hilbert_schmidt: Hilbert-Schmidt Norm Between Estimated Autocovariance...

In CovEsts: Nonparametric Estimators for Covariance Functions

Hilbert-Schmidt Norm Between Estimated Autocovariance Functions.

Description

This function computes the Hilbert-Schidmt norm between two estimated autocovariance functions.

Usage

hilbert_schmidt(est1, est2)

Arguments

est1	A numeric vector representing the first estimated autocovariance function.
est2	A numeric vector of the same length as est1 representing the second estimated autocovariance function

Details

This function computes the Hilbert-Schidmt norm between two estimated autocovariance functions. The Hilbert-Schmidt norm of a matrix

D = \left[(d_{i,j})_{1 \le i,j \le n}\right] = \left[ {\begin{array}{ccccc} D(h_{0}) & D(h_{1}) & \cdots & D(h_{n - 1}) & D(h_{n}) \\ D(h_{1}) & D(h_{0}) & \cdots & D(h_{n - 2}) & D(h_{n - 1}) \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ D(h_{n - 1}) & D(h_{n - 2}) & \cdots & D(h_{0}) & D(h_{1}) \\ D(h_{n}) & D(h_{n - 1}) & \cdots & D(h_{1}) & D(h_{0}) \\ \end{array}} \right] ,

over a set of lags \{h_{0}, h_{1}, \dots , h_{N} \}, where D(h) = \hat{C}_{1}(h) - \hat{C}_{2}(h), is defined as

{\left\Vert D \right\Vert}_{HS} = \sqrt{\sum_{i,j} d_{i, j}^{2}}.

Value

A numeric value representing the estimated Hilbert-Schmidt norm between two estimated autocovariance functions.

Examples

x <- seq(0, 5, by=0.1)
estCov1 <- exp(-x^2)
estCov2 <- exp(-x^2.1)
hilbert_schmidt(estCov1, estCov2)

CovEsts documentation built on Sept. 10, 2025, 10:39 a.m.