Generate Eigenvalue Sequence

Generates eigenvalue sequences with different decay patterns. These control the variance contribution of each mode in Karhunen-Loeve simulation.

Usage

eVal(M, type = c("linear", "exponential", "wiener"))

Arguments

M

Number of eigenvalues to generate.

type

Character. Type of eigenvalue decay:

linear: lambda_k = 1/k for k = 1, ..., M
exponential: lambda_k = exp(-k) for k = 1, ..., M
wiener: lambda_k = 1/((k - 0.5)*pi)^2, the Wiener process eigenvalues

Value

A numeric vector of length M containing the eigenvalues in decreasing order.

Details

The eigenvalues control how much variance each eigenfunction contributes to the simulated curves:

linear: Slow decay, higher modes contribute more variation. Produces rougher curves.
exponential: Fast decay, higher modes contribute very little. Produces smoother curves.
wiener: Specific decay matching Brownian motion. Use with Wiener eigenfunctions for true Brownian motion simulation.

Examples

# Compare decay patterns
lambda_lin <- eVal(20, "linear")
lambda_exp <- eVal(20, "exponential")
lambda_wie <- eVal(20, "wiener")

plot(1:20, lambda_lin, type = "b", log = "y", ylim = c(1e-10, 1),
     main = "Eigenvalue Decay Patterns", xlab = "k", ylab = expression(lambda[k]))
lines(1:20, lambda_exp, col = "red", type = "b")
lines(1:20, lambda_wie, col = "blue", type = "b")
legend("topright", c("Linear", "Exponential", "Wiener"),
       col = c("black", "red", "blue"), lty = 1, pch = 1)