# Unbiased Isotonic Regression Tree for Discovering Hidden Heterogeneity in Monotonicity Constraints
## 5 Case Study
### Results
To evaluate the prediction performance of the proposed UIRT, comparisons were made with the popular decision tree model CART and the most theoretically rigorous tree models MOB, IRT, and UIRT. All models were configured using the same tree control parameters to facilitate a comparison. The root mean squared error (RMSE) was used as a performance measure, and 10-fold cross-validation was conducted for each model. Figure 10 presents the RMSE values for the four models at each fold. In Figure 10, the \(y\)-axis represents the RMSE values, while the \(x\)-axis corresponds to the folds. Each bar indicates the RMSE for each model with the following color scheme: blue for MOB, green for CART, orange for IRT, and red for UIRT. The MOB yields a substantially higher RMSE value than other methods in every fold because MOB relies on simple linear regression at each terminal node. Linear regression is suitable for capturing global linear trends between a response and a predictor, but this structural assumption cannot capture piecewise monotonic data patterns, resulting  
Figure 7: The six subgroups identified by the proposed isotonic regression tree. in a higher MSE. In contrast, CART improves performance by dividing the predictor space into intervals and then using the mean value within each region for predictions. This step function-like segmentation allows CART to better account for local variations, leading to a lower RMSE compared to MOB. However, CART's reliance on mean values could oversimplify the relationship between a response and a predictor. This limitation makes it less effective than IRT, which employs isotonic regression to better capture the monotonic relationship. UIRT further enhances prediction accuracy by addressing selection bias during the splitting process. UIRT statistically identifies the most significant split variables and applies isotonic regression within each subgroup. This two-step approach eliminates the selection bias inherent in IRT and discovers meaningful heterogeneous monotonically increasing patterns. As a result, UIRT achieves the lowest RMSE by combining statistically sound splitting. Regarding the case of wind turbine 2, the resultant isotonic regression trees are almost identical to that of wind turbine 1 displayed in Figure 9. In other words, subgroups 1, 2, and 3 align with subgroups 4, 5, and 6, respectively, but differ slightly in the cutpoint for each split variable. When comparing the counterparts, the estimated power curves for wind turbine 2 are consistently above those of wind turbine 1. For example, for subgroup 1 (i.e., light blue) and subgroup 4 (i.e., dark blue), the dark blue curve reaches the saturation level earlier compared to the light blue curve. Also, in the intermediate range of wind speed (e.g., between approximately 7 m/s and 11 m/s), the dark blue line clearly shows higher power output compared to light blue. The earlier attainment of maximum power and the higher overall power generation at moderate speeds suggest that wind turbine 2 is better suited for efficient energy production, which is identified by the proposed method. Figure 8: The different monotonic increasing patterns associated with wind turbine 1.