Radial Basis Functions

What is a Radial Basis Function?

A Radial basis function is a function whose value depends only on the distance from the origin. In effect, the function must contain only real values. Alternative forms of radial basis functions are defined as the distance from another point denoted C, called a center.

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How does a Radial Basis Function work?

A Radial basis function works by defining itself by the distance from its origin or center. This is done by incorporating the absolute value of the function. Absolute values are defined as the value without its associated sign (positive or negative). For example, the absolute value of -4, is 4. Accordingly, the radial basis function is a function in which its values are defined as:  

{\\textstyle \\varphi (\\mathbf {x} )=\\varphi (\\left\\|\\mathbf {x} \\right\\|)}

The Gaussian variation of the Radial Basis Function, often applied in Radial Basis Function Networks, is a popular alternative. The formula for a Gaussian with a one-dimensional input is:

The Gaussian function can be plotted out with various values for Beta:

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Applications of the Radial Basis Function

Radial basis functions make up the core of the Radial Basis Function Network, or RBFN. This particular type of neural network is useful in cases where data may need to be classified in a non-linear way. RBFNs work by incorporating the Radial basis function as a neuron and using it as a way of comparing input data to training data. An input vector is processed by multiple Radial basis function neurons, with varying weights, and the sum total of the neurons produce a value of similarity. If input vectors match the training data, they will have a high similarity value. Alternatively, if they do not match the training data, they will not be assigned a high similarity value. Comparing similarity values with different classifications of data allows for non-linear classification.