The Koopman operator and lifting functions using radial basis functions - cconceptual question

This is a conceptual question on how to apply a radial basis function as a lifting function to approximate the Koopman operator.

Given a dataset of two variables, Xc and Xv, with 10 data points ("D" is a dataset, which is a 2 X 10 matrix):

D = [Xc(1) ......Xv(10);

Xv(1) .... Xv(10)]

==> Example: Let's say I select my Koopman lifting functions for Z as the identity functions, and their cross product:

Z(1) = [Xc(1); Xv(1); Xc(1)*Xv(1)]....

Z(10) = [Xc(10); Xv(10); Xc(10)*Xv(10)]

==> Example 2: If I make my lifting functions as all monomials up to order 2:

Z(1) = [Xc(1); Xv(1); Xc(1)*Xv(1); Xc(1)*Xc(1); Xv(1)*Xv(1)]....

Z(10) = [Xc(10); Xv(10); Xc(10)*Xv(10); Xc(10)*Xc(10); Xv(10)*Xv(10)]

Great! No problems so far.

==> Now, my real question is: how do I use a radial basis function to my dataset "D" as a lifting function? I know that one such RBF is the inverse quadratuc:

Y = 1 / (1 + (er)^2 )

How do I use this as a lifting function for my dataset D?

Answer 1

[Vatsal]

Hi,

I understand that you want to apply a radial basis function as a lifting function to approximate the Koopman operator.

To use the inverse quadratic radial basis function (RBF) as a lifting function for the dataset "D," you would apply the RBF to each data point in the dataset. The inverse quadratic RBF is given by the formula:

To apply this RBF as a lifting function to the dataset "D," you would perform the following steps:

• Calculate the Radial Distance: For each data point in "D," measure the distance from a chosen centre point. This distance can be calculated using a method like finding the straight-line distance (Euclidean distance) between each data point and the chosen centre.
• Apply the RBF: Use the calculated distances to apply the inverse quadratic RBF to each data point in "D." This involves using the formula for the inverse quadratic RBF to transform each distance into a new value.
• Create the Transformed Dataset: After applying the RBF to each data point, assemble a new dataset with the transformed values. This new dataset will represent the original data in a higher-dimensional space, capturing more complex relationships and interactions.

I hope this helps!