Fuzzy clustering of univariate and multivariate time series by genetic multiobjective optimization

Given a set of time series, it is of interest to discover subsets that share similar properties. For instance, this may be useful for identifying and estimating a single model that may fit conveniently several time series, instead of performing the usual identification and estimation steps for each one. On the other hand time series in the same cluster are related with respect to the measures assumed for cluster analysis and are suitable for building multivariate time series models. Though many approaches to clustering time series exist, in this view the most effective method seems to have to rely on choosing some features relevant for the problem at hand and seeking for clusters according to their measurements, for instance the autoregressive coe±cients, spectral measures or the eigenvectors of the covariance matrix. Some new indexes based on goodnessof-fit criteria will be proposed in this paper for fuzzy clustering of multivariate time series. A general purpose fuzzy clustering algorithm may be used to estimate the proper cluster structure according to some internal criteria of cluster validity. Such indexes are known to measure actually definite often conflicting cluster properties, compactness or connectedness, for instance, or distribution, orientation, size and shape. It is argued that the multiobjective optimization supported by genetic algorithms is a most effective choice in such a di±cult context. In this paper we use the Xie-Beni index and the C-means functional as objective functions to evaluate the cluster validity in a multiobjective optimization framework. The concept of Pareto optimality in multiobjective genetic algorithms is used to evolve a set of potential solutions towards a set of optimal non-dominated solutions. Genetic algorithms are well suited for implementing di±cult optimization problems where objective functions do not usually have good mathematical properties such as continuity, differentiability or convexity. In addition the genetic algorithms, as population based methods, may yield a complete Pareto front at each step of the iterative evolutionary procedure. The method is illustrated by means of a set of real data and an artificial multivariate time series data set.Fuzzy clustering, Internal criteria of cluster validity, Genetic algorithms, Multiobjective optimization, Time series, Pareto optimality

Fuzzy set methods for object recognition in space applications

Progress on the following tasks is reported: (1) fuzzy set-based decision making methodologies; (2) feature calculation; (3) clustering for curve and surface fitting; and (4) acquisition of images. The general structure for networks based on fuzzy set connectives which are being used for information fusion and decision making in space applications is described. The structure and training techniques for such networks consisting of generalized means and gamma-operators are described. The use of other hybrid operators in multicriteria decision making is currently being examined. Numerous classical features on image regions such as gray level statistics, edge and curve primitives, texture measures from cooccurrance matrix, and size and shape parameters were implemented. Several fractal geometric features which may have a considerable impact on characterizing cluttered background, such as clouds, dense star patterns, or some planetary surfaces, were used. A new approach to a fuzzy C-shell algorithm is addressed. NASA personnel are in the process of acquiring suitable simulation data and hopefully videotaped actual shuttle imagery. Photographs have been digitized to use in the algorithms. Also, a model of the shuttle was assembled and a mechanism to orient this model in 3-D to digitize for experiments on pose estimation is being constructed

Clustering under the line graph transformation: application to reaction network

BACKGROUND: Many real networks can be understood as two complementary networks with two kind of nodes. This is the case of metabolic networks where the first network has chemical compounds as nodes and the second one has nodes as reactions. In general, the second network may be related to the first one by a technique called line graph transformation (i.e., edges in an initial network are transformed into nodes). Recently, the main topological properties of the metabolic networks have been properly described by means of a hierarchical model. While the chemical compound network has been classified as hierarchical network, a detailed study of the chemical reaction network had not been carried out. RESULTS: We have applied the line graph transformation to a hierarchical network and the degree-dependent clustering coefficient C(k) is calculated for the transformed network. C(k) indicates the probability that two nearest neighbours of a vertex of degree k are connected to each other. While C(k) follows the scaling law C(k) ~ k(-1.1 )for the initial hierarchical network, C(k) scales weakly as k(0.08 )for the transformed network. This theoretical prediction was compared with the experimental data of chemical reactions from the KEGG database finding a good agreement. CONCLUSIONS: The weak scaling found for the transformed network indicates that the reaction network can be identified as a degree-independent clustering network. By using this result, the hierarchical classification of the reaction network is discussed

Information Theoretical Importance Sampling Clustering

A current assumption of most clustering methods is that the training data and future data are taken from the same distribution. However, this assumption may not hold in most real-world scenarios. In this paper, we propose an information theoretical importance sampling based approach for clustering problems (ITISC) which minimizes the worst case of expected distortions under the constraint of distribution deviation. The distribution deviation constraint can be converted to the constraint over a set of weight distributions centered on the uniform distribution derived from importance sampling. The objective of the proposed approach is to minimize the loss under maximum degradation hence the resulting problem is a constrained minimax optimization problem which can be reformulated to an unconstrained problem using the Lagrange method. The optimization problem can be solved by both an alternative optimization algorithm or a general optimization routine by commercially available software. Experiment results on synthetic datasets and a real-world load forecasting problem validate the effectiveness of the proposed model. Furthermore, we show that fuzzy c-means is a special case of ITISC with the logarithmic distortion, and this observation provides an interesting physical interpretation for fuzzy exponent $m$.Comment: 15 pages, 9 figure

A Survey on Soft Subspace Clustering

Subspace clustering (SC) is a promising clustering technology to identify clusters based on their associations with subspaces in high dimensional spaces. SC can be classified into hard subspace clustering (HSC) and soft subspace clustering (SSC). While HSC algorithms have been extensively studied and well accepted by the scientific community, SSC algorithms are relatively new but gaining more attention in recent years due to better adaptability. In the paper, a comprehensive survey on existing SSC algorithms and the recent development are presented. The SSC algorithms are classified systematically into three main categories, namely, conventional SSC (CSSC), independent SSC (ISSC) and extended SSC (XSSC). The characteristics of these algorithms are highlighted and the potential future development of SSC is also discussed.Comment: This paper has been published in Information Sciences Journal in 201

Clustering Stability: An Overview

A popular method for selecting the number of clusters is based on stability arguments: one chooses the number of clusters such that the corresponding clustering results are "most stable". In recent years, a series of papers has analyzed the behavior of this method from a theoretical point of view. However, the results are very technical and difficult to interpret for non-experts. In this paper we give a high-level overview about the existing literature on clustering stability. In addition to presenting the results in a slightly informal but accessible way, we relate them to each other and discuss their different implications