Clustering multivariate time series using energy distance
提出一种基于能量距离的多元时间序列聚类方法，无需模型假设，通过计算成分序列分布差异的相异矩阵，再应用层次聚类得到树状图，并用模拟和实际数据（如GDP、美国各州人口）验证了方法有效性。
A novel methodology is proposed for clustering multivariate time series data using energy distance defined in Székely and Rizzo (2013). Specifically, a dissimilarity matrix is formed using the energy distance statistic to measure the separation between the finite‐dimensional distributions for the component time series. Once the pairwise dissimilarity matrix is calculated, a hierarchical clustering method is then applied to obtain the dendrogram. This procedure is completely nonparametric as the dissimilarities between stationary distributions are directly calculated without making any model assumptions. In order to justify this procedure, asymptotic properties of the energy distance estimates are derived for general stationary and ergodic time series. The method is illustrated in a simulation study for various component time series that are either linear or nonlinear. Finally, the methodology is applied to two examples; one involves the GDP of selected countries and the other is the population size of various states in the U.S.A. in the years 1900–1999.