Raubitzek Sebastian, Neubauer Thomas, Friedrich Jan, Rauber Andreas
TU Wien, Information and Software Engineering Group, Favoritenstrasse 9-11/194, 1040 Vienna, Austria.
ForWind, Institute of Physics, University of Oldenburg, Küpkersweg 70, 26129 Oldenburg, Germany.
Entropy (Basel). 2022 May 17;24(5):718. doi: 10.3390/e24050718.
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 -interpolation, which is an abbreviation for se-ce-trajectory-smoothing 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.
我们提出了一种用于插值单变量时间序列数据的新方法。该方法结合了多点分数布朗桥、遗传算法以及用于从单变量时间序列数据重建相空间的塔肯斯定理。其基本思想是，首先生成一组不同的随机插值时间序列数据，其次，使用遗传算法在该组数据中找到能生成最平滑重建相空间轨迹的片段。在此，一条平滑轨迹曲线被发现沿着该曲线具有较低的二阶导数方差。为简单起见，我们将所开发的方法称为 -插值，它是“se-ce-trajectory-smoothing chastic interpolation”的缩写。我们用洛伦兹系统的单变量时间序列、五个非模型数据集对所提出的方法进行了测试和验证，并与三次样条插值和线性插值进行了比较。我们发现平滑准则能保证在已知模型和非模型数据上具有低误差。最后，我们对所讨论的非模型数据集进行了插值，并展示了相应改进后的相空间图。所提出的方法对于插值低采样时间序列数据集很有用，例如用于机器学习、回归分析或时间序列预测方法。此外，结果表明沿着给定相空间轨迹的二阶导数方差是用于非模型时间序列数据相空间分析的一个有价值的工具，并且我们期望它对未来的研究有用。
