Algorithmic Approaches for Assessing Irreversibility in Time Series: Review and Comparison
Abstract
1. Introduction
1.1. Applications
1.2. Assessing Irreversibility in Real-World Time Series
2. Numerical Methods
2.1. The BDS Statistic
2.2. Ramsey and Rothman’s Time Reversibility Test
2.3. The DFK Test
2.4. Permutation Patterns (Permp) Test
2.5. The Ternary Coding (TC) Test
2.6. Micro-Scale Trends (MSTrends) Test
2.7. Visibility Graphs
2.8. Local Clustering Coefficient
2.9. Additional Irreversibility Tests
3. Evaluation and Comparison
3.1. Preliminaries
3.2. Tests’ Performance on Synthetic Data
- the previously defined logistic map , with , and without observational noise;
- the Henon map, defined as , , with and ;
- the Generalised Autoregressive Conditional Heteroskedasticity (GARCH) model [85], defined as , with , , and being independent random numbers drawn from an uniform distribution ; and
- the three variables of the Lorenz chaotic system, a continuos system defined as , , and , with , and .
- time series composed of random number extracted from a normal distribution ;
- time series composed of random number extracted from a uniform distribution ;
- the x variable of the Arnold Cat map, a conservative (and hence reversible) chaotic map defined as: , .
- the Ornstein-Uhlenbeck process, i.e., a mean-reverting linear Gaussian process T [1].
3.3. Ensemble Testing
- Combining all eight tests into a single one (here denoted by Ens:all).
- Combine the two most effective tests, i.e., BDS and MSTrends (Ens:BDS_MSTrends).
- Combine the pairs of tests that yield the most independent results, such that the errors of one of them could be corrected by the other. These pairs would be Ramsey and PermP (Ens:Ramsey_PermP), and lCC and MSTrends (Ens:lCC_MSTrends).
3.4. Analysing Real-World Data: The Case of Human Electro-Encephalography
3.5. Computational Cost
4. Software
- Tests are organised in independent modules, whose names are reported in Table 2. Each test has a main function, always called GetPValue, which returns two results: the p-value of the test, and the statistic of the test when available. Besides the time series to be analysed, each test expects a set of parameters, which are also described in Table 2.
- Some tests include simplifications in their execution; to illustrate, the permutation patterns algorithm only considers patterns of size 3; and the micro-scale trends one only linear regressions of the data. The reader should nevertheless note that this will be a live library, and that such simplifications may disappear in the future. We therefore recommend to check the online documentation of the project.
- In order to simplify the use of the library, it includes an example program UsageExample.py, which creates a time series using a logistic map, and sequentially executes all available tests with different parameters. We recommend the interested reader to start by checking this program.
5. Discussion and Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Test | Parameter | Values |
---|---|---|
BDS | none | |
Ramsey | k | |
DFK | n | |
L | ||
Permutation patterns | 3 | |
Ternary Coding | Number of segments | 20 |
10th percentile of | ||
MSTrends | ||
d | 1 | |
Visibility graph | none | |
LocalCC | none |
Module (test) | Reference | Parameter | Meaning |
---|---|---|---|
BDS | [87,88] | none | |
Ramsey | [49] | kappa | The lag k |
DFK | [90] | n | Number of possible symbols |
L | Word length | ||
PermPatterns | [52,84,92,93] | none | |
TernaryCoding | [98] | segL | Length of the segments on which is calculated |
alpha | Threshold for symbolisation | ||
MSTrends | [99] | wSize | Length of overlapping windows |
wSize2 | Length of windows used to calculate higher central moments | ||
VisibilityGraph | [99] | none | |
LocalCC | [107] | none | |
Pomeau | [5] | numRnd | Number of random repetitions to extract the p-value |
tau | Embedding delay | ||
Diks | [108] | embD | Embedding dimension, or size of each vector |
dVar | Bandwidth of the analysis | ||
Ens_BDS_MSTrends | none |
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Zanin, M.; Papo, D. Algorithmic Approaches for Assessing Irreversibility in Time Series: Review and Comparison. Entropy 2021, 23, 1474. https://doi.org/10.3390/e23111474
Zanin M, Papo D. Algorithmic Approaches for Assessing Irreversibility in Time Series: Review and Comparison. Entropy. 2021; 23(11):1474. https://doi.org/10.3390/e23111474
Chicago/Turabian StyleZanin, Massimiliano, and David Papo. 2021. "Algorithmic Approaches for Assessing Irreversibility in Time Series: Review and Comparison" Entropy 23, no. 11: 1474. https://doi.org/10.3390/e23111474
APA StyleZanin, M., & Papo, D. (2021). Algorithmic Approaches for Assessing Irreversibility in Time Series: Review and Comparison. Entropy, 23(11), 1474. https://doi.org/10.3390/e23111474