Probabilistic Reduced-Dimensional Vector Autoregressive Modeling with
  Oblique Projections
Yanfang Mo S. Joe Qin
In this paper, we propose a probabilistic reduced-dimensional vector
autoregressive (PredVAR) model to extract low-dimensional dynamics from
high-dimensional noisy data. The model utilizes an oblique projection to
partition the measurement space into a subspace that accommodates the
reduced-dimensional dynamics and a complementary static subspace. An optimal
oblique decomposition is derived for the best predictability regarding
prediction error covariance. Building on this, we develop an iterative PredVAR
algorithm using maximum likelihood and the expectation-maximization (EM)
framework. This algorithm alternately updates the estimates of the latent
dynamics and optimal oblique projection, yielding dynamic latent variables with
rank-ordered predictability and an explicit latent VAR model that is consistent
with the outer projection model. The superior performance and efficiency of the
proposed approach are demonstrated using data sets from a synthesized Lorenz
system and an industrial process from Eastman Chemical.