# Explainable MRI-Based Ensemble Learnable Architecture for Alzheimer's Disease Detection
## 3 Methodology
### Ensemble Classifier  
In this study, two widely adopted ensemble approaches, stacking and hard voting, are utilized to maximize the predictive power of the framework. Both methods capitalize on the diverse feature extraction capabilities of multiple deep learning architectures, including Multiclassifier-CNN, DenseNet, MobileNetV2, VGG19, EfficientNet B0, ResNet, and ViT, to deliver a comprehensive and balanced classification output. These ensemble strategies ensure that the final predictions are not overly reliant on any single model's performance but instead represent a collective decision-making process, enhancing both accuracy and robustness in Alzheimer's disease classification.  
_Stacking_ is an ensemble method that enhances performance by combining predictions from multiple base models through a higher-level meta-model, which can be mathematically expressed as follows [26]:  
\[y_{\text{meta}}=g(f_{1}(x),f_{2}(x),\dots,f_{n}(x)) \tag{1}\]  
where:  
* \(f_{1},f_{2},\dots,f_{n}\) are the predictions from the base models for an input \(x\);
* \(g\) denotes the meta-model that combines these predictions to generate the final output \(y_{\text{meta}}\).  
As shown in Figure 2, the stack framework leverages the predictions of multiple pre-trained models as input features for a meta-model, integrating outputs from diverse architectures, including Multiclassifier-CNN, DenseNet, MobileNetV2, VGG19, EfficientNet B0, ResNet, and ViT. Each base model generates predictions for every input sample, capturing complementary features that contribute to a more robust final prediction.  
The predictions from the base models are reshaped into meta-features and combined in the second stage of the stack framework. These meta-features are passed to a logistic regression meta-model, which maps the combined predictions to the final output classes. The meta-model uses the aggregated information from all base models to enhance classification accuracy and generalization.  
_Hard voting_, introduced because it is a widely used ensemble learning method, aggregates predictions from multiple classifiers, where each independently produces a prediction, and the ensemble's final output is determined by the majority vote, mathematically expressed as follows [27]:  
\[\hat{y}=\text{mode}(\{y_{1},y_{2},\dots,y_{M}\}) \tag{2}\]  
where:  
* \(\hat{y}\) is the final ensemble prediction;
* \(\{y_{1},y_{2},\dots,y_{M}\}\) are predictions made by \(M\) classifiers;
* mode represents the majority label among the predictions.  
In Figure 3, the process begins with the initialization of the ensemble and the collection of predictions from the seven models. The ensemble includes architectures such as Multiclassifier-CNN, DenseNet, MobileNetV2, VGG19, EfficientNet B0, ResNet, and ViT, each providing independent predictions for the input data. Each model contributes its prediction, and the final decision is based on the majority vote (_argmax_) across the  
Figure 2: Stacking ensemble architecture for multi-model classification.  
models. This voting mechanism ensures that the ensemble captures the consensus among the individual models, reducing the likelihood of misclassification caused by the biases or errors of any single model.  
This method leverages the diversity of the included architectures, combining their unique strengths to enhance generalization and accuracy. By relying on majority consensus, the hard voting ensemble is particularly effective in handling challenging multi-class classification problems where individual models may have varying levels of confidence. The output of the ensemble represents the most agreed-upon prediction, improving the robustness of the classification system.