Reconstructing state spaces from multivariate data using variable delays

We study two methods for constructing a nonuniform embedding for multivariate data. A nonuniform embedding is a state space reconstruction which is more flexible than the common delay coordinates with fixed delays since it contains variable delays. Using these methods, we can extract causal relationships among many variables in a more suitable way. We demonstrate that the proposed methods can give more precise predictions and simpler models than some previous methods.

Figure 1

(Color online) Delays for predicting the east wind at the point with a circle obtained using the extension of the method of Judd and Mees. The dots show points of observation. Symbol $+(\times )$ shows points whose east (north) wind helps to predict the east wind at the point with a circle. The numbers shown are the magnitudes of the delays. For example, the number 1 means $10\min$.

Figure 2

(Color online) Delays for predicting east wind at the point with a circle obtained using cross validation. The dots show points of observation. Symbol $+(\times )$ shows points whose east (north) wind helps to predict the east wind at the point with a circle. The numbers shown are the magnitudes of the delays. For example, the number 1 means $10\min$.