Optimal state-space reconstruction using derivatives on projected manifold

Abstract

A paradigm for optimal state-space reconstruction with nonuniform time delays is proposed. A comparison based on a diffeomorphic measure and a smoothness cost function shows that the proposed methodology achieves a better reconstruction compared to a reconstruction based on time delays that are multiples of the first minimum of mutual information. It is also shown how the proposed methodology is a more reliable approach to determining the embedding dimension.

Stability of exoplanetary systems retrieved from scalar time series

“…Besides the classical time delay embedding there are various techniques that guarantee a successful trajectory reconstruction Lekscha & Donner (2018). Moreover, one can also find alternative parameter estimation of delay embedding (Small & Tse 2004;Hirata et al 2006;Nichkawde 2013) as well as different kind of reconstruction methods (Hirata & Aihara 2017;Uzal et al 2011;Carroll 2018a;Lekscha & Donner 2018) that can also be tested in dynamical astronomy.…”

Selection of Optimal Embedding Parameters Applied to Short and Noisy Time Series from Rössler System

“…A criterion for determining the C-C-1 method q parameter is suggested on paragraph IIId and improves the implementation of the C-C-1 method. A technic based on the statistic Rul'kov test is proposed in paragraph IV to measure the state space reconstruction quality [18] [19] [33].…”

“…The other ratio is the nearest-neighbor distance on the reconstructed state space to the distance between the corresponding points on the original state space. For a smooth mapping between the time series, the quotient of these two ratios should be close to unity [18] [19]. This paper also constitutes a sort of set-up in order to become familiar with…”

“…In the case of long free noise time series, as Takens embedding theorem ensures a topological equivalence between the original state space and the reconstructed one, the quality of the reconstruction is measured through the conservation of invariants such as the maximum Lyapunov exponent and the correlation dimension. In the case of limited noise free data set, we have used a technic based on a statistic approach similar to the Rul'kov et al test [18], by calculating the quotient F of two ratios. One ratio is the nearest-neighbor distance on the original state space to the distance on the corresponding points on the reconstructed state space.…”

Sparse model from optimal nonuniform embedding of time series

“…m is thus chosen when there are no false nearest neighbors. Recent investigations have revealed two major caveats with this approach [5]. 1.…”