fdars (version 0.3.3)

eVal: Generate Eigenvalue Sequence

Description

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"))

Value

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

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

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.

See Also

eFun, simFunData

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)