## 3 Results and Discussion
### Modeling the Color Appearance of Carpets  
Multiple linear regression analysis was performed to identify which test variables are affecting the CIE L*, a*, and b* values. The L*, a*, and b* values were assumed as individual dependent variables while using stepwise multiple linear regression under different illuminants for different colors. This is to address the change in every axis of the CIELAB scale. The independent variables for the regression analysis were the pile height, stitch rate, gauge of the carpet, and dye concentration. The consideration of dye concentration as an independent variable in the equations is due to addressing the shade variations. Illuminants and colors were addressed as test conditions rather than independent variables, as the interaction between dye/colorant molecules and illuminants' wavelengths varies depending on the energy absorption pattern of the corresponding dye/colorants. Hence, a set of 18 (3 \(\times\) 3 \(\times\) 2) regression equations were developed for three dependent variables (L*, a*, and b*) while considering three different illuminants and two different colors. For the regression analyses, \(p\)-values \(\leq\) 0.05 were considered statistically significant. At each stage of the stepwise regression process, the independent variables that had the weakest correlation with the dependent variables were eliminated from the model. This could only happen if the variable satisfied the elimination criterion, which is \(p\) > 0.05. The process ended when there were no longer any variables in the model that met the criteria for elimination (all variables in the model had \(p\)-values \(\leq\) 0.05). Several equations were derived by utilizing the significant parameters affecting the L*, a*, and b* values under different illuminants and colors, which are included in Table 7. In the equations, g = gauge of the carpet, s = stitch rate of the carpet, \(p\) = pile height of the carpet, and d = dye concentration. The summary of the regression analysis is presented in Table 7.  
Table 7 revealed that the R square values for statistical models were between 0.60-0.92, which means nearly 60-92% of the L*, a*, and b* values can be explained by the carpet features and the dye concentration. Here, out of 18 equations, only 3 equations have R\({}^{2}\) values less than 0.70. This is indicating a good amount of data variations can be explained by such equations especially considering the wide range color variability in lightness, chroma, and hue scales. These equations can be used for color-prediction model development. For lower R\({}^{2}\) < 0.70 values, other parameters can be considered to develop better color-predicting statistical models. Variations in the R square values could be associated with the corresponding dye particle used to dye the carpets, as it determines the absorption and reflection of certain wavelengths of incident lights. The residual sum of squares (RSS) for the generated models is <40. However, for the larger RSS values, the squared mean of the dependent variables was also found to be significantly higher. From the table, it is realized that dye concentration has the most significant impact on the L*, a*, and b* values. From the equations, it has been observed that increasing the dye/colorant molecules in the carpet structure will reduce the corresponding color scale values (\(\,\pm\)L*\(\,\)\(