## 4 Proposed Work
The normalization was performed column wise for each feature present in the feature set. The normalized value, \(N(x)\) was calculated using Equation (10), where \(x\) is an attribute, \(curr(x)\) denotes the value of an attribute corresponding to the current instance; and \(min(x)\) and \(max(x)\) denote the minimum and maximum values of an attribute, respectively, corresponding to all the instances. \[N(x)=\frac{curr(x)-min(x)}{max(x)-min(x)} \tag{10}\]

Thereafter, the features that had maximum and minimum values equal for all instances were filtered out as they were not considered to have significant contribution in the classification stage. The normalized values of the features were utilized to find out the cosine similarity [57] between each of two features from the feature set. Equation (11) measures the similarity between features \(p\) and \(q\), where they represent any two features from a feature set, \(p_{i}\) and \(q_{i}\) denote the values for features \(p\) and \(q\) respectively, corresponding to \(i\)th instance and \(n\) denotes the total number of instances. The values of the cosine similarities were stored in the form of a matrix for faster computation. \[similarity(p,q)=cos\theta=\frac{p.q}{\left\|p\right\|\theta\right\|}=\frac{ \sum_{i=1}^{n}p_{i}q_{i}}{\sqrt{\sum_{i=1}^{n}p_{i}^{2}\sum_{i=1}^{n}q_{i}^{2}}} \tag{11}\]

For instance, let there be a feature vector \(F=\{f_{1},\ f_{2},\ f_{3},\ldots f_{n}\}\), where \(f_{1},\ f_{2},\ f_{3},\ \ldots f_{n}\) are the features in a feature set with dimension \(n\), then the cosine similarity matrix is represented as given in Equation (12), where \(cos\theta_{a,b}\) is the cosine similarity between \(a\)th and \(b\)th feature and \(a,b\leq n\). \[S_{n\times n}=\left( \begin{array}{ccc}cos\theta_{1,\ 1}&\cdots&cos\theta_{1,\ n}\\ \vdots&\ddots&\vdots\\ cos\theta_{n,\ 1}&\cdots&cos\theta_{n,\ n}\end{array} \right) \tag{12}\]

For each feature subset, a new feature subset was created using HMCR and PAR. Either a feature is selected from the feature subset in HM (based on cosine similarity value) or a random feature is selected from the existing feature set. For example, let there be a feature subset \(S=\{f_{2},f_{8},f_{6},f_{5},f_{3}\}\) in HM, having a combination of features from the existing feature set say \(F=\{f_{1},f_{2},f_{3},f_{4},f_{5},f_{6},\ f_{7},f_{8},f_{9},f_{10}\}\). If the HMCR is satisfied in an iteration, we select a feature, say \(f_{5}\). Again, if PAR is satisfied then the feature \(f_{5}\) is replaced by selecting a cosine similar feature, say \(f_{2}\) and removing \(f_{5}\). Another option is to select a complete random feature, say \(f_{7}\) from the global feature set, if neither of the condition is satisfied. Suppose \(f_{2}\) has been used to replace \(f_{5}\) and for another feature say \(f_{3}\), again \(f_{2}\) is found to be its cosine similar feature. Therefore, in this case both \(f_{5}\) and \(f_{3}\) are replaced by a single feature \(f_{2}\); thus, reducing the feature dimension. This selection of features for replacement has been done n times to improvise the new feature subset from the existing feature subset in HM, where \(n\) is the dimension of \(S\) and the improvisation of the new feature subset carries on maximum number of iteration times. Thus, in each iteration a new improvised feature subset (with reduced or same dimension) was generated. The selection method of an optimal feature subset using SFHSA is explained in Algorithm 1 whereas the flowchart of the same is provided in Figure 5.

A decision is then made, whether the improvised subset is better in quality than the previous feature subset. For determining the quality of the feature subsets, mRMR [58] was applied. In evaluation of the feature subsets, mRMR has proved to be an efficient technique [33; 59; 60; 61]. The concept of mRMR involves maximizing the Relevancy-R (PCC [62] between the feature and the class) and minimizing the Redundancy-D (PCC between the features in the subset).