  In the last few decades, building regression models for non-scalar variables,
including time series, text, image, and video, has attracted increasing
interests of researchers from the data analytic community. In this paper, we
focus on a multivariate time series regression problem. Specifically, we aim to
learn mathematical mappings from multiple chronologically measured numerical
variables within a certain time interval S to multiple numerical variables of
interest over time interval T. Prior arts, including the multivariate
regression model, the Seq2Seq model, and the functional linear models, suffer
from several limitations. The first two types of models can only handle
regularly observed time series. Besides, the conventional multivariate
regression models tend to be biased and inefficient, as they are incapable of
encoding the temporal dependencies among observations from the same time
series. The sequential learning models explicitly use the same set of
parameters along time, which has negative impacts on accuracy. The
function-on-function linear model in functional data analysis (a branch of
statistics) is insufficient to capture complex correlations among the
considered time series and suffer from underfitting easily. In this paper, we
propose a general functional mapping that embraces the function-on-function
linear model as a special case. We then propose a non-linear
function-on-function model using the fully connected neural network to learn
the mapping from data, which addresses the aforementioned concerns in the
existing approaches. For the proposed model, we describe in detail the
corresponding numerical implementation procedures. The effectiveness of the
proposed model is demonstrated through the application to two real-world
problems.
