Reservoir Kernels and Volterra Series.

Area of Science:

• Machine Learning
• Signal Processing
• Time Series Analysis

Background:

• Fading memory filters are crucial for analyzing systems with decaying influence from past inputs.
• Volterra series expansion provides a powerful framework for modeling nonlinear systems.
• Kernel methods offer a flexible approach for function approximation in machine learning.

Purpose of the Study:

• To construct a universal kernel capable of approximating any causal and time-invariant filter within the fading memory category.
• To introduce the Volterra reservoir kernel, derived from the state-space representation of Volterra series.
• To demonstrate the computational feasibility and empirical performance of the Volterra reservoir kernel.

Main Methods:

• Construction of a universal kernel based on the reservoir functional of a Volterra series state-space representation.
• Characterization of the kernel map using explicit recursions for computability.
• Application of the representer theorem for estimation problems with the Volterra reservoir kernel.

Main Results:

• The Volterra reservoir kernel is shown to approximate any analytic fading memory filter.
• The kernel map is computable via explicit recursions, enabling practical application.
• Empirical validation demonstrates the kernel's effectiveness in a complex financial modeling task.

Conclusions:

• The Volterra reservoir kernel offers a universal and computable approach for approximating fading memory filters.
• This kernel provides a powerful tool for nonlinear system identification and time series analysis.
• The study highlights the potential of the Volterra reservoir kernel in demanding applications like financial econometrics.