The localities where forest inventories were carried out at Cangshan Mountain(25°34′~26°00′N,99°55′~100°12′E), Yunnan province, SW China, including Malong Peak, Foding Peak and Maer Peak. The three peaks are located on the eastern slope of Cangshan Mountain. The eastern slope of Cangshan Mountain belongs to subtropical climate, with an annual average temperature of 15°C and a dominant wind direction of southwest monsoon. The annual precipitation is abundant, with a rainfall of more than 1000 mm. However, the dry and wet seasons are distinct, and the rainfall is concentrated from May to October, accounting for 84% of the total annual rainfall (Yuan et al., 2008). The predominant tree species are Pinus yunnanensis, Pinus armandii Franch., and Tsuga dumosa (D. Don) Eichler. The typical soil of the area is red soil.

The predominant tree species Pinus yunnanensis were surveyed within three circular sample plots (one plot in each peak). The radius of the corresponding circular sample plot is 18 meters (Malong Peak), 30 meters (Foding Peak) and 20 meters (Maer Peak). In each tree, the crown radius (CR_i, m) was measured at a different height above crown top (CH_i, m, i.e., the vertical height from crown top to each crown radius) along the crown profile; the diameter (DBH, cm) was measured at breast height (1.3 m aboveground), to the nearest 0.1 cm, and total tree height (TH, m) was measured to the nearest 0.1 m. The height to the base of the live crown (HCB, m), and the crown width (CW, m, the average values of two measures taken at eastwest direction and north–south direction) were also measured to the nearest 0.1 m in each tree (see Figure 1).

CHi, crown height from treetop to each measurement point i (m); CRi, crown radius to each measurement point i (m).

The three data files were merged for extensive visual examination, screening, and outlier detection in an initial exploratory analysis before the pooled data were used for modeling. During the exploratory analysis, some obvious data errors were corrected. Tree crown measurements with relative crown height (RCH) than 1 were removed from the data set. Some trees were removed to ensure that each tree crown contained at least four crown radius measurements. Finally, the crown profile data used in this study were collected from 3,096 measured CR values of 516 trees from Pinus yunnanensis forests ranging in age from 16 to 45 years. A data summary is presented in Table 2.

Considering the multiple measurements for each sample tree crown, the relative crown heights from 0.1 to 1 m with an even interval of 0.1 m were selected to develop crown profile models. However, actual measurements of CH at each tree had non-equidistant space-steps. To overcome this difficulty, the crown profile data were used to obtain numerically interpolated values of CR using piecewise cubic Hermite interpolating polynomial (PCHIP) implemented in MATLAB (R2021b). This interpolation method was adopted because it had the characteristic of preserving shape well. Within each tree crown, CR_h1 … CR_h11 denote CR in m at the 11 specified relative crown heights (h1 … h30) from crown top, ranging from 0.1 to 1 at an even interval of 0.1.

Our research framework is illustrated in Figure 2. The whole framework comprises three parts: input features, model, and output. First, we need to extract the features we need from the data we have obtained. Then, we will divide the data into training and testing data, where the training data will be further split the validation data out. Next, the training and validation data are fed into the proposed LSTM and its variant algorithms for training. After the training process, we can put the testing data into the trained model, and it will output the predicted results.

The performance of the proposed algorithms was evaluated based on their balanced the determination coefficient (R², equation (4)), the mean squared error (MSE, equation (5)), the mean absolute deviation (MAE, equation (6)), and the Mean Deviation (ME, equation (7)) values. The computation time of the proposed algorithms was also recorded to indicate the speed of each method.

where y_i represents the observed value for the ith analytic tree i^th; y i ∧ is the predicted value of i^th observed value; n is the number of trees, y i ¯ is the mean value for the observed.

The data split ratio adopted by the test was 8:1:1. That is, 80% of original data were used as the training set to train the model parameters, 10% of the original data was used as the validation set, which was used for model optimization during model training, and another 10% of original data was used as the test set to test the forecasting effect of model (Ju et al., 2019). All of models were run on a computer with Windows 11 operating system, Intel Core i7 CPU @ 3.20 GHz and RAM of 16.00 GB.

Understanding why a model makes a certain prediction can be as crucial as the prediction’s accuracy in the research of machine learning modeling. Here, we used SHAP (Lundberg and Lee 2017) to interpret the predictions of ensemble or deep learning models. We calculated SHAP value using shap package in Python. SHAP overall process: (1) Select the training set as the basic data set, and the explanation model instantiates an interpreter; (2) Select the test set as the explanation sample, and then calculate the SHAP value corresponding to each feature of each sample in the explanation sample through the interpreter. The greater the SHAP value, the greater the feature contribution.

The parameters used in all the models were tuned by grid search or experiment, and the parameters of the ensemble learning model were the same as the parameters of the single model, which ensured the validity of ensemble learning. The optimizing parameters can ensure that all of parameters maintain the good performance in training and prediction. The optimized parameters of all the models used in the experiment are summarized in Table 3.

Figure 4 visualizes the training performance of Vanila LSTM. For Vanila LSTM model, the loss curves of the training and validation sets tended to be flat after parameter tuning, and the learning rate decreased to 0.0001. In the case of not amplifying the samples of the dataset, the model was fully trained at this time, but the effect of the training model had room for further improvement.

The variant algorithm of LSTM has been used to further improve model accuracy. The training results of different crown profile models based on LSTM and its variant algorithms are listed Table 4. We observed that the LSTM’s variant algorithms performed higher accuracy compared to molecular LSTM algorithm (Vanila LSTM). From Table 4 and Figure 5, the proposed Hybrid LSTM-LightGBM performs better than two other variant algorithms with the minimum value of MAE as 0.2336, MSE as 0.1089 and ME as 3.91E-16 and the maximum value of R² as 0.8656 for training datasets. And the better one of two other variant algorithms is Integrated LSTM-LightGBM with MAE as 0.2798, MSE as 0.1558, ME as -5.49E-12 and R² as 0. 8078.For validation datasets, the proposed Integrated LSTM-LightGBM realizes the best rank, followed by Hybrid LSTM-LightGBM. Moreover, from the analysis of training time, the Hybrid LSTM-LightGBM required slightly more computational time compared to Integrated LSTM-LightGBM. We also observed that LightGBM was the fastest algorithm among the proposed algorithms as it consumed the shortest computation time for training datasets in this study, whereas Stacked LSTMs-LightGBM was the most time-consuming algorithm, requiring, on average, approximately 719 times longer computational time than LightGBM due to large number of algorithm parameters (Table 4).

Table 5 shows the error for different model predictions. As seen from the four forecasting performance indices (R², ME, MAE and MSE) that the Integrated LSTM-LightGBM model had some degrees of advantages over LightGBM and Vanila LSTM, indicating that in the case of more complex data, LSTM’s variant algorithms were sufficient to learn crown profile features and it could predict accurately. Although the ensemble learning model (LightGBM) was not inferior to Vanila LSTM and its other variant algorithms in test results, the overall trend of the deep learning model was better. The comparable model performance on both validation and test sets indicated that the applied LSTM’s variant algorithms offered generalizability and transferability of the developed model to previously LSTM algorithm. Rank metric from testing results showed that Integrated LSTM-LightGBM performance achieved the best performance followed by LightGBM and Stacked LSTMs-LightGBM.

Considering that LSTM’s variant algorithms were composed of LightGBM and Vanila LSTM, the relative importance of potential influential predictors for predicting crown profile was investigated for each of LightGBM (Figure 6) and Vanila LSTM models (Figure 7). For beeswarm plot, each point is a result, its position on the x-axis represents the SHAP value of the feature, and the color represents the relative size of the feature, with red representing high and blue representing low.

For LightGBM model, LAG(CR) (CR_i-1 ), RCH and CW were found the most important predictors, while other variables were relatively insignificant variables (see Figure 6 feature importance bar plot). Furthermore, we found that LAG (CR), the most important feature, was basically positively correlated with CR (the model output); The RCH also had an obvious influence. Further analysis, if the RCH is large, the estimated value will be reduced significantly, because the leftmost point in the row of RCH is basically red in feature importance beeswarm plot; The SHAP value in the data point with large CW value is greater than 0, which belongs to positive correlation (see Figure 6 beeswarm plot).

For the Vanila LSTM model, the most important variables were found to be the Poly (CW, LCL, CH)³ and the original measurement tree factors (LAG(CR), DBH and TH), and the least important predictors were some derived factor (CLR, TSC, RCH, HCW, CFR and HCB), in that order (see Figure 7 left plot). The beeswarm plot (see Figure 7 right plot) shows that CH*LCL², CW*LCL², LCL³ and CW²*LCL are obviously positively correlated with predictors, while CH³ and CH²*CW are basically negatively correlated with predictors. The blue dots of CH*LCL², CW*LCL2, LCL³ and CW³ variables are mainly concentrated in the area where SHAP value is less than 0, indicating that when their values are small, the estimate of CR will be reduced. For feature DBH, CH, LCL and CW, most of the points are diffuse in SHAP=0, so it has no effect on most of them, only a small part.

The relative importance of influential predictors was different for the LightGBM and Vanila LSTM models. There were only three factors that contribute significantly to LightGBM model, while almost all factors contributed to Vanila LSTM model. This phenomenon can explain why deep learning outperforms ensemble learning when there are many features.

Previous studies have shown that machine learning has broad application in crown profile modeling. These machine learning includes MLP, SVR, RF, AdaBoost, GBDT and XGBoost (Tian et al., 2021; Chen et al., 2022). Among them, the ensemble learning algorithms can deal with complex nonlinear relationship and show strong prediction ability when predicting the crown profile (Chen et al., 2022).The purpose of this paper is to find the applicable model for crown profile prediction by comparing the ensemble and deep learning algorithms based on the same data format. So far, the deep learning algorithm has not been used to predict crown profile yet based on space-dependent data. Results from this study (see Table 4 and Table 5) showed that the Integrated LSTM-LightGBM model consistently obtained the best performance for crown profile prediction compared with LightGBM model. The overall trend of the deep learning model was better than the ensemble learning model.

An important finding was that there were only three feature variables that contribute significantly to LightGBM model, while almost all feature variables contributed to Vanila LSTM model (see Figures 6, 7). Two phenomena can be attributed to the finding. Firstly, the LSTM’s variant model was superior to the single LSTM model. The model relied on feature variables to develop its prediction algorithms. So, when the number of features contributed to model was less, the generalization ability of the whole model was therefore reduced and the accuracy of the predictor importance measurement was also affected. Secondly, the deep learning outperformed ensemble learning when there were many features (Zhang et al., 2018; Jan et al., 2019). The LSTM series was affected by more variables than LightGBM in this study. This finding supported the notion that more feature variables bring more room for improvement for LSTM series.

Previous results have shown that the machine learning is an effective technique in improving the crown profile prediction accuracy. However, understanding why a model makes a certain prediction can be as crucial as the prediction’s accuracy (Lundberg and Lee, 2017). Interpretable machine learning techniques can generally be grouped into two categories: intrinsic interpretability and post-hoc interpretability (Du et al., 2019). Intrinsic interpretability incorporates interpretability directly to their structures, including decision tree, rule-based model, linear model, attention model, etc. In contrast, the post-hoc interpretability requires selecting and training a black-box model (ensemble or deep learning) and applying interpretability methods (feature importance, partial dependency graph) to explanate after training (Molnar, 2020). Current explanations are usually given in the format of feature importance vectors, which are a complete causal attribution explanation (Molnar et al., 2018). The explanation audiences, such as developers or researchers, can utilize the statistical analysis of the feature importance distribution to debug the models (Du et al., 2019).

In this paper, we adopt a unified framework for interpreting predictions of ensemble or deep learning models, SHAP. The relative importance of influential predictors was different for the LightGBM and Vanila LSTM models. There are more factors that contribute significantly to Vanila LSTM model compared to LightGBM model. The phenomenon can be attributed to two causes. Firstly, over-fitting can be a problem, especially for the LightGBM model. The generalization ability of the whole model was therefore reduced and the accuracy of the predictor importance measurement was also affected. Secondly, the problem of non-convex optimization makes convergence to a local optimum possible when the parameters of the model are learned and adjusted. This will lead to the deviation of the predictor importance’s estimation (Huang et al., 2019).

One of the limitations of this paper is that whether it is LightGBM or LSTM series, whether it is a Vanila LSTM model or an LSTM’s variant model, their robustness is difficult to guarantee for abnormal data and false data, but compared to the ensemble and deep learning models, the model generated by deep learning is still better, but it will consume more computing resources. Fortunately, the computational complexity of these algorithms is relatively small in crown profile modelling, and the main purpose of this paper is to explore the performance of deep learning methods. However, if we can solve the above problems, it will be more conducive to the improvement of the algorithm, this part of the task will be placed in our follow-up work.

Age has a certain influence on the crown profile (Yu et al., 2021). However, age variable inputs have not been used in this paper to predict crown profile because it is difficult to obtain the age of each tree for the uneven aged forest in this study. In future research, age variable may be used for improving the model predictions of crown profile. Moreover, current models may be biased when applied at a large scale as the crown profile is largely influenced by site quality, stand density, spatial structure (such as mingling, neighborhood comparison, and uniform angle index), and random variabilities caused by various stochastic factors that vary from stands to stands. Thus, the prediction bias can be reduced through the integration of all kinds of variability into the crown profile models (Sharma et al., 2017).

Several important model-agnostic interpretability methods such as Partial Dependence Plot (PDP), Individual Conditional Expectation (ICE), Permuted Feature Importance, Global Surrogate, Local Surrogate (LIME), SHAP Value exist, and while none of them are perfect, they can help researchers interpret the results of even very complex machine learning models (Zhou, 2019). For this study, we only used a method (SHAP Value) to interpret the predictions of ensemble or deep learning models. In the future work, multiple model-agnostic methods may be adopted, each of them represents a step toward more fully understanding machine learning models. As machine learning becomes more and more ubiquitous, grasping how these models find answers will be crucial to improving their performance and reliability.