Detecting determinism in short time series using a quantified averaged false nearest neighbors approach

We propose a criterion to detect determinism in short time series. This criterion is based on the estimation of the parameter $E_2$ defined by the averaged false neighbors method for analyzing time series [Cao, Physica D 110, 43 (1997)]. Using surrogate data testing with several chaotic and stochastic simulated time series, we show that the variation coefficient of $E_2$ over a few values of the embedding dimension $d$ defines a suitable statistic to detect determinism in short data sequences. This result holds for a time series generated by a high-dimensional chaotic system such as the Mackey-Glass one. Different decreasing lengths of the time series are included in the numerical experiments for both synthetic and real-world data. We also investigate the robustness of the criterion in the case of deterministic time series corrupted by additive noise.

Figure 1

Cosine values of the angles between successive tangent vectors obtained from a PSR performed on a $1000$-point Lorenz time series (upper frame). The embedding dimension is set to $7$ and the lag time is $4$. A similar figure obtained for an iteratively refined surrogate of the same time series is shown in the lower frame. In this case, the CTM for the original time series is $0.0053$. The surrogate CTM is $0.1071$. The CTM quantifies the irregularity of the cosine fluctuations. The smoothness of the Lorenz system implies cosine values that are close to $1$ with very weak irregularity. The surrogate procedure adds a random component which creates important fluctuations of the cosine values.

Figure 2

Percentage of FNN’s as a function of the embedding dimension $d$ obtained with a $1000$-point Ikeda time series. The choice of the threshold $R_{tol}$ has an important influence on the behavior of the method.

Figure 3

AFN parameter $E_2$ as a function of the embedding dimension obtained with a $1000$-point WGN time series (with lag time $\tau =1$).

Figure 4

AFN parameter $E_2$ as a function of the embedding dimension obtained with a $1000$-point simulated colored noise time series generated by an autoregressive $\mathrm{AR}\left(1\right)$ process (defined by $x_n=0.95x_{n-1}+{\eta }_n$, where ${\eta }_n$ are samples of a centered WGN with unit standard deviation). The lag time is $\tau =1$.

Figure 5

AFN parameter $E_2$ as a function of the embedding dimension obtained with a $1000$-point $x$-component Ikeda time series (with lag time $\tau =1$).

Figure 6

AFN parameter $E_2$ as a function of the embedding dimension obtained with a $1000$-point $x$-component Lorenz time series (with lag time $\tau =8$).

Figure 7

A $1000$-point length simulated Lorenz time series (upper frame) and one of its iteratively refined surrogates (lower frame) obtained by the IAAFT surrogate method.

Figure 8

A $1000$-point length simulated Rössler time series (upper frame) and one of its iteratively refined surrogates (lower frame) obtained by the IAAFT surrogate method.

Figure 9

A $1000$-point length simulated colored noise time series generated by an autoregressive $\mathrm{AR}\left(1\right)$ process (defined by $x_n=0.95x_{n-1}+{\eta }_n$, where ${\eta }_n$ are samples of a centered WGN series with unit standard deviation, in the upper frame) and one of its iteratively refined surrogates (lower frame) obtained by the IAAFT surrogate method.

Figure 10

$1250$-point length simulated Mackey-Glass time series (in the upper frame) and one of its iteratively refined surrogates (lower frame).

Figure 11

Human postural sway time series recorded during quiet standing on a force platform. The COP trajectory along the anteroposterior direction is plotted against time.