# Hierarchical Tensor Decomposition of Module Partition for the Mechanical Simulation
## 3 Simulation Study
This score is determined by taking the harmonic mean of the Precision and the Recall, as \(\text{F1}=2\dfrac{\text{Precision }\times\text{Recall}}{\text{Precision }+\text{Recall}}\). The four existing methods and the proposed HTD method are applied in simulated data, as shown in Figure 2, to extract the foreground cluster region and sparse spots from **X**. The region of foreground clusters and sparse spots is labeled in Figure 6a,b, respectively, where the yellow section represents the foreground cluster region, the green sections represent the sparse spots, and the blue section represents the background with noise. The RMSE, Precision, Recall, and F1 score of the proposed HTD, Local thresholding, RPCA, and SSD methods are reported in Table 1. According to Equation (3), the elements belonging to sparse spots in **S** are extracted and binarized. Figure 5: Results of the simulation data using the proposed method and benchmark methods. Given TP, the Precision is determined by FP. Low Precision indicates a high FP, which means few sparse spots are correctly categorized as sparse spots. Consequently, a higher Precision algorithm is preferred. Similarly, given TP, the recall is determined by FN. Table 1 depicts the local thresholding method's superior performance in recall, but poor performance in precision. It is primarily due to its heightened sensitivity to large amounts of noise. Additionally, local thresholding using multiple thresholds may result in a mix of foreground cluster regions and sparse spots, as shown in Figure 7a. In Figure 7b, the highlighted green regions on **X** are extracted sparse elements based on SSD. SSD models the background and the sparse component with basis expansion. As shown in Figure 5c, the basis function, such as a B-spline basis, smooths the background \(\hat{\textbf{B}}\). Then, the RMSE of \(\hat{\textbf{B}}\) is computed as 0.0078. However, there was a false identification of the surrounding areas as part of the sparse component, which included foreground cluster regions and sparse spots. This caused low precision. Based on HTD, the background and foreground cluster regions layers are decomposed from **X**. \(\text{RMSE}_{\textbf{B}_{1}}\) and \(\text{RMSE}_{\textbf{B}_{2}}\) are calculated, as shown in Table 1. To extract the foreground cluster region, the postprocessing of \(\hat{\textbf{B}}_{2}\) is applied based on Equation (4). The region to be detected in \(\hat{\textbf{B}}_{2}\) has a higher intensity value than the background. Therefore, the element \((i,j)\) satisfying \(\hat{\textbf{B}}_{n}(i,j)>f_{a_{n}}\) is considered as foreground cluster element. \(v_{\text{max}}=0\), \(h=2.1\). Table 1 demonstrates that the RPCA achieves a better precision for sparse spot extraction; however, its recall requires to be improved. This is because RPCA is sensitive to the high-intensity value element. It is more clear, as shown in Figure 7c, that the extracted sparse elements by RPCA are highlighted with green color on **X**. As shown in Figure 8, the proposed HTD extracts the foreground cluster region and sparse spots with the most efficient performance, among all the methods. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & **Local Thresholding** & **SSD** & **RPCA** & & **HTD** & \\ \hline  & \(\hat{\textbf{B}}\) & \(\hat{\textbf{S}}\) & \(\hat{\textbf{B}}\) & \(\hat{\textbf{S}}\) & \(\hat{\textbf{L}}\) & \(\hat{\textbf{S}}\) & \(\hat{\textbf{B}}_{1}\) & \(\hat{\textbf{B}}_{2}\) & \(\hat{\textbf{S}}\