A simple statistical approach to prediction in open high dimensional chaotic systems

TL;DR

This paper introduces a simple linear regression-based method for predicting high-dimensional open chaotic systems, such as climate and financial markets, by leveraging multiple low-dimensional embeddings.

Contribution

It proposes a novel approach combining linear response theory with multiple embeddings to improve prediction in high-dimensional chaotic systems, addressing the challenge of no close neighbors.

Findings

Effective prediction of regional rainfall several seasons ahead.

Successful application to stock index forecasting.

Theoretical support for linear response in high-dimensional chaos.

Abstract

Two recent papers on prediction of chaotic systems, one on multi-view embedding1 , and the second on prediction in projection2 provide empirical evidence to support particular prediction methods for chaotic systems. Multi-view embedding1 is a method of using several multivariate time series to come up with an improved embedding based predictor of a chaotic time series. Prediction in projection2 discusses how much smaller embeddings can provide useful prediction even though they may not be able to resolve the dynamics of the system. Both papers invoke a nearest neighbor3, or Lorenz method of Analogue (LMA)4 approach to estimation. However with open high dimensional chaotic systems there may be no very close nearest neighbor trajectories in a history, so in this paper we add in the thread of linear response theory5, although our approach is quite simple assuming that linear regressions6 on multiple embeddings5 will be sufficient. The approach is thus to create linear response models5 of multiple1 low dimensional embeddings2 to provide practical methods of predicting in high dimensional, open chaotic systems. Some theory is developed, mostly around two unproven conjectures to suggest methods of prediction, and they are applied to prediction in the earth's climate system (predicting regional rainfall in a small region) multiple seasons ahead and to power a simulated automated trading program applied to three stock indexes, the Dow Jones industrial average, the Dow Jones transportation average, and the Dow Jones utility average, based on data from 1929 through 20077