## 3 Methodology
### Ensemble Learning  
Ensemble learning is a technique that combines the predictions of multiple individual models to make more accurate and robust predictions. It is widely used in machine  
Figure 1: Illustration of the proposed methodology.  
learning and has shown promising results in various domains, including Alzheimer's disease classification. In this paper, an ensemble of transfer learning models is proposed using two different methods: simple averaging and weighted averaging. By combining the predictions of these models, we aim to improve the extraction of sparse patterns and features from MRI images for Alzheimer's disease diagnosis. These learning models are used for identifying Alzheimer's disease. The proposed methodology's block diagram is shown in Figure 2. Our transfer learning models are input with 2D image slices.  
Combining the judgements from several different models yields superior outcomes to those produced by a single model [47]. The generalization capacity of an ensemble, comprising multiple learners, is often considerably stronger than that of an individual learner, especially when it comes to weak learners. While an ensemble of weak learners can theoretically achieve good performance, in practice, people tend to prefer strong learners for various reasons. These reasons include minimizing the number of individual learners and leveraging the ability to reuse previously learned information about the strong learners [47].  
In this paper, we utilise heterogeneous ensembles which have individual learners called component learners or simply individual learners. Unlike homogeneous ensembles, where all individual learners are of the same type, such as employing the same classification models, heterogeneous ensembles consist of diverse individual learners and learning algorithms, without relying on a single base learner or learning algorithm.  
In order to compare and circumvent these problems, different ensemble models have been developed along with simple averaging. Weighted averaging provides various weights to various base learners. The goals were to determine which ensemble method had a strong correlation with the accuracy of the ensemble, which type of averaging had a high capacity to produce classification models that were not only accurate but also more diverse, and which types of ensembles were most likely to be able to enhance classification performance.  
Simple Averaging: The most straightforward method to combine the output is by averaging them together. Let our ensemble contain \(T\) individual learners \(\{h_{1},h_{2},h_{3},\ldots\ldots,h_{T}\}\), where \(h_{i}(x)\) is the result of \(h_{i}\) on \(x\) number of samples. The equation can be described as  
Figure 2: Graphical abstract of the proposed study.  
\[H(x)=\frac{1}{T}{\sum_{i=1}^{T}{h_{i}(x)}}, \tag{3}\]  
This approach has the benefit being easily implemented [28], but it does not take into account that some models may perform better than others.  
Weighted Averaging: The transfer learning model which has lower classification error is given a smaller weight when combined for ensembling, as described by [29]. The values of weights are assigned manually with the help of multiplying with a factor \(\alpha\) to each model.  
\[H(x)={\sum_{i=1}^{T}{\alpha_{i}h_{i}(x)}}, \tag{4}\]  
where \(\alpha\) is the weight of an individual learner \(h_{i}\), and \(\alpha_{i}\geq 0\), \(\sum_{i=1}^{T}{\alpha_{i}}=1\). It is shown in [30] that optimal choice for alpha is  
\[\alpha_{i=}\frac{{\sum_{j}{C_{ij}}^{-1}}}{{\sum_{k}{\sum_{j}{C_{kj}}^{-1}}}}, \tag{5}\]