## 3 Proposed Methodology
### Data Preprocessing
PCA starts with the computation of the covariance matrix \(\Sigma\) of the standardized data Xstd. The covariance matrix expresses the covariance between each pair of features in the data such that the following holds:

\[\sum=\frac{1}{n-1}X_{std}^{T}X_{std} \tag{7}\]

PCA entails determining the eigenvectors and eigenvalues of the covariance matrix. It reveals the extent of variance in the data along the newly established feature axes. The eigenvectors are sorted by their eigenvalues in descending order. The first few eigenvectors (principal components) are selected based on the desired number of features or a set threshold for the amount of variance to be retained. Finally, the original standardized data Xstd is projected onto these principal components to obtain the transformed feature set. \[X_{pca}=X_{std}\times W \tag{8}\]

where W is the matrix containing the selected eigenvectors. Some eigenvalues and eigenvectors are shown in Table 4.

The final feature vector is a combination of both fMRI and EEG features, ready to be input into a machine learning classifier for PD diagnosis. This vector encapsulates information about both the structural/functional aspects of the brain (from fMRI) and the electrical activity (from EEG). This integrated approach aims to leverage the strengths of both imaging techniques, offering a more comprehensive and nuanced view of the brain's functioning, which is particularly beneficial in PD diagnosis processing. ### Classification Using LightGBM

In this research, LightGBM [40], a highly efficient gradient-boosting framework, has been adopted for the classification of PD. The PD diagnosis features that are extracted from MRI and EEG data, the LightGBM model, can be finely tuned to differentiate between PD (positive) and non-PD (negative) cases. This binary classification task requires setting the objective parameter to binary. \begin{table}
\begin{tabular}{l l} \hline \hline
**Eigenvalues** & **Eigenvector** \\ \hline
0.92 & [0.5,\(-\)0.3,0.1,0.7] \\ \hline
0.78 & [\(-\)0.1,0.6,\(-\)0.7,0.2] \\ \hline
0.63 & [0.7,0.2,0.4,\(-\)0.5] \\ \hline
0.55 & [\(-\)0.2,\(-\)0.5,0.6,0.7] \\ \hline
0.41 & [0.3,0.7,0.5,0.2] \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Eigenvalues and Eigenvector. Algorithm 3 gives a detailed description of the LightGBM-based PD classification. The dataset (x, y) is split into training and testing sets. The LightGBM model parameters are defined, including the objective, boosting type, and other parameters like the number of leaves, max depth, and fractions for feature and data sampling. The model is trained using the specified parameters and training data. The final Predictions are made on the test set and then converted to binary outputs. ```
Input: - X [features_matrix] // Matrix of features extracted from MRI and EEG - y [labels_vector] // Vector of labels (PD or non-PD) Output: - Model based Diagnosis 1. Data Preparation X_train, X_test, y_train, y_test - split_data(X, y, test_size)
2. Create LightGBM Dataset train_data - create_dataset(X_train, y_train)
3. Set LightGBM Parameters params - { 'objective': 'binary', 'boosting_type': ['gbdt', 'goss'], 'num_leaves': set_number_of_leaves(),'max_depth': set_max_depth(),'min_data_in_leaf': set_min_data_in_leaf(), 'feature_fraction': set_feature_fraction(),'bagging_fraction': set_bagging_fraction() }
4. Model Training model - lgb.train(params, train_data)
5. Model Predictions y_pred - model.predict(X_test)
6. Transform Probabilities to Binary Output (for binary classification) y_pred_binary - convert_to_binary(y_pred)
```

**Algorithm 3** LightGBM for PD Classification

## 4 Experimental Results and Evaluation

This section provides an in-depth analysis of the performance evaluation of the proposed model, with each aspect elaborated in detail in the following subsections. ### Dataset Description

Two distinct types of datasets have been used for the analysis of the proposed work. Table 5 shows the distinctive detail of each dataset.