# Weighted \(z\)-Distance-Based Clustering and Its Application to Time-Series Data
## 1 Introduction  
In the field of artificial intelligence, clustering techniques play a very important role [1; 2]. Clustering is an unsupervised learning technology, with the purpose of forming meaningful clusters for the unlabeled data instances under consideration. Intuitively, similar instances are to be grouped in a cluster and different instances grouped in different clusters. Clustering has been widely utilized in a variety of applications, such as revealing the internal structure of the data [3], deriving segmentation of the data [4; 5], preprocessing the data for other artificial intelligence (AI) techniques [6; 7], business intelligence [1; 8], and knowledge discovery in data [9; 10]. For example, in electronic text processing [11; 12; 13], clustering is used to reduce the dimensionality in order to improve the efficiency of the processing, or to ease the curse of dimensionality encountered for high-dimensional problems. In the recommendation applications in e-commerce [14], the size of the information matrix is reduced by clustering to enhance the efficiency of making recommendations. In power systems, clustering helps predict the trend of electricity demand in the future [15]. In stock market forecasting and social media data analysis [1; 16], clustering technology is an important and indispensable core key. Therefore, developing good clustering technology is a very critical issue. Many types of clustering algorithms have been proposed [2; 17]. Similarity or distance measures are core components to cluster similar data into the same clusters, while dissimilar or distant data are placed into different clusters [18]. Centroid-based clustering [19; 20; 21; 22; 23; 24; 25; 26] groups data instances in an exclusive way. If an instance belongs to a definite cluster, it could not be included in another cluster. K-means is one such algorithm, well-known in the AI community. To use it, the user has to provide the desired number of clusters, \(K\), in advance. Each instance is assigned to the nearest cluster center. Then the \(K\) cluster centers are re-estimated. This process is repeated until the cluster centers are stable. Self-organizing mapping (SOM) employs a set of representatives. When a vector is presented, all representatives compete with each other. The winner is updated so as to move toward the vector. Hierarchical clustering [27; 28; 29; 30; 31; 32] creates a hierarchical decomposition of the set of data instances using some criteria. Two strategies, bottom-up and top-down, are adopted in hierarchical algorithms. The user usually has to decide how many and what clusters are most desirable from the offered hierarchy of clusters. Distribution-based clustering [33; 34; 35; 36; 37; 38] is based on distribution models. Fuzzy C-means uses fuzzy sets to cluster instances, and data instances are bound to each cluster by means of a membership function. Therefore, each instance may belong to several clusters with different degrees of membership. Gaussian mixture model-Expectation maximization (GMM-EM) uses a completely probabilistic approach. Each cluster is mathematically represented by a parametric distribution, e.g., a Gaussian, and the entire data set is therefore modeled by a mixture of these distributions. Density-based algorithms [39; 40; 41], e.g., Density-based spatial clustering of applications with noise (DBSCAN), consider clusters as dense regions in the instance space. Most of them do not impose any shape restrictions on the resulting clusters. However, they differ in the way the density is defined. Subspace clustering [42; 43; 44; 45; 46; 47], e.g., Clustering in quest (CLIQUE), searches for clusters within the various subspaces of the feature space. It reveals the clusters as well as the subspaces where they reside. Also, message passing [48; 49], a recent development in computer science and statistical physics, has led to the creation of new types of clustering algorithms. A special type of clustering is time-series clustering.