Prediction in Projection

TL;DR

This paper demonstrates that simple near-neighbor forecast models using incomplete two-dimensional embeddings can accurately predict dynamical systems, often outperforming full embeddings, which is valuable for real-time forecasting.

Contribution

It shows that incomplete embeddings can be effectively used for accurate predictions, challenging the assumption that full embeddings are necessary for reliable forecasting.

Findings

Two-dimensional embeddings can produce accurate forecasts.

Incomplete embeddings sometimes outperform full embeddings.

Simple near-neighbor methods are effective for real-time prediction.

Abstract

Prediction models that capture and use the structure of state-space dynamics can be very effective. In practice, however, one rarely has access to full information about that structure, and accurate reconstruction of the dynamics from scalar time-series data---e.g., via delay-coordinate embedding---can be a real challenge. In this paper, we show that forecast models that employ incomplete embeddings of the dynamics can produce surprisingly accurate predictions of the state of a dynamical system. In particular, we demonstrate the effectiveness of a simple near-neighbor forecast technique that works with a two-dimensional embedding. Even though correctness of the topology is not guaranteed for incomplete reconstructions like this, the dynamical structure that they capture allows for accurate predictions---in many cases, even more accurate than predictions generated using a full embedding. This could be very useful in the context of real-time forecasting, where the human effort required to produce a correct delay-coordinate embedding is prohibitive.