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

Louvain-like Methods for Community Detection in Multi-Layer Networks

In many complex systems, entities interact with each other through complicated patterns that embed different relationships, thus generating networks with multiple levels and/or multiple types of edges. When trying to improve our understanding of those complex networks, it is of paramount importance to explicitly take the multiple layers of connectivity into account in the analysis. In this paper, we focus on detecting community structures in multi-layer networks, i.e., detecting groups of well-connected nodes shared among the layers, a very popular task that poses a lot of interesting questions and challenges. Most of the available algorithms in this context either reduce multi-layer networks to a single-layer network or try to extend algorithms for single-layer networks by using consensus clustering. Those approaches have anyway been criticized lately. They indeed ignore the connections among the different layers, hence giving low accuracy. To overcome these issues, we propose new community detection methods based on tailored Louvain-like strategies that simultaneously handle the multiple layers. We consider the informative case, where all layers show a community structure, and the noisy case, where some layers only add noise to the system. We report experiments on both artificial and real-world networks showing the effectiveness of the proposed strategies.Comment: 16 pages, 4 figure

Multi-criteria Anomaly Detection using Pareto Depth Analysis

We consider the problem of identifying patterns in a data set that exhibit anomalous behavior, often referred to as anomaly detection. In most anomaly detection algorithms, the dissimilarity between data samples is calculated by a single criterion, such as Euclidean distance. However, in many cases there may not exist a single dissimilarity measure that captures all possible anomalous patterns. In such a case, multiple criteria can be defined, and one can test for anomalies by scalarizing the multiple criteria using a linear combination of them. If the importance of the different criteria are not known in advance, the algorithm may need to be executed multiple times with different choices of weights in the linear combination. In this paper, we introduce a novel non-parametric multi-criteria anomaly detection method using Pareto depth analysis (PDA). PDA uses the concept of Pareto optimality to detect anomalies under multiple criteria without having to run an algorithm multiple times with different choices of weights. The proposed PDA approach scales linearly in the number of criteria and is provably better than linear combinations of the criteria.Comment: Removed an unnecessary line from Algorithm

Clustering Solutions of Multiobjective Function Inlining Problem

Hard real time-systems are often small devices operating on batteries that must react within a given deadline, so they must satisfy their timing, code size, and energy consumption requirements. Since these three objectives contradict each other, compilers for real-time systems go towards multiobjective optimizations which result in sets of trade-off solutions. A system designer can use the solution sets to choose the most suitable system configuration. Evolutionary algorithms can find trade-off solutions but the solution set might be large which complicates the task of the system designer. We propose to divide the solution set into clusters, so the system designer chooses the most suitable cluster and examines a smaller subset in detail. In contrast to other clustering techniques, our method guarantees that the sizes of all clusters are less than a predefined limit. Our method clusters a set by using any existing clustering method, divides clusters with sizes exceeding the predefined size into smaller clusters, and reduces the number of clusters by merging small clusters. The method guarantees that the final clusters satisfy the size constraint. We demonstrate our approach by considering a well-known compiler-based optimization called function inlining. It substitutes function calls by the function bodies which decreases the execution time and energy consumption of a program but increases its code size