RBFTransformer

A custom transformer to apply Radial Basis Functions (RBFs) to time series data.

This transformer allows the user to apply various RBF kernels such as Gaussian, multiquadric, and inverse multiquadric to time series data. The transformation generates new features that augment the input data based on the distances between time points and specified center points, offering a flexible and non-linear feature representation for machine learning models.

This implementation is inspired by the RepeatingBasisFunction transformer from the scikit-lego package: https://github.com/koaning/scikit-lego/blob/main/sklego/preprocessing/repeatingbasis.py

Mathematical Background: Consider a time series with time stamps \(t_1, t_2, \dots, t_N\)-

The transformation computes the kernel distance between the time points and a set of predefined “center points” \(c_1, \dots, c_K\). For each time point \(t_i\), the RBF is computed between \(t_i\) and every center point \(c_k\), producing a matrix of transformed values. Each kernel function depends on the distance between a time point and a center point.

Mathematically, the transformation for the Gaussian RBF is defined as:

\(\phi(t_i, c_k) = \exp(-\gamma (t_i - c_k)^2)\)

where \(\gamma\) is a scaling factor controlling the spread of the RBF.

Additional types of RBFs are available:

• Multiquadric:

  \(\phi(t_i, c_k) = \sqrt{1 + \gamma (t_i - c_k)^2}\)

• Inverse Multiquadric:

  \(\phi(t_i, c_k) = \frac{1}{\sqrt{1 + \gamma (t_i - c_k)^2}}\)

These transformations produce new features for each time point, enhancing the representational power of the data by adding non-linear transformations that account for proximity to center points.