## 4 Comparison of Prediction Accuracy
### Discussion  
After studying the characteristics of three different anisotropic strength criteria, this paper evaluates the prediction performance of different strength criteria using the root mean square error (RMSE), square of the correlation coefficient, and mean absolute error. The RMSE, square of the correlation coefficient, and mean absolute error for the fitting of strength in 12 different types of rocks are depicted in Figures 15, 16 and 17, respectively. Figure 14: Experimental data of Tournemire Shale fit with [PERSON]’s model. Figure 13: Experimental data of Penhyn Slate fit with plane of patchy weakness model. Figure 15: Comparison of RMSE predicated using SPW, PPW, and Pariseau criteria for various anisotropic rocks. A comparative evaluation of predictive accuracy among the three criteria, as illustrated in Figure 15, reveals variations in RMSE performance across different anisotropic rock formations. The SPW criterion demonstrates the largest prediction errors for ten out of twelve lithological groups, with exceptions observed only in Vaca Muerta shale and Green River Shale 1. This limited predictive capability, as previously discussed, stems from its inability to account for complex hybrid failure mechanisms that involve concurrent bedding-parallel sliding and cross-bedding shear failure. In contrast, the PPW criterion achieves superior predictive precision, yielding the lowest RMSE values for seven distinct stratified rock types, while the Pariseau criterion exhibits optimal performance for the remaining five formations. Although both the PPW and Pariseau criteria substantially outperform the SPW criterion in prediction fidelity, a nuanced comparison of their RMSE  
Figure 16: Comparison of R\({}^{2}\) predicted using SPW, PPW, and Pariseau criteria for various anisotropic rocks. Figure 17: Comparison of MAE predicted using SPW, PPW, and Pariseau criteria for various anisotropic rocks. distributions suggests marginally greater practical reliability of the PPW criterion for heterogeneous layered rock systems. This hierarchical performance pattern underscores the critical importance of incorporating multi-modal failure considerations in anisotropic strength modeling, particularly for lithologies exhibiting pronounced mechanical contrasts between bedding planes and matrix materials. Assessing the predictive accuracy of different anisotropic strength criteria, the square of the correlation coefficient was used as an indicator, as shown in Figure 16. Analysis reveals that except for Bossier shale and Penrhyn Slate, the square of the correlation coefficient for strength prediction by SPW, PPW, and Pariseau criteria exceeds 0.9 for various rock types. This indicates the strong predictive capability of these three anisotropic strength criteria for the characteristics of layered rock strength. Statistically, the PPW criterion yields the highest R\({}^{2}\) values for predicting the strength of seven different types of rock, while the Pariseau criterion achieves the highest R\({}^{2}\) values for the remaining five types of layered rock strength prediction. Consequently, the PPW criterion demonstrates reasonably good predictive capabilities for anisotropic rock strength. A complementary quantitative assessment through mean absolute error (MAE) evaluation, as visualized in Figure 17, reinforces the hierarchical performance pattern observed in RMSE analysis. The SPW criterion demonstrates statistically superior MAE magnitudes across eight lithological groups, corroborating its overestimation tendencies in strength predictions. In marked contrast, the PPW criterion achieves precision dominance with minimized MAE values for nine distinct stratified formations, whereas the Pariseau criterion shows optimal performance for three residual cases.