# Generation of Sparse Antennas and Scatterers Based on Optimal Current Grid Approximation
## 3 Verification of the Modeling Results for Scattering Plate Using MoM with PBF Based on WG Model
In this section, we also discuss the influence of \(T\) on the FS scattering characteristics. To do this, we first obtained matrix \(M_{\mathrm{r}}\) for the 61 \(\times\) 61 = 3721 sparse structures and then changed the value of \(T\) to create different FSs. Logically, when applying various \(T\) values, the final structure will have various shapes, and consequently, their scattering characteristics will also differ. Since the BSCS contains many values (3721), comparing the BSCS between all the obtained sparse structures (for example, if the \(T\) change step is 10, we would get approximately 372 FSs) and the original WG structure based on their graphs is nearly impossible. Therefore, to analyze the similarity of the BSCSs for FSs and the original WG structure as \(T\) changes, we calculated the similarity coefficients for the BSCS for each FS and the BSCS for WG and analyzed them using different correlation coefficients to determine the overall assessment. Specifically, we used the following coefficients to evaluate the similarity: the Pearson, Spearman's rank, Cosine, Euclidean correlation coefficient, and standard deviation of the difference between two vectors. The Pearson correlation coefficient reflects the linear correlation between two vectors [68; 69; 70] (in this case, the BSCS for each FS and WG):  
\[corr(a,b)\ =\ \frac{\sum\limits_{i\ =\ 1}^{n}\left(X_{a,i}-\overline{X_{a}} \right)\left(Y_{b,j}-\overline{Y_{b}}\right)}{\sqrt{\sum\limits_{i\ =\ 1}^{n}\left(X_{a,i}-\overline{X_{a}}\right)^{2}\sum\limits_{i\ =\ 1}^{n}\left(Y_{b,j}-\overline{Y_{b}}\right)^{2}}} \tag{1}\]  
In this equation, \(a\) and \(b\) are the vectors being considered (BSCS for each FS and WG), \(n\) is the total number of elements in each vector, \(X_{a,j}\) and \(Y_{b,i}\) are the \(i\)-th elements \(X\), \(Y\) of vectors \(a\) and \(b\), and \(\overline{X_{a}}\), \(\overline{Y_{b}}\) are the mean values of vectors \(a\) and \(b\). [PERSON]'s rank correlation coefficient is similar to the Pearson coefficient in assessing the degree of correlation between two vectors. If the value is close to +1, it indicates a strong positive correlation. Conversely, if the value is close to -1, it signifies a strong negative correlation [71]. However, one of the primary benefits of [PERSON]'s coefficient benefits is its non-parametric nature, which means it does not require the assumption of a linear relationship between variables [72; 73]. Additionally, [PERSON]'s correlation coefficient is less sensitive to strong outliers compared to the Pearson correlation coefficient [74]:  
\[corr(a,b)\ =\ 1-\frac{6\sum d^{2}}{n(n^{2}-1)} \tag{2}\]  
where \(d\) is the difference between the two ranks of two vectors \(a\) and \(b\). The Cosine coefficient measures the similarity between two vectors by calculating the angle between them [75; 76]. The value of this coefficient also ranges from -1 to +1 [76]. Coefficients with values within this range will be collectively referred to as group 1 coefficients. \[corr(a,b)\ =\ \frac{a.b}{|a||b|} \tag{3}\]  
where \(|a|\ =\ \sqrt{X_{1}^{2}+X_{2}^{2}+\ldots+X_{n}^{2}}\), \(|b|\ =\ \sqrt{Y_{1}^{2}+Y_{2}^{2}+\ldots+Y_{n}^{2}}\). The Euclidean coefficient [77; 78] is commonly used to compare the differences between the element values of vectors. This coefficient is easy to understand and intuitive, reflecting the actual distance between points in a multidimensional space:  
\[corr(a,b)\ =\ \sqrt{\sum_{i\ =\ 1}^{n}\left(X_{i}-Y_{i}\right)^{2}} \tag{4}\]  
The standard deviation of the difference between the two vectors indicates the degree of dispersion of the values in the difference vector of the two BSCS vectors:  
\[corr(a,b)\ =\ \text{std}(A_{i}-B_{i}) \tag{5}\]  
A low standard deviation suggests that the elements in the two vectors are relatively close to each other, meaning that their similarity is high. Conversely, a high standard deviation indicates a significant difference between the elements, meaning the two vectors are not similar.