## 2 The Proposed Methodology
### Feature Selection Using a Genetic Algorithm  
A GA is grounded in the concepts of biological evolution and natural selection. The principles of natural selection are believed to govern the evolution of living species. Over generations, living organisms develop characteristics that would enable them to thrive amid the adversities of their environments. A GA works much the same way, as it iteratively refines a given solution by progressively selecting better candidate solutions, while discarding inferior choices. Doing so, it mimics the mechanisms of biological evolution, namely crossover and mutations. A fitness or objective function is used to estimate the quality of each solution. A GA takes as input a set of m-dimensional vectors in \(R^{m}\) as given in Equation (1):  
\[\{X_{1}^{(m)},X_{2}^{(m)},...,X_{k}^{(m)}\}, \tag{1}\]  
where  
\[X_{i}^{(m)}=[x_{1},x_{2},...,x_{m-1},x_{m}]. \tag{2}\]  
The GA generates an \(n\)-dimensional subset of vectors in \(R^{n}\), as given in Equation (3):  
\[\{X_{1}^{(n)},X_{2}^{(n)},...,X_{k}^{(n)}\}, \tag{3}\]  
where  
\[X_{i}^{(n)}=[x_{1},x_{2},...,x_{n-1},x_{n}]. \tag{4}\]  
The GA just reduces the dimensionality of each vector in the set given in Equation (1) without affecting the cardinality of the set, i.e., \(n\ll m\). The dimensions selected by the GA are basically those SMART attributes, which minimize the fitness function. Thus, \(X_{i}^{(n)}\) represents a vector selected by the GA in the optimal subset of selected feature vectors. Equation (5) shows the fitness function proposed in this study:  
\[F=\frac{\overline{C}}{\overline{S}}, \tag{5}\]  
Here, \(\overline{C}\) denotes the average compactness of the classes, as given in Equation (6),  
\[\overline{C}=\frac{1}{L}\sum_{i}^{L}C_{i}, \tag{6}\]  
and \(\overline{S}\), as given in Equation (7):  
\[\overline{S}=\frac{2}{L(L-1)}\sum_{i\
eq j}^{L}S_{ij}, \tag{7}\]  
represents the average value of the separability among different classes, while \(L\) is the total number of classes in a given problem. In this study, the number of classes is two, i.e., healthy and failed hard disks. The notions of compactness of a class and the separability of two classes are illustrated in Figure 2 in a three-dimensional space. The compactness of a class measures how well different instances of that class are clustered together, whereas separability between two classes estimates how easy it is to separate the clusters formed by the instances of those two classes. The proposed fitness function is intuitively similar to the notion of Fisher ratio and, in some studies, it has been used in combination with Fisher ratio for feature selection [34]. The GA iterates to find a subset of features that would minimize the fitness function, as given in Equation (5), which is a ratio of the average values of these two quantities. The compactness of a given class is estimated by calculating the distance, mostly the Euclidean distance, between the feature vectors of the class and the class mean or centroid. The mean or centroid, \(\mu^{(i)}\), of a class \(i\), which has a total of \(N\) instances, can be determined using Equation (8):  
\[\mu^{(i)}=\frac{1}{N}\sum_{j=1}^{N}X_{j}. \tag{8}\]  
The mean value of the Euclidean norm given in Equation (9), which measures the distance of an instance of the class from the class centroid, can be used as an estimate for the compactness of class \(i\),  
\[C_{i}=\frac{1}{N}\sum_{j=1}^{N}\|X_{j}-\mu^{(i)}\|. \tag{9}\]  
The Euclidean distance between the centroids of classes \(i\) and \(j\), as given in Equation (10), can be used as an estimate for the separability between these two classes:  
\[S_{ij}=\|\mu^{(i)}-\mu^{(j)}\|. \tag{10}\]  
The proposed two-tier approach for feature selection using a GA and feature significance scores is illustrated in \[X_{i}^{(m)}=[x_{1},x_{2},...,x_{m-1},x_{m}]. \tag{2}\]0 The GA requires the SMART attributes or features to be encoded into chromosomes. New chromosomes are generated from old ones using the mechanism of mutations and crossovers.