Radial Basis Function: Working, Types, and Advantages

In this blog, we will understand the concept of the Radial Basis Function, along with its examples, types, workings, architecture, and related advantages in the domain of neural networks.

Given below are the following topics we are going to understand:

• What is Radial Basis Function?
• Why Do We Need Radial Basis Functions?
• How Do Radial Basis Functions Work?
• Types of Radial Basis Function
• RBF Network Architecture
• Advantages of Radial Basis Function
• Conclusion
• FAQ’s

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What is Radial Basis Function?

Radial Basis Function is defined as the mathematical function that takes real-valued input and provides the real-valued output determined by the distance between the input value and a fixed point projected in space. This fixed point is positioned at an imaginary location within the spatial context.

This type of transformation function is widely used in various machine learning and deep learning algorithms, such as SVM, Artificial Neural Networks, etc. In the context of machine learning and computational mathematics, RBFs are utilized as activation functions in neural networks or as a basic function for interpolation and approximation. 

The most commonly used Radial Basis Function in machine learning is the Gaussian Radial Basis Function. We will learn more about it in the upcoming section.

Example of Radial Basis Function 

Let us now understand the Radial Basis Function with the help of an example. Now, suppose you want to improve the accuracy of option pricing by considering more sophisticated models that can capture non-linear relationships and varying volatility. Radial Basis Functions offer a suitable solution in this context.

For simplicity, let’s focus on predicting the option price based on the current stock price (S), time to expiration (T), and implied volatility (σ). These factors interact in a complex, non-linear manner. Traditional linear models may struggle to capture the complicated dependencies between these variables.

In an RBF-based approach, each combination of stock price, time to expiration, and implied volatility is associated with a radial basis function. The radial basis function is high when the input values are close to the center (representing a specific combination of S, T, and σ) and decreases as the distance increases.

The RBF-based option pricing model demonstrates superior adaptability to the intricacies of financial markets compared to traditional linear models. It effectively handles non-linear relationships and varying volatility, offering a more nuanced and accurate representation of market dynamics. The centers of the radial basis functions would represent key market conditions, and the model learns the appropriate weights during the training process to accurately predict option prices based on these conditions.

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Why Do We Need Radial Basis Functions?

One primary reason for the need of RBFs is their ability to efficiently capture complex, non-linear relationships within data. Unlike simpler linear models, RBFs can model complex patterns and dependencies by transforming input data into a high-dimensional space, where these relationships become more apparent. 

Radial Basis Functions (RBFs) play a crucial role in various fields, including machine learning, signal processing, and numerical analysis, due to their unique mathematical properties and versatile applications. This characteristic makes RBFs particularly valuable in tasks such as pattern recognition, interpolation, and approximation, where complex and non-linear relationships are prevalent. 

How Do Radial Basis Functions Work?

Radial Basis Functions in neural networks are almost similar to K-Nearest Neighbor models in terms of conceptuality, though the implementation of both models is quite different. The Radial Basis Function (RBF) operates by assigning weights to input vectors based on their distances from predefined centers or prototypes in the input space.

The RBF calculates the Euclidean distance between the input vector and the center, squares it, divides it by 2σ², and applies an exponential function. The result is a weighted output that reflects the proximity of the input to the center. 

In the context of Radial Basis Function Networks (RBFNs), multiple RBFs with different centers are used, and their outputs are combined to produce the network’s final output. During training, the RBFN adjusts the parameters (centers and widths) to fit the training data, often utilizing techniques such as k-means clustering or optimization algorithms. 

Furthermore, RBFs provide a flexible means of capturing similarities between input patterns, making them useful for tasks like pattern recognition, regression, and interpolation.

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Types of Radial Basis Function

There are several types of Radial Basis Functions (RBFs), each with its own characteristics and mathematical formulations. Some common types include:

• Gaussian Radial Basis Function: It has a bell-shaped curve and is often employed in various applications due to its simplicity and effectiveness. It is represented as:

• Multiquadric Radial Basis Function: It provides a smooth interpolation and is commonly used in applications like meshless methods and radial basis function interpolation. It is defined as:

• Inverse Multiquadric Radial Basis Function:  This type of function is similar to the Multiquadric RBF but has the inverse in the denominator, resulting in a different shape. Here is the formula for this function:

• Thin Plate Spline Radial Basis Function: The Thin Plate Spline RBF is defined as ϕ(r)= r²log(r) is the Euclidean distance between the input and the center. This RBF is often used in applications involving thin-plate splines, which are used for surface interpolation and deformation.

• Cubic Radial Basis Function: The Cubic RBF is defined as ϕ(r)= r³  where r is the Euclidean distance. It has cubic polynomial behavior and is sometimes used in interpolation.

RBF Network Architecture

The architecture of an RBFN generally consists of three layers: an input layer, a hidden layer with radial basis functions, and an output layer. Here’s a breakdown of the architecture:

• Input Layer: The input layer comprises nodes that depict the features or dimensions of the input data, with each node corresponding to an element in the input vector.
• Hidden Layer: The hidden layer consists of nodes connected to radial basis functions, where each node functions as a prototype or center for an RBF.
• Output Layer: The output layer generates the final network output, usually calculated as a linear combination of the activations originating from the hidden layer.

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Advantages of Radial Basis Function

Let us now explore the various advantages of the Radial Basis Function, which are provided in the following points:

• RBF showcases a strong resistance to input noise.
• By using the Radial Basis Function, it is quite easy to solve the problems that exist in datasets that have complex non-linear distributions such as logarithmic functions, trigonometric functions, power functions, and Gaussian function (normal distribution).
• After utilizing the Radial Basis Function, hidden patterns in the distribution can be generalized in a better way.
• Handling one hidden layer is quite easy in RBF.
• If you use RBF, you can interpret the exact meaning of each node present in the hidden layer of the Radial Basis Function network.

Conclusion

As advancements continue, RBF networks may find increased relevance in solving real-world problems across diverse domains, making them a promising area for ongoing research and development.

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