# Hierarchical Tensor Decomposition of Module Partition for the Mechanical Simulation
## 3 Simulation Study
As a result, \(\mathbf{X}\) is divided into four components, \(\mathbf{\hat{B}}_{1}\), \(\mathbf{\hat{B}}_{2}\), \(\mathbf{\hat{B}}_{3}\), and \(\mathbf{\hat{S}}\). Figure 2: The simulation data with two smooth layers, sparse spots layer, and a noise layer and the scree plot of the simulation data. In order to demonstrate the good performance of the HTD method, we compare and analyze the simulation data in Figure 2 against the four existing methods, i.e., the Niblack local thresholding SSD, RPCA, and CP decomposition. All the algorithms were coded in MATLAB R2020a and run on a desktop with a CPU of i7 9750H, RAM of 16 GB, and GPU of RTX2060. As shown in Figure 5a-e, the green solid wireframe represents the decomposed result for HTD, local thresholding, RPCA, SSD, and CP decomposition, respectively. The proposed HTD method is clearly superior to other benchmark methods in terms of performance. Evidently, the intensity variations of \(\hat{\mathbf{B}}_{1},\hat{\mathbf{B}}_{2}\), and \(\hat{\mathbf{S}}\), in Figure 5a, are approximately equal to the ground truths for the corresponding components. For example, in Figure 5a, the color bar for \(\hat{\mathbf{B}}_{1}\) is about \((20,45)\), and the ground truth is about \((15,50)\); the color bar for \(\hat{\mathbf{B}}_{2}\) is about \((0,50)\), and the ground truth is about \((0,70)\); the color bar for \(\hat{\mathbf{S}}\) is about \((0,80)\), and the ground truth is about \((0,80)\). In Figure 5b, the local thresholding method mixed the background with high-intensity variation areas, foreground cluster regions, and sparse spots into one component. The poor performance of this method is because it is sensitive to the local window size, the bias [24], and noises. In Figure 5c, the \(\hat{\mathbf{B}}\) extraction shows reasonable performance, and the intensity variation corresponds well with the ground truth; however, the foreground cluster regions and sparse spots are combined into one component. This issue was caused by the SSD identifying regions with high intensity variations as anomalies. According to Figure 5d, the background and foreground cluster region are merged into the extraction component \(\hat{\mathbf{L}}\). The reason for such a problem is that a low-rank two-way tensor is the sum of two-way tensors formed by the outer product of two orthogonal vectors. Therefore, data with a low rank property does not demonstrate smoothness and low intensity variation features. Furthermore, the intensity variation \(\hat{\mathbf{S}}\) depicted in Figure 5d is sensitive to noise with high-intensity values. In Figure 5e, the CP method is used to decompose the simulation data into four components, and each  
Figure 4: The scree plot of the simulation data. (**a**) is for Figure 2; (**b**) is for Figure 3.  
Figure 3: The simulation data with three smooth layers, sparse spots layer, and a noise layer and the scree plot of the simulation data. component has a linear relationship between the rows and columns. However, the CP method cannot interpret the relationship between extraction components and ground truth. In qualitative analysis, as previously mentioned, the proposed algorithm is superior to the benchmark methods. To more effectively illustrate the performance of the proposed algorithm, Precision, Recall, and F1 are introduced. Precision is defined by the TP and the FP, as \(\text{Precision }=\dfrac{\text{TP}}{\text{TP}+\text{FP}}\). Recall is defined by the TP and the FN, as \(\text{Recall }=\dfrac{\text{TP}}{\text{TP}+\text{FN}}\). Precision evaluates the percentage of correct component extraction among all elements that are classified as a correct component by the proposed method. Recall evaluates the percentage of all the actual component elements that are correctly extracted. The Recall is sensitive to the FP, whereas Precision is sensitive to the FP. To evaluate the overall performance of the proposed method for deterministic component extraction, the F1 score is introduced.