Piecewise aggregate representations and lower-bound distance functions for multivariate time series

Dimensionality reduction is one of the most important methods to improve the efficiency of the techniques that are applied to the field of multivariate time series data mining. Due to multivariate time series with the variable-based and time-based dimensions, the reduction techniques must take both of them into consideration. To achieve this goal, we use a center sequence to represent a multivariate time series so that the new sequence can be seen as a univariate time series. Thus two sophisticated piecewise aggregate representations, including piecewise aggregate approximation and symbolization applied to univariate time series, are used to further represent the extended sequence that is derived from the center one. Furthermore, some distance functions are designed to measure the similarity between two representations. Through being proven by some related mathematical analysis, the proposed functions are lower bound on Euclidean distance and dynamic time warping. In this way, false dismissals can be avoided when they are used to index the time series. In addition, multivariate time series with different lengths can be transformed into the extended sequences with equal length, and their corresponding distance functions can measure the similarity between two unequal-length multivariate time series. The experimental results demonstrate that the proposed methods can reduce the dimensionality, and their corresponding distance functions satisfy the lower-bound condition, which can speed up the calculation of similarity search and indexing in the multivariate time series datasets.