Efficient Interpretable Nonlinear Modeling for Multiple Time Series
Kevin Roy Luis Miguel Lopez-Ramos Baltasar Beferull-Lozano
Predictive linear and nonlinear models based on kernel machines or deep
neural networks have been used to discover dependencies among time series. This
paper proposes an efficient nonlinear modeling approach for multiple time
series, with a complexity comparable to linear vector autoregressive (VAR)
models while still incorporating nonlinear interactions among different
time-series variables. The modeling assumption is that the set of time series
is generated in two steps: first, a linear VAR process in a latent space, and
second, a set of invertible and Lipschitz continuous nonlinear mappings that
are applied per sensor, that is, a component-wise mapping from each latent
variable to a variable in the measurement space. The VAR coefficient
identification provides a topology representation of the dependencies among the
aforementioned variables. The proposed approach models each component-wise
nonlinearity using an invertible neural network and imposes sparsity on the VAR
coefficients to reflect the parsimonious dependencies usually found in real
applications. To efficiently solve the formulated optimization problems, a
custom algorithm is devised combining proximal gradient descent, stochastic
primal-dual updates, and projection to enforce the corresponding constraints.
Experimental results on both synthetic and real data sets show that the
proposed algorithm improves the identification of the support of the VAR
coefficients in a parsimonious manner while also improving the time-series
prediction, as compared to the current state-of-the-art methods.