# Forecasting and Granger Modelling with Non-linear Dynamical Dependencies

## TL;DR

This paper introduces a novel kernel-based method for forecasting multivariate time series that captures non-linear dependencies and uncovers hidden dynamic relationships, outperforming traditional linear models.

## Contribution

It develops a new approach for learning vector-valued functions with multiple matrix-valued kernels directly from data, enabling better modeling of non-linearities and dynamic dependencies.

## Key findings

- Superior predictive performance on non-linear series
- Effective recovery of hidden dynamic relationships
- Alternative to existing graphical Granger methods

## Abstract

Traditional linear methods for forecasting multivariate time series are not able to satisfactorily model the non-linear dependencies that may exist in non-Gaussian series. We build on the theory of learning vector-valued functions in the reproducing kernel Hilbert space and develop a method for learning prediction functions that accommodate such non-linearities. The method not only learns the predictive function but also the matrix-valued kernel underlying the function search space directly from the data. Our approach is based on learning multiple matrix-valued kernels, each of those composed of a set of input kernels and a set of output kernels learned in the cone of positive semi-definite matrices. In addition to superior predictive performance in the presence of strong non-linearities, our method also recovers the hidden dynamic relationships between the series and thus is a new alternative to existing graphical Granger techniques.