scholarly journals Permutation Entropy Based on Non-Uniform Embedding

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 612 ◽  
Author(s):  
Mei Tao ◽  
Kristina Poskuviene ◽  
Nizar Alkayem ◽  
Maosen Cao ◽  
Minvydas Ragulskis
Keyword(s):  
Time Series ◽  
Digital Image ◽  
Real World ◽  
Computational Experiments ◽  
Permutation Entropy ◽  
Coordinate Space ◽  
Entropy Measure ◽  
Uniform Attractor ◽  
Uniform Embedding

A novel visualization scheme for permutation entropy is presented in this paper. The proposed scheme is based on non-uniform attractor embedding of the investigated time series. A single digital image of permutation entropy is produced by averaging all possible plain projections of the permutation entropy measure in the multi-dimensional delay coordinate space. Computational experiments with artificially-generated and real-world time series are used to demonstrate the advantages of the proposed visualization scheme.

scholarly journals On the Behaviour of Weighted Permutation Entropy on Fractional Brownian Motion in the Univariate and Multivariate Setting

2021 ◽  
Vol 34 (1) ◽  
Author(s):  
Marisa Mohr ◽  
Florian Wilhelm ◽  
Ralf Möller
Keyword(s):  
Time Series ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Real World ◽  
Multivariate Time Series ◽  
Permutation Entropy ◽  
Qualitative Behaviour ◽  
Univariate Time Series ◽  
Real World Applications ◽  
Real World Problems

The estimation of the qualitative behaviour of fractional Brownian motion is an important topic for modelling real-world applications. Permutation entropy is a well-known approach to quantify the complexity of univariate time series in a scalar-valued representation. As an extension often used for outlier detection, weighted permutation entropy takes amplitudes within time series into account. As many real-world problems deal with multivariate time series, these measures need to be extended though. First, we introduce multivariate weighted permutation entropy, which is consistent with standard multivariate extensions of permutation entropy. Second, we investigate the behaviour of weighted permutation entropy on both univariate and multivariate fractional Brownian motion and show revealing results.

scholarly journals Analysing nystagmus waveforms: a computational framework

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Richard V. Abadi ◽  
Ozgur E. Akman ◽  
Gemma E. Arblaster ◽  
Richard A. Clement
Keyword(s):  
Time Series ◽  
Periodic Orbit ◽  
Late Onset ◽  
Periodic Oscillation ◽  
Permutation Entropy ◽  
Entropy Measure ◽  
Unstable Periodic Orbit ◽  
Additional Tool ◽  
Orbit Analysis ◽  
Over Time

AbstractWe present a new computational approach to analyse nystagmus waveforms. Our framework is designed to fully characterise the state of the nystagmus, aid clinical diagnosis and to quantify the dynamical changes in the oscillations over time. Both linear and nonlinear analyses of time series were used to determine the regularity and complexity of a specific homogenous phenotype of nystagmus. Two-dimensional binocular eye movement recordings were carried out on 5 adult subjects who exhibited a unilateral, uniplanar, vertical nystagmus secondary to a monocular late-onset severe visual loss in the oscillating eye (the Heimann-Bielschowsky Phenomenon). The non-affected eye held a central gaze in both horizontal and vertical planes (± 10 min. of arc). All affected eyes exhibited vertical oscillations, with mean amplitudes and frequencies ranging from 2.0°–4.0° to 0.25–1.5 Hz, respectively. Unstable periodic orbit analysis revealed only 1 subject exhibited a periodic oscillation. The remaining subjects were found to display quasiperiodic (n = 1) and nonperiodic (n = 3) oscillations. Phase space reconstruction allowed attractor identification and the computation of a time series complexity measure—the permutation entropy. The entropy measure was found to be able to distinguish between a periodic oscillation associated with a limit cycle attractor, a quasiperiodic oscillation associated with a torus attractor and nonperiodic oscillations associated with higher-dimensional attractors. Importantly, the permutation entropy was able to rank the oscillations, thereby providing an objective index of nystagmus complexity (range 0.15–0.21) that could not be obtained via unstable periodic orbit analysis or attractor identification alone. These results suggest that our framework provides a comprehensive methodology for characterising nystagmus, aiding differential diagnosis and also permitting investigation of the waveforms over time, thereby facilitating the quantification of future therapeutic managements. In addition, permutation entropy could provide an additional tool for future oculomotor modelling.

Phenomenological Modelling

2018 ◽  
Author(s):  
Ray Huffaker ◽  
Marco Bittelli ◽  
Rodolfo Rosa
Keyword(s):  
Nonlinear Dynamics ◽  
Time Series ◽  
Real World ◽  
Biological Diversity ◽  
False Positives ◽  
World Systems ◽  
Economic Activities ◽  
Essential Services ◽  
Low Dimension ◽  
Convergent Cross Mapping

Detecting causal interactions among climatic, environmental, and human forces in complex biophysical systems is essential for understanding how these systems function and how public policies can be devised that protect the flow of essential services to biological diversity, agriculture, and other core economic activities. Convergent Cross Mapping (CCM) detects causal networks in real-world systems diagnosed with deterministic, low-dimension, and nonlinear dynamics. If CCM detects correspondence between phase spaces reconstructed from observed time series variables, then the variables are determined to causally interact in the same dynamic system. CCM can give false positives by misconstruing synchronized variables as causally interactive. Extended (delayed) CCM screens for false positives among synchronized variables.

scholarly journals Time-Delay Identification Using Multiscale Ordinal Quantifiers

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 969
Author(s):  
Miguel C. Soriano ◽  
Luciano Zunino
Keyword(s):  
Time Series ◽  
Time Delay ◽  
Real World ◽  
Propagation Speed ◽  
Real World Data ◽  
External Observer ◽  
The North ◽  
Weather Phenomenon ◽  
The North Atlantic Oscillation ◽  
Finite Propagation Speed

Time-delayed interactions naturally appear in a multitude of real-world systems due to the finite propagation speed of physical quantities. Often, the time scales of the interactions are unknown to an external observer and need to be inferred from time series of observed data. We explore, in this work, the properties of several ordinal-based quantifiers for the identification of time-delays from time series. To that end, we generate artificial time series of stochastic and deterministic time-delay models. We find that the presence of a nonlinearity in the generating model has consequences for the distribution of ordinal patterns and, consequently, on the delay-identification qualities of the quantifiers. Here, we put forward a novel ordinal-based quantifier that is particularly sensitive to nonlinearities in the generating model and compare it with previously-defined quantifiers. We conclude from our analysis on artificially generated data that the proper identification of the presence of a time-delay and its precise value from time series benefits from the complementary use of ordinal-based quantifiers and the standard autocorrelation function. We further validate these tools with a practical example on real-world data originating from the North Atlantic Oscillation weather phenomenon.

Comparing different approaches to compute Permutation Entropy with coarse time series

2019 ◽  
Vol 513 ◽  
pp. 635-643 ◽  
Author(s):  
Francisco Traversaro ◽  
Nicolás Ciarrocchi ◽  
Florencia Pollo Cattaneo ◽  
Francisco Redelico
Keyword(s):  
Time Series ◽  
Permutation Entropy

scholarly journals Time-Series of Distributions Forecasting in Agricultural Applications: An Intervals’ Numbers Approach

2021 ◽  
Vol 5 (1) ◽  
pp. 12
Author(s):  
Christos Bazinas ◽  
Eleni Vrochidou ◽  
Chris Lytridis ◽  
Vassilis Kaburlasos
Keyword(s):  
Time Series ◽  
Accurate Prediction ◽  
Computational Experiments ◽  
Stationary Time Series ◽  
Neural Architecture ◽  
Recursive Scheme ◽  
Stationary Time ◽  
Agricultural Applications ◽  
Data Statistics

This work represents any distribution of data by an Intervals’ Number (IN), hence it represents all-order data statistics, using a “small” number of L intervals. The INs considered are induced from images of grapes that ripen. The objective is the accurate prediction of grape maturity. Based on an established algebra of INs, an optimizable IN-regressor is proposed, implementable on a neural architecture, toward predicting future INs from past INs. A recursive scheme tests the capacity of the IN-regressor to learn the physical “law” that generates the non-stationary time-series of INs. Computational experiments demonstrate comparatively the effectiveness of the proposed techniques.

NONLINEAR GATED EXPERTS FOR TIME SERIES: DISCOVERING REGIMES AND AVOIDING OVERFITTING

1995 ◽  
Vol 06 (04) ◽  
pp. 373-399 ◽  
Author(s):  
ANDREAS S. WEIGEND ◽  
MORGAN MANGEAS ◽  
ASHOK N. SRIVASTAVA
Keyword(s):  
Time Series ◽  
Real World ◽  
Markov Models ◽  
Multilayer Perceptrons ◽  
Time Step ◽  
Local Complexity ◽  
Previous State ◽  
Segmentation Task ◽  
Gating Network ◽  
Update Rules

In the analysis and prediction of real-world systems, two of the key problems are nonstationarity (often in the form of switching between regimes), and overfitting (particularly serious for noisy processes). This article addresses these problems using gated experts, consisting of a (nonlinear) gating network, and several (also nonlinear) competing experts. Each expert learns to predict the conditional mean, and each expert adapts its width to match the noise level in its regime. The gating network learns to predict the probability of each expert, given the input. This article focuses on the case where the gating network bases its decision on information from the inputs. This can be contrasted to hidden Markov models where the decision is based on the previous state(s) (i.e. on the output of the gating network at the previous time step), as well as to averaging over several predictors. In contrast, gated experts soft-partition the input space, only learning to model their region. This article discusses the underlying statistical assumptions, derives the weight update rules, and compares the performance of gated experts to standard methods on three time series: (1) a computer-generated series, obtained by randomly switching between two nonlinear processes; (2) a time series from the Santa Fe Time Series Competition (the light intensity of a laser in chaotic state); and (3) the daily electricity demand of France, a real-world multivariate problem with structure on several time scales. The main results are: (1) the gating network correctly discovers the different regimes of the process; (2) the widths associated with each expert are important for the segmentation task (and they can be used to characterize the sub-processes); and (3) there is less overfitting compared to single networks (homogeneous multilayer perceptrons), since the experts learn to match their variances to the (local) noise levels. This can be viewed as matching the local complexity of the model to the local complexity of the data.

The modified permutation entropy-based independence test of time series

2018 ◽  
Vol 48 (10) ◽  
pp. 2877-2897
Author(s):  
Emad Ashtari Nezhad ◽  
Yadollah Waghei ◽  
G. R. Mohtashami Borzadaran ◽  
H. R. Nilli Sani ◽  
Hadi Alizadeh Noughabi
Keyword(s):  
Time Series ◽  
Permutation Entropy ◽  
Independence Test
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