List of basic functions

Polyharmonic splines

Biharmonic2D

The basic function for biharmonic spline interpolation of 2D data. It can also be used for interpolation of any dimensional data.

$$ ϕ_\mathtt{bh2}(r) = s ⋅ \left(\sqrt{r^2 + c^2}\right)^2 \ln \left(\sqrt{r^2 + c^2}\right). $$

$ϕ_\mathtt{bh2}(‖⋅‖)$ is conditionally positive definite of order 2 on every $ℝ^d$. Thus, the interpolant requires the polynomial part of degree ≥ 1.

Biharmonic3D

The basic function for biharmonic spline interpolation of 3D data. It can also be used for interpolation of any dimensional data.

$$ ϕ_\mathtt{bh3}(r) = -s ⋅ \sqrt{r^2 + c^2}. $$

$ϕ_\mathtt{bh3}(‖⋅‖)$ is conditionally positive definite of order 1 on every $ℝ^d$. Thus, the interpolant requires the polynomial part of degree ≥ 0.

Triharmonic2D

The basic function for triharmonic spline interpolation of 2D data. It can also be used for interpolation of any dimensional data.

$$ ϕ_\mathtt{th2}(r) = -s ⋅ \left(\sqrt{r^2 + c^2}\right)^4 \ln \left(\sqrt{r^2 + c^2}\right). $$

$ϕ_\mathtt{th2}(‖⋅‖)$ is conditionally positive definite of order 3 on every $ℝ^d$. Thus, the interpolant requires the polynomial part of degree ≥ 2.

Triharmonic3D

The basic function for triharmonic spline interpolation of 3D data. It can also be used for interpolation of any dimensional data.

$$ ϕ_\mathtt{th3}(r) = s ⋅ \left(\sqrt{r^2 + c^2}\right)^3. $$

$ϕ_\mathtt{th3}(‖⋅‖)$ is conditionally positive definite of order 2 on every $ℝ^d$. Thus, the interpolant requires the polynomial part of degree ≥ 1.

Covariance functions

CovCubic

$$ C_\mathtt{cub}(r) = s ⋅ \begin{cases} 1 - 7 ⋅ \left(\frac{r}{a}\right)^2 + \frac{35}{4} ⋅ \left(\frac{r}{a}\right)^3 - \frac{7}{2} ⋅ \left(\frac{r}{a}\right)^5 + \frac{3}{4} ⋅ \left(\frac{r}{a}\right)^7 & \text{if }\left(\frac{r}{a}\right) ≤ 1, \\ 0 & \text{otherwise}. \end{cases} $$

$C_\mathtt{cub}(‖⋅‖)$ is positive definite on $ℝ^d$ up to $d = 3$.

CovExponential

The exponential model.

$$ C_\mathtt{exp}(r) = s ⋅ \exp\left(-3 ⋅ \left(\frac{r}{a}\right)\right). $$

$C_\mathtt{exp}(‖⋅‖)$ is positive definite on every $ℝ^d$.

CovGaussian

The Gaussian model.

$$ C_\mathtt{gau}(r) = s ⋅ \exp\left(-3 ⋅ \left(\frac{r}{a}\right)^2\right). $$

$C_\mathtt{gau}(‖⋅‖)$ is positive definite on every $ℝ^d$.

CovGeneralizedCauchy{3,5,7,9}

The generalized Cauchy model of order $α = 3, 5, 7, 9$.

$$ C_\mathtt{gcα}(r) = s ⋅ \left(1 + A ⋅ \left(\frac{r}{a}\right)^2\right)^{-α/2}. $$

The parameter $A$ is chosen so that $C_\mathtt{gcα}(1) ≈ 0.0442$.

$C_\mathtt{gcα}(‖⋅‖)$ is positive definite on every $ℝ^d$.

CovSpherical

$$ C_\mathtt{sph}(r) = s ⋅ \begin{cases} 1 - \frac{3}{2} ⋅ \left(\frac{r}{a}\right) + \frac{1}{2} ⋅ \left(\frac{r}{a}\right)^3 & \text{if }\left(\frac{r}{a}\right) ≤ 1, \\ 0 & \text{otherwise}. \end{cases} $$

$C_\mathtt{sph}(‖⋅‖)$ is positive definite on $ℝ^d$ up to $d = 3$.

CovSpheroidal{3,5,7,9}

The spheroidal model of order $α = 3, 5, 7, 9$.

$$ C_\mathtt{spα}(r) = s ⋅ \begin{cases} 1 - A ⋅ \left(\frac{r}{a}\right) & \text{if }\left(\frac{r}{a}\right) ≤ R_0, \\ B ⋅ \left(1 + C ⋅ \left(\frac{r}{a}\right)^2\right)^{-α/2} & \text{otherwise}. \end{cases} $$

The parameters $R_0$, $A$, $B$ and $C$ are chosen so that the function is smooth and $C_\mathtt{spα}(1) ≈ 0.0373$.

$C_\mathtt{spα}(‖⋅‖)$ is positive definite on $ℝ^d$ up to $d = 4$, according to a numerical analysis.

The spheroidal model is a piecewise function consists of the linear and the Cauchy models. It resembles to the spherical model while has smoother shape ( $C_\mathtt{spα}(r)$ is of class $C^2$ on $(0, ∞)$ ). However, it does not have a geometrical interpretation as the spherical model does. See this article for details.

The nugget effect model

The nugget effect model is not implemented as an RBF. Instead, you can combine it with any other RBFs by calling the function Model::set_nugget. The nugget effect model is included in fitting but excluded from evaluation to keep the interpolant continuous.

$$ C_\mathtt{nug}(r) = s ⋅ \begin{cases} 1 & \text{if }r = 0, \\ 0 & \text{otherwise}. \end{cases} $$

$C_\mathtt{nug}(‖⋅‖)$ is positive definite on every $ℝ^d$.