Use of False Nearest Neighbours for Selecting Variables and Embedding Parameters for State Space Reconstruction

Anna Krakovská,Hana Budáčová,Kristína Mezeiová

doi:10.1155/2015/932750

Anna Krakovská, Hana Budáčová + Show 1 more

Open Access

https://doi.org/10.1155/2015/932750

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Abstract

If data are generated by a system with a d-dimensional attractor, then Takens’ theorem guarantees that reconstruction that is diffeomorphic to the original attractor can be built from the single time series in 2d+1-dimensional phase space. However, under certain conditions, reconstruction is possible even in a space of smaller dimension. This topic is very important because the size of the reconstruction space relates to the effectiveness of the whole subsequent analysis. In this paper, the false nearest neighbour (FNN) methods are revisited to estimate the optimum embedding parameters and the most appropriate observables for state space reconstruction. A modification of the false nearest neighbour method is introduced. The findings contribute to evidence that the length of the embedding time window (TW) is more important than the reconstruction delay time and the embedding dimension (ED) separately. Moreover, if several time series of the same system are observed, the choice of the one that is used for the reconstruction could also be critical. The results are demonstrated on two chaotic benchmark systems.

Highlights

State space reconstruction is usually an unavoidable step before the analysis of a time series in terms of dynamical systems theory
To be able to take a position on whether the combinations of parameters that are promising according to the False First Nearest Neighbour (FFNN) method are optimal for the D2 computation, we evaluated the error in the dimension estimates over the possible combinations
We evaluated the prediction error for different combinations of embedding parameters and compared the resulting maps with the maps that were obtained by the FFNN method

Summary

Introduction

State space reconstruction is usually an unavoidable step before the analysis of a time series in terms of dynamical systems theory. For noise-free data of unlimited length, the existence of a diffeomorphism between the original attractor and the reconstructed image is guaranteed for almost any choice of delay and a sufficiently high embedding dimension. The search for the proper dimension is based on a step-by-step expansion of the reconstruction space while simultaneously following some proper diffeomorphism invariant that is expected to stay constant after reaching the sufficient embedding dimension As examples of such invariants, the correlation dimension, largest Lyapunov exponent, predictability indices, or percentage of false nearest neighbours have previously been mentioned. The most commonly used method for selecting the embedding parameters consists of the first minimum of the mutual information to estimate the time delay and the FNN test to find the sufficient embedding dimension.

Data and Methods

Results

Discussion and Conclusions