Supported methods

Currently, three different interpolation methods are available; Radial Basis Functions, Inverse Distance Weighting and Nearest Neighbor.

Radial Basis Functions

For radial basis function interpolation, the interpolated value at some point $\mathbf{x}$ is given by

\[u(\mathbf{x}) = \displaystyle \sum_{i = 1}^{N}{ w_i \phi(||\mathbf{x} - \mathbf{x_i}||)}\]

where $||\mathbf{x} - \mathbf{x_i}||$ is the distance between $\mathbf{x}$ and $\mathbf{x}_i$, and $\phi(r)$ is one of the basis functions defined below.

To use radial basis function interpolation, pass one of the available basis functions as method to interpolate.

If a GeneralizedRadialBasisFunction is used, an additional polynomial term is added in order for the resulting matrix to be positive definite:

\[u(\mathbf{x}) = \displaystyle \sum_{i = 1}^{N}{ w_i \phi(||\mathbf{x} - \mathbf{x_i}||)} + \mathbf{P}(\mathbf{x})\mathbf{λ}\]

where $\mathbf{P}(\mathbf{x})$ is the matrix defining a complete homogeneous symmetric polynomial of degree degree, and $\mathbf{λ}$ is a vector containing the polynomial coefficients.

Available basis functions

Multiquadratic
\[ϕ(r) = \sqrt{1 + (ɛr)^2}\]
InverseMultiquadratic
\[ϕ(r) = \frac{1}{\sqrt{1 + (ɛr)^2}}\]
Gaussian
\[ϕ(r) = e^{-(ɛr)^2}\]
InverseQuadratic
\[ϕ(r) = \frac{1}{1 + (ɛr)^2}\]
Polyharmonic spline
\[ϕ(r) = \begin{cases} \begin{align*} &r^k & k = 1, 3, 5, ... \\ &r^k \mathrm{ln}(r) & k = 2, 4, 6, ... \end{align*} \end{cases}\]
ThinPlate spline
A thin plate spline is the special case $k = 2$ of the polyharmonic splines. ThinPlate() is a shorthand for Polyharmonic(2).
GeneralizedMultiquadratic
\[ϕ(r) = \left(1 + (ɛr)^2\right)^\beta\]
The generalzized multiquadratic results in a positive definite system for polynomials of degree $m \geq \lceil\beta\rceil$.
GeneralizedPolyharmonic spline
\[ϕ(r) = \begin{cases} \begin{align*} &r^k & k = 1, 3, 5, ... \\ &r^k \mathrm{ln}(r) & k = 2, 4, 6, ... \end{align*} \end{cases}\]
The generalized polyharmonic spline results in a positive definite system for polynomials of degree
\[\begin{cases} \begin{align*} m &\geq \left\lceil\frac{\beta}{2}\right\rceil & k = 1, 3, 5, ... \\ m &= \beta + 1 & k = 2, 4, 6, ... \end{align*} \end{cases}\]

Inverse Distance Weighting

Also called Shepard interpolation, the basic version computes the interpolated value at some point $\mathbf{x}$ by

\[u(\mathbf{x}) = \begin{cases} \frac{\displaystyle \sum_{i = 1}^{N}{ w_i(\mathbf{x}) u_i } } { \displaystyle \sum_{i = 1}^{N}{ w_i(\mathbf{x}) } }, & \text{if } ||\mathbf{x} - \mathbf{x_i}|| \neq 0 \text{ for all } i \\ u_i, & \text{if } ||\mathbf{x} - \mathbf{x_i}|| = 0 \text{ for some } i \end{cases}\]

where $||\mathbf{x} - \mathbf{x_i}||$ is the distance between $\mathbf{x}$ and $\mathbf{x}_i$, and $w_i(\mathbf{x}) = \frac{1}{||\mathbf{x} - \mathbf{x_i}||^P}$.

This model is selected by passing a Shepard object to interpolate.

Nearest Neighbor

Nearest neighbor interpolation produces piecewise constant interpolations by returning the data value of the nearest sample point.

To use nearest neighbor interpolation, pass a NearestNeighbor object to interpolate.