# Unbiased Isotonic Regression Tree for Discovering Hidden Heterogeneity in Monotonicity Constraints
## 5 Case Study
### Results  
In constructing the proposed UIRT, the power output was used as a response variable, and wind speed served as a predictor following the practice of wind turbine energy analysis. All other variables, such as the type of wind turbine (e.g., WT1 and WT2), wind direction, air density, turbulence intensity, and wind shear, functioned as split variables in the tree. The significance level for split variable selection was set to 0.05, the maximum tree depth was set to 3, and the minimum sample size per node was set to 500. Figure 7 shows six different subgroups characterized by the type of turbine, turbulence intensity, and air density. The first subgroup is associated with wind turbine 1; the turbulence intensity is less than equal to the value of 1.18; and air density is no greater than 0.37. The second subgroup shares the same characteristic types of wind turbine and turbulence intensity as the first subgroup, but its air density is greater than 0.37. The third subgroup is defined by turbulence intensity above the value of 1.18 under wind turbine 1. The fourth subgroup, linked to wind turbine 2, shows a turbulence intensity of 1.15 or less and air density of up to 0.29. The fifth subgroup has the same wind turbine type and turbulence intensity conditions as the first subgroup but has an air density exceeding 0.29. The sixth subgroup is characterized by a turbulence intensity greater than 1.15 while still associated with wind turbine 2. It was found that identical influential environmental factors are related to each turbine, but their cutpoints are different. Figure 6: (**a**) The schematic representation of the main components of a wind turbine, and (**b**) an example of a real wind turbine. The distinct increasing patterns between wind power and wind speed for wind turbine 1 and wind turbine 2 are represented in Figures 5 and 6, respectively. For each plot in Figures 8 and 9, the \(x\)-axis denotes wind speed, and the \(y\)-axis has normalized power output. For subgroup 1 in Figure 8, the estimated power curve colored with light blue closely adheres to the overall trend in the data. Specifically, the isotonic regression follows the dense concentration of points in the lower and middle wind speed ranges, providing a smooth approximation of power output as wind speed increases. This smooth isotonic fit implies that this model can capture gradual transitions in power output as wind speed changes, avoiding abrupt jumps. A similar pattern was also detected in subgroup 3 in Figure 8 (i.e., light green line). However, the estimated power output in subgroup 3 was marginally higher than that in subgroup 1, with the largest difference observed when the wind speed reached approximately 12.5 m/s. This suggests that for a fixed wind speed, more power can potentially be produced when the turbulence intensity level exceeds 1.18 on wind turbine 1. In contrast, the estimated power curve in subgroup 2 has a more piecewise-constant representation. In the red curve, although it seems to poorly fit the points above the curve, this is a consequence of isotonic regression's monotonically increasing constraint. The model must balance between fitting both the points above and below the curve. As a result, the curve adjusts downward to better fit the points below the curve to maintain increasing property. This trade-off causes the model to prioritize the fit of the lower observations at the cost of underfitting the upper observations. This downward adjustment is a strategic choice by the model to adhere to the monotonic constraint across the entire dataset. However, when the wind speed is relatively low, the model appears to provide a much better fit. The isotonic regression in this lower range of wind speed effectively captures the initial rise in power output as the wind speed increases, which may indicate that this model is well-suited for capturing turbine behavior in lower wind speed conditions.