The researchers compared quadratic regression, Lasso, random forest and multilayer perceptron models using 110 valid samples for each ink color. Under random five-fold cross-validation, they evaluated predictive accuracy with the coefficient of determination, R², and root mean square error, and used paired t-tests to compare models. The stability of estimated optimal solid densities was assessed with the coefficient of variation, CV%, across folds. These analyses concerned the relationship between measured process variables and solid-density outcomes in the collected printing data, rather than a comparison of printing presses, ink formulations or operating environments.

Under random five-fold cross-validation, random forest and multilayer perceptron outperformed the parametric models—quadratic regression and Lasso—for black, cyan and magenta inks. The two flexible models did not differ significantly from one another for those three colors in paired tests, with reported p-values above 0.05. Yellow ink produced a different ranking: the multilayer perceptron achieved the best performance, with an R² of 0.9770, and was statistically superior to random forest under the same validation design, p = 0.0296. The result therefore supports a color-dependent comparison rather than a general claim that one algorithm dominated every ink.

The initial ranking did not survive a shift to blocked cross-validation. Lag-1 autocorrelation in the sequentially collected process data was 0.96–0.98, indicating that nearby observations were highly related. The researchers added temporal-block and spatial-block strategies to reduce data leakage that could occur when random splitting distributes strongly correlated observations between training and test sets. Under blocked validation, mean R² values were negative for most models and ink colors, limiting the interpretability of comparisons among the algorithms. The substantial decline in predictive metrics showed that performance under randomly divided folds was highly dependent on the validation strategy.

Despite that deterioration, the estimated optima remained comparatively insensitive to fold structure, with deviations of 0.0034 to 0.0152 across all four ink colors. The recommended optimal solid densities were 1.67 for black ink, estimated by random forest; 1.44 for cyan ink, also estimated by random forest; 1.42 for magenta ink, likewise estimated by random forest; and 0.99 for yellow ink, estimated by the multilayer perceptron. Response surfaces fitted under random five-fold validation contained broad high-quality plateaus for cyan and magenta inks and a multi-peak pattern for yellow ink. Because these features emerged under a validation design whose predictive advantage was not retained under blocked validation, their physical interpretation requires caution. Within the tested dataset, however, the stability of the estimated solid densities was markedly less sensitive to fold design than R², RMSE and inter-model rankings, offering a practical measure of optimization robustness without proving validity beyond the study’s printing conditions.

Subject of Research: Comparison of four models for estimating optimal solid density in four-color offset printing

Article Title: Comparative study of quadratic regression, Lasso, random forest, and multilayer perceptron for modeling optimal solid density in offset printing

Article References: Qiu, B., Li, L., Liu, X., Sheng, X., Jiang, Y., Liu, X., & Ling, Y.-L. (2026). Comparative study of quadratic regression, Lasso, random forest, and multilayer perceptron for modeling optimal solid density in offset printing. PLOS One, 21(10), e0361203. https://doi.org/10.1371/journal.pone.0361203

Image Credits: AI Generated

DOI: 10.1371/journal.pone.0361203

Keywords: offset printing, solid density, random forest, multilayer perceptron, quadratic regression, Lasso, cross-validation, data leakage, process optimization, ink colors, Comparative, quadratic