Description Usage Arguments Details Value Examples
View source: R/generation_kernel.R
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.
1 2 | generation_kernel(type = "sobolev", parameter = NULL, domain,
thres = 0.99, return.derivatives = FALSE)
|
type |
string. Type of kernel. Three possible choices implemented:
|
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 |
domain |
vector. |
thres |
scalar. Threshold to identify the significant
eigenvalues of the kernel. The number of significant eigennvalues ∑_{j = 1}^J θ_j ≥q \textrm{thres} ∑_{j = 1}^{∞} θ_j. Default is 0.99. |
return.derivatives |
bool. If |
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.
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.
1 2 3 4 5 6 7 8 9 10 11 12 13 | # 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
|
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