## 3 Methodology
### Feature Ranking and Engineering Leveraging TOPSIS Methodology  
In order to rank the current entered and extracted features in terms of their effectiveness on the system performance, the TOPSIS methodology has been utilized. A similar procedure in [35; 44; 46] has been used to implement and run the TOPSIS methodology. First of all, the decision matrix \(D\) has been built (see Equation (1)), in which each given column corresponds to the assessment metrics, and each given row reflects the feature. The assessment metrics consist of the main measurement components, e.g., correlation,  
Figure 2: Dataset generation and collection using dynamic pattern recognition.  
variance, and entropy. Note that the aim of leveraging the TOPSIS methodology is to recognize only the most affected features in order to refine the accuracy of the system.  
\[D=\begin{bmatrix}x_{11}&x_{12}&\cdots&x_{1n}\\ x_{21}&x_{22}&\cdots&x_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ x_{m1}&x_{m2}&\cdots&x_{mm}\end{bmatrix} \tag{1}\]  
Next, to ensure the features' comparability, the \(D\) has been normalized leveraging the following formula, as seen in Equation (2):  
\[r_{ij}=\frac{x_{ij}}{\sqrt{\sum_{i=1}^{m}x_{ij}^{2}}}\ \ \ \ \ \forall i,j \tag{2}\]  
After \(D\) is normalized, weights \(\mathbf{w_{j}}\) have been applied to normalized values in order to drive the weighted normalized matrix, as shown in Equation (3):  
\[v_{ij}=w_{j}\cdot r_{ij} \tag{3}\]  
The worst and possible values for each metric are represented by negative-ideal (\(A-\)) and ideal (\(A+\)) formulas, respectively, as follows [46], in Equations (4) and (5):  
\[A^{+}=\left\{\max(v_{ij})\middle|j\in J,\ \min\ (v_{ij})\middle|j\in J^{\prime}\right\} \tag{4}\]  
\[A^{-}=\left\{\min(v_{ij})\middle|j\in J,\ \max\ (v_{ij})\middle|j\in J^{\prime}\right\} \tag{5}\]  
\(J^{\prime}\) signifies non-beneficial criteria, while \(J\) refers to beneficial criteria.  
In order to find the actual distance of each extracted feature from the negative-ideal and ideal solutions, the separation measures (\(S-\) and \(S+\)) have been performed, as follows, in Equation (6):  
\[S_{i}^{+}=\sqrt{\sum_{j=1}^{n}\left(v_{ij}-A_{j}^{+}\right)^{2}},\ S_{i}^{-}= \sqrt{\sum_{j=1}^{n}\left(v_{ij}-A_{j}^{-}\right)^{2}} \tag{6}\]  
All given features have been ranked based on their values in \(C_{i}\), in which higher values reflect high impact [47]. The relative closeness (\(C_{i}\)) of each feature to the ideal solution has been calculated, as follows, in Equation (7):  
\[C_{i}=\frac{S_{i}^{-}}{S_{i}^{+}+S_{i}^{-}} \tag{7}\]  
It is worth mentioning that the velocity, trajectory curvature, and touch pressure have been proven to represent the most valuable and effective features based on our experimental results obtained from the TOPSIS technique. Such valuable features have been demonstrated to refine the ML performance by increasing the model accuracy and mitigating the complexity of the authentication design compared to implementing the technique without incorporating the TOPSIS methodology. The choice of TOPSIS as the feature selection method was driven by its computational simplicity, ease of interpretability, and suitability for real-time mobile authentication scenarios. Unlike other MCDM methods, such as AHP and VIKOR, which may require extensive pairwise comparisons or introduce complex weighting schemes, TOPSIS provides a straightforward and scalable ranking process based on distance metrics. Additionally, while embedded feature selection methods within machine learning algorithms (e.g., L1 regularization) are widely used, they often lack cross-model transparency, making them less effective for a hybrid ensemble setup. The iterative approach adopted in this study refines the ranked feature set by incorporating feedback from the classification performance, thus enhancing both the selection quality and the final model accuracy.