  Highway traffic states data collected from a network of sensors can be
considered a high-dimensional nonlinear dynamical system. In this paper, we
develop a novel data-driven method -- anti-circulant dynamic mode decomposition
with sparsity-promoting (circDMDsp) -- to study the dynamics of highway traffic
speed data. Particularly, circDMDsp addresses several issues that hinder the
application of existing DMD models: limited spatial dimension, presence of both
recurrent and non-recurrent patterns, high level of noise, and known mode
stability. The proposed circDMDsp framework allows us to numerically extract
spatial-temporal coherent structures with physical meanings/interpretations:
the dynamic modes reflect coherent spatial bases, and the corresponding
temporal patterns capture the temporal oscillation/evolution of these dynamic
modes. Our result based on Seattle highway loop detector data showcases that
traffic speed data is governed by a set of periodic components, e.g., mean
pattern, daily pattern, and weekly pattern, and each of them has a unique
spatial structure. The spatiotemporal patterns can also be used to
recover/denoise observed data and predict future values at any timestamp by
extrapolating the temporal Vandermonde matrix. Our experiments also demonstrate
that the proposed circDMDsp framework is more accurate and robust in data
reconstruction and prediction than other DMD-based models.
