# On the Coordinate Change to the First-Order Spline Kernel for Regularized Impulse Response Estimation

## TL;DR

This paper explores new kernels derived from coordinate changes of the first-order spline kernel for regularized impulse response estimation, revealing properties like maximum entropy and sparse inverse Gram matrices, with spectral analysis and numerical validation.

## Contribution

It introduces novel kernels based on alternative coordinate changes, extending the properties of the first-order spline kernel for improved impulse response estimation.

## Key findings

- New kernels inherit maximum entropy property
- Inverse Gram matrices are sparse
- Spectral analysis confirms kernel properties

## Abstract

The so-called tuned-correlated kernel (sometimes also called the first-order stable spline kernel) is one of the most widely used kernels for the regularized impulse response estimation. This kernel can be derived by applying an exponential decay function as a coordinate change to the first-order spline kernel. This paper focuses on this coordinate change and derives new kernels by investigating other coordinate changes induced by stable and strictly proper transfer functions. It is shown that the corresponding kernels inherit properties from these coordinate changes and the first-order spline kernel. In particular, they have the maximum entropy property and moreover, the inverse of their Gram matrices has sparse structure. In addition, the spectral analysis of some special kernels are provided. Finally, a numerical example is given to show the efficacy of the proposed kernel.