How Does PNNs Work?
PNNs operate by estimating the probability density function (PDF) of each class using the Parzen window technique. The process can be broken down into the following steps:
- Training Phase: During training, the network stores the training samples and their corresponding class labels.
- Pattern Matching: When a new input vector is presented, the network computes the similarity between the input vector and each training sample using a kernel function, typically a Gaussian function.
- Probability Estimation: The network then estimates the PDF for each class by summing the kernel outputs for all training samples belonging to that class.
- Classification: Finally, the network assigns the input vector to the class with the highest estimated probability.
The probability density function for a class [Tex]C_k[/Tex] is given by:
[Tex]P(C_k | x) = \frac{N_k}{\sum_{i=1}^{N_k} K(x, x_i)}[/Tex]
Where:
- (x) is the input vector.
- (x_i) are the training samples belonging to class (C_k).
- (N_k) is the number of training samples in class (C_k).
- (K) is the kernel function, often a Gaussian function.
The Gaussian kernel function is defined as:
[Tex]K(x, x_i) = \exp \left( -\frac{\|x – x_i\|^2}{2\sigma^2} \right) [/Tex]
- This equation calculates the probability that an input vector (x) belongs to class (C_k).
- It does this by considering the similarity between the input vector and all the training samples in class (C_k) using the kernel function.
The Gaussian kernel function measures the similarity between two vectors based on their Euclidean distance. The parameter (\sigma) controls the width of the Gaussian, determining how far away two points can be while still being considered similar.
Probabilistic Neural Networks: A Statistical Approach to Robust and Interpretable Classification
Probabilistic Neural Networks (PNNs) are a class of artificial neural networks that leverage statistical principles to perform classification tasks. Introduced by Donald Specht in 1990, PNNs have gained popularity due to their robustness, simplicity, and ability to handle noisy data. This article delves into the intricacies of PNNs, providing a detailed explanation, practical examples, and insights into their applications.
Table of Content
- What is Probabilistic Neural Network (PNN)?
- Bayes’ Rule in Probabilistic Neural Network
- How Does PNNs Work?
- Implementation of Probabilistic Neural Network
- Advantages and Disadvantages of PNNs
- Use-Cases and Applications of PNN