# Prediction of Diabetes Using Statistical and Machine Learning Modelling Techniques
## 3 Methodologies  
### Models Overview  
This section provides a brief overview of the models used for diabetes prediction, including Multiple Linear Regression (MLR), Bayesian Linear Regression (BLR), Support Vector Regression (SVR), Artificial Neural Network (ANN), and the Adaptive Neuro-Fuzzy Inference System (ANFIS). It highlights the mathematical foundations of these models and their unique operational characteristics. These models were selected based on their ability to handle regression tasks with varying levels of complexity. MLR and BLR serve as interpretable benchmarks, offering a foundation for comparison. SVR was chosen for its ability to mitigate overfitting through margin optimisation, while ANN effectively captures complex nonlinear relationships. ANFIS was selected for its hybrid nature, combining rule-based inference with neural adaptability. This diverse selection ensures a balanced evaluation of accuracy, interpretability, and computational efficiency, facilitating a comprehensive comparison between traditional regression techniques and advanced machine learning approaches to identify the most effective model. 1. Multiple Linear Regression  
Multiple Linear Regression is one of the most common types of linear regression analysis. It is an extended form of simple linear regression, with a relationship between more than two variables [21]. In predictive analysis, this technique describes the relationship between one dependent (response) variable and two or more independent (predictor) variables. The general model of Multiple Linear Regression is  
\[Y=\beta_{0}+\beta_{1}X_{1}+\beta_{2}X_{2}+\ldots+\beta_{n}X_{n} \tag{1}\]  
where \(Y\) is the dependent variable; \(\beta_{0},\beta_{1},\beta_{2},\ldots,\beta_{n}\) are the coefficients; and \(X_{1},X_{2},\ldots,X_{n}\) are the independent variables. 2. Bayesian Linear Regression  
Bayesian Linear Regression is based on a generative method, which is different from a discriminant one which depends on Bayesian inference to build linear regression models [22]. Once the model is specified, the posterior distribution of parameters and forecasts of the model are computed by the method. This statistical analysis enables the method to define the complexity of the model through training, which produces a model with few possibilities to overfit. In contrast to the simple linear regression model, the responses in Bayesian Linear Regression are assumed as samples from the probability distribution, for example the normal (Gaussian) distribution, which is  
\[Y\sim N\left(\beta^{T}X,\sigma^{2}\right) \tag{2}\]  
The product of the parameters \(\beta\) and the inputs \(X\) is the mean of the Gaussian, where the normal deviation is \(\sigma\). As well as the responses, in Bayesian models the parameters are also supposed to be sampled from a distribution. The aim is to define the posterior probability distribution for the parameters of the model with given \(X\) inputs and \(Y\) outputs, as in Equation (3):  
\[P(\beta|Y,X)=\frac{P(Y\mid\beta,X)P(\beta|X)}{P(Y|X)} \tag{3}\]  
The result obtained from modelling by Bayesian Linear Regression is not a single estimate, but rather a distribution range which can be used to produce inferences regarding new observations. This distribution enables the determination of uncertainty in the model, which is considered one of the advantages of Bayesian modelling methods. When the volume of data increases, the uncertainty of the result declines, presenting a better level of certainty in the approximation [23]. 3. Support Vector Regression  
Support Vector Machine (SVM) is a popular method developed by Vapnik. The generalised concepts of SVM have been applied to regression problems such as modelling and prediction and accordingly called Support Vector Regression (SVR). SVR has been effectively utilised to deal with forecasting issues in many areas as diverse as pharmacology, economics, and power systems analysis.