## 2 Background Fundamentals
### Variance Inflation Factor (VIF)  
Predictive models deal with various influential parameters that improve the efficiency of the decision-making process. The variance inflation factor (VIF) determines the correlation between the independent variables. The independent variables are represented as \(a_{i}\), \(1\leq i\leq n\). The VIF threshold, \((\tau)\), is indicated as the threshold value for attribute selection. Algorithm 1 shows the process of attribute selection using VIF. Let us explain the process of VIF using a sample information system, represented in Table 1 and in Example 1.  
The variance inflation factor for the \(i\)th attribute is shown in Equation (1):  
\[VIF=\frac{1}{1-{R_{l}}^{2}} \tag{1}\]  
```
Parameters:
1. \([A]_{m\times n}\): Input matrix of dimension \(m\times n\).
2. \(a_{i}\): attribute \(a_{i}\in[A]_{m\times n}\), where \(1\leq i\leq n\).
3. \(R_{i}^{2}\): R-square for each attribute \(a_{i}\).
4. VIF: variance inflation factor.
5. \(\tau\): VIF threshold value, (assumed based on the data pattern.)
``` Input: Input matrix is \([A]_{m\times n}\). Output: Reduced dataset is \([A_{R}]_{p\times q}\). Procedure to calculate the VIF value
1: Load input matrix \([A]_{m\times n}\).
2:\(a_{i}\in[A]_{m\times n}\) where \(i=1,2,...,n\).
3: Calculate R-square \({R_{l}}^{2}\) for each attribute \(a_{i}\).
4: Calculate \(VIF_{i}=\frac{1}{1-{R_{l}}^{2}}\).
5:if\(VIF_{i}\geq\tau\)then.
6: Discard the corresponding attribute, \(a_{i}\).
7:\(i=i+1\).
8: Go to 5.
9:else
10: Corresponding attributes have no redundancy.
11:endif ```
**Example 1**.: _A sample information system is given in Table 1. where the attributes are chlorine, Cl, represented as \((a_{1})\), magnesium, Mg, represented as \((a_{2})\), manganese, Mn, represented as \((a_{3})\), iron, Fe, represented as \((a_{4})\), calcium, Ca, represented as \((a_{5})\), Zinc, Zn, represented as \((a_{6})\), and copper, Cu, represented as \((a_{7})\)._  
On considering the attribute \(a_{1}\)(Chlorine), the VIF is calculated using Equation (1) as follows:  
\[VIF_{a_{1}}=\frac{1}{1-0.860}=7.120\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Cl** & **Mg** & **Mn** & **Fe** & **Ca** & **Zn** & **Cu** \\ \((a_{1})\) & \((a_{2})\) & \((a_{3})\) & \((a_{4})\) & \((a_{5})\) & \((a_{6})\) & \((a_{7})\) \\
**(ppm)** & **(ppm)** & **(ppm)** & **(ppm)** & **(ppm)** & **(ppm)** & **(ppm)** \\ \hline
0.22 & 31.33 & 5.3 & 19.98 & 38.2 & 2.67 & 2 \\
0.67 & 42.44 & 4.7 & 6.01 & 35.8 & 4.97 & 3 \\
0.22 & 43.13 & 9.4 & 8.89 & 17 & 3.11 & 2 \\
0.35 & 22.45 & 9.5 & 12.25 & 11.3 & 6.34 & 1 \\
0.7 & 52.87 & 9.8 & 3.2 & 32.5 & 4.81 & 4 \\
0.13 & 43.36 & 0.9 & 17.1 & 26.2 & 7.32 & 4 \\
0.8 & 29.19 & 9.2 & 8.5 & 27.1 & 2.64 & 3 \\
0.44 & 34.56 & 2.7 & 6.62 & 15.6 & 1.37 & 2 \\
0.1 & 37.23 & 4.4 & 16.21 & 15.3 & 2.22 & 1 \\
0.9 & 41.78 & 4.1 & 1.98 & 11.02 & 3.71 & 2 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Information System.  
Thus, a summary is given in Table 2.  
From Table 2, it can be seen that the attributes \(a_{2},a_{3},a_{4},a_{5}\), and \(a_{6}\) have a VIF value \(\tau\) less than the considered threshold of 5. The information system is now reduced with \(a_{2},a_{3},a_{4},a_{5},a_{6}\) attributes. Again, the reduced information system checked with the VIF values and found there is no change in the attributes to be reduced. Thus, the reduced information system is given in Table 3.