Causal, Correlation and Automatic Relevance Determination Machines for Granger Causality

Abstract

In this chapter, the causal, correlation and automatic relevance determination (ARD) machines are built to model non-linear Granger causality and are compared. The multi-layer perceptron (MLP) neural network trained using the scaled gradient method is used to build both the causal, correlation and ARD machines. The causal machine is a feed-forward MLP with past data being used as inputs to predict future values of a chaotic time series. The correlation machine is the auto-associative model which maps the past and present values to predict itself and a Nelder-Mead optimization method is used to predict future values. The ARD machine is an MLP which models the input-output relationship via Bayesian approach by penalizing the prior coefficient associated with each input variable. When these methods were tested, it was found that the correlation machine models Granger causality better than the causal machine while the ARD performs well. The complex nature of the correlation machine (inference using auto-associative neural networks) compared to the MLP causal machine renders the correlation machine more sensitive to Granger causality than the causal machine.