Invariant subspace learning for time series data based on dynamic time warping distance

Low-dimensional and compact representation of time series data is of importance for mining and storage. In practice, time series data are vulnerable to various temporal transformations, such as shift and temporal scaling, however, which are unavoidable in the process of data collection. If a learning algorithm directly calculates the difference between such transformed data based on Euclidean distance, the measurement cannot faithfully reflect the similarity and hence could not learn the underlying discriminative features. In order to solve this problem, we develop a novel subspace learning algorithm based on dynamic time warping (DTW) distance which is an elastic distance defined in a DTW space. The algorithm aims to minimize the reconstruction error in the DTW space. However, since DTW space is a semi-pseudo metric space, it is difficult to generalize common subspace learning algorithms for such semi-pseudo metric spaces. In this work, we introduce warp operators with which DTW reconstruction error can be approximated by reconstruction error between transformed series and their reconstructions in a subspace. The warp operators align time series data with their linear representations in the DTW space, which is in particular important for misaligned time series, so that the subspace can be learned to obtain an intrinsic basis (dictionary) for the representation of the data. The warp operators and the subspace are optimized alternatively until reaching equilibrium. Experiments results show that the proposed algorithm outperforms traditional subspace learning algorithms and temporal transform-invariance based methods (including SIDL, Kernel PCA, and SPMC et. al), and obtains competitive results with the state-of-the-art algorithms, such as BOSS algorithm.