Delay-Embedding Spatio-Temporal Dynamic Mode Decomposition

Spatio-temporal dynamic mode decomposition (STDMD) is an extension of dynamic mode decomposition (DMD) designed to handle spatio-temporal datasets. It extends the framework so that it can analyze data that have both spatial and temporal variations. This facilitates the extraction of spatial structures along with their temporal evolution. The STDMD method extracts temporal and spatial development information simultaneously, including wavenumber, frequencies, and growth rates, which are essential in complex dynamic systems. We provide a comprehensive mathematical framework for sequential and parallel STDMD approaches. To increase the range of applications of the presented techniques, we also introduce a generalization of delay coordinates. The extension, labeled delay-embedding STDMD allows the use of delayed data, which can be both time-delayed and space-delayed. An explicit expression of the presented algorithms in matrix form is also provided, making theoretical analysis easier and providing a solid foundation for further research and development. The novel approach is demonstrated using some illustrative model dynamics.