generation_kernel: 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 kernel

Description

Given a particular type of kernel, to be chosen among (gaussian, exponential and sobolev), it returns the evaluation of the eigenfunctions of the kernel on the grid domain and the correspondent eigenvalues.

Usage

generation_kernel(type = "sobolev", parameter = NULL, domain,
  thres = 0.99, return.derivatives = FALSE)

Arguments

type	string. Type of kernel. Three possible choices implemented: gaussian, exponential and sobolev. For the other types of kernel, define manually the eigenfunctions and eigenvectors of the kernel.
parameter	scalar. Value of the characteristic parameter of the kernel. It is the σ parameter of the Gaussian and the Exponential kernel, as introduced in kernlab; and the σ parameter of the Sobolev kernel as in the sobolev_kernel_generation function.
domain	vector. m-length vector of the abscissa grid for the definition of the kernel.
thres	scalar. Threshold to identify 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 also the matrix of the evaluation of the derivatives of the eigenfunctions on the time domain. Default is FALSE.

Details

Here the list of the kernel defined in this function

• gaussian

  k(x, x') = \exp(-σ \| x- x'\|^2)

• exponential

  k(x, x') = \exp(-σ \| x- x'\|)

• sobolev, the kernel associated to the norm in the H^1 space

  \| f \|^2 = \int_{D} f(t)^2 dt + \frac{1}{σ} \int_{D} f'(t)^2 dt

  where D is the one-dimensional domain and f' is the first derivative of the function.

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. derivatives is the (m-1) \times J matrix of the derivatives of the eigenfunctions evaluated on the time domain.

Examples

# definition of the kernel
type_kernel <-  'sobolev'
param_kernel <-  8
T_domain <- seq(0, 1, length = 50)
kernel_here <- generation_kernel ( type = type_kernel,
                                   parameter = param_kernel,
                                   domain = T_domain,
                                   thres = 0.99,
                                   return.derivatives = TRUE)

eigenvalues <- kernel_here$eigenval
eigenvectors <- kernel_here$eigenvect
der <- kernel_here$derivatives

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