On the Statistical Properties of Multiscale Permutation Entropy: Characterization of the Estimator’s Variance
Permutation Entropy (PE) and Multiscale Permutation Entropy (MPE) have been extensively used in the analysis of time series searching for regularities. Although PE has been explored and characterized, there is still a lack of theoretical background regarding MPE. Therefore, we expand the available M...
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| Format: | Article |
| Language: | English |
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MDPI AG
2019-04-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/21/5/450 |
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| author | Antonio Dávalos Meryem Jabloun Philippe Ravier Olivier Buttelli |
| author_facet | Antonio Dávalos Meryem Jabloun Philippe Ravier Olivier Buttelli |
| author_sort | Antonio Dávalos |
| collection | DOAJ |
| description | Permutation Entropy (PE) and Multiscale Permutation Entropy (MPE) have been extensively used in the analysis of time series searching for regularities. Although PE has been explored and characterized, there is still a lack of theoretical background regarding MPE. Therefore, we expand the available MPE theory by developing an explicit expression for the estimator’s variance as a function of time scale and ordinal pattern distribution. We derived the MPE Cramér−Rao Lower Bound (CRLB) to test the efficiency of our theoretical result. We also tested our formulation against MPE variance measurements from simulated surrogate signals. We found the MPE variance symmetric around the point of equally probable patterns, showing clear maxima and minima. This implies that the MPE variance is directly linked to the MPE measurement itself, and there is a region where the variance is maximum. This effect arises directly from the pattern distribution, and it is unrelated to the time scale or the signal length. The MPE variance also increases linearly with time scale, except when the MPE measurement is close to its maximum, where the variance presents quadratic growth. The expression approaches the CRLB asymptotically, with fast convergence. The theoretical variance is close to the results from simulations, and appears consistently below the actual measurements. By knowing the MPE variance, it is possible to have a clear precision criterion for statistical comparison in real-life applications. |
| first_indexed | 2024-04-11T21:33:10Z |
| format | Article |
| id | doaj.art-8dddb308cf6d4343952e370f1c6a9da7 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-11T21:33:10Z |
| publishDate | 2019-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-8dddb308cf6d4343952e370f1c6a9da72022-12-22T04:01:50ZengMDPI AGEntropy1099-43002019-04-0121545010.3390/e21050450e21050450On the Statistical Properties of Multiscale Permutation Entropy: Characterization of the Estimator’s VarianceAntonio Dávalos0Meryem Jabloun1Philippe Ravier2Olivier Buttelli3Laboratoire Pluridisciplinaire de Recherche en Ingénierie des Systèmes, Mécanique, Énergétique (PRISME), University of Orléans, 45100 Orléans, INSA-CVL, FranceLaboratoire Pluridisciplinaire de Recherche en Ingénierie des Systèmes, Mécanique, Énergétique (PRISME), University of Orléans, 45100 Orléans, INSA-CVL, FranceLaboratoire Pluridisciplinaire de Recherche en Ingénierie des Systèmes, Mécanique, Énergétique (PRISME), University of Orléans, 45100 Orléans, INSA-CVL, FranceLaboratoire Pluridisciplinaire de Recherche en Ingénierie des Systèmes, Mécanique, Énergétique (PRISME), University of Orléans, 45100 Orléans, INSA-CVL, FrancePermutation Entropy (PE) and Multiscale Permutation Entropy (MPE) have been extensively used in the analysis of time series searching for regularities. Although PE has been explored and characterized, there is still a lack of theoretical background regarding MPE. Therefore, we expand the available MPE theory by developing an explicit expression for the estimator’s variance as a function of time scale and ordinal pattern distribution. We derived the MPE Cramér−Rao Lower Bound (CRLB) to test the efficiency of our theoretical result. We also tested our formulation against MPE variance measurements from simulated surrogate signals. We found the MPE variance symmetric around the point of equally probable patterns, showing clear maxima and minima. This implies that the MPE variance is directly linked to the MPE measurement itself, and there is a region where the variance is maximum. This effect arises directly from the pattern distribution, and it is unrelated to the time scale or the signal length. The MPE variance also increases linearly with time scale, except when the MPE measurement is close to its maximum, where the variance presents quadratic growth. The expression approaches the CRLB asymptotically, with fast convergence. The theoretical variance is close to the results from simulations, and appears consistently below the actual measurements. By knowing the MPE variance, it is possible to have a clear precision criterion for statistical comparison in real-life applications.https://www.mdpi.com/1099-4300/21/5/450Multiscale Permutation Entropyordinal patternsestimator varianceCramér–Rao Lower Boundfinite-length signals |
| spellingShingle | Antonio Dávalos Meryem Jabloun Philippe Ravier Olivier Buttelli On the Statistical Properties of Multiscale Permutation Entropy: Characterization of the Estimator’s Variance Entropy Multiscale Permutation Entropy ordinal patterns estimator variance Cramér–Rao Lower Bound finite-length signals |
| title | On the Statistical Properties of Multiscale Permutation Entropy: Characterization of the Estimator’s Variance |
| title_full | On the Statistical Properties of Multiscale Permutation Entropy: Characterization of the Estimator’s Variance |
| title_fullStr | On the Statistical Properties of Multiscale Permutation Entropy: Characterization of the Estimator’s Variance |
| title_full_unstemmed | On the Statistical Properties of Multiscale Permutation Entropy: Characterization of the Estimator’s Variance |
| title_short | On the Statistical Properties of Multiscale Permutation Entropy: Characterization of the Estimator’s Variance |
| title_sort | on the statistical properties of multiscale permutation entropy characterization of the estimator s variance |
| topic | Multiscale Permutation Entropy ordinal patterns estimator variance Cramér–Rao Lower Bound finite-length signals |
| url | https://www.mdpi.com/1099-4300/21/5/450 |
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