A rough-and-ready cluster-based approach for extracting finite-time coherent sets from sparse and incomplete trajectory data

Abstract

We present a numerical method to identify regions of phase space that are approximately retained in a mobile compact neighbourhood over a finite time duration. Our approach is based on spatio-temporal clustering of trajectory data. The main advantages of the approach are the ability to produce useful results (i) when there are relatively few trajectories and (ii) when there are gaps in observation of the trajectories as can occur with real data. The method is easy to implement, works in any dimension, and is fast to run.Keywords: coherent sets, Lagrangian coherent structure, spatio-temporal clustering.Coherent features in time-dependent dynamical systems are difficult to identify, and considerable effort has been put into the development of identification algorithms. Most approaches require knowledge of the dynamical system or highresolution trajectory information, which in applications may not be available. We present a trajectory-based method that is aimed squarely at the situation where the available information is poor: there are few trajectories, the available trajectories do not span the full time duration under consideration, and there are missing observations within trajectories. As our method is very simple to implement and fast to run, it also provides a rapid "first cut" coherent structure analysis even in situations where the full dynamical system or high-resolution trajectory data is available.

Identification of individual coherent sets associated with flow trajectories using coherent structure coloring

“…It is important to emphasize that this approach is effective for identifying structures associated with individual flow trajectories, but is not intended to identify the full set of coherent structures in a flow, as is a common goal with other clustering and spectral graph theory methods 6,8,11 . Application of the method described here to the full set of Lagrangian trajectories could potentially be used to identify the full set of coherent structures associated with those trajectories.…”

“…This unique definition leads to corresponding changes to the conventional adjacency matrix and its analysis, as described in this paper. The more versatile definition of coherence renders the problem of identifying all of the coherent structures in the flow as ill-posed, making the present method distinct from, and complementary to, other coherent structure identification algorithms [6][7][8]10,11 . We show that it is generally more useful to consider an individual Lagrangian particle trajectory, and to identify the set of other trajectories that are coherent with the reference trajectory of interest.…”

“…Each coherent structure corresponds to a separate eigenvector. This method allows for detection of an arbitrary number of structures in the flow, unlike fuzzy c-means clustering 6 , but is dependent on the eigengap heuristic to determine the number of structures. The location of this gap can be extremely sensitive to the number of particles tracked, the initial location of the particles in the flow field, and any applied sparsification of the adjacency matrix.…”

“…In these cases, the assumptions inherent in the calculation of the deformation gradient are not valid, especially the ansatz of initially closely-spaced trajectories. In these cases, other techniques must be used to accurately detect regions of fluid coherence 6,7 .…”

Network-based study of Lagrangian transport and mixing

“…Very recently, spatio-temporal clustering algorithms have been proven to be very effective for the extraction of coherent sets from sparse and possibly incomplete trajectory data (see, e.g., Froyland and Padberg-Gehle, 2015;Hadjighasem et al, 2016;Banisch and Koltai, 2017;Schlueter-Kuck and Dabiri, 2017). Here, distance measures between trajectories are used to define groups of trajectories that remain close and/or behave similarly in the time span under investigation.…”