Radial Basis Function Networks are a type of local learning method that combines ideas from a number of approaches in machine learning.

This is another approach of learning to approximate a function. It's a global approximation (linear combination of localised approximations) and is similar to distance-weighted regression, except it's "eager" not "lazy".

# Diagram

# Learned Hypothesis

The hypothesis has the form of:

(1)Where $x_u$ is an instance of *X* and the Kernel function K() decreases as the distance d() increases. *k* (hidden units) is a user-provided constant that specifies the number of kernel functions to be included

## Kernel Function

It's common to set the kernel function as

(2)We can use this to approximate any function with arbitrarily small error if the *k* is sufficiently large and the kernel widths $\sigma ^2$ can be individually specified.

# Training Radial Basis Function Networks

1.

- Set
*k*(num hidden units) - Set $x_u$ and $\sigma_u^2$ for each hidden unit
*u*

2.

- Train upper-level function to set weights $w_u$
- First choose variance (and perhaps mean) for each Ku
- Then hold Ku fixed, and train linear output layer – efficient methods to fit linear function

- Fit the data to minimise squared error (like in linear models)

3. Figure out which subsets to use for each kernel function

- Scatter them uniformly throughout the instance space