## 3 Methodology
### A Concise Information on Factor Analysis  
Factor analysis is a well-known multivariate statistical technique for data reduction. In its heart, it allows the determination of the overlapping measures of the common property among a variable set [52]. This measurement overlap is then clustered together into a single explaining variable, which is called a factor. Therefore, the initial stage of every factor analysis is to reduce multiple variables into a few explaining factors. In this study, these variables are the specifically chosen soil properties that define the soil plasticity characteristics. However, reducing the multiple variables into a few factors also reduces the amount of data explained. Therefore, it is crucial to have an explained variance as high as possible. [PERSON] [53] suggested the factors should explain the variability of at least 70% for a meaningful factor analysis.  
If the factor analysis is a part of a subsequent analysis, such as in this study, then factor scores are generally used. The factor scores can be assigned to the individual soil property based on a weighted fashion. Consequently, these scores can be summed by various methods, weighted on an individual property or even used in regression analysis [54]. Considering this study is aimed at measuring plasticity potential, a higher potentiallyplastic soil would yield a higher factor score and vice versa. Eventually, the factor scores represent each soil and act as the new soil property value for future analysis.  
Another fundamental aspect of factor analysis is its division into two distinguished parts. Exploratory factor analysis is the most used type that allows a researcher to reduce the number of variables by their common measures. The researcher might not be interested in the underlying relations between these variables. The confirmatory factor analysis requires a much more sophisticated research aspect. The researcher can perform an exploratory approach during the initial stage and then choose the variables purposefully to confirm the research hypothesis. In this study, both approaches were used accordingly.  
The Principal Logic behind Factor Analysis  
The main philosophy behind factor analysis is to derive an uncorrelated and linear combination of the original variables to explain the original variable as much as possible [55]. These linear combinations are the \"factors\".  
Figure 3: The research flow chart.  
A factor analysis aims at partitioning the original variables into meaningful groups, which are the factors, so that each group explains a proportion of total variance within the data. This partitioning of factors is initiated by extracting the most explaining factor that accounts for the most variance. This is a progressive type of analysis until all the variance is explained by the extracted factors. Therefore, it is a crucial point to select the factors that have meaning in terms of the research and explains the data set as much as possible.  
The first factor always explains the most variance in the data set and the last factor has the least explaining capability. The mathematical expression of these factors can be defined by the following equations (Equations (1) and (2)).  
\[F_{1}=w_{11}a_{1}+w_{12}a_{2}+\cdots+w_{1k}a_{k}+\mu_{1}+f_{1} \tag{1}\]  
\[\vdots\]  
\[F_{j}=w_{j1}a_{j}+w_{j2}a_{2}+\cdots+w_{jk}a_{k}+\mu_{j}+f_{j} \tag{2}\]  
Here \(F\) is the notation for each factor, \(w\) is the loading of the variable within the given factor. \(w\) can be taught as factor score coefficient or factor weights. \(a\) represents the individual variables in the data set, \(\mu\) is the mean of the variable and the \(f\) is the specific factors. \(j\) and \(k\) are the subscripts that define the corresponding factor and variable, respectively. Considering the equations above, the weight within the factor is the most important coefficient in a research question. It allows labelling a factor based on research interest.