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