generation_kernel_periodic: Definition of the eigenfunctions and eigenvalues of the...

In ardeeshany/AFSSEN: Adaptive Function on Scalar Elastic Net

Definition of the eigenfunctions and eigenvalues of the periodic kernel

Description

Given the period p and the smoothing parameter σ, it returns the evaluation of the eigenfunctions of the kernel on the grid domain and the correspondent eigenvalues.

Usage

generation_kernel_periodic(period = NULL, parameter = NULL, domain,
  thres = 0.99, return.derivatives = FALSE)

Arguments

period	scalar. Period of the kernel.
parameter	scalar. Parameter to tune the smoothness level of the kernel. The σ parameter introduced in Details.
domain	vector. m-length vector for the abscissa grid of the kernel.
thres	scalar. Threshold for the identification of the significant eigenvalues of the kernel. The number of significant eigennvalues J is the minimum J s.t. ∑_{j = 1}^J θ_j ≥q \textrm{thres} ∑_{j = 1}^{∞} θ_j. Default is 0.99.
return.derivatives	bool. If TRUE the function returns the matrix of the derivatives of the selected eigenfunctions evaluated on the time domain. Default is FALSE.

Details

The periodic kernel of period p defined in this function is

K(x, y) = σ^2 \exp ≤ft\{ -2/σ \sin^2≤ft(\frac{π | x - y|}{p} \right)\right\}.

where σ is the smoothing parameter.

Value

list containing

• eigenvect m \times J matrix of the eigenfunctions of the kernel evaluated on the domain.

• eigenval J-length vector of the eigenvalues of the kernel

• derivatives. if return.derivatives = TRUE. m-1 \times J matrix of the derivatives of the eigenfunctions.

Examples

param_kernel <- 8
T_domain <- seq(0, 1, length = 50)
kernel_here <- generation_kernel_periodic(period = 1/2,
                                          parameter = param_kernel,
                                          domain = T_domain,
                                          thres = 1-10^{-16},
                                          return.derivatives = TRUE)
names(kernel_here)

ardeeshany/AFSSEN documentation built on Aug. 28, 2022, 2:22 p.m.