Modeling multivariate time series on manifolds with skew radial basis functions

Area of Science:

• Computational Mathematics
• Data Science
• Nonlinear Dynamics

Background:

• Developing accurate empirical models for high-dimensional, sparse data is challenging.
• Existing methods often require manual parameter tuning and struggle with complex data structures.
• Radial basis functions offer a flexible framework for nonlinear function approximation.

Discussion:

• The presented algorithm iteratively refines nonlinear empirical mappings using radial and skew radial basis functions.
• A statistical hypothesis test, considering correlations in multivariate ranges, guides function placement and model refinement.
• Local fitting and autocorrelation analysis determine model parameters and function scales automatically.

Key Insights:

• The method effectively handles scattered or sparse data, constructing models from high-dimensional domains to multivariate ranges.
• Compactly supported skew radial basis functions enhance model accuracy, order, and convergence.
• The algorithm automatically determines model parameters and the number of basis functions, minimizing ad hoc input.

Outlook:

• The approach can be extended to higher-dimensional ranges, yielding reduced-order models by leveraging data correlations.
• The methodology is applicable to diverse problems, including modeling data on manifolds and predicting chaotic time series.
• Future work may explore further optimizations and applications in complex systems modeling.