## 2 Materials and Methods
### Data Analysis  
To analyze the heavy metals in the 14 sediment samples, pricipal component analysis (PCA) and partial least squares regression (PLSR) were utilized in this study. PCA is a multivariate discrimination method that reduces the dimension of a group of data and enables transformation with a reduced number of variables to intuitively distribute the data [16]. PLSR is a multivariate regression method that maximizes the covariance between a response variable (\(y\)) and a predictor matrix (\(X\)) by defining orthogonal and linear combinations from the original predictor variables (\(x_{i}\)). These results can be interpreted in terms of the variable loadings and weights on the most explanatory factors and regression  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Sample Number**} & \multicolumn{3}{c}{**Reference Concentration**} \\ \cline{2-4}  & **Cd** & **Cr** & **Pb** \\  & **(mg/kg)** & **(mg/kg)** & **(mg/kg)** \\ \hline M1 & 0.04 & 196.17 & 13.08 \\ M2 & 0.28 & 12.98 & 9.42 \\ M3 & 0.10 & 17.33 & 9.41 \\ M4 & 0.33 & 40.75 & 16.97 \\ M5 & 0.26 & 29.46 & 14.05 \\ M6 & 0.27  
coefficients of the individual predictors. The main advantage of PLSR over standard multiple linear regression is that it can handle multidimensional and collinear datasets [17; 18]. To use the full range of informative data and eliminate noise, a data-processing method is essential for generating a robust calibration model. In this study, the SG derivative was applied to obtain a better relationship from the measured data. The SG derivative provides a simplified least squares procedure for simultaneously smoothing and differentiating data [19]. Numerical derivation of a vector that includes a smoothing step is well known in data preprocessing [20]. The SG derivative can effectively eliminate spectrum noise because the derivatives are calculated for the fitted polynomial of each point. The smoothing method of linear least squares can accurately fit the subsets of adjacent data with a certain polynomial order. Smoothing and transformation of digitized data from a continuous spectrum to a first or second derivative may be appropriate preprocessing techniques [21]. The first-order derivative and second-order polynomial with five points were selected in the SG derivative as the appropriate mode in this study to avoid calculation error and excessive smoothness. The performance of the model in quantitatively measuring the sediments was evaluated in terms of the coefficient of determination (\(R^{2}\)), root mean square error of calibration (RMSEC), and root mean square error of cross validation (RMSECV). The \(R^{2}\) is defined as the proportion of variance explained by the linear regression model. It is a measure of the success of predicting a dependent variable from independent variables and is calculated as follows [21]:  
\[R^{2}=1-\frac{\sum_{i=1}^{n}(\hat{y}_{i}-y_{i})^{2}}{\sum_{i=1}^{n}(\overline{ y}_{i}-y_{i})^{2}}=1-\frac{\text{SSE}}{\text{SST}} \tag{1}\]  
where \(n\), \(y_{i}\), \(\hat{y}_{i}\), and \(\overline{y}_{i}\) are the number of observations, measured value, predicted responses, and average of \(y\), respectively. SSE is the sum of the square error, and SST is the sum of the square total. Therefore, the coefficient of determination ranges from 0 to 1. As the value approaches 1, the model is said to be well fitted. The root mean square error (RMSE) is generally employed in model evaluation studies. It has been used as a standard statistical metric to measure models in climatic and environmental studies [22]. The RMSE is calculated as follows:  
\[\text{RMSE}(\text{V})=\sqrt{\frac{\sum_{i=1}^{n}(\hat{y}_{i}-y_{i})^{2}}{n}} \tag{2}\]  
where \(\hat{y}_{i}\), \(y_{i}\), and \(n\) are the reference concentration of the \(i\)th sample, predicted concentration of the \(i\)th sample, and total number of samples, respectively. This can be used for calibration (RMSEC) and cross-validation (RMSECV).