Order pattern recurrence plots: unveiling determinism buried in noise 1,2
2
1
Shuixiu Lu , Zongwei Luo , Guoqiang Zhang , Sebastian Oberst 1
1∗
Centre for Audio, Acoustics and Vibration (CAAV), University of Technology Sydney, NSW 2007, Australia 2 Department of Computer Science and Engineering, Southern University of Science and Technology, China
Introduction
Periodic, chaotic and stochastic dynamics
Both a stochastic system (random noise, high dimensional) and a chaotic system (deterministic, here of low dimension) generate irregular time series and a wideband power spectrum [1]. Pseudo-randomness of the chaotic system complicates to accurately distinguish between noise and chaos. Deterministic mechanisms govern a chaotic system, which allows differentiating between both systems. Here, we introduce order pattern recurrence plots (OPRP) to identify noise and chaos; we use OPRP and forbidden symbols to detect the deterministic signals from the driven noise, additive noise and linearly increasing noise.
A stochastic system can visit every possible symbol state which is up to D! such that the embedding dimension is D. However, the chaotic system has forbidden ordinal patterns which lead to a shrinking number of symbol states. As a result, in a fixed time window, the probability of the recurrence of symbols in a chaotic regime is higher than that of a stochastic system.
Materials and methods Bandt-Pompe methodology [2] encodes a consecutive time series to a symbolic sequence with an embedding dimension D. (xi , xi+1 , . . . , xi−D+1 ) → si
Figure 1: Increasing value of the embedding dimension of the OPRP: (a) enlarging the vertical distance, (b) reduction of the diagonal oriented lines but preservation of a dense plot, and (c) dispersion of the ordinal patterns with growing sparsity. (NS: number of the symbol states, iterations (t): 200,000).
(1)
where si is both an ordinal pattern and a symbol state. When the number of symbol states is strictly less than D!, the missing states are called forbidden symbols. An OPRP [3, 4] visualises the recurrence pattern of the ordinal patterns. ( 1, si = sj Ri,j (D) = (2) 0, si 6= sj Driven noise xt+1 = f (xt )+aζt where ζt is an iid random variable, a > 0 is a signal-tonoise ratio, representing the noise level.
Influence of the noise on the chaotic dynamics The amplitude of driven noise is small. Both the OPRP and the recurrence plot can unveil determinism when the signal-to-noise is high enough (Fig.2(a)). For a chaotic system with additive noise, the noise allows the forbidden symbols that are supposed to be ruled out to be accepted. As the amplitude of noise signal increases, the number of forbidden symbols decreases and eventually all ordinal patterns are accepted. Whereas the uniform noise has no state where the recurrence rate is larger than 0.2%, the chaotic signal under study has many states above 0.2% (Fig.2(b), Fig.1(c)). Linearly increasing noise causes the recurrence plot to pale away from the line of identity [4]. However, in the present case the OPRP forms diagonal oriented recurrence pattern owing to 88.75% of the symbols being forbidden. The noise signal weakens the recurrence rate of the downward ordinal patterns with an initial symbol code being 5 and it intensifies the recurrence rate of the symbols including 25 (Fig.2(c), Fig1(b)).
Additive noise is defined as xt+1 = yt+1 +aζt , where yt+1 = f (yt ) Linearly increasing noise is defined xt+1 = yt+1 + at, where yt+1 = f (yt )
as
Conclusion Monitoring the embedding dimension can be used to qualify chaotic versus stochastic dynamics. An optimal dimension can preserve the diagonal lines in a chaotic systems and lead to thinly distributed single isolated points with rare diagonal lines in a stochastic system. 1. Linearly increasing noise intensifies the density of the OPRP and diagonal lines. 2. The sensitivity of the forbidden symbols to additive noise depends on the properties of the chaotic system. The visibility of forbidden symbols in a chaotic system are sensitive to noise.
Figure 2: Influence of noise: (a) weak with dominant chaotic signal, a = 2.24%, (b) strong with the failure to detect the determinism, a = 50%, (c) weak on the OPRP,a = 1%. ζ follows an uniform distribution. Used parameters: threshold value = 0.1, embedding dimension m = 1, and time delay τ = 1, t = 200, 000(a, b), t = 20, 000(c).
References [1] O. A. Rosso, H. A. Larrondo, M. T. Martin, A. Plastino, and M. A. Fuentes. Distinguishing noise from chaos. Physical Review Letters, 99(15):154102, 2007. [2] Christoph Bandt and Bernd Pompe. Permutation entropy: a natural complexity measure for time series. Physical Review Letters, 88(17):174102, 2002. [3] Andreas Groth. Visualization of coupling in time series by order recurrence plots. Physical Review E, 72(4):046220, 2005. [4] Norbert Marwan, M Carmen Romano, Marco Thiel, and Jürgen Kurths. Recurrence plots for the analysis of complex systems. Physics Reports, 438(5-6):237– 329, 2007.
Figure 3: Change of NS under different levels of additive noise: (a) S-shape increase of NS in the periodic system; (b, c) logarithmic growth of NS in the chaotic system. *Corresponding author:
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