K means algorithm

In this paper a framework for maximum a posteriori (MAP) estimation of hidden Markov models (HMM) is presented. Three key issues of MAP estimation, namely the choice of prior distribution family, the specification of the parameters of prior densities and the evaluation of the MAP estimates, are addressed. Using HMMs with Gaussian mixture state observation densities as an example, it is assumed that the prior densities for the HMM parameters can be adequately represented as a product of Dirichlet and normal-Wishart densities. The classical maximum likelihood estimation algorithms, namely the forward-backward algorithm and the segmental k-means algorithm, are expanded and MAP estimation formulas are developed. Prior density estimation issues are discussed for two classes of applications: parameter smoothing and model adaptation, and some experimental results are given illustrating the practical interest of this approach. Because of its adaptive nature, Bayesian learning is shown to serve as a unified approach for a wide range of speech recognition applications.

Kernel k-means is an extension of the standard kmeans clustering algorithm that identifies nonlinearly separable clusters. In order to overcome the cluster initialization problem associated with this method, in this work we propose the global kernel k-means algorithm, a deterministic and incremental approach to kernel-based clustering. Our method adds one cluster at each stage through a global search procedure consisting of several executions of kernel k-means from suitable initializations. This algorithm does not depend on cluster initialization, identifies nonlinearly separable clusters and, due to its incremental nature and search procedure, locates near optimal solutions avoiding poor local minima. Furthermore a modification is proposed to reduce the computational cost that does not significantly affect the solution quality. We test the proposed methods on artificial data and also for the first time we employ kernel k-means for MRI segmentation along with a novel kernel. The proposed methods compare favorably to kernel k-means with random restarts.

This paper deals with learning first-order logic rules from data lacking an explicit classification predicate. Consequently, the learned rules are not restricted to predicate definitions as in supervised inductive logic programming. First-order logic offers the ability to deal with structured, multi-relational knowledge. Possible applications include first-order knowledge discovery, induction of integrity constraints in databases, multiple predicate learning, and learning mixed theories of predicate definitions and integrity constraints. One of the contributions of our work is a heuristic measure of confirmation, trading off novelty and satisfaction of the rule. The approach has been implemented in the Tertius system. The system performs an optimal best-first search, finding the k most confirmed hypotheses, and includes a non-redundant refinement operator to avoid duplicates in the search. Tertius can be adapted to many different domains by tuning its parameters, and it can deal either with individual-based representations by upgrading propositional representations to first-order, or with general logical rules. We describe a number of experiments demonstrating the feasibility and flexibility of our approach.

In this paper, we aim to compare empirically four initialization methods for the K-Means algorithm: random, Forgy, MacQueen and Kaufman. Although this algorithm is known for its robustness, it is widely reported in the literature that its performance depends upon two key points: initial clustering and instance order. We conduct a series of experiments to draw up (in terms of mean, maximum, minimum and standard deviation) the probability distribution of the square-error values of the final clusters returned by the K-Means algorithm independently on any initial clustering and on any instance order when each of the four initialization methods is used. The results of our experiments illustrate that the random and the Kaufman initialization methods outperform the rest of the compared methods as they make the K-Means more effective and more independent on initial clustering and on instance order. In addition, we compare the convergence speed of the K-Means algorithm when using each of the four initialization methods. Our results suggest that the Kaufman initialization method induces to the K-Means algorithm a more desirable behaviour with respect to the convergence speed than the random initialization method.

In this paper, we present a novel algorithm for performing k-means clustering. It organizes all the patterns in a k-d tree structure such that one can find all the patterns which are closest to a given prototype efficiently. The main intuition behind our approach is as follows. All the prototypes are potential candidates for the closest prototype at the root level. However, for the children of the root node, we may be able to prune the candidate set by using simple geometrical constraints. This approach can be applied recursively until the size of the candidate set is one for each node.

Akfmct-An iterative descent algorithm based on a Lagrangian formulation is introduced for designing vector quantizers having minimum distortion subject to an entropy constraint. These entropy-constrained vector quantizers (ECVQ's) can be used in tandem with variable rate noiseless coding systems to provide locally optimal variable rate block source coding with respect to a fidelity criterion. Experiments on sampled speech and on synthetic sources with memory indicate that for waveform coding at low rates (about 1 bit/sample) under the squared error distortion measure, about 1.6 dB improvement in the signal-to-noise ratio can be expected over the best scalar and lattice quantizers when block entropy-coded with blocklength 4. Even greater gains are made over other forms of entropy-coded vector quantizers. For pattern recognition, it is shown that the ECVQ algorithm is a generalization of the k-means and related algorithms for estimating cluster means, in that the ECVQ algorithm estimates the prior cluster probabilities as well. Experiments on multivariate Gaussian distributions show that for clustering problems involving classes with widely different priors, the ECVQ outperforms the k-means algorithm in both likelihood and probability of error.

We explore the idea of evidence accumulation for combining the results of multiple clusterings. Initially, n d−dimensional data is decomposed into a large number of compact clusters; the K-means algorithm performs this decomposition, with several clusterings obtained by N random initializations of the K-means. Taking the cooccurrences of pairs of patterns in the same cluster as votes for their association, the data partitions are mapped into a co-association matrix of patterns. This n × n matrix represents a new similarity measure between patterns. The final clusters are obtained by applying a MST-based clustering algorithm on this matrix. Results on both synthetic and real data show the ability of the method to identify arbitrary shaped clusters in multidimensional data.

The fast evolution of digital video has brought many new multimedia applications and, as a consequence, has increased the amount of research into new technologies that aim at improving the effectiveness and efficiency of video acquisition, archiving, cataloging and indexing, as well as increasing the usability of stored videos. Among possible research areas, video summarization is an important topic that potentially enables faster browsing of large video collections and also more efficient content indexing and access. Essentially, this research area consists of automatically generating a short summary of a video, which can either be a static summary or a dynamic summary. In this paper, we present VSUMM, a methodology for the production of static video summaries. The method is based on color feature extraction from video frames and k-means clustering algorithm. As an additional contribution, we also develop a novel approach for the evaluation of video static summaries. In this evaluation methodology, video summaries are manually created by users. Then, several user-created summaries are compared both to our approach and also to a number of different techniques in the literature. Experimental results show -with a confidence level of 98% -that the proposed solution provided static video summaries with superior quality relative to the approaches to which it was compared.

We introduced a multilayer perceptron neural network (MLPNN) based classification model as a diagnostic decision support mechanism in the epilepsy treatment. EEG signals were decomposed into frequency sub-bands using discrete wavelet transform (DWT). The wavelet coefficients were clustered using the K-means algorithm for each frequency sub-band. The probability distributions were computed according to distribution of wavelet coefficients to the clusters, and then used as inputs to the MLPNN model. We conducted five different experiments to evaluate the performance of the proposed model in the classifications of different mixtures of healthy segments, epileptic seizure free segments and epileptic seizure segments. We showed that the proposed model resulted in satisfactory classification accuracy rates.

A genetic algorithm-based efficient clustering technique that utilizes the principles of K-Means algorithm is described in this paper. The algorithm called KGA-clustering, while exploiting the searching capability of K-Means, avoids its major limitation of getting stuck at locally optimal values. Its superiority over the K-Means algorithm and another genetic algorithm-based clustering method, is extensively demonstrated for several artificial and real life data sets. A real life application of the KGA-clustering in classifying the pixels of a satellite image of a part of the city of Mumbai is provided.

Abstract— Emergence of modern techniques for scientific data collection has resulted in large scale accumulation of data per-taining to diverse fields. Conventional database querying methods are inadequate to extract useful information from huge data banks. Cluster analysis is one ...

Most existing data stream classification techniques ignore one important aspect of stream data: arrival of a novel class. We address this issue and propose a data stream classification technique that integrates a novel class detection mechanism into traditional classifiers, enabling automatic detection of novel classes before the true labels of the novel class instances arrive. Novel class detection problem becomes more challenging in the presence of concept-drift, when the underlying data distributions evolve in streams. In order to determine whether an instance belongs to a novel class, the classification model sometimes needs to wait for more test instances to discover similarities among those instances. A maximum allowable wait time T c is imposed as a time constraint to classify a test instance. Furthermore, most existing stream classification approaches assume that the true label of a data point can be accessed immediately after the data point is classified. In reality, a time delay T l is involved in obtaining the true label of a data point since manual labeling is time consuming. We show how to make fast and correct classification decisions under these constraints and apply them to real benchmark data. Comparison with state-of-the-art stream classification techniques prove the superiority of our approach.

In this paper, we propose a method based on the Laplacian in the frequency domain for video text detection. Unlike many other approaches which assume that text is horizontallyoriented, our method is able to handle text of arbitrary orientation. The input image is first filtered with Fourier-Laplacian. K-means clustering is then used to identify candidate text regions based on the maximum difference. The skeleton of each connected component helps to separate the different text strings from each other. Finally, text string straightness and edge density are used for false positive elimination. Experimental results show that the proposed method is able to handle graphics text and scene text of both horizontal and non-horizontal orientation.

A fundamental challenge in the design of wireless sensor networks (WSNs) is to maximize their lifetimes. Data aggregation has emerged as a basic approach in WSNs in order to reduce the number of transmissions of sensor nodes, and hence minimizing the overall power consumption in the network. We study optimal data aggregation in WSNs. Data aggregation is affected by several factors, such as the placement of aggregation points, the aggregation function, and the density of sensors in the network. The determination of an optimal selection of aggregation points is thus extremely important. We present exact and approximate algorithms to find the minimum number of aggregation points in order to maximize the network lifetime. Our algorithms use a fixed virtual wireless backbone that is built on top of the physical topology. We also study the tradeoffs between energy savings and the potential delay involved in the data aggregation process. Numerical results show that our approach provides substantial energy savings.

We compare the three basic algorithms for model-based clustering on high-dimensional discretevariable datasets. All three algorithms use the same underlying model: a naive-Bayes model with a hidden root node, also known as a multinomial-mixture model. In the first part of the paper, we perform an experimental comparison between three batch algorithms that learn the parameters of this model: the Expectation-Maximization (EM) algorithm, a "winner take all" version of the EM algorithm reminiscent of the K-means algorithm, and modelbased agglomerative clustering. We find that the EM algorithm significantly outperforms the other methods, and proceed to investigate the effect of various initialization methods on the final solution produced by the EM algorithm. The initializations that we consider are (1) parameters sampled from an uninformative prior, (2) random perturbations of the marginal distribution of the data, and (3) the output of agglomerative clustering. Although the methods are substantially different, they lead to learned models that are similar in quality.

We study the problem of clustering data objects whose locations are uncertain. A data object is represented by an uncertainty region over which a probability density function (pdf) is defined. One method to cluster uncertain objects of this sort is to apply the UK-means algorithm, which is based on the traditional K-means algorithm. In UK-means, an object is assigned to the cluster whose representative has the smallest expected distance to the object. For arbitrary pdf, calculating the expected distance between an object and a cluster representative requires expensive integration computation. We study various pruning methods to avoid such expensive expected distance calculation.

Kernel Methods are algorithms that, by replacing the inner product with an appropriate positive definite function, implicitly perform a nonlinear mapping of the input data into a high dimensional feature space. In this paper, we present a kernel method for clustering inspired by the classical K-Means algorithm in which each cluster is iteratively refined using a one-class Support Vector Machine. Our method, which can be easily implemented, compares favorebly with respect to popular clustering algorithms, like K-Means, Neural Gas and Self Organizing Maps, on a synthetic dataset and three UCI real data benchmarks (IRIS data, Wisconsin breast cancer database, Spam database).

A full-rank under-determined linear system of equations Ax = b has in general infinitely many possible solutions. In recent years there is a growing interest in the sparsest solution of this equation-the one with the fewest non-zero entries, measured by x 0 . Such solutions find applications in signal and image processing, where the topic is typically referred to as "sparse representation". Considering the columns of A as atoms of a dictionary, it is assumed that a given signal b is a linear composition of few such atoms. Recent work established that if the desired solution x is sparse enough, uniqueness of such a result is guaranteed. Also, pursuit algorithms, approximation solvers for the above problem, are guaranteed to succeed in finding this solution.

Clustering techniques have received attention in many fields of study such as engineering, medicine, biology and data mining. The aim of clustering is to collect data points. The K-means algorithm is one of the most common techniques used for clustering. However, the results of K-means depend on the initial state and converge to local optima. In order to overcome local optima obstacles, a lot of studies have been done in clustering. This paper presents an efficient hybrid evolutionary optimization algorithm based on combining Modify Imperialist Competitive Algorithm (MICA) and K-means (K), which is called K-ICA, for optimum clustering N objects into K clusters. The new Hybrid K-ICA algorithm is tested on several data sets and its performance is compared with those of MICA, ACO, PSO, Simulated Annealing (SA), Genetic Algorithm (GA), Tabu Search (TS), Honey Bee Mating Optimization (HBMO) and K-means. The simulation results show that the proposed evolutionary optimization algorithm is robust and suitable for handling data clustering.

Reconstructed daily mean sea level pressure patterns of the North Atlantic–European region are classified for the period 1850 to 2003 to explore long-term changes of the atmospheric circulation and its impact on long-term temperature variability in the central European region. Commonly used k-means clustering algorithms resulted in classifications of low quality because of methodological deficiencies leading to local optima by chance for complex datasets. In contrast, a newly implemented clustering scheme combining the concepts of simulated annealing and diversified randomization (SANDRA) is able to reduce substantially the influence of chance in the cluster assignment, leading to partitions that are noticeably nearer to the global optimum and more stable. The differences between conventional cluster analysis and the SANDRA scheme are significant for subsequent analyses of single clusters—in particular, for trend analysis. Conventional indices used to determine the appropriate numbe...

Objective: There is accumulated evidence that mesial temporal lobe seizures are preceded by a preictal transition that evolves over minutes to hours. In the present study, we investigated these possible preictal changes in long-term intracranial recordings of five patients by a measure of phase synchronization. In order to clearly distinguish preictal changes from all the other interictal states, we developed an automatic extraction of representative patterns of interictal synchronization activity. This reference library was used to classify the successive synchronization patterns of long-term recordings into groups of similar patterns. Altered states of brain synchronization were identified as deviating from patterns in the reference library and were evaluated relative to the times of seizure onset in terms of sensitivity and specificity.

We present a decision theoretic formulation of product partition models (PPMs) that allows a formal treatment of different decision problems such as estimation or hypothesis testing and clustering methods simultaneously. A key observation in our construction is the fact that PPMs can be formulated in the context of model selection. The underlying partition structure in these models is closely related to that arising in connection with Dirichlet processes. This allows a straightforward adaptation of some computational strategies-originally devised for nonparametric Bayesian problems-to our framework. The resulting algorithms are more flexible than other competing alternatives that are used for problems involving PPMs. We propose an algorithm that yields Bayes estimates of the quantities of interest and the groups of experimental units. We explore the application of our methods to the detection of outliers in normal and Student t regression models, with clustering structure equivalent to that induced by a Dirichlet process prior. We also discuss the sensitivity of the results considering different prior distributions for the partitions.

This paper describes a technique for clustering homogeneously distributed data in a peer-to-peer environment like sensor networks. The proposed technique is based on the principles of the K-Means algorithm. It works in a localized asynchronous manner by communicating with the neighboring nodes. The paper offers extensive theoretical analysis of the algorithm that bounds the error in the distributed clustering process compared to the centralized approach that requires downloading all the observed data to a single site. Experimental results show that, in contrast to the case when all the data is transmitted to a central location for application of the conventional clustering algorithm, the communication cost (an important consideration in sensor networks which are typically equipped with limited battery power) of the proposed approach is 0020-0255/$ -see front matter Ó Sciences 176 (2006Sciences 176 ( ) 1952Sciences 176 ( -1985 www.elsevier.com/locate/ins significantly smaller. At the same time, the accuracy of the obtained centroids is high and the number of samples which are incorrectly labeled is also small.

Kernel (or similarity) matrix plays a key role in many machine learning algorithms such as kernel methods, manifold learning, and dimension reduction. However, the cost of storing and manipulating the complete kernel matrix makes it infeasible for large problems. The Nyström method is a popular sampling-based low-rank approximation scheme for reducing the computational burdens in handling large kernel matrices. In this paper, we analyze how the approximating quality of the Nyström method depends on the choice of landmark points, and in particular the encoding powers of the landmark points in summarizing the data. Our (non-probabilistic) error analysis justifies a "clustered Nyström method" that uses the k-means clustering centers as landmark points. Our algorithm can be applied to scale up a wide variety of algorithms that depend on the eigenvalue decomposition of kernel matrix (or its variant), such as kernel principal component analysis, Laplacian eigenmap, spectral clustering, as well as those involving kernel matrix inverse such as least-squares support vector machine and Gaussian process regression. Extensive experiments demonstrate the competitive performance of our algorithm in both accuracy and efficiency.

This paper presents a novel optimized kernel k-means algorithm (OKKC) to combine multiple data sources for clustering analysis. The algorithm uses an alternating minimization framework to optimize the cluster membership and kernel coefficients as a non-convex problem. In the proposed algorithm, the problem to optimize the cluster membership and the problem to optimize the kernel coefficients are all based on the same Rayleigh quotient objective, therefore the proposed algorithm converges locally. OKKC has a simpler procedure and lower complexity than other algorithms proposed in the literature. Simulated and real-life data fusion applications are experimentally studied, and the results validate that the proposed algorithm has comparable performance, moreover, it is more efficient on large scale data sets. 1

In this paper, an evolutionary clustering technique is described that uses a new point symmetry-based distance measure. The algorithm is therefore able to detect both convex and non-convex clusters. Kd-tree based nearest neighbor search is used to reduce the complexity of finding the closest symmetric point. Adaptive mutation and crossover probabilities are used. The proposed GA with point symmetry (GAPS) distance based clustering algorithm is able to detect any type of clusters, irrespective of their geometrical shape and overlapping nature, as long as they possess the characteristic of symmetry. GAPS is compared with existing symmetry-based clustering technique SBKM, its modified version, and the well-known K-means algorithm. Sixteen data sets with widely varying characteristics are used to demonstrate its superiority. For real-life data sets, ANOVA and MANOVA statistical analyses are performed.

Digital data explosion mandates the development of scalable tools to organize the data in a meaningful and easily accessible form. Clustering is a commonly used tool for data organization. However, many clustering algorithms designed to handle large data sets assume linear separability of data and hence do not perform well on real world data sets. While kernel-based clustering algorithms can capture the non-linear structure in data, they do not scale well in terms of speed and memory requirements when the number of objects to be clustered exceeds tens of thousands. We propose an approximation scheme for kernel k-means, termed approximate kernel k-means, that reduces both the computational complexity and the memory requirements by employing a randomized approach. We show both analytically and empirically that the performance of approximate kernel k-means is similar to that of the kernel k-means algorithm, but with dramatically reduced run-time complexity and memory requirements.

In mapping the k-means algorithm to FPGA hardware, we examined algorithm level transforms that dramatically increased the achievable parallelism. We apply the k-means algorithm to multi-spectral and hyper-spectral images, which have tens to hundreds of channels per pixel of data. Kmeans is an iterative algorithm that assigns assigns to each pixel a label indicating which o f K clusters the pixel belongs to.

The adaptation of energy production to demand has been traditionally very important for utilities in order to optimize resource consumption. This is especially true also in microgrids where many intelligent elements have to adapt their behaviour depending on the future generation and consumption conditions. However, traditional forecasting has been performed only for extremely large areas, such as nations and regions. This work aims at presenting a solution for Short-Term Load Forecasting (STLF) in microgrids, based on a three-stage architecture which starts with pattern recognition by a Self-Organizing Map (SOM), a clustering of the previous partition via kmeans algorithm, and finally demand forecasting for each cluster with a MultiLayer Perceptron. Model validation was performed with data from a microgrid-sized environment provided by the Spanish company Iberdrola.

For multiple-objective optimization problems, a common solution methodology is to determine a Pareto optimal set. Unfortunately, these sets are often large and can become difficult to comprehend and consider. Two methods are presented as practical approaches to reduce the size of the Pareto optimal set for multiple-objective system reliability design problems. The first method is a pseudo-ranking scheme that helps the decision maker select solutions that reflect his/her objective function priorities. In the second approach, we used data mining clustering techniques to group the data by using the k-means algorithm to find clusters of similar solutions. This provides the decision maker with just k general solutions to choose from. With this second method, from the clustered Pareto optimal set, we attempted to find solutions which are likely to be more relevant to the decision maker. These are solutions where a small improvement in one objective would lead to a large deterioration in at least one other objective. To demonstrate how these methods work, the well-known redundancy allocation problem was solved as a multiple objective problem by using the NSGA genetic algorithm to initially find the Pareto optimal solutions, and then, the two proposed methods are applied to prune the Pareto set. r

This paper presents an impedance-based structural health monitoring (SHM) technique considering temperature effects. The temperature variation results in significant impedance variations, particularly a frequency shift in the impedance, which may lead to erroneous diagnostic results of real structures. In order to minimize the effects of the temperature variation, a previously proposed temperature compensation technique based on the crosscorrelation between the reference-impedance data and a concurrent impedance data is revisited. In this study, cross-correlation coefficient (CC) after an effective frequency shift, which is defined as the frequency shift causing two impedance data to have the maximum correlation, is utilized. Validation of the proposed technique was carried out on a lab-sized steel truss bridge member under a temperature varying environment. Further, an outlier analysis providing the optimal decision limits has been carried out for more systematic damage detection. It has been found that the threshold level shall be properly selected considering the daily temperature range and the minimum target damage level for detection. It has been demonstrated that the proposed effective frequency shift method can be effectively used in the automated continuous SHM of critical structural members under temperature variations. http://mc.manuscriptcentral.com/jimss

This paper introduces an efficient detection of brain tumor from cerebral MRI images. The methodology consists of three steps: enhancement, segmentation and classification. To improve the quality of images and limit the risk of distinct regions fusion in the segmentation phase an enhancement process is applied. We adopt mathematical morphology to increase the contrast in MRI images. Then we apply Wavelet Transform in the segmentation process to decompose MRI images. At last, the k-means algorithm is implemented to extract the suspicious regions or tumors. Some of experimental results on brain images show the feasibility and the performance of the proposed approach.

We present a genetic algorithm for selecting centers to seed the popular k-means method for clustering. Using a novel crossover operator that exchanges neighboring centers, our GA identifies superior partitions using both benchmark and large simulated data sets.

Image segmentation remains one of the major challenges in image analysis. Many segmentation algorithms have been developed for various applications. Unsatisfactory results have been encountered in some cases, for many existing segmentation algorithms. In this paper, we introduce three modified versions of the conventional moving k-means clustering algorithm called the fuzzy moving k-means, adaptive moving k-means and adaptive fuzzy moving k-means algorithms for image segmentation application. Based on analysis done using standard images (i.e. original bridge and noisy bridge) and hard evidence on microscopic digital image (i.e. segmentation of Sprague Dawley rat sperm), our final segmentation results compare favorably with the results obtained by the conventional k-means, fuzzy c-means and moving k-means algorithms. The qualitative and quantitative analysis done proved that the proposed algorithms are less sensitive with respect to noise. As such, the occurrence of dead centers, center redundancy and trapped center at local minima problems can be avoided. The proposed clustering algorithms are also less sensitive to initialization process of clustering value. The final center values obtained are located within their respective groups of data. This enabled the size and shape of the object in question to be maintained and preserved. Based on the simplicity and capabilities of the proposed algorithms, these algorithms are suitable to be implemented in consumer electronics products such as digital microscope, or digital camera as post processing tool for digital images.

A new method of color image segmentation is proposed in this paper. It's based on K-means algorithm in HSI space and has the advantage over those based on the RGB space. Both the hue and the intensity components are fully utilized. In the process of hue clustering, the special cyclic property of the hue component is taken into consideration. The paper gives the definition of the distance and the center in the hue space, based on which the hue-clustering algorithm is implemented. Utilized in medical image processing, the new method has got a good performance.

In this paper, we present a fast k-means clustering algorithm (FKMCUCD) using the displacements of cluster centers to reject unlikely candidates for a data point. The computing time of our proposed algorithm increases linearly with the data dimension d, whereas the computational complexity of major available kd-tree based algorithms increases exponentially with the value of d. Theoretical analysis shows that our method can reduce the computational complexity of full search by a factor of SF and SF is independent of vector dimension. The experimental results show that compared to full search, our proposed method can reduce computational complexity by a factor of 1.37-4.39 using the data set from six real images. Compared with the filtering algorithm, which is among the available best algorithms of k-means clustering, our algorithm can effectively reduce the computing time. 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.

In recent years, spectral clustering method has gained attentions because of its superior performance compared to other traditional clustering algorithms such as K-means algorithm. The existing spectral clustering algorithms are all off-line algorithms, i.e., they can not incrementally update the clustering result given a small change of the data set. However, the capability of incrementally updating is essential to some applications such as real time monitoring of the evolving communities of websphere or blogsphere. Unlike traditional stream data, these applications require incremental algorithms to handle not only insertion/deletion of data points but also similarity changes between existing items. This paper extends the standard spectral clustering to such evolving data by introducing the incidence vector/matrix to represent two kinds of dynamics in the same framework and by incrementally updating the eigenvalue system. Our incremental algorithm, initialized by a standard spectral clustering, continuously and efficiently updates the eigenvalue system and generates instant cluster labels, as the data set is evolving. The algorithm is applied to a blog data set. Compared with recomputation of the solution by standard spectral clustering, it achieves similar accuracy but with much lower computational cost. Close inspection into the blog content shows that the incremental approach can discover not only the stable blog communities but also the evolution of the individual multi-topic blogs.

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.