Permutation Entropy of Weakly Noise-Affected Signals

We analyze the permutation entropy of deterministic chaotic signals affected by a weak observational noise. We investigate the scaling dependence of the entropy increase on both the noise amplitude and the window length used to encode the time series. In order to shed light on the scenario, we perform a multifractal analysis, which allows highlighting the emergence of many poorly populated symbolic sequences generated by the stochastic fluctuations. We finally make use of this information to reconstruct the noiseless permutation entropy. While this approach works quite well for Hénon and tent maps, it is much less effective in the case of hyperchaos. We argue about the underlying motivations.

Download Full-text

Discriminating chaotic and stochastic time series using permutation entropy and artificial neural networks

Scientific Reports ◽

10.1038/s41598-021-95231-z ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

B. R. R. Boaretto ◽

R. C. Budzinski ◽

K. L. Rossi ◽

T. L. Prado ◽

S. R. Lopes ◽

...

Keyword(s):

Time Series ◽

Flicker Noise ◽

Permutation Entropy ◽

Temporal Correlations ◽

Systems Research ◽

Open Questions ◽

Chaotic Signals ◽

Artificial Neural ◽

The Difference ◽

Artificial Neural Network Ann

AbstractExtracting relevant properties of empirical signals generated by nonlinear, stochastic, and high-dimensional systems is a challenge of complex systems research. Open questions are how to differentiate chaotic signals from stochastic ones, and how to quantify nonlinear and/or high-order temporal correlations. Here we propose a new technique to reliably address both problems. Our approach follows two steps: first, we train an artificial neural network (ANN) with flicker (colored) noise to predict the value of the parameter, $$\alpha$$ α , that determines the strength of the correlation of the noise. To predict $$\alpha$$ α the ANN input features are a set of probabilities that are extracted from the time series by using symbolic ordinal analysis. Then, we input to the trained ANN the probabilities extracted from the time series of interest, and analyze the ANN output. We find that the $$\alpha$$ α value returned by the ANN is informative of the temporal correlations present in the time series. To distinguish between stochastic and chaotic signals, we exploit the fact that the difference between the permutation entropy (PE) of a given time series and the PE of flicker noise with the same $$\alpha$$ α parameter is small when the time series is stochastic, but it is large when the time series is chaotic. We validate our technique by analysing synthetic and empirical time series whose nature is well established. We also demonstrate the robustness of our approach with respect to the length of the time series and to the level of noise. We expect that our algorithm, which is freely available, will be very useful to the community.

Download Full-text

Analysis of Noise and Velocity in GNSS EPN-Repro 2 Time Series

Remote Sensing ◽

10.3390/rs13142783 ◽

2021 ◽

Vol 13 (14) ◽

pp. 2783

Author(s):

Sorin Nistor ◽

Norbert-Szabolcs Suba ◽

Kamil Maciuk ◽

Jacek Kudrys ◽

Eduard Ilie Nastase ◽

...

Keyword(s):

Time Series ◽

Visual Inspection ◽

Flicker Noise ◽

Vertical Component ◽

Likelihood Estimation ◽

Noise Model ◽

Noise Amplitude ◽

Allan Deviation ◽

Power Spectral ◽

Position Time Series

This study evaluates the EUREF Permanent Network (EPN) station position time series of approximately 200 GNSS stations subject to the Repro 2 reprocessing campaign in order to characterize the dominant types of noise and amplitude and their impact on estimated velocity values and associated uncertainties. The visual inspection on how different noise model represents the analysed data was done using the power spectral density of the residuals and the estimated noise model and it is coherent with the calculated Allan deviation (ADEV)-white and flicker noise. The velocities resulted from the dominant noise model are compared to the velocity obtained by using the Median Interannual Difference Adjusted for Skewness (MIDAS). The results show that only 3 stations present a dominant random walk noise model compared to flicker and powerlaw noise model for the horizontal and vertical components. We concluded that the velocities for the horizontal and vertical component show similar values in the case of MIDAS and maximum likelihood estimation (MLE), but we also found that the associated uncertainties from MIDAS are higher compared to the uncertainties from MLE. Additionally, we concluded that there is a spatial correlation in noise amplitude, and also regarding the differences in velocity uncertainties for the Up component.

Download Full-text

Edge Permutation Entropy: An Improved Entropy Measure for Time-Series Analysis

IECON 2019 - 45th Annual Conference of the IEEE Industrial Electronics Society ◽

10.1109/iecon.2019.8927449 ◽

2019 ◽

Cited By ~ 1

Author(s):

Zhiqiang Huo ◽

Yu Zhang ◽

Lei Shu ◽

Xiaowen Liao

Keyword(s):

Time Series ◽

Time Series Analysis ◽

Permutation Entropy ◽

Entropy Measure ◽

Series Analysis

Download Full-text

Comparing different approaches to compute Permutation Entropy with coarse time series

Physica A Statistical Mechanics and its Applications ◽

10.1016/j.physa.2018.08.021 ◽

2019 ◽

Vol 513 ◽

pp. 635-643 ◽

Cited By ~ 4

Author(s):

Francisco Traversaro ◽

Nicolás Ciarrocchi ◽

Florencia Pollo Cattaneo ◽

Francisco Redelico

Keyword(s):

Time Series ◽

Permutation Entropy

Download Full-text

Modeling Noisy Time Series: Physiological Tremor

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127498001157 ◽

1998 ◽

Vol 08 (07) ◽

pp. 1505-1516 ◽

Cited By ~ 51

Author(s):

J. Timmer

Keyword(s):

Time Series ◽

Stochastic Systems ◽

State Space Model ◽

Physiological Tremor ◽

Space Model ◽

Deterministic Systems ◽

Estimated Parameters ◽

Noisy Time Series ◽

Observational Noise ◽

Small Amplitude Oscillation

Empirical time series often contain observational noise. We investigate the effect of this noise on the estimated parameters of models fitted to the data. For data of physiological tremor, i.e. a small amplitude oscillation of the outstretched hand of healthy subjects, we compare the results for a linear model that explicitly includes additional observational noise to one that ignores this noise. We discuss problems and possible solutions for nonlinear deterministic as well as nonlinear stochastic processes. Especially we discuss the state space model applicable for modeling noisy stochastic systems and Bock's algorithm capable for modeling noisy deterministic systems.

Download Full-text

Multifractal Analysis of Hydrologic Data Using Wavelet Methods and Fluctuation Analysis

Discrete Dynamics in Nature and Society ◽

10.1155/2017/3148257 ◽

2017 ◽

Vol 2017 ◽

pp. 1-18 ◽

Cited By ~ 3

Author(s):

Tongzhou Zhao ◽

Liang Wu ◽

Dehua Li ◽

Yiming Ding

Keyword(s):

Time Series ◽

Water Level ◽

Multifractal Analysis ◽

Simulation Models ◽

Northern China ◽

Fluctuation Analysis ◽

Multifractal Detrended Fluctuation Analysis ◽

Wavelet Leaders ◽

Detailed Characteristic ◽

Wavelet Methods

We study the multifractal properties of water level with a high-frequency and massive time series using wavelet methods (estimation of Hurst exponents, multiscale diagram, and wavelet leaders for multifractal analysis (WLMF)) and multifractal detrended fluctuation analysis (MF-DFA). The dataset contains more than two million records from 10 observation sites at a northern China river. The multiscale behaviour is observed in this time series, which indicates the multifractality. This multifractality is detected via multiscale diagram. Then we focus on the multifractal analysis using MF-DFA and WLMF. The two methods give the same conclusion that at most sites the records satisfy the generalized binomial multifractal model, which is robust for different times (morning, afternoon, and evening). The variation in the detailed characteristic parameters of the multifractal model indicates that both human activities and tributaries influence the multifractality. Our work is useful for building simulation models of the water level of local rivers with many observation sites.

Download Full-text