Clustering using PK-D: A connectivity and density dissimilarity

We present a new dissimilarity, which combines connectivity and density information. Usually, connectivity and density are conceived as mutually exclusive concepts; however, we discuss a novel procedure to merge both information sources. Once we have calculated the new dissimilarity, we apply MDS in order to find a low dimensional vector space representation. The new data representation can be used for clustering and data visualization, which is not pursued in this paper. Instead we use clustering to estimate the gain from our approach consisting of dissimilarity + MDS. Hence, we analyze the partitions’ quality obtained by clustering high dimensional data with various well known clustering algorithms based on density, connectivity and message passing, as well as simple algorithms like k-means and Hierarchical Clustering (HC). The quality gap between the partitions found by k-means and HC alone compared to k-means and HC using our new low dimensional vector space representation is remarkable. Moreover, our tests using high dimensional gene expression and image data confirm these results and show a steady performance, which surpasses spectral clustering and other algorithms relevant to our work.     

Clustering data with measurement errors

Traditional clustering methods assume that there is no measurement error, or uncertainty, associated with data. Often, however, real world applications require treatment of data that have such errors. In the presence of measurement errors, well-known clustering methods like k-means and hierarchical clustering may not produce satisfactory results.In this article, we develop a statistical model and algorithms for clustering data in the presence of errors. We assume that the errors associated with data follow a multivariate Gaussian distribution and are independent between data points. The model uses the maximum likelihood principle and provides us with a new metric for clustering. This metric is used to develop two algorithms for error-based clustering, hError and kError, that are generalizations of Ward's hierarchical and k-means clustering algorithms, respectively.We discuss types of clustering problems where error information associated with the data to be clustered is readily available and where error-based clustering is likely to be superior to clustering methods that ignore error. We focus on clustering derived data (typically parameter estimates) obtained by fitting statistical models to the observed data. We show that, for Gaussian distributed observed data, the optimal error-based clusters of derived data are the same as the maximum likelihood clusters of the observed data. We also report briefly on two applications with real-world data and a series of simulation studies using four statistical models: (1) sample averaging, (2) multiple linear regression, (3) ARIMA models for time-series, and (4) Markov chains, where error-based clustering performed significantly better than traditional clustering methods.     

Clustering data with measurement errors

Traditional clustering methods assume that there is no measurement error, or uncertainty, associated with data. Often, however, real world applications require treatment of data that have such errors. In the presence of measurement errors, well-known clustering methods like k-means and hierarchical clustering may not produce satisfactory results.In this article, we develop a statistical model and algorithms for clustering data in the presence of errors. We assume that the errors associated with data follow a multivariate Gaussian distribution and are independent between data points. The model uses the maximum likelihood principle and provides us with a new metric for clustering. This metric is used to develop two algorithms for error-based clustering, hError and kError, that are generalizations of Ward's hierarchical and k-means clustering algorithms, respectively.We discuss types of clustering problems where error information associated with the data to be clustered is readily available and where error-based clustering is likely to be superior to clustering methods that ignore error. We focus on clustering derived data (typically parameter estimates) obtained by fitting statistical models to the observed data. We show that, for Gaussian distributed observed data, the optimal error-based clusters of derived data are the same as the maximum likelihood clusters of the observed data. We also report briefly on two applications with real-world data and a series of simulation studies using four statistical models: (1) sample averaging, (2) multiple linear regression, (3) ARIMA models for time-series, and (4) Markov chains, where error-based clustering performed significantly better than traditional clustering methods.     

A fuzzy c-means-type algorithm for clustering of data with mixed numeric and categorical attributes employing a probabilistic dissimilarity functional

Gath–Geva (GG) algorithm is one of the most popular methodologies for fuzzy c-means (FCM)-type clustering of data comprising numeric attributes; it is based on the assumption of data deriving from clusters of Gaussian form, a much more flexible construction compared to the spherical clusters assumption of the original FCM. In this paper, we introduce an extension of the GG algorithm to allow for the effective handling of data with mixed numeric and categorical attributes. Traditionally, fuzzy clustering of such data is conducted by means of the fuzzy k-prototypes algorithm, which merely consists in the execution of the original FCM algorithm using a different dissimilarity functional, suitable for attributes with mixed numeric and categorical attributes. On the contrary, in this work we provide a novel FCM-type algorithm employing a fully probabilistic dissimilarity functional for handling data with mixed-type attributes. Our approach utilizes a fuzzy objective function regularized by Kullback–Leibler (KL) divergence information, and is formulated on the basis of a set of probabilistic assumptions regarding the form of the derived clusters. We evaluate the efficacy of the proposed approach using benchmark data, and we compare it with competing fuzzy and non-fuzzy clustering algorithms.     

Finding groups in data: Cluster analysis with ants

We present in this paper a modification of Lumer and Faieta’s algorithm for data clustering. This approach mimics the clustering behavior observed in real ant colonies. This algorithm discovers automatically clusters in numerical data without prior knowledge of possible number of clusters. In this paper we focus on ant-based clustering algorithms, a particular kind of a swarm intelligent system, and on the effects on the final clustering by using during the classification different metrics of dissimilarity: Euclidean, Cosine, and Gower measures. Clustering with swarm-based algorithms is emerging as an alternative to more conventional clustering methods, such as e.g. k-means, etc. Among the many bio-inspired techniques, ant clustering algorithms have received special attention, especially because they still require much investigation to improve performance, stability and other key features that would make such algorithms mature tools for data mining.As a case study, this paper focus on the behavior of clustering procedures in those new approaches. The proposed algorithm and its modifications are evaluated in a number of well-known benchmark datasets. Empirical results clearly show that ant-based clustering algorithms performs well when compared to another techniques.     

Subquadratic Approximation Algorithms for Clustering Problems in High Dimensional Spaces

One of the central problems in information retrieval, data mining, computational biology, statistical analysis, computer vision, geographic analysis, pattern recognition, distributed protocols is the question of classification of data according to some clustering rule. Often the data is noisy and even approximate classification is of extreme importance. The difficulty of such classification stems from the fact that usually the data has many incomparable attributes, and often results in the question of clustering problems in high dimensional spaces. Since they require measuring distance between every pair of data points, standard algorithms for computing the exact clustering solutions use quadratic or “nearly quadratic” running time; i.e., O(dn ^2?α(d)) time where n is the number of data points, d is the dimension of the space and α(d) approaches 0 as d grows. In this paper, we show (for three fairly natural clustering rules) that computing an approximate solution can be done much more efficiently. More specifically, for agglomerative clustering (used, for example, in the Alta Vista? search engine), for the clustering defined by sparse partitions, and for a clustering based on minimum spanning trees we derive randomized (1 + ∈) approximation algorithms with running times Õ(d ² n ^2?γ) where γ > 0 depends only on the approximation parameter ∈ and is independent of the dimension d.     

Scale-invariant clustering with minimum volume ellipsoids

This paper develops theory and algorithms concerning a new metric for clustering data. The metric minimizes the total volume of clusters, where the volume of a cluster is defined as the volume of the minimum volume ellipsoid (MVE) enclosing all data points in the cluster. This metric is scale-invariant, that is, the optimal clusters are invariant under an affine transformation of the data space. We introduce the concept of outliers in the new metric and show that the proposed method of treating outliers asymptotically recovers the data distribution when the data comes from a single multivariate Gaussian distribution. Two heuristic algorithms are presented that attempt to optimize the new metric. On a series of empirical studies with Gaussian distributed simulated data, we show that volume-based clustering outperforms well-known clustering methods such as k-means, Ward's method, SOM, and model-based clustering.     

Software cost estimation based on modified <Emphasis Type="Italic">K</Emphasis>-Modes clustering Algorithm

Unsupervised technique like clustering may be used for software cost estimation in situations where parametric models are difficult to develop. This paper presents a software cost estimation model based on a modified K-Modes clustering algorithm. The aims of this paper are: first, the modified K-Modes clustering which is an enhancement over the simple K-Modes algorithm using a proper dissimilarity measure for mixed data types, is presented and second, the proposed K-Modes algorithm is applied for software cost estimation. We have compared our modified K-Modes algorithm with existing algorithms on different software cost estimation datasets, and results showed the effectiveness of our proposed algorithm.     

基于不相似性度量优化的密度峰值聚类算法

密度峰值聚类（clustering by fast search and find of density peaks,简称DPC）是一种基于局部密度和相对距离属性快速寻找聚类中心的有效算法.DPC通过决策图寻找密度峰值作为聚类中心,不需要提前指定类簇数,并可以得到任意形状的簇聚类.但局部密度和相对距离的计算都只是简单依赖基于距离度量的相似度矩阵,所以在复杂数据上DPC聚类结果不尽如人意,特别是当数据分布不均匀、数据维度较高时.另外,DPC算法中局部密度的计算没有统一的度量,根据不同的数据集需要选择不同的度量方式.第三,截断距离d_c的度量只考虑数据的全局分布,忽略了数据的局部信息,所以d_c的改变会影响聚类的结果,尤其是在小样本数据集上.针对这些弊端,提出一种基于不相似性度量优化的密度峰值聚类算法（optimized density peaks clustering algorithm based on dissimilarity measure,简称DDPC）,引入基于块的不相似性度量方法计算相似度矩阵,并基于新的相似度矩阵计算样本的K近邻信息,然后基于样本的K近邻信息重新定义局部密度的度量方法.经典数据集的实验结果表明,基于不相似性度量优化的密度峰值聚类算法优于DPC的优化算法FKNN-DPC和DPC-KNN,可以在密度不均匀以及维度较高的数据集上得到满意的结果;同时统一了局部密度的度量方式,避免了传统DPC算法中截断距离d_c对聚类结果的影响.     

Clustering ellipses for anomaly detection

Comparing, clustering and merging ellipsoids are problems that arise in various applications, e.g., anomaly detection in wireless sensor networks and motif-based patterned fabrics. We develop a theory underlying three measures of similarity that can be used to find groups of similar ellipsoids in p-space. Clusters of ellipsoids are suggested by dark blocks along the diagonal of a reordered dissimilarity image (RDI). The RDI is built with the recursive iVAT algorithm using any of the three (dis) similarity measures as input and performs two functions: (i) it is used to visually assess and estimate the number of possible clusters in the data; and (ii) it offers a means for comparing the three similarity measures. Finally, we apply the single linkage and CLODD clustering algorithms to three two-dimensional data sets using each of the three dissimilarity matrices as input. Two data sets are synthetic, and the third is a set of real WSN data that has one known second order node anomaly. We conclude that focal distance is the best measure of elliptical similarity, iVAT images are a reliable basis for estimating cluster structures in sets of ellipsoids, and single linkage can successfully extract the indicated clusters.     

结合结构相似度的自适应多尺度SAR图像变化检测

目的 结合高斯核函数特有的性质,提出一种基于结构相似度的自适应多尺度SAR图像变化检测算法。方法 本文提出的算法包括差异图像获取、高斯多尺度分解、基于结构相似性的最优尺度选择、特征矢量构造以及模糊C均值分类。首先,通过对多时相SAR图像进行对数比运算获取差异图像,然后,利用基于图像的结构相似度估计高斯多尺度变换的最优尺度,继而在该最优尺度参数下逐像素构建变化检测特征矢量,最后通过模糊C均值聚类方法实现变化像素与未变化像素的分离,生成最终的变化检测结果图。结果 在两组真实的SAR图像数据上测试本文算法,正确检测率分别达到0.9952和0.9623,Kappa系数分别为0.8200和0.8540,相比传统算法有了较大的提高。结论 本文算法充分利用了尺度信息,对噪声的鲁棒性有所提高。实测SAR数据的实验结果表明,本文算法可以智能获取最优分解尺度,显著提高了SAR图像变化检测性能。     

A dissimilarity function for geospatial polygons

Similarity plays an important role in many data mining tasks and information retrieval processes. Most of the supervised, semi-supervised, and unsupervised learning algorithms depend on using a dissimilarity function that measures the pair-wise similarity between the objects within the dataset. However, traditionally most of the similarity functions fail to adequately treat all the spatial attributes of the geospatial polygons due to the incomplete quantitative representation of structural and topological information contained within the polygonal datasets. In this paper, we propose a new dissimilarity function known as the polygonal dissimilarity function (PDF) that comprehensively integrates both the spatial and the non-spatial attributes of a polygon to specifically consider the density, distribution, and topological relationships that exist within the polygonal datasets. We represent a polygon as a set of intrinsic spatial attributes, extrinsic spatial attributes, and non-spatial attributes. Using this representation of the polygons, PDF is defined as a weighted function of the distance between two polygons in the different attribute spaces. In order to evaluate our dissimilarity function, we compare and contrast it with other distance functions proposed in the literature that work with both spatial and non-spatial attributes. In addition, we specifically investigate the effectiveness of our dissimilarity function in a clustering application using a partitional clustering technique (e.g. \(k\) -medoids) using two characteristically different sets of data: (a) Irregular geometric shapes determined by natural processes, i.e., watersheds and (b) semi-regular geometric shapes determined by human experts, i.e., counties.     

A fuzzy relational clustering algorithm based on a dissimilarity measure extracted from data.

One of the critical aspects of clustering algorithms is the correct identification of the dissimilarity measure used to drive the partitioning of the data set. The dissimilarity measure induces the cluster shape and therefore determines the success of clustering algorithms. As cluster shapes change from a data set to another, dissimilarity measures should be extracted from data. To this aim, we exploit some pairs of points with known dissimilarity value to teach a dissimilarity relation to a feed-forward neural network. Then, we use the neural dissimilarity measure to guide an unsupervised relational clustering algorithm. Experiments on synthetic data sets and on the Iris data set show that the relational clustering algorithm based on the neural dissimilarity outperforms some popular clustering algorithms (with possible partial supervision) based on spatial dissimilarity.     

Extensions to the k-Means Algorithm for Clustering Large Data Sets with Categorical Values

The k-means algorithm is well known for its efficiency in clustering large data sets. However, working only on numeric values prohibits it from being used to cluster real world data containing categorical values. In this paper we present two algorithms which extend the k-means algorithm to categorical domains and domains with mixed numeric and categorical values. The k-modes algorithm uses a simple matching dissimilarity measure to deal with categorical objects, replaces the means of clusters with modes, and uses a frequency-based method to update modes in the clustering process to minimise the clustering cost function. With these extensions the k-modes algorithm enables the clustering of categorical data in a fashion similar to k-means. The k-prototypes algorithm, through the definition of a combined dissimilarity measure, further integrates the k-means and k-modes algorithms to allow for clustering objects described by mixed numeric and categorical attributes. We use the well known soybean disease and credit approval data sets to demonstrate the clustering performance of the two algorithms. Our experiments on two real world data sets with half a million objects each show that the two algorithms are efficient when clustering large data sets, which is critical to data mining applications.     

A semi-supervised regression model for mixed numerical and categorical variables

In this paper, we develop a semi-supervised regression algorithm to analyze data sets which contain both categorical and numerical attributes. This algorithm partitions the data sets into several clusters and at the same time fits a multivariate regression model to each cluster. This framework allows one to incorporate both multivariate regression models for numerical variables (supervised learning methods) and k-mode clustering algorithms for categorical variables (unsupervised learning methods). The estimates of regression models and k-mode parameters can be obtained simultaneously by minimizing a function which is the weighted sum of the least-square errors in the multivariate regression models and the dissimilarity measures among the categorical variables. Both synthetic and real data sets are presented to demonstrate the effectiveness of the proposed method.     

Locally linear metric adaptation with application to semi-supervised clustering and image retrieval

Many computer vision and pattern recognition algorithms are very sensitive to the choice of an appropriate distance metric. Some recent research sought to address a variant of the conventional clustering problem called semi-supervised clustering, which performs clustering in the presence of some background knowledge or supervisory information expressed as pairwise similarity or dissimilarity constraints. However, existing metric learning methods for semi-supervised clustering mostly perform global metric learning through a linear transformation. In this paper, we propose a new metric learning method that performs nonlinear transformation globally but linear transformation locally. In particular, we formulate the learning problem as an optimization problem and present three methods for solving it. Through some toy data sets, we show empirically that our locally linear metric adaptation (LLMA) method can handle some difficult cases that cannot be handled satisfactorily by previous methods. We also demonstrate the effectiveness of our method on some UCI data sets. Besides applying LLMA to semi-supervised clustering, we have also used it to improve the performance of content-based image retrieval systems through metric learning. Experimental results based on two real-world image databases show that LLMA significantly outperforms other methods in boosting the image retrieval performance.     

Incremental K-clique clustering in dynamic social networks

Clustering entities into dense parts is an important issue in social network analysis. Real social networks usually evolve over time and it remains a problem to efficiently cluster dynamic social networks. In this paper, a dynamic social network is modeled as an initial graph with an infinite change stream, called change stream model, which naturally eliminates the parameter setting problem of snapshot graph model. Based on the change stream model, the incremental version of a well known k-clique clustering problem is studied and incremental k-clique clustering algorithms are proposed based on local DFS (depth first search) forest updating technique. It is theoretically proved that the proposed algorithms outperform corresponding static ones and incremental spectral clustering algorithm in terms of time complexity. The practical performances of our algorithms are extensively evaluated and compared with the baseline algorithms on ENRON and DBLP datasets. Experimental results show that incremental k-clique clustering algorithms are much more efficient than corresponding static ones, and have no accumulating errors that incremental spectral clustering algorithm has and can capture the evolving details of the clusters that snapshot graph model based algorithms miss.     

Determining the number of clusters using information entropy for mixed data

In cluster analysis, one of the most challenging and difficult problems is the determination of the number of clusters in a data set, which is a basic input parameter for most clustering algorithms. To solve this problem, many algorithms have been proposed for either numerical or categorical data sets. However, these algorithms are not very effective for a mixed data set containing both numerical attributes and categorical attributes. To overcome this deficiency, a generalized mechanism is presented in this paper by integrating Rényi entropy and complement entropy together. The mechanism is able to uniformly characterize within-cluster entropy and between-cluster entropy and to identify the worst cluster in a mixed data set. In order to evaluate the clustering results for mixed data, an effective cluster validity index is also defined in this paper. Furthermore, by introducing a new dissimilarity measure into the k-prototypes algorithm, we develop an algorithm to determine the number of clusters in a mixed data set. The performance of the algorithm has been studied on several synthetic and real world data sets. The comparisons with other clustering algorithms show that the proposed algorithm is more effective in detecting the optimal number of clusters and generates better clustering results.     

On computing the diameter of a point set in high dimensional Euclidean space

We consider the problem of computing the diameter of a set of n points in d-dimensional Euclidean space under Euclidean distance function. We describe an algorithm that in time O(dnlogn+n²) finds with high probability an arbitrarily close approximation of the diameter. For large values of d the complexity bound of our algorithm is a substantial improvement over the complexity bounds of previously known exact algorithms. Computing and approximating the diameter are fundamental primitives in high dimensional computational geometry and find practical application, for example, in clustering operations for image databases.