Interpolating Strange Attractors via Fractional Brownian Bridges

Area of Science:

• Time Series Analysis
• Computational Physics
• Data Science

Background:

• Accurate interpolation of univariate time series is crucial for subsequent analysis.
• Existing methods like cubic spline and linear interpolation may not capture complex dynamics effectively.
• Reconstructing phase space from univariate data is challenging, especially with limited sampling.

Purpose of the Study:

• To develop a novel stochastic interpolation method for univariate time series data.
• To improve the accuracy and quality of interpolated time series, particularly for low-sampled datasets.
• To validate the effectiveness of the proposed method against existing techniques and demonstrate its utility in phase space analysis.

Main Methods:

• Combining multi-point fractional Brownian bridges with a genetic algorithm.
• Utilizing Takens' theorem for phase space reconstruction.
• Employing a genetic algorithm to identify stochastic interpolations yielding the smoothest phase space trajectories, defined by low variance in second derivatives.

Main Results:

• The proposed phase-space-trajectory-smoothing stochastic interpolation (PhaSpaSto) method demonstrated low interpolation errors on both model (Lorenz system) and non-model datasets.
• PhaSpaSto outperformed cubic spline and linear interpolation in accuracy.
• The method successfully generated improved phase space portraits for non-model datasets.

Conclusions:

• The variance of second derivatives along a phase space trajectory is a valuable metric for assessing trajectory smoothness and data quality.
• PhaSpaSto is effective for interpolating low-sampled time series data, benefiting applications like machine learning and regression analysis.
• The method offers a promising tool for phase space analysis of non-model time series data.