## 3 Methodology
The outputs from the neurons in the output layer are denoted as  
\[\hat{y_{1}}^{(out)}\text{ for the prediction from }a_{1}^{(out)}, \tag{4}\]  
\[\hat{y_{2}}^{(out)}\text{ for the prediction from }a_{2}^{(out)}, \tag{5}\]  
\[\hat{y_{3}}^{(out)}\text{ for the prediction from }a_{3}^{(out)}, \tag{6}\]  
The arrows represent the flow of information through the network, from the inputs to the hidden layer and finally to the output layer, where predictions are made based on the learned weights and biases. Training an MLP involves an iterative optimization of its weights and biases to minimize a predefined loss function, typically achieved through the back-propagation and gradient descent methods. During training, the network learns to adjust its parameters by comparing its predictions with the ground truth labels, thereby iteratively refining its ability to generalize to unseen data. Despite their effectiveness, MLPs are not without limitations. They may suffer from over-fitting, especially when dealing with high-dimensional data sets or insufficient training samples. Additionally, interpreting the learned representations within the hidden layers can be challenging due to the network's black-box nature. Nonetheless, MLPs remain a cornerstone in the domain of deep learning, owing to their capacity to learn complex relationships within data and their applicability across diverse domains. In the context of predicting customer churn, MLPs offer a powerful tool for uncovering patterns and trends within customer data, enabling businesses to proactively identify and retain at-risk customers. Through the meticulous tuning of architecture and training parameters, MLPs stand poised to deliver accurate and actionable insights, driving informed decision-making in customer relationship management. The RBF network is also a type of neural network characterized by its distinctive architecture and the use of radial basis functions for processing information. Unlike traditional feed-forward networks like the MLP, RBF networks exhibit a unique structure that enables them to excel in certain types of tasks, particularly in function approximation and pattern recognition. At the heart of the RBF network lies a series of interconnected layers, including an input layer, a hidden layer, and an output layer. However, unlike MLPs, RBF networks possess a single hidden layer containing a set of neurons, each associated with a radial basis function. These radial basis functions, often Gaussian functions, serve as activation functions for the hidden layer neurons, transforming the input data into a higher-dimensional feature space. The input layer of the RBF network comprises neurons equal to the number of features in the training data set. These inputs are then fed into the hidden layer, where the radial basis functions compute the similarity between the input data and a set of prototype vectors, also known as centroids. The activation of each hidden neuron corresponds to the proximity of the input data to these centroids, effectively mapping the input space into a nonlinear feature space. Once the input data have been transformed by the radial basis functions, the resulting representations are passed to the output layer for further processing. In classification tasks, the output layer typically employs a linear combination of its inputs to produce class predictions, while in regression tasks, it outputs continuous values based on the learned representations. Training an RBF network involves two main steps: the determination of the centroids and the adjustment of the weights connecting the hidden and output layers. The centroids are often initialized using clustering algorithms such as k-means, while the weights are optimized using techniques like least squares or gradient descent. A key advantage of RBF networks lies in their ability to capture complex nonlinear relationships within data while maintaining a relatively simple and interpretable architecture.