Interpolating Strange Attractors via Fractional Brownian Bridges
We present a novel method for interpolating univariate time series data. The proposed method combines multi-point fractional Brownian bridges, a genetic algorithm, and Takens’ theorem for reconstructing a phase space from univariate time series data. The basic idea is to first generate a population...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-05-01
|
| Series: | Entropy |
| Subjects: | |
| Online Access: | https://www.mdpi.com/1099-4300/24/5/718 |
| _version_ | 1797499970855632896 |
|---|---|
| author | Sebastian Raubitzek Thomas Neubauer Jan Friedrich Andreas Rauber |
| author_facet | Sebastian Raubitzek Thomas Neubauer Jan Friedrich Andreas Rauber |
| author_sort | Sebastian Raubitzek |
| collection | DOAJ |
| description | We present a novel method for interpolating univariate time series data. The proposed method combines multi-point fractional Brownian bridges, a genetic algorithm, and Takens’ theorem for reconstructing a phase space from univariate time series data. The basic idea is to first generate a population of different stochastically-interpolated time series data, and secondly, to use a genetic algorithm to find the pieces in the population which generate the smoothest reconstructed phase space trajectory. A smooth trajectory curve is hereby found to have a low variance of second derivatives along the curve. For simplicity, we refer to the developed method as <i>PhaSpaSto</i>-interpolation, which is an abbreviation for <b>pha</b>se-<b>spa</b>ce-trajectory-smoothing <b>sto</b>chastic interpolation. The proposed approach is tested and validated with a univariate time series of the Lorenz system, five non-model data sets and compared to a cubic spline interpolation and a linear interpolation. We find that the criterion for smoothness guarantees low errors on known model and non-model data. Finally, we interpolate the discussed non-model data sets, and show the corresponding improved phase space portraits. The proposed method is useful for interpolating low-sampled time series data sets for, e.g., machine learning, regression analysis, or time series prediction approaches. Further, the results suggest that the variance of second derivatives along a given phase space trajectory is a valuable tool for phase space analysis of non-model time series data, and we expect it to be useful for future research. |
| first_indexed | 2024-03-10T03:55:08Z |
| format | Article |
| id | doaj.art-5a6e5ef0aaf44acdb1e7a361c7017d8b |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T03:55:08Z |
| publishDate | 2022-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-5a6e5ef0aaf44acdb1e7a361c7017d8b2023-11-23T10:56:08ZengMDPI AGEntropy1099-43002022-05-0124571810.3390/e24050718Interpolating Strange Attractors via Fractional Brownian BridgesSebastian Raubitzek0Thomas Neubauer1Jan Friedrich2Andreas Rauber3TU Wien, Information and Software Engineering Group, Favoritenstrasse 9-11/194, 1040 Vienna, AustriaTU Wien, Information and Software Engineering Group, Favoritenstrasse 9-11/194, 1040 Vienna, AustriaForWind, Institute of Physics, University of Oldenburg, Küpkersweg 70, 26129 Oldenburg, GermanyTU Wien, Information and Software Engineering Group, Favoritenstrasse 9-11/194, 1040 Vienna, AustriaWe present a novel method for interpolating univariate time series data. The proposed method combines multi-point fractional Brownian bridges, a genetic algorithm, and Takens’ theorem for reconstructing a phase space from univariate time series data. The basic idea is to first generate a population of different stochastically-interpolated time series data, and secondly, to use a genetic algorithm to find the pieces in the population which generate the smoothest reconstructed phase space trajectory. A smooth trajectory curve is hereby found to have a low variance of second derivatives along the curve. For simplicity, we refer to the developed method as <i>PhaSpaSto</i>-interpolation, which is an abbreviation for <b>pha</b>se-<b>spa</b>ce-trajectory-smoothing <b>sto</b>chastic interpolation. The proposed approach is tested and validated with a univariate time series of the Lorenz system, five non-model data sets and compared to a cubic spline interpolation and a linear interpolation. We find that the criterion for smoothness guarantees low errors on known model and non-model data. Finally, we interpolate the discussed non-model data sets, and show the corresponding improved phase space portraits. The proposed method is useful for interpolating low-sampled time series data sets for, e.g., machine learning, regression analysis, or time series prediction approaches. Further, the results suggest that the variance of second derivatives along a given phase space trajectory is a valuable tool for phase space analysis of non-model time series data, and we expect it to be useful for future research.https://www.mdpi.com/1099-4300/24/5/718time series interpolationphase space reconstructionTakens’ theoreminterpolationstochastic interpolationgenetic algorithm |
| spellingShingle | Sebastian Raubitzek Thomas Neubauer Jan Friedrich Andreas Rauber Interpolating Strange Attractors via Fractional Brownian Bridges Entropy time series interpolation phase space reconstruction Takens’ theorem interpolation stochastic interpolation genetic algorithm |
| title | Interpolating Strange Attractors via Fractional Brownian Bridges |
| title_full | Interpolating Strange Attractors via Fractional Brownian Bridges |
| title_fullStr | Interpolating Strange Attractors via Fractional Brownian Bridges |
| title_full_unstemmed | Interpolating Strange Attractors via Fractional Brownian Bridges |
| title_short | Interpolating Strange Attractors via Fractional Brownian Bridges |
| title_sort | interpolating strange attractors via fractional brownian bridges |
| topic | time series interpolation phase space reconstruction Takens’ theorem interpolation stochastic interpolation genetic algorithm |
| url | https://www.mdpi.com/1099-4300/24/5/718 |
| work_keys_str_mv | AT sebastianraubitzek interpolatingstrangeattractorsviafractionalbrownianbridges AT thomasneubauer interpolatingstrangeattractorsviafractionalbrownianbridges AT janfriedrich interpolatingstrangeattractorsviafractionalbrownianbridges AT andreasrauber interpolatingstrangeattractorsviafractionalbrownianbridges |