# Interval-Based LDA Algorithm for Electrocardiograms for Individual Verification
## 2 Materials and Methods
### Verification  
In addition to the biometric data for subject A, the template database includes information about the identifier related to the subject A. The term \"biometric data\" is referred to as the \(\left\{ \begin{array}{l}\phi^{A}(i,v)\end{array} \right|\forall i\in\text{ training set}\left\{ \begin{array}{l}\text{{y}i\in\text{training set}\left\{\begin{array}{l}\text{{y}i\in\text{training set}\left\{\begin{array}{l}\text{{y}i\in\text{training set}\left\{\begin{array}{l}\text{{y}i\in\text{training set}\left\}\right\}\end{array}}}}}}\end{array}\right.\end{array}\right.\end{array} \right\}\) stored along with the \(\text{{TH}}_{\phi}(i,v)\) and \(\text{{TH}}_{\theta}(i)\) in the template database. During the verification stage, the system validates the identity claimed by an unknown person by checking the similarity of the person's biometric data to the enrolled data retrieved by the identifier input by subject A. Consider an unknown subject B to be verified. If an unknown person B is detected by the biometric system during the verification, first, a biometric input is extracted from the person by calculating the feature vector, \(\left\{ \begin{array}{l}\phi_{c}^{B}(i,v)\end{array} \right|\forall i,c\in \text{verification mode}\left\{ \begin{array}{l}\text{{y}i\in\text{verification mode}\left\{\begin{array}{l}\text{{y}i\in\text{verification mode}\left\{\begin{array}{l}\text{{y}i\in\text{verification mode}\left\}\right.\end{array}}}}}\end{array}\right.\end{array} \right.\). Subsequently, the system calculates the difference between the unknown input and templates, resulting in the amplitude and angle distances in the same manner as during enrollment. Next, the system calculates the similarity to determine how close the subjects A and B are. The computation is carried out by applying a similarity threshold to the distances and normalizing the resulting value to the range [0,1], where 0 means \"no match at all\" and 1 means \"perfect match\". In this paper, two similarity measures are considered, and those are the similarities of the amplitude and angle distances, denoted as \(S_{\phi}\) and \(S_{\theta}\), respectively. Subsequently, we define a matching score S to quantify the match between the unknown and template subject by concatenating the two similarities into \(S=0.5S_{\phi}+0.5S_{\theta}\), where both similarities are equally weighted. Given a matching score S, the system checks whether the score \(S\) is above a certain threshold S\({}_{\text{TH}}\) for the signature classification. A detailed explanation of the verification is given below. **Step 1:** First, consider an unknown subject B to be verified and let its ACP be \(\Phi_{c}^{B}(m)\), which belongs to class \(c\). It can be represented by the set of the ACPlets of  
\[\Phi_{c}^{B}(m):\text{ }\Phi_{c}^{B}(m)=\left\{ \begin{array}{l}\phi_{c}^{B}(1,v),\phi_{c}^{B}(2,v),\cdots,\phi_{c}^{B}(i,v),\cdots,\text{ }\phi_{c}^{B}(Nl,v)\end{array} \right\} \tag{21}\]  
**Step 2:** Compute the amplitude distance \(\Delta\varphi_{c}^{B}(i,v)\) between  
\[\overline{\phi}^{A}(i,v)\text{ and }\phi_{c}^{B}(i,v),\text{ given by : }\Delta\phi_{c}^{B}(i,v)=\left|\overline{\phi}^{A}(i,v)-\phi_{c}^{B}(i,v)\right| \tag{22}\]  
where this distance shows how close the amplitudes of \(\overline{\phi}^{A}(i,v)\) and \(\phi_{c}^{B}(i,v)\) are. Smaller \(\Delta\varphi_{c}^{B}(i,v)\) leads to a higher possibility that the unknown subject B is to be subject A. **Step 3:** Calculate the similarity of the amplitude distance: \(S_{\phi}\)  
The similarity of the amplitude distance (\(S_{\phi}\)) can be obtained by applying the threshold of the amplitude similarity to \(\Delta\varphi_{c}^{B}(i,v)\) as follows:  
\[ \begin{array}{c}for\;i=1\;:NI\\ for\;v=0\;:L-1\\ \Omega(i,v)=\frac{TH_{\phi}(i,v)}{\Delta\varphi_{c}^{B}(i,v)}\text{ for a given class}\\ \Omega(i,v)=0\text{ if }\Delta\varphi_{c}^{B}(i,v)>TH_{\phi_{max}}(i,v)\\ \Omega(i,v)=1\text{ if }\Delta\varphi_{c}^{B}(i,v)\leq TH_{\phi}(i,v)\\ \end{array}  \tag{23}\]  
end  
\[