# Weighted \(z\)-Distance-Based Clustering and Its Application to Time-Series Data
## 1 Introduction
A sequence composed of continuous, real-valued temporal data is known as a time series. Time-series data are of interest in various areas, e.g., engineering, business, finance, economics, and healthcare. There are good surveys and reviews published on time-series clustering [50; 51; 52]. In [50], time-series clustering methods are grouped into three major categories, raw-data-based, feature-based, and model-based approaches. Raw-data-based clustering methods work with raw data, either in the time or frequency domain. The two time series being compared are normally sampled at the same interval. Clustering based on raw data implies working with high-dimensional space. It is also not desirable to work directly with the raw data that are highly noisy. Feature-based clustering methods have been proposed to address these concerns. Although most feature extraction methods are generic in nature, the extracted features are usually application dependent. Model-based clustering methods consider each time series being generated by some kind of model or by a mixture of underlying probability distributions. Time series are considered similar when the models characterizing individual series are similar. However, model-based approaches usually have scalability problems, and its performance reduces when the clusters are close to each other. In [51], time-series clustering approaches are classified into six groups: partitioning, hierarchical, grid-based, model-based, density-based clustering, and multi-step clustering algorithms. A partitioning clustering method, e.g., K-means, makes \(k\) groups from unlabeled objects in the way that each group contains at least one object. Hierarchical clustering is an approach of cluster analysis that makes a hierarchy of clusters using agglomerative or divisive algorithms. Model-based clustering, e.g., SOM, assumes a model for each cluster, and finds the best fit of data to that model. In density based clustering [41], clusters are subspaces of dense objects which are separated by subspaces in which objects have low density. The grid-based methods quantize the space into a finite number of the cells that form a grid, and then perform clustering on the grid's cells [53]. A multi-step approach presented in [54] uses a three-phase method: (1) pre-clustering of time series; (2) purifying and summarization; and (3) merging, to construct the clusters based on similarity in shape. [PERSON] and [PERSON] proposed a self-constructing clustering (SCC) algorithm [20] which has been applied in various applications [4; 6; 13; 14; 16; 55]. SCC is an exclusive clustering method. For a given training set, SCC performs only one training cycle on the training instances. Initially, no clusters exist. Training instances are considered one by one. If an instance is closer enough to an existing cluster, the instance is assigned to the most suitable cluster. Otherwise, a new cluster is created and the instance is assigned to it. SCC offers several advantages. First, the algorithm runs through the training instances only one time, so it is fast. Second, the distributions of the data are statistically characterized. Third, the similarity threshold, instead of the number of clusters, is specified in advance by the user. However, once an instance is assigned to a cluster, the assignment will not be changed afterwards. The clusters produced may depend on the order the instances are considered, and assignment errors are more likely to occur. As a result, the accuracy of the result can be low. Also, all dimensions are equally weighted in the clustering process, which may not be suitable in certain applications, e.g., time-series clustering [56; 57; 58]. In this paper, we propose improvements to the SCC method to overcome its shortcomings. Two or more training cycles on the instances are performed. In each cycle, training instances are considered one by one. An instance can be added into or removed from a cluster, and thus it is allowed to be re-assigned to another cluster.