## 3 Proposed Model
### User Authentication Using Classification Algorithm  
In this section, the user authentication process using different classifier models (SVM, LR and RF) is described. Our main goal is to make our system faster and real time system for the future. As we know Deep learning is very efficient and provides better accuracy. But to make a proper deep learning model, we need huge amount of data. However, in this type of single person identification or authentication system, huge amount of person data collection is complex. In this research, we are working with one dimensional single data and our dataset is limited. Hence, if we can extract the good or optimal features which can classify the classes very well, then we can use less computation overhead classifier to get a good result. Considering this point or concept in mind, we chooses the less computational classifiers (SVM, LR, RF) which can classify the classes very well with less amount of data in the training process. If we can provide good features from the data, then these classifiers can train the model very well and can provide good and stable accuracy result with less computational overhead. To check the robustness of the proposed model, in the authentication phase, the selected optimal features are validated using different classifiers to identify the writes. Figure 8 represents the user authentication process.  
Figure 8: Block Diagram of User Authentication Process.  
Figure 7: Process of objective value calculation: (**a**) Within-class compactness (**b**) between-class distance.  
A short description of the 3 ML classifiers is given below:  
Logistic Regression (LR): The logistic regression is one of the widely used linear and statistical models for discriminant analysis. Logistic regression can occasionally lead to outperforming other sophisticated nonlinear models such as ensemble learners or support vector machines due to simplicity and interoperability [35].  
\[\pi_{i}=\beta_{0}+\beta_{1}+\ \cdots+\beta_{n}X_{n} \tag{4}\]  
For a binary classifier, the logistic regression model is expressed by summing over the linear combinations of input features and a corresponding weight (\(w\)) plus a bias term (\(b\)) for each instance as shown in Equations (5) and (6).  
\[P\Big{(}y^{(i)}=1\Big{|}x^{(i)},\ w\Big{)}\ =1-\frac{1}{1+exp^{(w^{T}x^{(i)}+b)}} \tag{5}\]  
\[P\Big{(}y^{(i)}=0\Big{|}x^{(i)},\ w\Big{)}\ =1-\frac{1}{1+exp^{(w^{T}x^{(i)}+b)}} \tag{6}\]  
Random Forest (RF): Random forests is an ensemble learning algorithm for classification, regression, and other tasks that operates by constructing a multitude of decision trees at training time. RF is comprised of n collections of de-correlated decision trees. It uses multiple trees to average (regression) or compute majority votes (classification) in the terminal leaf nodes when making a prediction [35]. The following equation is needed for the calculation of RF model:  
\[Entropy=-plog_{2}(p)\ -\ qlog_{2}(q) \tag{7}\]  
Support Vector Machine (SVM): SVM is an efficient and popular and machine learning technique which is widely used for classification due to its simplicity and computational efficiency. Different kernel techniques are there for SVM. The RBF non-linear kernel is formulated as Equation (8).  
\[\mathrm{k}\big{(}\mathrm{s}\mathrm{v}_{i},\mathrm{s}\mathrm{v}_{j}\big{)}\ =\exp\Bigg{(}-\frac{\|\mathrm{s}\mathrm{v}_{i}-\mathrm{s}\mathrm{v}_{j}\|^{2} }{2\sigma^{2}}\Bigg{)} \tag{8}\]  
where \(\mathrm{k}\big{(}\mathrm{s}\mathrm{v}_{i},\mathrm{s}\mathrm{v}_{j}\big{)}\) is the kernel function, and \(\mathrm{s}\mathrm{v}_{i}\) and \(\mathrm{s}\mathrm{v}_{j}\) are the input data, and parameter \(\sigma\) is a set by the user. The \(\sigma\) used here to determine the width of the kernel function k. Note that, if \(\sigma\) values are small, then overtraining may occur. Again, if \(\sigma\) values are large, then the basis function puts an oval around the points without describing their shapes or patterns [36].