Abstract
Data assimilation techniques, such as ensemble Kalman filtering, have been
shown to be a highly effective and efficient way to combine noisy data with a
mathematical model to track and forecast dynamical systems. However, when
dealing with high-dimensional data, in many situations one does not have a
model, so data assimilation techniques cannot be applied. In this paper, we use
dynamic mode decomposition to generate a low-dimensional, linear model of a
dynamical system directly from high-dimensional data, which is defined by
temporal and spatial modes, that we can then use with data assimilation
techniques such as the ensemble Kalman filter. We show how the dynamic mode
decomposition can be combined with the ensemble Kalman filter (which we call
the DMDEnKF) to iteratively update the current state and temporal modes as new
data becomes available. We demonstrate that this approach is able to track time
varying dynamical systems in synthetic examples, and experiment with the use of
time-delay embeddings. We then apply the DMDEnKF to real world seasonal
influenza-like illness data from the USA Centers for Disease Control and
Prevention, and find that for short term forecasting, the DMDEnKF is comparable
to the best mechanistic models in the ILINet competition.