## 4 Proposed Geometric Similarity Evaluation Methodology  
Simulation-based ADS validation is more time-efficient and cost-effective than real-vehicle testing. However, when calculating the credibility of simulation for multiple speed conditions, especially across numerous scenarios, substantial resources are still consumed. Considering the countless situations in which ADSs must operate, a systematic approach is required to improve validation efficiency. Herein, we propose an efficient credibility verification methodology based on geometric similarity evaluation, as illustrated in Figure 4. In Step 0, three evaluation metrics--parameter-based reliability (\(R_{m}\)), scenario-based reliability (\(R_{k}\)), and dynamics-based fidelity (\(F_{k}\))--are utilized to determine credibility. In Step 1, we evaluate credibility under different speed conditions (low, medium, and high) within the same scenario and visualize the derived evaluation metrics through a spider chart. The geometric similarity between the shapes is analyzed using the Procrustes, Frechet, and Hausdorff distance techniques. These techniques evaluate the efficiency of the verification process by quantitatively determining whether speed is a dominant factor in the credibility assessment for a given scenario. In Step 2, we additionally evaluate the geometric similarity between different scenarios. This involves verifying whether a particular scenario can represent many other scenarios during credibility assessment. The proposed assessment framework ensures the credibility of the XILS platforms used to test ADSs while significantly reducing the verification time and cost compared with traditional independent verification approaches. The complete methodology is detailed in Sections 4.1 and 4.2. The Procrustes distance quantifies the similarity between two sets of points by optimally aligning them to their common center via rotation, translation, and scaling, before minimizing the sum of the squared errors between the points [55]. This least-squares criterion aligns each configuration's centroid and allows structural differences to be analyzed via linear transformations, enabling global shape comparison without requiring explicit point-to-point correspondences. Specifically, optimal rotation and scaling solutions are obtained by performing an eigen-decomposition of the transformation matrix between the two configurations. \[ \begin{split} f_{6}&=\underset{\lambda,\mathcal{R}, \mathcal{R}}{min}\|y_{1}-\big{(}\lambda y_{2}\mathcal{R}+\mathbf{1}\mathbf{1}^{ \top}\big{)}\|_{F}^{2},\\ \text{where }&\big{\{}\mathbf{1}\in\mathbb{R}^{n\times 1 }\big{\}},\ \big{\{}\mathbf{t}^{\top}\in\mathbb{R}^{1\times m}\big{\}},\ \big{\{}\mathbf{1}\mathbf{t}^{\top}\in\mathbb{R}^{n\times m}\big{\}}.\end{split}  \tag{37}\]  
In Equation (37), the Procrustes distance \(f_{6}\) quantifies similarity based on the global alignment between two point sets \(y_{1},y_{2}\in\mathbb{R}^{n\times m}\). Here, \(n\) denotes the number of points in each shape, \(m\) denotes the dimensionality of the space in which each point resides, and each row of the matrix corresponds to a single \(m\)-dimensional point. Procrustes alignment involves applying the scaling factor \(\lambda\), the rotation matrix \(\mathcal{R}\), and the translation vector \(t\) to \(y_{2}\), in that order, which transforms it into \(\lambda y_{2}R+\mathbf{1}\mathbf{1}^{\top}\). We compute the sum of the squared Frobenius norms of the differences between corresponding points in the transformed point set and \(y_{1}\); subsequently, we define the minimal value of this sum as the Procrustes distance \(f_{6}\). Here, \(\mathbf{1}\in\mathbb{R}^{n\times 1}\) is a column vector of ones, and multiplying it by \(\mathbf{t}^{\top}\in\mathbb{R}^{1\times m}\) ensures that the same translation vector is applied uniformly to all points. A smaller optimized Frobenius norm indicates that the two point sets are globally well aligned, implying structural similarity.