Selecting embedding delays: An overview of embedding techniques and a new method using persistent homology

Abstract

Delay embedding methods are a staple tool in the field of time series analysis and prediction. However, the selection of embedding parameters can have a big impact on the resulting analysis. This has led to the creation of a large number of methods to optimize the selection of parameters such as embedding lag. This paper aims to provide a comprehensive overview of the fundamentals of embedding theory for readers who are new to the subject. We outline a collection of existing methods for selecting embedding lag in both uniform and non-uniform delay embedding cases. Highlighting the poor dynamical explainability of existing methods of selecting non-uniform lags, we provide an alternative method of selecting embedding lags that includes a mixture of both dynamical and topological arguments. The proposed method, Significant Times on Persistent Strands (SToPS), uses persistent homology to construct a characteristic time spectrum that quantifies the relative dynamical significance of each time lag. We test our method on periodic, chaotic, and fast-slow time series and find that our method performs similar to existing automated non-uniform embedding methods. Additionally, [Formula: see text]-step predictors trained on embeddings constructed with SToPS were found to outperform other embedding methods when predicting fast-slow time series.

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“…Let x t be a continuous univariate times series. Then we can construct a state-space representation of the process as in Equation 2.12, if we choose an appropriate dimension d of the presumed underlying dynamical system and an embedding delay τ [23; 24]. The existence and calculations of the embedding dimension and delay are ensured by Taken’s embedding theorem [25].…”

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“…A three dimensional non-uniform delay embedding was used to reconstruct the phase space of the ECG dynamics. Delays were individually selected with the SToPS [52] method based on the first 20000 data points corresponding to the healthy dynamics prior to the onset of VF. The attractor network was then constructed using the first half of the time prior to VF onset as training data.…”

“…The permutation entropy was calculated by converting the time series into a symbolic sequence with each sequence corresponding to an ordered sequence of observations ranked by magnitude [16,41]. This conversion utilised 7 uniform lags of size equal to the first lag calculated from SToPS [52] in the delay embedding.…”

“…where x(t), ⃗ x(t) are the original time series and reconstructed delay vector respectively and (τ 1 , ..., τ n ) are selected time lags. The non-uniform delay embedding method is a generalisation of the uniform delay embedding often used for phase space reconstruction [52]. The latter method is simpler in its formulation and only requires the selection of two hyperparameters, the delay lag τ and embedding dimension m. The resulting delay vector is then constructed as:…”

“…To address this, we use the SToPS algorithm, a persistent homology based algorithm proposed by Tan et al [52], that identifies dynamically relevant time scales from the recurrence behaviour of chaotic and non-stationary time series. Instead of outputting a collection of time lags, SToPS calculates the significance of each potential time lag to construct a spectrum of time scales from which individual lag values may be selected to construct a delay vector.…”