eFun: Generate Eigenfunction Basis

Description

Evaluates orthonormal eigenfunction bases at specified argument values. These eigenfunctions can be used for Karhunen-Loeve simulation.

Usage

eFun(argvals, M, type = c("Fourier", "Poly", "PolyHigh", "Wiener"))

Value

A matrix of dimension length(argvals) x M containing the eigenfunction values. Each column is an eigenfunction, normalized to have unit L2 norm on [0, 1].

Arguments

argvals

Numeric vector of evaluation points in [0, 1].

M

Number of eigenfunctions to generate.

type

Character. Type of eigenfunction system:

Fourier: Fourier basis: 1, sqrt(2)sin(2pikt), sqrt(2)cos(2pikt)

Poly

Orthonormal Legendre polynomials of degrees 0, 1, ..., M-1

PolyHigh

Orthonormal Legendre polynomials starting at degree 2

Wiener

Wiener process eigenfunctions: sqrt(2)*sin((k-0.5)pit)

Details

The eigenfunctions are orthonormal with respect to the L2 inner product: integral(phi_j * phi_k) = 1 if j == k, 0 otherwise.

Fourier: Suitable for periodic data. First function is constant.
Poly: Orthonormalized Legendre polynomials. Good for smooth data.
PolyHigh: Legendre polynomials starting at degree 2, useful when linear and constant components are handled separately.
Wiener: Eigenfunctions of the Brownian motion covariance. Eigenvalues decay as 1/((k-0.5)*pi)^2.

Examples

Run this code

t <- seq(0, 1, length.out = 100)

# Generate Fourier basis
phi_fourier <- eFun(t, M = 5, type = "Fourier")
matplot(t, phi_fourier, type = "l", lty = 1,
        main = "Fourier Eigenfunctions", ylab = expression(phi(t)))

# Generate Wiener eigenfunctions
phi_wiener <- eFun(t, M = 5, type = "Wiener")
matplot(t, phi_wiener, type = "l", lty = 1,
        main = "Wiener Eigenfunctions", ylab = expression(phi(t)))

# Check orthonormality (should be identity matrix)
dt <- diff(t)[1]
gram <- t(phi_fourier) %*% phi_fourier * dt
round(gram, 2)