## 3 Proposed Methodology
### Data Preprocessing
The adaptive wavelet function can then be represented as follows:

\[\psi a,b,\theta(t)=\frac{1}{\sqrt{a}}\psi\theta\binom{t-b}{a} \tag{3}\]

Here, \(\psi\theta\)(t) is the adaptive mother wavelet, which changes according to \(\theta\)(t), a function of time. The FAWT of the EEG signal is computed to obtain a time-frequency representation. Features are then extracted from this representation, such as the energy, power, or frequency-specific components within specific time windows. These features capture the dynamic changes in the EEG signal, which are crucial for identifying PD. Some extracted features are shown in Table 2.

#### 3.3.2 MRI Feature Extraction

For the extraction of MRI-related features, a Functional MRI (fMRI) based measure has been used. fMRI is a time series extraction for each voxel or Region of Interest (ROI). The extracted time series is represented by Ti(t) where Ti(t) is the BOLD signal over time t for the \(i\)th voxel or ROI. The functional connectivity has been computed by using the Pearson correlation coefficient between the time series of different ROIs or voxels such that the following holds:

\[C_{ij}=\frac{\sum_{\text{c}}(\tau_{i}(t)-\tau_{\text{U}})-(\tau_{j}(t)-\tau_{j}) }{\sqrt{\sum_{\text{c}}(\tau_{i}(t)-\tau_{\text{U}})^{2}-\sum_{\text{c}}(\tau_{ j}(t)-\tau_{j})^{2}}} \tag{4}\]

where \(C_{ij}\) is the correlation coefficient between ROI i and ROI j, Ti(t) and Tj(t) are the time series for ROIs i and j, respectively, and \(\overline{\tau_{i}}\), \(\overline{\tau_{i}}\) are their mean values over time. The functional connectivity graph has been formed by using graph metrics like degree, betweenness centrality, etc., to characterize the network, where nodes represent ROIs and edges represent the strength of connectivity (e.g., correlation). In the end, the General Linear Model (GLM) [39] models the BOLD signal in each voxel as a linear combination of explanatory variables (like task conditions) and confounds:

\[Ti(t)=\beta 0+\beta 1X1(t)+...+\beta nXn(t)+\epsilon i(t) \tag{5}\]

where \(X1(t),...,Xn(t)\) are the explanatory variables, \(\beta 0,...,\beta n\) are the coefficients, and \(\epsilon\)i(t) is the error term. All the obtained features are standardized to the fMRI feature vector so that it is on a similar scale to the EEG feature vector. This step is crucial for effective data fusion and model training. Some extracted features from MRI are shown in Figure 4:0 \begin{table}
\begin{tabular}{l c} \hline
**Features** & **Values** \\ \hline Delta Band Power & 10 \(\mu\)V\({}^{2}\) \\ \hline Alpha Band Power & 15 \(\mu\)V\({}^{2}\) \\ \hline Beta Band Power & 20 \(\mu\)V\({}^{2}\) \\ \hline Theta/Beta Ratio & 0.75 \\ \hline Spectral Entropy & 0.85 \\ \hline Phase Locking Value & 0.65 \\ \hline Cross-Frequency Coupling Strength & 0.55 \\ \hline \end{tabular}
\end{table} 
Table 2: Extracted features from EEG. \begin{table}
\begin{tabular}{l c} \hline
**Features** & **Values** \\ \hline Resting-State Functional Connectivity & 0.65 \\ \hline Regional Homogeneity (ReHo) & 1.25 \\ \hline Low-Frequency Fluctuations (ALFF) & 0.85 \\ \hline Regional Functional Connectivity Patterns & 0.7 \\ \hline Task-Based Activation Patterns & 1.2 \\ \hline \end{tabular}
\end{table} 
Table 3: Extracted features from MRI. ### Feature Fusion

The extracted features from both MRI and EEG datasets are concatenated to form a single, high-dimensional feature vector for each patient. In this work, the most prominent approach based on Principle Component Analysis (PCA) has been applied to future fusion. PCA is a statistical technique employed to diminish the dimensions of extensive datasets, maintaining the majority of the initial variance in the process. The fused feature matrix X, where each row represents a patient and each column a feature (either from MRI or EEG). If there are n patients and p features, then X is an n \(\times\) p matrix. PCA is sensitive to the variances of the initial variables. \[X_{std}=\frac{X-\mu}{\sigma} \tag{6}\]

where \(\mu\) is the mean and \(\sigma\) shows the standard deviation.