# Unbiased Isotonic Regression Tree for Discovering Hidden Heterogeneity in Monotonicity Constraints
## 6 Conclusions and Discussion  
The isotonic regression tree is useful for detecting subgroups with heterogeneous monotonic constraints between a predictor and a response variable. Simulation and real data analysis were carried out in this study to examine selection bias when selecting split variables. The results revealed that conventional isotonic regression tree is susceptible to the selection bias problem. To nullify the bias, this study proposes an unbiased isotonic regression tree, and the proposed UIRT successfully reduces the selection bias for split variable selection. The application of the proposed tree to a real dataset demonstrates that the proposed tree can identify meaningful subgroups with different monotonically increasing patterns. Several interesting open problems remain in this study. One potential problem is related to a chi-square test for selecting continuous split variables. As a chi-square test is inherently designed for a categorical variable, the continuous variables are necessarily divided into quantile-based segments to accommodate this limitation. Such discretization  
Figure 10: Results of prediction performance of the four methods. Figure 9: The different monotonic increasing patterns associated with wind turbine 2.  
or ad hoc approximation of possible change points does not make use of the full model's information over split variables and, thus, may reduce the sensitivity of the model to detect different monotonically increasing patterns hidden in the data. The same issue applies to the residuals in conducting a chi-square test, as only their signs are considered rather than their actual distributions. This study plans to explore how to overcome such discretization in future research. Another problem is how to handle multiple predictors in isotonic regression trees with unbiased split variable selection. It can be considered that multiple predictors in regression are common pratices and offer valuable insights for interpretation. However, in isotonic regression, it becomes challenging to determine how to accommodate monotonic constraints across multiple variables, and the computational complexity of finding an optimal monotonic solution grows substantially. In particular, this complexity can further intensify within a tree-based context due to split variable selection steps. Several relevant studies could provide a foundational approach to addressing the complexity and optimization challenges in isotonic regression [58, 59]. Lastly, the current isotonic regression tree is limited in that it can only handle constraints that are either purely monotonically increasing or decreasing. While the proposed model can identify different degrees of monotonically increasing/decreasing patterns over subgroups, it may fail to detect the case where the mixture of monotonically increasing and decreasing patterns exists. This limitation restricts the model's capability to capture more complex, mixed constraints within the same dataset, highlighting a gap in its applicability to diverse data scenarios. Future research could develop a variable selection method that can handle the mixture of monotonically increasing and decreasing constraints, which could be present in real-world scenarios. The proposed method could hold significant potential to transform practical applications. In the context of learning regression trees, unbiased split variable selection considers all covariates in a fair fashion, irrespective of the number of potential cutpoints in split variables. The proposed UIRT enhances model interpretability and responsibility by prioritizing variables based on their genuine relevance to the target variable rather than the size of their optimization space. Furthermore, UIRT mitigates the risk of overfitting, particularly in datasets containing mixed variable types, and improves generalization by avoiding excessive reliance on variables with a higher number of potential splits.