it works well for categorical data sets however the performance of k modes clustering is especially sensitive to the selection of initial cluster centers therefore choosing the proper initial cluster centers is a key step for k modes clustering in this paper we consider the initialization of k modes clustering from the view of outlier detection we present two different initialization algorithms for k modes clustering where the first is based on the traditional distance based outlier detection technique and the second is based on the partition entropy based outlier detection technique by using the above two outlier detection techniques to calculate the degree of outlierness of each object our algorithms can guarantee that the chosen initial cluster centers are not outliers moreover during the process of initialization we adopt a new distance metric weighted matching distance metric to calculate the distance between two objects described by categorical attributes experimental results on several uci data sets demonstrate the effectiveness of our initialization algorithms for k modes clustering

new heuristics for two bounded degree spanning tree problems a vertex v of a connected graph g v e is called a branch vertex if its degree is greater than two pertaining to branch vertices this paper studies two optimization problems having roots in the domain of optical networks the first one referred to as mbv seeks a spanning tree t of g with the minimum number of branch vertices whereas the second problem referred to as mds seeks a spanning tree t of g with the minimum degree sum of branch vertices both mbv and mds are np hard two heuristics approaches are presented for each problem the first approach is a problem specific heuristic whereas the latter one is a hybrid ant colony optimization algorithm computational results show the effectiveness of our proposed approaches

a general framework of hierarchical clustering and its applications hierarchical clustering problem is a traditional topic in computer science which aims to discover a consistent hierarchy of clusters with different granularities one of the most important open questions on hierarchical clustering is the identification of the meaningful clustering levels in the hierarchical structure in this paper we answer this question from algorithmic point of view in particular we derive a quantitative analysis on the impact of the low level clustering costs on high level clusters when agglomerative algorithms are run to construct the hierarchy this analysis enables us to find meaningful clustering levels which are independent of the clusters hierarchically beneath it we thus propose a general agglomerative hierarchical clustering framework which automatically constructs meaningful clustering levels this framework is proven to be generally applicable to any k clustering problem in any relaxed metric space in which strict triangle inequality is relaxed within some constant factor to fully utilize the hierarchical clustering framework we conduct some case studies on k median and k means clustering problems in both of which our framework achieves better approximation factor than the state of the art methods we also extend our framework to handle the data stream clustering problem which allows only one scan on the whole data set by incorporating our framework into guha s data stream clustering algorithm the clustering quality is greatly enhanced with only small extra computation cost incurred the extensive experiments show that our proposal is superior to the distance based agglomerative hierarchical clustering and data stream clustering algorithms on a variety of data sets

an adaptive unsupervised approach toward pixel clustering and color image segmentation this paper proposes an adaptive unsupervised scheme that could find diverse applications in pattern recognition as well as in computer vision particularly in color image segmentation the algorithm named ant colony fuzzy c means hybrid algorithm afha adaptively clusters image pixels viewed as three dimensional data pieces in the rgb color space the ant system as algorithm is applied for intelligent initialization of cluster centroids which endows clustering with adaptivity considering algorithmic efficiency an ant subsampling step is performed to reduce computational complexity while keeping the clustering performance close to original one experimental results have demonstrated afha clustering s advantage of smaller distortion and more balanced cluster centroid distribution over fcm with random and uniform initialization quantitative comparisons with the x means algorithm also show that afha makes a better pre segmentation scheme over x means we further extend its application to natural image segmentation taking into account the spatial information and conducting merging steps in the image space extensive tests were taken to examine the performance of the proposed scheme results indicate that compared with classical segmentation algorithms such as mean shift and normalized cut our method could generate reasonably good or better image partitioning which illustrates the method s practical value

the k means type algorithms versus imbalanced data distributions k means is a partitional clustering technique that is well known and widely used for its low computational cost the representative algorithms include the hard k means and the fuzzy k means however the performance of these algorithms tends to be affected by skewed data distributions i e imbalanced data they often produce clusters of relatively uniform sizes even if input data have varied cluster sizes which is called the uniform effect in this paper we analyze the causes of this effect and illustrate that it probably occurs more in the fuzzy k means clustering process than the hard k means clustering process as the fuzzy index m increases the uniform effect becomes evident to prevent the effect of the uniform effect we propose a multicenter clustering algorithm in which multicenters are used to represent each cluster instead of one single center the proposed algorithm consists of the three subalgorithms the fast global fuzzy k means best m plot and grouping multicenter algorithms they will be respectively used to address the three important problems 1 how are the reliable cluster centers from a dataset obtained 2 how are the number of clusters which these obtained cluster centers represent determined 3 how is it judged as to which cluster centers represent the same clusters the experimental studies on both synthetic and real datasets illustrate the effectiveness of the proposed clustering algorithm in clustering balanced and imbalanced data

evolution equations invariant under two dimensional space time schrodinger group the most general second order evolution equation t f x t x x xx xx 0 invariant under the galilei galilei similitude and schrodinger groups in two dimensions is constructed a preliminary step is a classification of all possible realizations of the corresponding lie algebras of vector fields in r2 c parametrized by x t and applications of this study include the investigation of nonlinear alternatives to quantum mechanics and nonrelativistic classical field theories among the schrodinger invariant equations in particular are found integrable equations linearizable by contact transformations

a method for solving the multiple ellipses detection problem in this paper the multiple ellipses detection problem on the basis of a data points set coming from a number of ellipses in the plane not known in advance is considered an ellipse is considered as a mahalanobis circle with some positive definite matrix a very efficient method for solving this problem is proposed this method very successfully combines the well known direct least squares method and the ransac algorithm with a realistic statistical model of multiple ellipses in the plane the method is illustrated and tested on numerous synthetic and real world applications the method was also compared with other similar methods in the case when a data points set comes from a number of ellipses with clear edges the proposed method gives results similar to other known methods however when a data points set comes from a number of ellipses with noisy edges the proposed method performs significantly better than the other methods we should emphasize the advantage and utility of the proposed methods in a variety of applications such as medical image analysis and ultrasound image segmentation highlightsransac based method for solving the multiple ellipses detection problem is proposed the proposed method shows high efficiency the method has the potential to solve real time applications data is considered as a realization of a realistic statistical model pseudocodes of all proposed algorithms are provided

on new galilei invariant equations in two dimensional spacetime we study realizations of galilei groups acting as transformation lie groups in the space of two independent variables and one dependent variable classification of realizations of the lie algebras and within the class of lie vector fields is carried out utilizing the classification results we have constructed the full sets of second order scalar differential equations in two dimensional spacetime invariant under the lie algebras and

fast global k means clustering using cluster membership and inequality in this paper we present a fast global k means clustering algorithm by making use of the cluster membership and geometrical information of a data point this algorithm is referred to as mfgkm the algorithm uses a set of inequalities developed in this paper to determine a starting point for the jth cluster center of global k means clustering adopting multiple cluster center selection mcs for mfgkm we also develop another clustering algorithm called mfgkm mcs mcs determines more than one starting point for each step of cluster split while the available fast and modified global k means clustering algorithms select one starting point for each cluster split our proposed method mfgkm can obtain the least distortion while mfgkm mcs may give the least computing time compared to the modified global k means clustering algorithm our method mfgkm can reduce the computing time and number of distance calculations by a factor of 3 78 5 55 and 21 13 31 41 respectively with the average distortion reduction of 5 487 for the statlog data set compared to the fast global k means clustering algorithm our method mfgkm mcs can reduce the computing time by a factor of 5 78 8 70 with the average reduction of distortion of 30 564 using the same data set the performances of our proposed methods are more remarkable when a data set with higher dimension is divided into more clusters

mgrs a multi granulation rough set the original rough set model was developed by pawlak which is mainly concerned with the approximation of sets described by a single binary relation on the universe in the view of granular computing the classical rough set theory is established through a single granulation this paper extends pawlak s rough set model to a multi granulation rough set model mgrs where the set approximations are defined by using multi equivalence relations on the universe a number of important properties of mgrs are obtained it is shown that some of the properties of pawlak s rough set theory are special instances of those of mgrs moreover several important measures such as accuracy email protected quality of email protected and precision of email protected are presented which are re interpreted in terms of a classic measure based on sets the marczewski steinhaus metric and the inclusion degree measure a concept of approximation reduct is introduced to describe the smallest attribute subset that preserves the lower approximation and upper approximation of all decision classes in mgrs as well finally we discuss how to extract decision rules using mgrs unlike the decision rules and rules from pawlak s rough set model the form of decision rules in mgrs is or several pivotal algorithms are also designed which are helpful for applying this theory to practical issues the multi granulation rough set model provides an effective approach for problem solving in the context of multi granulations

the lidskii mirsky wielandt theorem additive and multiplicative versions we use a simple matrix splitting technique to give an elementary new proof of the lidskii mirsky wielandt theorem and to obtain a multiplicative analog of the lidskii mirsky wielandt theorem which we argue is the fundamental bound in the study of relative perturbation theory for eigenvalues of hermitian matrices and singular values of general matrices we apply our bound to obtain numerous bounds on the matching distance between the eigenvalues and singular values of matrices our results strengthen and generalize those in the literature

frames of translates with prescribed fine structure in shift invariant spaces abstract for a given finitely generated shift invariant fsi subspace w l 2 r k we obtain a simple criterion for the existence of shift generated sg bessel sequences e f induced by finite sequences of vectors f w n that have a prescribed fine structure i e such that the norms of the vectors in f and the spectra of s e f are prescribed in each fiber of spec w t k we complement this result by developing an analogue of the so called sequences of eigensteps from finite frame theory in the context of sg bessel sequences that allows for a detailed description of all sequences with prescribed fine structure then given 1 n 0 we characterize the finite sequences f w n such that f i 2 i for 1 i n and such that the fine spectral structure of the shift generated bessel sequences e f has minimal spread i e we show the existence of optimal sg bessel sequences with prescribed norms in this context the spread of the spectra is measured in terms of the convex potential p w induced by w and an arbitrary convex function r r

a tabu search approach to the clustering problem in this paper we consider the problem of clustering m objects into c clusters the objects are represented by points in an n dimensional euclidean space and the objective is to classify these m points into c clusters such that the distance between points within a cluster and its center which is to be found is minimized the problem is a nonconvex program that has many local minima it has been studied by many researchers and the most well known algorithm for solving it is the k means algorithm in this paper we develop a new algorithm for solving this problem based on a tabu search technique preliminary computational experience on the developed algorithm are encouraging and compare favorably with both the k means and the simulated annealing algorithms

fast global k means clustering based on local geometrical information the fast global k means fgkm clustering algorithm is one of the most effective approaches for resolving the local convergence of the k means clustering algorithm numerical experiments show that it can effectively determine a global or near global minimizer of the cost function however the fgkm algorithm needs a large amount of computational time or storage space when handling large data sets to overcome this deficiency a more efficient fgkm algorithm namely fgkm a is developed in this paper in the development we first apply local geometrical information to describe approximately the set of objects represented by a candidate cluster center on the basis of the approximate description we then propose an acceleration mechanism for the production of new cluster centers as a result of the acceleration the fgkm a algorithm not only yields the same clustering results as that of the fgkm algorithm but also requires less computational time and fewer distance calculations than the fgkm algorithm and its existing modifications the efficiency of the fgkm a algorithm is further confirmed by experimental studies on several uci data sets

maximum certainty data partitioning abstract problems in data analysis often require the unsupervised partitioning of a dataset into clusters many methods exist for such partitioning but most have the weakness of being model based most assuming hyper ellipsoidal clusters or computationally infeasible in anything more than a three dimensional data space we re consider the notion of cluster analysis in information theoretic terms and show that minimisation of partition entropy can be used to estimate the number and structure of probable data generators

optimal dual frames for erasures for any given frame for the purpose of encoding in a finite dimensional hilbert space we investigate its dual frames that are optimal for erasures for the purpose of decoding we show that in general the canonical dual is not necessarily optimal moreover optimal dual frames are not necessarily unique we present some sufficient conditions under which the canonical dual is the unique optimal dual frame for the erasure problem as an application we get that the canonical dual is the only optimal dual when either the frame is induced by a group representation or the frame is uniform tight

aggregation pheromone density based data clustering ants bees and other social insects deposit pheromone a type of chemical in order to communicate between the members of their community pheromone that causes clumping or clustering behavior in a species and brings individuals into a closer proximity is called aggregation pheromone this article presents a new algorithm called apc for clustering data sets based on this property of aggregation pheromone found in ants an ant is placed at each location of a data point and the ants are allowed to move in the search space to find points with higher pheromone density the movement of an ant is governed by the amount of pheromone deposited at different points of the search space more the deposited pheromone more is the aggregation of ants this leads to the formation of homogenous groups of data the proposed algorithm is evaluated on a number of well known benchmark data sets using different cluster validity measures results are compared with those obtained using two popular standard clustering techniques namely average linkage agglomerative and k means clustering algorithm and with an ant based method called adaptive time dependent transporter ants for clustering atta c experimental results justify the potentiality of the proposed apc algorithm both in terms of the solution clustering quality as well as execution time compared to other algorithms for a large number of data sets

fast searching algorithm for vector quantization using projection and triangular inequality in this paper a new and fast searching algorithm for vector quantization is presented two inequalities one used for terminating the searching process and the other used to delete impossible codewords are presented to reduce the distortion computations our algorithm makes use of a vector s features mean value edge strength and texture strength to reject many unlikely codewords that cannot be rejected by other available approaches experimental results show that our algorithm is superior to other algorithms in terms of computing time and the number of distortion calculations compared with available approaches our method can reduce the computing time and the number of distortion computations significantly compared with the best method of reducing distortion computation our algorithm can further reduce the number of distortion calculations by 29 to 58 4 compared with the best encoding algorithm for vector quantization our approach also further reduces the computing time by 8 to 47 7

k harmonic means data clustering with simulated annealing heuristic abstract clustering procedures partition a set of objects into clusters such that objects in the same cluster are more similar to each other than objects in different clusters according to some predefined criteria clustering is a popular data analysis and data mining technique since clustering problem have np complete nature the larger the size of the problem the harder to find the optimal solution and furthermore the longer to reach a reasonable results one of the most used techniques for clustering is based on k means such that the data is partitioned into k clusters although k means algorithm is easy to implement and works fast in most situations it suffers from two major drawbacks one is sensitivity to initialization and the other is convergence to local optima it is seen from the studies k harmonic means clustering solves the problem of initialization but since its greedy search nature the second problem convergence to local optima still remains in this paper we develop a new algorithm for solving this problem based on a simulated annealing technique simulated annealing k harmonic means clustering sakhmc the experiment results on the iris and the other well known data illustrate the robustness of the sakhmc clustering algorithm

nonsmooth nonconvex optimization approach to clusterwise linear regression problems abstract clusterwise regression consists of finding a number of regression functions each approximating a subset of the data in this paper a new approach for solving the clusterwise linear regression problems is proposed based on a nonsmooth nonconvex formulation we present an algorithm for minimizing this nonsmooth nonconvex function this algorithm incrementally divides the whole data set into groups which can be easily approximated by one linear regression function a special procedure is introduced to generate a good starting point for solving global optimization problems at each iteration of the incremental algorithm such an approach allows one to find global or near global solution to the problem when the data sets are sufficiently dense the algorithm is compared with the multistart spath algorithm on several publicly available data sets for regression analysis

ant intelligence inspired blind data detection for ultra wideband radar sensors given the computational complexity and sophisticated implementation of traditionally parametric channel estimators it has been gradually recognized that the existing data detection methodologies based on the finite impulse response fir propagation channel modeling may become infeasible for ultra wideband uwb radar sensors especially in some large scale distributed scenarios by exploiting the implicit information involved in the received signals in this investigation we present a non parametric uwb data detection scheme for the distributed radar sensor networks a novel characteristic representation is suggested first from a pattern classification point of view a group of quantitative features are then extracted by making full use of the inherent property of uwb propagations thus uwb data detection is formulated as a pattern classification problem in a multidimensional feature space by thoroughly utilizing the self similarity of the representative patterns the ant swarm intelligence inspired clustering algorithm with the new designed ant movement strategy is adopted to perform unsupervised data detections the developed scheme is independent of any a priori modeling information which essentially avoids the expensive parametric estimators and thus enables practically feasible realizations to alleviate the computational burden the principle component analysis pca is further employed to compress the feature space the simulation results validate the new algorithm which is superior to the other popular non parametric data analysis schemes

a novel ant based clustering algorithm using the kernel method a novel ant based clustering algorithm integrated with the kernel ack method is proposed there are two aspects to the integration first kernel principal component analysis kpca is applied to modify the random projection of objects when the algorithm is run initially this projection can create rough clusters and improve the algorithm s efficiency second ant based clustering is performed in the feature space rather than in the input space the distance between the objects in the feature space which is calculated by the kernel function of the object vectors in the input space is applied as a similarity measure the algorithm uses an ant movement model in which each object is viewed as an ant the ant determines its movement according to the fitness of its local neighbourhood the proposed algorithm incorporates the merits of kernel based clustering into ant based clustering comparisons with other classic algorithms using several synthetic and real datasets demonstrate that ack method exhibits high performance in terms of efficiency and clustering quality

a new measure of uncertainty based on knowledge granulation for rough sets in rough set theory accuracy and roughness are used to characterize uncertainty of a set and approximation accuracy is employed to depict accuracy of a rough classification although these measures are effective they have some limitations when the lower upper approximation of a set under one knowledge is equal to that under another knowledge to overcome these limitations we address in this paper the issues of uncertainty of a set in an information system and approximation accuracy of a rough classification in a decision table an axiomatic definition of knowledge granulation for an information system is given under which these three measures are modified theoretical studies and experimental results show that the modified measures are effective and suitable for evaluating the roughness and accuracy of a set in an information system and the approximation accuracy of a rough classification in a decision table respectively and have a much simpler and more comprehensive form than the existing ones