Using missing dispersion patterns to detect determinism and nonlinearity in time series data

Abstract

Kulp et al. proposed the number of missing ordinal patterns as a test statistic to detect the nonlinearity of time series in 2017. Inspired by the article, we propose a novel method called the number of missing dispersion patterns (NMDPs), which refers to the number of dispersion patterns that do not appear in the series after being symbolized using the Rostaghi and Azami method. The proposed method can distinguish between deterministic and stochastic dynamics well. By comparing the statistical difference between the NMDP results of the original series and the NMDP results of the wavelet iterative amplitude adjustment Fourier transform surrogate series, NMDP is demonstrated as a useful statistic for testing series nonlinearity. In this paper, we first apply NMDP to the model time series and verify its effectiveness as a test for determinism as well as nonlinearity and its ability to distinguish noise-contaminated series from purely stochastic series. We then apply NMDP to different stock index time series. The experiment results show that our method may be a suitable tool for quantifying the complexity of the inherent structure of the market under different development levels.

Appendices

Appendix A: The effect of class ${\varvec{c}}$ on the $\varvec{R_{\mathrm{{MDP}}}}$ results

Fig. 18

$R_{\mathrm{{MDP}}}$ results for the noisy series of the logistic map with $r=3.720$ and $\sigma = 0.01$

Fig. 19

$R_{\mathrm{{MDP}}}$ results for the noisy series of the logistic map with $r=3.720$ and $\sigma = 0.05$

Fig. 20

$R_{\mathrm{{MDP}}}$ results for the noisy series of the logistic map with $r=3.720$ and $\sigma = 0.1$

We consider the $R_{\mathrm{{MDP}}}$ results for the series of the logistic map without Gaussian white noise (Fig. 17) and the noisy series of the logistic map with $\sigma =0.01$ (Fig. 18), $\sigma =0.05$ (Fig. 19) and $\sigma =0.1$ (Fig. 20). It is important to note that c should not be too small as the data values will be too poorly classified to characterize the amplitude values of the series. Furthermore, a too large c will cause the $R_{\mathrm{{MDP}}}$ results to be overly sensitive to the noise present in the series and increase the computation. We can see that as c increases, $R_{\mathrm{{MDP}}}$ increases. The $R_{\mathrm{{MDP}}}$ for embedding dimension $m=5$ and $m=6$ are close to each other and almost identical at $c=6$. By contrast, we find that $c=5$ is adequate to describe the dynamics of the series and not overly sensitive to the noise present in the series. Therefore, we used $c=5$ in the full text of this study. However, the setting of the c value is flexible and can be determined according to the needs of the research.

Fig. 21

Evolution of six indices and the corresponding missing ordinal patterns (MOP) under embedding dimension $D = 5$, for a time window of 200 elements (upper magenta line) and 100 elements (lower black line)

Appendix B: Deterministic test using the $\varvec{R_\mathrm{{MOP}}}$ under six indices

See Fig. 21.