## 2 Methodology
### Breast Lesions Screening with a GRA-Based Classifier  
As shown in Figure 4, in the classification layer, a GRA-based classifier is a fully connected multilayer network, consisting of an input layer, GRA-based layer, summation layer, and output layer. Its function is based on the similarity level to automatically identify possibilities, including Nor, B, and M classes. Its pattern recognition scheme uses Gaussian-based gray indicators to separate the normality (Nor) level from the two abnormality levels, which perform the pattern recognition tasks by using straightforward mathematical operations without optimization/fine-tuning algorithms, hyperparameter assignments, and iteration computations [40, 48, 49]. The Gaussian functions are used to measure the similarity level between a testing dataset, \(x_{0}\), and training datasets, \(x_{k}\), which are represented as \(x_{0}\) = [\(x_{1}\)(0), \(x_{2}\)(0), \(x_{3}\)(0), \(\dots\), \(x_{i}\)(0), \(\dots\), \(x_{100}\)(0)] and \(x_{k}\) = [\(x_{1}\)(\(k\)), \(x_{2}\)(\(k\)), \(x_{3}\)(\(k\)), \(\dots\), \(x_{i}\)(\(k\)), \(\dots\), \(x_{100}\)(\(k\))], \(k\) = 1, 2, 3, \(\dots\), \(K\), respectively. The similarity level can be measured by the Euclidean distance (\(ED\))  
\[ED(k)=\sqrt{\sum_{i=1}^{100}{(\Delta d_{i}(k))^{2}}},\ d_{i}(k)=x_{i}(0)-x_{i}(k ),\ \ i=1,\ 2,\ 3,\ \dots,\ 100 \tag{12}\]  
where parameter, \(d_{i}(k)\), is the difference between a testing dataset and \(K\) training datasets; \(K\) is the number of training datasets. The function of the gray grade, \(g(k)\), can be defined as follows [49, 50]:  
\[g(k)=\exp(-\frac{1}{2}(\frac{ED(k)^{2}}{\sigma^{2}})),\ k\ =1,\ 2,\ 3,\ \dots,\ K \tag{13}\]  
\[\sigma^{2}=(\Delta d_{\text{max}}-\Delta d_{\text{min}})^{2},\ \left\{ \begin{array}{l}\Delta d_{\text{min}}=\min_{\forall i \forall k}(\Delta d_{i}(k))\\ \Delta d_{\text{max}}=\max_{\forall i\forall k}(\Delta d_{i}(k))\end{array}  \right.,\ \ (\Delta d_{\text{max}}-\Delta d_{\text{min}})\
eq 0 \tag{14}\]  
where \(\sigma\) is the standard deviation which can be automatically determined by the term \"\((\Delta d_{\text{max}}\ -\Delta d_{\text{min}})\)\"; \(\Delta d_{\text{max}}\) and \(\Delta d_{\text{min}}\) are the maximum and minimum difference values, respectively; and \(K\) comparative data are created by training datasets, \(x_{k}\), including (1)  
Figure 6: Downsampling 1D feature signals for normality (Nor) and abnormality levels (benign (B) and malignant (M)) as preliminary class identification. (**a**) Feature signals for normality level; (**b**) feature signals for abnormality levels. Nor class, (2) B class, (3) and M class. Then, the GRA-based classifier's output can be normalized as follows:  
\[y_{j}=\frac{\sum\limits_{k=1}^{K}w_{kj}g(k)}{\sum\limits_{k=1}^{K}g(k)}=\sum \limits_{k=1}^{K}w_{kj}s_{j},\;s_{j}=\frac{g(k)}{\sum\limits_{k=1}^{K}g(k)},\;j =1,\;2,\;\ldots,\;m \tag{15}\]  
\[Y_{j}=\left\{ \begin{array}{ll}1,\;y_{j}\geq 0.50\\ 0,\;\;y_{j}<0.50\end{array} \right. \tag{16}\]  
where parameters, \(w_{kj}\), are the connected weighting values as the desired class referring to the input feature signal between the GRA-based layer and summation layer, which can be set by \(K\times 3\) (\(m=3\), three classes in this study) output training data, as encoded by value \"1\" or value \"0\"; the output pattern vector for three classes can be encoded as [_Nor_, \(B\), _M_] = [0/1, 0/1, 0/1] and binary encoding as (1) _Class-Nor_: [1, 0, 0], (2) _Class-B_: [0, 1, 0], and (3) _Class-M_: [0, 0, 1]. The classifier's output, \(Y_{j}\), \(j=1\), \(2\), \(3\). can be decided by the threshold value at \"value 0.5\" to identify the disease present (for value 1) or disease absent (for value 0).