Prediction in projection

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 delaycoordinate 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. LEAD PARAGRAPHPrediction models constructed from state-space dynamics have a long and rich history, dating back to roulette and beyond. A major stumbling block in the application of these models in real-world situations is the need to reconstruct the dynamics from scalar time-series data-e.g., via delay-coordinate embedding. This procedure, which is the topic of a large and active body of literature, involves estimation of two free parameters: the dimension m of the reconstruction space and the delay, τ , between the observations that make up the coordinates in that space. Estimating good values for these parameters is not trivial; it requires the proper mathematics, attention to the data requirements, computational effort, and expert interpretation of the results of the calculations. This is a major challenge if one is interested in real-time forecasting, especially when the systems involved operate a) Electronic mail: joshua.garland@colorado.edu b) Electronic mail: lizb@colorado.edu on fast time scales. In this paper, we show that the full effort of delay-coordinate embedding is not always necessary when one is building forecast models, and can indeed be overkill. Using synthetic time-series data generated from the Lorenz-96 atmospheric model and real data from a computer performance experiment, we demonstrate that a two-dimensional embedding of scalar timeseries data from a dynamical system gives simple forecast methods enough traction to generate accurate predictions of the future course of those dynamics-sometimes even more accurate than predictions created using the full embedding. Since incomplete embeddings do not preserve the topology of the full dynamics, this is interesting from a mathematical standpoint. It is also potentially useful in practice. This reduced-order forecasting strategy involves only one free parameter (τ ), good values for which, we believe, can be estimated 'on the fly' using information-theoretic and/or machine-learning algorithms. As suc...

The intrinsic predictability of ecological time series and its potential to guide forecasting

“…), which can then be used to forecast system dynamics (Lorenz , Farmer and Sidorowich , Sauer et al. , Casdagli and Eubank , Smith , Weigend and Gershenfeld , Garland and Bradley , , Garland et al. ).…”

The intrinsic predictability of ecological time series and its potential to guide forecasting

“…The foundation for linking intrinsic and realized predictability lies in information theory and builds on research demonstrating a relationship between PE and FE for complex computer systems (Garland et al 2015). Information theory was originally developed by Claude Shannon as a mathematical description of communication (Shannon 1948) but has since been applied across many disciplines.…”

“…PE approximates and is inversely related to intrinsic predictability by quantifying how quickly the system generates new information. Time series with low permutation entropy have high redundancy and are expected to have high intrinsic predictability (Garland et al 2015).…”

“…By contrast, the intrinsic predictability of a system is an absolute measure that represents the highest achievable predictability (Lorenz 1995, Beckage et al 2011. The intrinsic predictability of a system can be approximated with model-free measure of time series complexity, such as Lyapunov exponents or permutation entropy (Boffetta et al 2002, Bandt & Pompe 2002, Garland et al 2015. In principle, intrinsic predictability has the potential to indicate whether the model, data, or system are limiting realized predictability.…”

“…Nonlinearity: When the deterministic processes governing system dynamics depend on the state of the system. Permutation entropy (PE): permutation entropy is a measure of the complexity of a time series (Bandt & Pompe, 2002) that is negatively correlated with a system's predictability (Garland et al 2015). Permutation entropy quantifies the combined new and lost information.…”