  Modern time series datasets are often high-dimensional, incomplete/sparse,
and nonstationary. These properties hinder the development of scalable and
efficient solutions for time series forecasting and analysis. To address these
challenges, we propose a Nonstationary Temporal Matrix Factorization (NoTMF)
model, in which matrix factorization is used to reconstruct the whole time
series matrix and vector autoregressive (VAR) process is imposed on a properly
differenced copy of the temporal factor matrix. This approach not only
preserves the low-rank property of the data but also offers consistent temporal
dynamics. The learning process of NoTMF involves the optimization of two factor
matrices and a collection of VAR coefficient matrices. To efficiently solve the
optimization problem, we derive an alternating minimization framework, in which
subproblems are solved using conjugate gradient and least squares methods. In
particular, the use of conjugate gradient method offers an efficient routine
and allows us to apply NoTMF on large-scale problems. Through extensive
experiments on Uber movement speed dataset, we demonstrate the superior
accuracy and effectiveness of NoTMF over other baseline models. Our results
also confirm the importance of addressing the nonstationarity of real-world
time series data such as spatiotemporal traffic flow/speed.
