# Random Forest-Based Prediction of the Optimal Solid Ink Density in Offset Lithography
## 3 Materials and Methods
### Optimal Solid Ink Density Matching Model  
In the offset printing process, increasing the ink output from the ink rollers leads to an increase in substrate ink thickness and ink density, resulting in higher color saturation and more vivid colors in the printed material [19] However, as the Lamb-Bill equation demonstrates, once a certain value is reached, further increases in ink thickness no longer occur. Continuing to increase the ink amount may cause disproportionate dot expansion, resulting in gradation issues and other problems in the printed image. Therefore, during the pre-press color correction stage, it is crucial to find a method to determine the optimal solid ink density, ensuring that the density adjustment leads to the best possible print quality. Traditional solid ink density determination in printed materials primarily relies on the printing parameters of relative contrast (K value), typically calculating the optimal solid ink density based on the density difference ratio at 75% or 80%. This method lacks control over the visual perception of printed materials and is being gradually phased out by printing companies. Relying solely on density control is insufficient to ensure consistent color quality. Integrating chromaticity control helps mitigate color deviations caused by variations in the printing environment, paper types, ink concentrations, and other influencing factors. Moreover, chromaticity control enables precise adjustments across different paper and ink types, ensuring uniform color quality across multiple print batches. This paper aims to utilize the international standard ISO 12647-2:2013, which specifies the colorimetric value requirements for printing solid ink color blocks, to control the optimal solid ink density. In printing, solid ink color blocks refer to areas of printed material where a single, uniform ink coverage is applied without the use of halftone screens or gradients. These blocks represent 100% ink density and are used in color management and quality control to assess color accuracy and consistency in the printing process. To investigate the relationship between ink density and chromaticity coordinates, the Pearson correlation coefficient is calculated to quantify the degree of association between the variables. The Pearson correlation coefficient is a statistical measure that quantifies the strength and direction of a linear relationship between two variables. A key property of this coefficient is that it remains unaffected by changes in the position or scale of the variables. This property allows it to be effectively applied in analyzing the relationship between abstract chromaticity parameters and controllable ink density values. Let the L, a, and b chromaticity coordinate values be denoted as X = {\(x_{1}\), \(x_{2}\), \(\ldots\), \(x_{i}\)}, and the corresponding solid ink density values as Y = {\(y_{1}\), \(y_{2}\), \(\ldots\), \(y_{i}\)}. The Pearson correlation coefficient is defined as shown in Equation (11). \[r=\frac{\sum{(x_{i}-\overline{x})(y_{i}-\overline{y})}}{\sqrt{\left(\sum{(x_{ i}-\overline{x})}^{2}\right)\left(\sum{(y_{i}-\overline{y})}^{2}\right)}} \tag{11}\]  
The Pearson correlation coefficient r, as defined by the equation, ranges between \(-1\) and \(1\). The closer the r value is to either extreme, the stronger the linear relationship between the two variables. When the correlation between the variables is weak, the r value approaches zero. In general, an absolute value of r greater than zero (\(|\textbf{r}|>0\,|\)) is considered indicative of a linear relationship between the variables. The Pearson correlation coefficients were computed by substituting the four data sets--L, a, b, and Ds--into Equation (11), as shown in Table 3.  
The results show that the Pearson coefficients for the L values exceed 0.8, while those for the a and b values are approximately 0.8. Although the L, a, and b chromaticity coordinates exhibit strong linear relationships with solid ink density, the correlations are not perfectly positive.