# Radial Basis Function

A radial basis function(RBF) is a real-valued function whose value depends only on the input and its distance from some fixed point (c) or the origin. The function satisfies the criteria below:

The distance is usually the Euclidean distance between two points. The Euclidean distance is calculated as follows:

The sum of radial basis functions is typically used to approximate the given function.

**But what does this mean?**

||x – x_{n}|| is the radial part of the function, since its value depends on some distance from a fixed center. The basis function is the mathematical representation of the gaussian function. Now, you have points (x_{n},y_{n}) **∈** D, where D is a dataset. All the points in D affect a hypothesis h(x). This means that all the points in the dataset affect and influence a machine learning model. But in our case, it affects h(x) specifically based on ||x – x_{n}|| , i.e the distance

**What do we do with the equation?**

We find the w_{n}s or weights that minimize the error between h(x) and y_{n}, so we basically try to equate them. This is called interpolation.

This is a representation of the gaussian distribution. You can observe above, an RBF is symmetric around the center if we take the topmost point as the center.

## Applications of Radial Basis Function

Radial basis functions serve as a glue for many topics of machine learning. RBFs are often used for classification.

### Relationship to nearest neighbor method

In the nearest neighbor(NN) method, you classify points according to their features. But is not going to be a smooth process. You thus consider only K nearest neighbors(KNN) to smoothen the classification boundaries. When we use RBF with K centers, we can assign more weight to the nearest points and less weight to the points that are far away. This makes the classification smoother than NN and KNN.

#### RBF with K centers

##### How to choose the centers?

We choose K number of clusters, where **K << N**. We get K number of centers for K clusters.

Our main objective is to minimize the distance between x_{n} and the closed center u_{k}. We choose a cluster around the center u_{k} and split the data points D into K clusters. We choose points that are nearest to the center. We minimize ||x_{n}-u_{k}||^{2} of all the points in each cluster where x_{n} are points nearest to the center u_{k}.

Mathematically, we express the equation as follows:

w is the weight we aim to calculate.

##### How to choose the weights?

We minimize the equation for h(x) and approximately equate it to y_{n}.

We can treat gamma as a learnable parameter and use the EM approach(see wiki) to solve it. i.e

- We fix the value of gamma and solve for w
_{n} - We fix w
_{n}, and try to minimize the error by adjusting gamma. We can calculate different gamma for different centers

### Relationship to Neural Networks

Radial Basis Function network is an artificial neural network with an input layer, a hidden layer, and an output layer. It is similar to 2-layer networks, but we replace the activation function with a **Gaussian radial basis function**. RBF centers can be selected randomly from a set of data by using supervised learning, or they can be defined by using k-means clusters using unsupervised learning[1]. We take each input vector and feed it into each basis. Then, we do a simple weighted sum to get our approximated function value at the end. This is the approximation is compared to the training data and then assigned a value of similarity. We train these using backpropagation like any neural network.

Some points about RBF Networks:

- Radial basis function networks are distinguished from other neural networks due to their universal approximation and faster learning speed
- They are used in image classification
- It is used for the non-linear classification of data

You can see the code in python in this Python Machine Learning post.

## RBF in Kernels

Radial Basis Function kernels are popular kernels in machine learning for non-linear data. It is the most commonly used kernel in Support Vector Machines(see our post).

## References

[1] Xanthoula Eirini Pantazi, Dimitrios Moshou, Dionysis Bochtis, Chapter 2 – Artificial intelligence in agriculture, Intelligent Data Mining and Fusion Systems in Agriculture, Academic Press, 2020, Pages 17-101, ISBN 9780128143919, https://doi.org/10.1016/B978-0-12-814391-9.00002-9.

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