Projection from diffusion-mapped delay coordinates

Patterns and Simulations

By T. Berry, R. Cressman, Z. Greguric Ferencek, T. Sauer

To achieve attractor reconstruction and dimension reduction from video data, we use Diffusion-Mapped Delay Coordinates (DMDC), a combination of weighted delay coordinates and diffusion maps. Exponentially weighting the delay coordinates tends to emphasize the most stable time scales; it remains to project down from (extremely) high-dimensional reconstruction space. Although most reasonable projections preserve topology, we find that projecting with diffusion maps tends to do the least damage to the geometry of the system, compared to using proper orthogonal decompositions or their generalizations.

The first video shows the results of DMDC applied to a video of a meandering spiral wave. The data consists of the video alone; no equations are used. The first two DMDC modes are localized near the tip of the spiral and pick out the vertical and horizontal component of the slow-moving tip location. Thus the lowest modes ignore the components of highest variance (from the spiral arm) and separate out the slowest time scale of the dynamics. On the other hand, SVD is greedy for variance, and its lowest modes concentrate there.

Projecting to slow time scales can be thought of as a way to achieve noise reduction, if the noise is high frequency. The second video shows that DMDC is superior to SVD applied to delay coordinates at projecting away white noise applied to a video.

The third video is an application of the method to extract the dynamical time scales from video of an experiment. The raw data is a photo of a nematic liquid crystal with area about 50 square micrometers. Equations of motion are not known. The subvideos show dynamics of successively faster time scales as projected from DMDC.

These videos are taken from: T. Berry, R. Cressman, Z. Greguric Ferencek, T. Sauer, Time-scale separation from diffusion-mapped delay coordinates, SIAM J. Appl. Dyn. Sys. 12, 618 - 649 (2013).

Author Institutional AffiliationGeorge Mason University
Author Email
Author Postal MailDepartment of Mathematics George Mason University
KeywordsDimension reduction, delay coordinates, diffusion maps

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