We introduce negative binomial linear dynamical system (NBDS) to jointly model daily COVID-19 deaths and cases reported at all 50 US States and D.C. and provide forecast in a purely data driven manner.
Linear dynamical systems, which assume the observations are Gaussian distributed given the underlying latent dynamics, are widely used to model real-valued multivariate time series and predict future observations. However, their Gaussian assumption severely limits their ability to model sequentially observed count vectors, which take non-negative integer values, usually follow right-skewed distributions, and are often over-dispersed (variance larger than the mean). For this reason, we propose to model count time series with discrete dynamic systems (DDS), which link the count observations to the underlying latent dynamics via appropriate distributions defined over non-negative intergers.
In particular, we consider graph gamma process (GGP) NBDS, a nonparametric Bayesian hierarchical model recently proposed by Kalantari and Zhou (2020). Generated from the GGP is an infinite dimensional directed sparse random graph, which is constructed by taking the OR operation of countably infinite binary adjacency matrices that share the same set of countably infinite nodes. Each of these adjacency matrices is associated with a weight to indicate its overall activation strength, and places a finite number of edges between a finite subset of nodes belonging to the same community. We use the generated random graph, whose number of nonzero-degree nodes is finite, to define both the sparsity patterns and dimensions of the latent state transition matrices of NBDS. We show how the formed overlapping communities in the latent space can provide interpretable latent transition patterns of the COVID-19 time series, providing good predictive performance and uncertainty estimation.