Basis Functions

The basis function determines how each node's influence changes with distance. EBF provides 12 basis functions, selected by passing a string name to the basis parameter:

model = ebf.EBF(n_nodes=10, basis='gaussian')

Gallery

The chart below shows all basis functions evaluated with unit weight (\(a_1 = 1\)) over a range of radii.

Reference Table

Growing Functions

These functions increase without bound as distance grows. They produce smooth surfaces that can extrapolate beyond the data range (for better or worse).

Name	Expression	When to Use
linear	\(a_1 \cdot r\)	Simple problems; piecewise-linear feel
quadratic	\(a_1 \cdot r^2\)	Smooth parabolic influence
cubic	\(a_1 \cdot r^3\)	Very smooth; similar to natural splines
multiquadric	\(a_1 (\sqrt{r^2+1} - 1)\)	Default. Good general choice; grows sub-linearly
cosh	\(a_1 \cdot \cosh(\sqrt{r^2+\varepsilon})\)	Exponential growth at large \(r\); use with care

Decaying Functions

These functions decay toward zero at large distances. Nodes have localized influence — far-away nodes don't affect the prediction. Good when you want the model to avoid wild extrapolation outside the data hull.

Name	Expression	When to Use
gaussian	\(a_1 \cdot e^{-r^2}\)	Classic localized influence; decays very quickly
inv_multiquadric	\(a_1 / \sqrt{r^2+1}\)	Complement to multiquadric; gentle decay
inv_quadratic	\(a_1 / (1+r^2)\)	Cauchy-like; heavier tails than Gaussian
inv_cosh	\(a_1 / \cosh(\sqrt{r^2+\varepsilon})\)	Sech-like decay; complement to cosh
matern32	\(a_1 (1+\sqrt{3}r) e^{-\sqrt{3}r}\)	C1-smooth; popular for physical/engineering data
matern52	\(a_1 (1+\sqrt{5}r+\frac{5}{3}r^2) e^{-\sqrt{5}r}\)	C2-smooth; smoother than Matern 3/2

Special Functions

These have logarithmic behavior near zero.

Name	Expression	When to Use
thin_plate	\(a_1 \cdot r^2 \ln(r^2)\)	Classic thin-plate spline; C1 smooth

Notes

• \(r\) is the non-Euclidean distance from the input point to a node (not the squared distance). In the code, the basis functions receive \(r^2\) directly and compute \(r\) internally where needed.

• \(\varepsilon\) is a small numerical stability offset (default 1e-8), set via the eps parameter on the EBF constructor. Only cosh and inv_cosh use it. thin_plate uses tf.math.xlogy which handles the \(r = 0\) case natively. All other functions have no singularity at \(r = 0\).