# Building Tree Allometry Relationships Based on TLS Point Clouds and Machine Learning Regression
## 3 Results  
Table 1 shows the statistics of goodness-of-fit (R\({}^{2}\)) for the six tested allometric models in the case of only including H as explanatory variable for estimating DBH. Specifically, it represents the mean value, the standard deviation, and the range of R\({}^{2}\) for the 100 repetitions performed, pointing out that individual tree-based models like DTR performed significantly worse (\(p<0.05\)) than linear regression or ensemble machine learning regression algorithms. In fact, small changes in the learning sample can cause dramatic changes in the built tree derived from individual tree-based models, and so the estimated results can be unstable and inaccurate. This is the reason why most recent studies have adopted bagging and boosting ensemble algorithms [25; 26]. Traditional linear regression turned out to be very competitive, providing results statistically similar to those yielded by sophisticated ensemble boosting algorithms, while, surprisingly, RFR worked significantly worse than boosting or linear regression methods, showing a high variability in prediction when varying training samples. It is important to note that ensemble learning is a branch of machine learning that builds and combines multiple learners to improve the outcomes of the learning process. In the case of bagging methods, such as RFR, they apply bootstrap samples randomly generated from the original dataset to train tree models and then aggregate the ensembles to obtain final predictions by majority voting. In this sense, the RFR algorithm usually improves predictions by decreasing the variance and avoiding overfitting, which is more recommended when developing models that include several explanatory variables (multivariate models). The statistics of goodness-of-fit corresponding to the multivariate allometric models, in which DBH depends on H and CD, are shown in Table 2. First, it should be noted that the prediction results were clearly better than those provided by the bivariate allometric models presented in Table 1, especially in the case of machine learning methods. Except linear regression, they also showed lower variability in R\({}^{2}\) when varying training samples, which points to a greater stability of the machine learning models tested in the case of multivariate regression than in the bivariate. Quite the opposite occurred with traditional linear regression, where the inclusion of the CD variable slightly improved the mean value of R\({}^{2}\), but also increased its standard deviation. This result indicates that machine learning regression methods are able to identify complex relationships between covariates not found using conventional regression-based approaches. In this regard, GBoost has been rated as one of the most competitive methods for learning problems when it comes to noisy data and complex non-linear dependencies [27]. GBoost and CatBoost boosting regression algorithms performed significantly better (\(p<0.05\)) than traditional linear regression and DTR, also showing high stability to the variation of training samples. These similar results between GBoost and CatBoost were expected as CatBoost is a member of the family of gradient boosting decision tree machine learning ensemble techniques. AdaBoost and RFR were statistically situated between the very good results offered by GBoost and CatBoost and the good results offered by linear regression, providing predictions not significantly different from those provided by linear regression. In this way, boosting methods, such as GBoost, CatBoost, and AdaBoost, are qualified as sequential ensemble algorithms that converts weak learners to strong learners by paying the most attention to the samples with the highest prediction errors, so increasing their weights in the next iteration and improving prediction accuracy by decreasing bias [27]. The bias of the multivariate models tested in this study was very low, as can be qualitatively appreciated in Figure 3. Indeed, the multivariate DTR model provided bias