scipy.interpolate._rbfinterp:RBFInterpolator

Notes

An RBF is a scalar valued function in N-dimensional space whose value at $x$ can be expressed in terms of $r=||x-c||$, where $c$ is the center of the RBF.

An RBF interpolant for the vector of data values $d$, which are from locations $y$, is a linear combination of RBFs centered at $y$ plus a polynomial with a specified degree. The RBF interpolant is written as

where $K(x,y)$ is a matrix of RBFs with centers at $y$ evaluated at the points $x$, and $P(x)$ is a matrix of monomials, which span polynomials with the specified degree, evaluated at $x$. The coefficients $a$ and $b$ are the solution to the linear equations

and

where $\lambda$ is a non-negative smoothing parameter that controls how well we want to fit the data. The data are fit exactly when the smoothing parameter is 0.

The above system is uniquely solvable if the following requirements are met:

• $P(y)$ must have full column rank. $P(y)$ always has full column rank when degree is -1 or 0. When degree is 1, $P(y)$ has full column rank if the data point locations are not all collinear (N=2), coplanar (N=3), etc.

• If kernel is 'multiquadric', 'linear', 'thin_plate_spline', 'cubic', or 'quintic', then degree must not be lower than the minimum value listed above.

• If smoothing is 0, then each data point location must be distinct.

When using an RBF that is not scale invariant ('multiquadric', 'inverse_multiquadric', 'inverse_quadratic', or 'gaussian'), an appropriate shape parameter must be chosen (e.g., through cross validation). Smaller values for the shape parameter correspond to wider RBFs. The problem can become ill-conditioned or singular when the shape parameter is too small.

The memory required to solve for the RBF interpolation coefficients increases quadratically with the number of data points, which can become impractical when interpolating more than about a thousand data points. To overcome memory limitations for large interpolation problems, the neighbors argument can be specified to compute an RBF interpolant for each evaluation point using only the nearest data points.

Array API Standard Support

RBFInterpolator has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

====================  ====================  ====================
Library               CPU                   GPU
====================  ====================  ====================
NumPy                 ✅                     n/a                 
CuPy                  n/a                   ✅                   
PyTorch               ✅                     ✅                   
JAX                   ✅                     ✅                   
Dask                  ⛔                     n/a                 
====================  ====================  ====================

Only the default neighbors=None is Array API compatible. If a non-default value of neighbors is given, the behavior is NumPy -only.

See dev-arrayapi for more information.