# An Intelligent Multimodal Biometric Authentication Model for Personalised Healthcare Services
## 2 Background and Related Works
### Machine Learning Algorithms
#### 2.1.1. SVM  
We have used SVM along with other classification algorithms. It constructs a hyperplane in multidimensional space to differentiate different classes [39]. In this work, we have two classes, AUTH to indicate the data of authenticated users and NAUTH, to indicate the signal data of intruder SVM generates optimal hyperplane in an iterative manner that is used to minimise any error in estimation. The core idea of using SVM is to find a maximum marginal hyperplane that best divides the signal dataset into AUTH and NAUTH classes. The SVM applies a decision function \(f(X)>0\) or \(f(X)<0\), to separate the input examples into two classes, where \(X=(x^{1}...,x^{d})\) with \(d\) being the dimension. The size of the training set \(N\) can be derived as \(y^{i},x^{i}\), with \(i=1,...,N,x^{i}\in R^{n}\) as the input part for the \(i\)-_th_ example, and the class label of \(y^{i}\in-1,1\). SVM maps \(X^{i}\) to a higher dimensional feature space that depends on a nonlinear function \(\phi(X)\), optimising the separated hyperplane through maximisation of the margin with the following quadratic equation:  
\[Maximise:\sum_{i=1}^{n}\alpha_{i}-\frac{1}{2}\sum_{j=1}^{n}\alpha_{i}y_{i}K(K _{i},X_{j})\alpha_{j}y_{j} \tag{1}\]  
\[Subject\ to:\sum_{i=1}^{n}\alpha_{i}y_{i}=0,0<=\alpha_{i}<=C(i=1,...,n), \tag{2}\]  
where \(\alpha>=0\) and it will be equal to 0 after optimisation. \(C\) expresses a degree of losing constraint where it is a positive constant that was chosen by the user. This means that having a large value number of \(C\) would result in more accurate classification during the learning phase. If \(K(x,X^{{}^{\prime}})\) is the kernel function which is inner a product defined by \(K(x,X^{{}^{\prime}})=\phi(X)\), Then the SVM decision function is narrated as:  
\[f(x)=\sum_{x_{i}\in SV}\alpha_{i}y_{i}K(X_{i},X), \tag{3}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Study** & **Signal Type** & **Method** & **Results** \\ \hline
[27] & ECG, Fingerprint & Score Fusion & Accuracy: ECG: 90.0\%, \\  & & Score Fusion & ECG + Fingerprint: \\  & & & EER: 0.1\% \\ \hline
[29] & ECG, Finger Vein & SVM, KNNs, RF, NB & \\  & & and ANN & EER: 0.12\%, 1.40\% \\ \hline
[28] & PPG & CWT and DLDA & EER: 0.5–6\% \\ \hline
[30] & ECG and Finger print & CNN and QG-MSVM & EER: 0.14\%, 0.10\%, 0.40\% and 0.32\% \\ \hline
[33] & ECG, PPG & Cross-Correlation Function (CCF) & Accuracy: PPG: 99.98\%, \\  & & Function (CCF) & ECG: 88.79\% \\ \hline
[34] & PPG & Naïve Bayes classifier & Achieved a recognition \\  & & & rate of 98.65\%, 97.76\%, \\  & & & and 99.69\% \\ \hline
[35] & PPG & Decision Tree, KNN, & Achieved accuracy rate of \\  & & Random Forest & 93\%, 98\%, and 99\% \\  & & & respectively \\ \hline
[36] & PPG & Gradient boosting tree & Accuracy: over 90\% and \\  & & (GBT) & false detection rate: 4\% \\ \hline
[32] & Fused ECG and PPG & CNN & Accuracy: 98.9\% \\ \hline
[37] & PPG & CNN & AUC of 78.2\% and 83.2\% \\ \hline
[38] & ECG, PPG, GSR & Classifiers & Equal Error Rate (EER): 0.05 for 25 subjects \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparisons with other Machine Learning-based Biometric Authentication Studies.  
with a common polynomial kernel that can be defined as follows:  
\[K(X,X^{{}^{\prime}})=<\phi(X),\phi(X^{{}^{\prime}})>. \tag{4}\]