The 2d folded Matern covariance function

folded.matern.covariance.2d evaluates the 2d folded Matern covariance function over an interval \([0,L]\times [0,L]\).

Usage

folded.matern.covariance.2d(
  h,
  m,
  kappa,
  nu,
  sigma,
  L = 1,
  N = 10,
  boundary = c("neumann", "dirichlet", "periodic", "R2")
)

Arguments

h, m

Vectors with two coordinates.

kappa

Range parameter.

nu

Shape parameter.

sigma

Standard deviation.

L

The upper bound of the square \([0,L]\times [0,L]\). By default, L=1.

N

The truncation parameter.

boundary

The boundary condition. The possible conditions are "neumann" (default), "dirichlet", "periodic" or "R2".

Value

The correspoding covariance.

Details

folded.matern.covariance.2d evaluates the 1d folded Matern covariance function over an interval \([0,L]\times [0,L]\) under different boundary conditions. For periodic boundary conditions $$C_{\mathcal{P}}((h_1,h_2),(m_1,m_2)) = \sum_{k_2=-\infty}^\infty \sum_{k_1=-\infty}^{\infty} (C(\|(h_1-m_1+2k_1L,h_2-m_2+2k_2L)\|),$$ for Neumann boundary conditions $$C_{\mathcal{N}}((h_1,h_2),(m_1,m_2)) = \sum_{k_2=-\infty}^\infty \sum_{k_1=-\infty}^{\infty} (C(\|(h_1-m_1+2k_1L,h_2-m_2+2k_2L)\|)+C(\|(h_1-m_1+2k_1L, h_2+m_2+2k_2L)\|)+C(\|(h_1+m_1+2k_1L,h_2-m_2+2k_2L)\|)+ C(\|(h_1+m_1+2k_1L,h_2+m_2+2k_2L)\|)),$$ and for Dirichlet boundary conditions: $$C_{\mathcal{D}}((h_1,h_2),(m_1,m_2)) = \sum_{k_2=-\infty}^\infty \sum_{k_1=-\infty}^{\infty} (C(\|(h_1-m_1+2k_1L,h_2-m_2+2k_2L)\|)- C(\|(h_1-m_1+2k_1L,h_2+m_2+2k_2L)\|)-C(\|(h_1+m_1+2k_1L, h_2-m_2+2k_2L)\|)+C(\|(h_1+m_1+2k_1L,h_2+m_2+2k_2L)\|)),$$ where \(C(\cdot)\) is the Matern covariance function: $$C(h) = \frac{\sigma^2}{2^{\nu-1}\Gamma(\nu)} (\kappa h)^\nu K_\nu(\kappa h).$$

We consider the truncation for \(k_1,k_2\) from \(-N\) to \(N\).

Examples

h <- c(0.5, 0.5)
m <- c(0.5, 0.5)
folded.matern.covariance.2d(h, m, kappa = 10, nu = 1 / 5, sigma = 1)
#> [1] 1.000043