Strong noise estimation in cubic splines

TL;DR

This paper introduces a method for accurately identifying and estimating the most significant noise component in cubic spline data, even with varying noise levels, using penalized least squares.

Contribution

It provides a novel approach to detect and estimate the dominant noise in spline data regardless of smoothing parameters.

Findings

The method accurately recovers the position of the most important noise.

It correctly estimates the sign of the dominant noise.

The approach works for all smoothing parameters.

Abstract

The data $(y_i,x_i)\in$ $\textbf{R}\times[a,b]$, $i=1,\ldots,n$ satisfy $y_i=s(x_i)+e_i$ where $s$ belongs to the set of cubic splines. The unknown noises $(e_i)$ are such that $var(e_I)=1$ for some $I\in \{1, \ldots, n\}$ and $var(e_i)=\sigma^2$ for $i\neq I$. We suppose that the most important noise is $e_I$, i.e. the ratio $r_I=\frac{1}{\sigma^2}$ is larger than one. If the ratio $r_I$ is large, then we show, for all smoothing parameter, that the penalized least squares estimator of the $B$-spline basis recovers exactly the position $I$ and the sign of the most important noise $e_I$.