Scalable Gromov–Wasserstein Based Comparison of Biological Time Series

A time series is an extremely abundant data type arising in many areas of scientific research, including the biological sciences. Any method that compares time series data relies on a pairwise distance between trajectories, and the choice of distance measure determines the accuracy and speed of the time series comparison. This paper introduces an optimal transport type distance for comparing time series trajectories that are allowed to lie in spaces of different dimensions and/or with differing numbers of points possibly unequally spaced along each trajectory. The construction is based on a modified Gromov–Wasserstein distance optimization program, reducing the problem to a Wasserstein distance on the real line. The resulting program has a closed-form solution and can be computed quickly due to the scalability of the one-dimensional Wasserstein distance. We discuss theoretical properties of this distance measure, and empirically demonstrate the performance of the proposed distance on several datasets with a range of characteristics commonly found in biologically relevant data. We also use our proposed distance to demonstrate that averaging oscillatory time series trajectories using the recently proposed Fused Gromov–Wasserstein barycenter retains more characteristics in the averaged trajectory when compared to traditional averaging, which demonstrates the applicability of Fused Gromov–Wasserstein barycenters for biological time series. Fast and user friendly software for computing the proposed distance and related applications is provided. The proposed distance allows fast and meaningful comparison of biological time series and can be efficiently used in a wide range of applications.