## 2 C-POD Method and Modal Sorting Method
### Entropy-Controlled Euclidean-to-Probability Mapping Modal Sorting Method  
One notable advantage of the POD method is its capacity to quantify the contribution of each mode to the reconstruction of the original data. This contribution is indicated by the singular values, which represent the projection strength of the original data matrix onto the orthogonal modes. Larger singular values imply a more significant contribution from the corresponding mode in reconstructing the data, thereby identifying it as a dominant mode that captures the primary operational characteristics of the system. In contrast, the C-POD method does not involve singular value decomposition during its construction, rendering it infeasible to assess mode contributions based on singular value magnitudes. However, in the initialization phase, we solve the eigenvalue problem of the snapshot correlation matrix--whose eigenvalues are mathematically related to the singular values of the original data matrix. These eigenvalues could theoretically be used to rank modes by importance. Yet, because C-POD replaces orthogonal POD modes with clustering centroids--which are neither orthogonal nor ordered--the original eigenvalue-based ranking becomes inconsistent and inapplicable. To address this, we propose a novel probabilistic mode sorting method: the entropy-controlled Euclidean-to-probability mapping (ECEPM), which offers a robust and interpretable way to evaluate mode significance within the C-POD framework. This probabilistic mode sorting framework incorporates both the spatial proximity between modes and snapshots, and the consistency of modal contributions. It effectively replaces energy-based sorting in POD by offering a statistical interpretation of mode importance, thereby enabling robust feature extraction and mode separation in the clustering-based dimensionality reduction method. Euclidean distance: The Euclidean distance quantifies the average proximity between each clustering mode and the dataset samples. A smaller distance suggests that the corresponding mode plays a significant role across a broader range of system states, thus representing a dominant dynamic feature. This measure offers clear geometric and physical interpretability. 1. Euclidean distance matrix construction: Assume the database matrix \(\mathbf{D}\in R^{m\times n}\), clustering bases \(\mathbf{  
and consistent influence on specific samples, making it more reliable for mode ranking. After obtaining the Euclidean distance matrix \(\mathbf{C}\in R^{b\times n}\), the goal is to transform it into a weight probability mapping matrix \(\mathbf{W}\in R^{b\times n}\), where \(\mathbf{W}_{ij}\) represents the probability value of the \(i\)-th clustering base for the \(j\)-th sample. A higher probability value indicates a greater contribution of that clustering base to the current sample value, while satisfying \(\sum_{i=1}^{b}\mathbf{W}_{ij}=1\). 1. Take the negative of the distance matrix \(\mathbf{C}\) and shift each column, to avoid overflow in exponential calculations, resulting in the matrix \(\mathbf{Z}\): \[\mathbf{Z}=-\mathbf{C}-\mathbf{M}\odot\mathbf{1}_{b}^{\mathrm{T}}\] (16) where \(\mathbf{M}=[\mathbf{M}_{1},\mathbf{M}_{2},\dots,\mathbf{M}_{k}]\), \(\mathbf{M}_{j}=\max_{i}(-\mathbf{D})_{ij}\), and \(\odot\) represents the outer product. 2. Perform the exponential operation on the shifted matrix and introduce the temperature parameter \(\tau>0\), resulting in: \[\mathbf{E}=\exp(\frac{\mathbf{Z}}{\tau})\] (17)
3. Normalize the exponential matrix column-wise to obtain the final probability weight matrix \(\mathbf{W}\): \[\mathbf{W}=\mathbf{E}\varnothing(\mathbf{1}_{n}\mathbf{1}_{n}^{\mathrm{T}}\mathbf{E})\] (18) where \(\varnothing\) represents element-wise division, \(\mathbf{1}_{n}^{\mathrm{T}}\mathbf{E}\) is the sum of each column, and \(\mathbf{1}_{n}\mathbf{1}_{n}^{\mathrm{T}}\mathbf{E}\) extends the column sum to a matrix where all columns are identical.