# Machine Learning for the Estimation of Diameter Increment in Mixed and Uneven-Aged Forests

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## Abstract

**:**

^{2}), root mean square error (RMSE), Akaike information criterion (AIC), and Bayesian information criterion (BIC) were utilized to evaluate the models. RMSE and R

^{2}of the MLP and ANFIS models were estimated for the four groups of beech ((1.61 and 0.23) and (1.57 and 0.26)), hornbeam ((1.42 and 0.13) and (1.49 and 0.10)), chestnut-leaved oak ((1.55 and 0.28) and (1.47 and 0.39)), and other species ((1.44 and 0.32) and (1.5 and 0.24)), respectively. Despite the low coefficient of determination, the correlation test in both techniques was significant at a 0.01 level for all four groups. In this study, we also determined optimal network parameters such as number of nodes of one or multiple hidden layers and the type of membership functions for modeling the diameter increment in the Hyrcanian forests. Comparison of the results of the two techniques showed that for the groups of beech and chestnut-leaved oak, the ANFIS technique performed better and that the modeling techniques have a deep relationship with the nature of the tree species.

## 1. Introduction

^{2}) of the model using field data were 0.38 and 0.60 for the increments of the basal area and volume, respectively. Other studies have also highlighted the successful use of ANNs in forest model development, including predicting the growth and mortality of trees (e.g., [27,28,29]). In the Hyrcanian uneven-aged and mixed forests, Bayat et al. [15] estimated forest tree height using an ANN and ANFIS and concluded that these methods have good ability for tree height estimation.

## 2. Materials and Methods

#### 2.1. Study Area and Data

#### 2.2. Determination of the Model Inputs

^{2}), basal area of the largest tree (BAL) (m

^{2}/ha), Shannon–Weiner’s species diversity index (H

_{S}) (Equation (1)) [33], size diversity index (H

_{d}) (Equation (2)), and the mean of basal area (BA) in sample plot (m

^{2}) (Table 2).

#### 2.3. Model Development

#### 2.3.1. Multilayer Perceptron Artificial Neural Network (MLP)

- (i)
- The number of neurons in input layer: The determination of inputs number for each species group was obtained by trial and error. Thus, the combination with the lowest mean squared error (MSE) values was selected as the final combination of the model. The tested combinations are shown in Table 4. Therefore, according to each species group, the suitable and normalized inputs were entered into the network. Then, these inputs, using hyperbolic tangent (tansig) function [26], were passed from hidden layer to the output layer.
- (ii)
- The number of neurons in hidden layer: In MLP, the nodes of hidden layer are a structural sensitive parameter. This means that very low nodes could lead to ill training and high nodes lead to over-fitting [40]. In this study, the structures of MLP with 1 to 30 neurons in the hidden layer were analyzed and their prediction accuracy was evaluated.
- (iii)
- The number of neurons in output layer: According to the aim of this study (diameter increment estimation) and number of parameters, the neuron’s number in the output layer is equal to 1. The outputs of hidden layer are eventually passed through the output layer with a linear transfer function (purelin) to provide the final output of the model.

No. | Model Inputs |
---|---|

1 | Natural logarithm of DBH (cm), square of DBH (m), BAL |

2 | Natural logarithm of DBH (cm), square of DBH (m), BAL, the average of BA at sample plot |

3 | Natural logarithm of DBH (cm), square of DBH (m), BAL, H_{s} |

4 | Natural logarithm of DBH (cm), square of DBH (m), BAL, H_{d} |

5 | Natural logarithm of DBH (cm), square of DBH (m), BAL, average BA at sample plot, H_{s} |

6 | Natural logarithm of DBH (cm), square of DBH (m), BAL, average BA at sample plot, H_{d} |

7 | Natural logarithm of DBH (cm), square of DBH (m), BAL, H_{s}, H_{d} |

8 | Natural logarithm of DBH (cm), square of DBH (m), BAL, average BA at sample plot, H_{s}, H_{d} |

#### 2.3.2. Neuro-Fuzzy Inference System (ANFIS)

#### 2.3.3. Model Evaluation

^{2}) (Equation (3)), root mean square error (RMSE) (Equation (4)), Akaike information criterion (AIC) (Equation (5)), Bayesian information criterion (BIC) (Equation (6)), and correlation coefficient (r) were used. The coefficient of determination (R

^{2}) is the proportion of the variance in the dependent variable that can be explained by the independent variables [49,50]. The root mean square error (RMSE) is the standard deviation of the residuals, which is the difference between the surveyed data and the fitted model [51,52].

_{par}, and SSE are observed values, estimated values, the average of observed values, the average of estimated values, total number of observations, number of inputs used in the model, and sum of the squared error (SSE), respectively. The best model has the highest value of R

^{2}and the lowest of three other criteria (RMSE, AIC, and BIC).

## 3. Results

^{2}and Pearson correlation coefficient (r)) than the network validation. In addition, the correlation values of the two techniques (ANN and ANFIS) for all species groups in all data sets were significant at a 0.01 level. The best combination of input variables was the same for both techniques (Table 5).

#### 3.1. Beech Group

^{2}). However, by comparing the four data sets in terms of the three evaluation criteria of RMSE, AIC, and BIC, a different result is obtained, so that the lowest RMSE of training and validation were seen in groups C and B, respectively. Further, group C had the least value of AIC and BIC criteria and showed better performance (Table 6).

^{2}of both validation and training, data set C provided the best results. Moreover, by comparing four data sets in terms of the three evaluation criteria of RMSE, AIC, and BIC, data set C had the lowest value and presented the best result and performance (Table 6).

#### 3.2. Hornbeam Group

^{2}, and C had the lowest RMSE, AIC, and BIC. Thus, for the hornbeam species, data set B generally provides the best outcome (Table 7).

#### 3.3. Chestnut-Leaved Oak Group

^{2}of the training was approximately twice the R

^{2}of the validation data. In both MLP and ANFIS models, in terms of correlation, RMSE, BIC, and AIC, it can be said that the data sets of A, B, B, and B provide the best results, respectively. Additionally, data set B also had the best performance in both models (Table 8).

#### 3.4. Other Species Group

^{2}in training. For the other three evaluation criteria (RMSE, AIC, and BIC), the best performance was for the data set D. Finally, we can say that data set D provides the best performance because the validation provides better R

^{2}than the training (Table 9). In the ANFIS model, data set C presented the best result in terms of all evaluation criteria (Table 9).

## 4. Discussion

_{d}, and average of BA were significant. These variables are associated with hornbeam (DBH, natural logarithm of DBH, and BAL) and chestnut-leaved oak, and other species (DBH and natural logarithm of DBH, BAL, and H

_{s}) showed a different pattern. The effectiveness of the two indices, size diversity and species diversity of Shannon–Wiener, on the increment rate were consistent with the results of Liang [57]. In the research for oak forests, an individual-tree model was presented, and it was concluded that the indices affecting the diameter increment include tree size, basal area, tree diameter, volume inventory, and site index [58]. The results of Lhotka and Loewenstein [59] on the development of individual-tree diameter growth models for the Missouri forest stands in the United States showed that the BAL parameter was effective for all species, while the patterns of remaining predictors were different for other species. The results of Lhotka and Loewenstein [59] were in line with the findings of the present study, except in one case. Unlike their findings, in our study the species composition (species diversity index) was not effective for all species groups. In general, the BAL variable has been used as the most important competition variable in previous growth and increment modeling studies (e.g., [60,61,62,63,64]). In this study, we incorporated all the measured independent variables, i.e., DBH, BAL, H

_{S}

_{,}H

_{d}

_{,}BA, number of trees per hectare, slope, aspect, and altitude into the modeling process. However, only DBH, BAL, BA, H

_{S}, and H

_{d}were significant and considered in the final model. Other variables were removed from the final model because they were insignificant in terms of correlation because the study area was relatively consistent in some respects (e.g., slope). If a larger area is selected for study, these variables may become meaningful [30,40]. However, variables of size diversity and species diversity have not been closely evaluated, except for few studies (e.g., [1,57]). Given the fact that the Hyrcanian forests have high biodiversity (species diversity) and high size diversity [30], consideration of these two indices in new studies is most important.

^{2}) of the model showed the lowest value compared to the other tree groups. This means that the inputs of the model did not properly account for the variation in the diameter increment. It is recommended that hybrid models be fitted for this group. The low R

^{2}values can be attributed to a large number of heterogeneous data used in this study. However, the values were statistically significant and allowed for identifying major growth driving factors. For future works, other factors (e.g., soil properties) from a large-scale area might be incorporated into the models for measuring, and perhaps improving, their performance.

## 5. Conclusions

^{2}of the MLP and ANFIS models were estimated for the four groups of beech ((1.61 and 0.23) and (1.57 and 0.26)), hornbeam ((1.42 and 0.13) and (1.49 and 0.10)), chestnut-leaved oak ((1.55 and 0.28) and (1.47 and 0.39)), and other species ((1.44 and 0.32) and (1.5 and 0.24)), respectively. Despite the low coefficient of determination, the correlation test in both techniques was significant at a 0.01 level for all four groups. In general, the ANFIS worked better when the data from the oak and beech groups were used, probably because these groups represent the dense-covered areas with trees. On the other hand, the ANN performed better with the hornbeam and other species groups that represent a wide but less-covered area with trees. Our study also provides information on optimizing and adjusting the parameters necessary for the application of machine learning in developing prediction models for the estimation of the diameter increment in the Hyrcanian forests. We found that modeling techniques have a deep relationship with the nature of the tree species. The results provide guidance for future studies in the same area (Hyrcanian forest) or elsewhere in the diverse forests of the world.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The location of study area in northern Iran and network of permanent sample plots in the form of red dots.

**Figure 2.**Changes rate of Pearson correlation coefficient and RMSE for training and validation data (four data sets) of beech species.

**Figure 3.**The changes rate of Pearson correlation coefficient and RMSE for training and validation data (four data sets) of hornbeam species.

**Figure 4.**The changes rate of Pearson correlation coefficient and RMSE for training and validation data (four data sets) of chestnut-leaved oak species.

**Figure 5.**The change rate of Pearson correlation coefficient and RMSE for training and validation data (four data sets) of other species.

Groups of Trees | Year | Maximum | Average | Standard Deviation |
---|---|---|---|---|

Beech | 2003 | 178 | 39.1 | 29.1 |

2012 | 180 | 41.8 | 36.1 | |

Chestnut-leaved oak | 2003 | 108 | 22.7 | 13.3 |

2012 | 115 | 25.9 | 18.3 | |

Hornbeam | 2003 | 133 | 23.9 | 17.5 |

2012 | 136 | 26.1 | 19.5 | |

Other species | 2003 | 186 | 34.3 | 25.6 |

2012 | 188 | 37.5 | 26 |

Variable | Minimum | Maximum | Standard Deviation |
---|---|---|---|

The number of trees per hectare | 20 | 1220 | 241 |

DBH (cm) | 7.5 | 188 | 24.7 |

Average of BA in sample plot (m^{2}) | 0.02 | 0.633 | 0.1 |

BAL (m^{2}/ha) | 0 | 52 | 8.4 |

H_{d} in sample plot | 0 | 2.468 | 0.314 |

H_{s} in sample plot | 0 | 1.8 | 0.663 |

^{2}/ha), size diversity index (H

_{d}), and Shannon–Weiner’s species diversity index (H

_{S}).

Data Set | Validation | Training | ||
---|---|---|---|---|

A | 1 | 2 | 3 | 4 |

B | 2 | 4 | 3 | 1 |

C | 3 | 3 | 2 | 1 |

D | 4 | 4 | 2 | 1 |

Species Group | Input Variables |
---|---|

Beech | Natural logarithm of DBH (cm), square of DBH (m), BAL, the average of BA, H _{d} |

Hornbeam | Natural logarithm of DBH (cm), square of DBH (m), BAL |

Chestnut-leaved oak | Natural logarithm of DBH (cm), square of DBH (m), BAL, H_{s} |

Other species | Natural logarithm of DBH (cm), square of DBH (m), BAL, H_{s} |

Data Set | Type | MLP | ANFIS | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

r | R^{2} | RMSE | BIC | AIC | r | R^{2} | RMSE | BIC | AIC | ||

A | Training | 0.47 | 0.22 | 1.61 | - | - | 0.52 | 0.27 | 1.58 | - | - |

Validation | 0.42 | 0.18 | 1.68 | - | - | 0.35 | 0.12 | 1.71 | - | - | |

Total | 0.44 | 0.20 | 1.64 | 19,345 | 19,278 | 0.48 | 0.23 | 1.62 | 19,217 | 19,189 | |

B | Training | 0.45 | 0.204 | 1.66 | - | - | 0.54 | 0.29 | 1.51 | - | - |

Validation | 0.41 | 0.17 | 1.59 | - | - | 0.35 | 0.12 | 1.84 | - | - | |

Total | 0.45 | 0.204 | 1.64 | 19,332 | 19,265 | 0.49 | 0.24 | 1.6 | 19,150 | 19,122 | |

C | Training | 0.48 | 0.23 | 1.60 | - | - | 0.53 | 0.28 | 1.58 | - | - |

Validation | 0.40 | 0.16 | 1.83 | - | - | 0.40 | 0.16 | 1.62 | - | - | |

Total | 0.47 | 0.22 | 1.63 | 19,331 | 19,234 | 0.50 | 0.25 | 1.59 | 19,136 | 19,108 | |

D | Training | 0.47 | 0.22 | 1.631 | - | - | 0.51 | 0.26 | 1.57 | - | - |

Validation | 0.42 | 0.18 | 1.619 | - | - | 0.41 | 0.17 | 1.7 | - | - | |

Total | 0.46 | 0.21 | 1.632 | 19,309 | 19,242 | 0.49 | 0.24 | 1.6 | 19,191 | 19,163 | |

Final model | 0.48 | 0.23 | 1.613 | 19,256 | 19,189 | 0.51 | 0.26 | 1.57 | 19,117 | 19,089 |

Data Set | Type | MLP | ANFIS | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

r | R^{2} | RMSE | BIC | AIC | r | R^{2} | RMSE | BIC | AIC | ||

A | Training | 0.35 | 0.12 | 1.45 | - | - | 0.32 | 0.10 | 1.57 | - | - |

Validation | 0.33 | 0.11 | 1.38 | - | - | 0.3 | 0.09 | 1.38 | - | - | |

Total | 0.35 | 0.12 | 1.42 | 27,474 | 27,426 | 0.32 | 0.10 | 1.52 | 27,871 | 27,853 | |

B | Training | 0.37 | 0.14 | 1.42 | - | - | 0.35 | 0.12 | 1.57 | - | - |

Validation | 0.33 | 0.11 | 1.46 | - | - | 0.3 | 0.09 | 1.46 | - | - | |

Total | 0.36 | 0.13 | 1.43 | 27,468 | 27,419 | 0.33 | 0.11 | 1.54 | 27,948 | 27,930 | |

C | Training | 0.34 | 0.11 | 1.41 | - | - | 0.3 | 0.09 | 1.51 | - | - |

Validation | 0.26 | 0.07 | 1.53 | - | - | 0.26 | 0.07 | 1.51 | - | - | |

Total | 0.32 | 0.10 | 1.42 | 27,473 | 27,424 | 0.28 | 0.08 | 1.51 | 27,814 | 27,796 | |

D | Training | 0.33 | 0.11 | 1.42 | - | - | 0.33 | 0.11 | 1.58 | - | - |

Validation | 0.23 | 0.05 | 1.44 | - | - | 0.28 | 0.08 | 1.43 | - | - | |

Total | 0.32 | 0.10 | 1.44 | 27,521 | 27,473 | 0.32 | 0.10 | 1.54 | 27,941 | 27,923 | |

Final model | 0.36 | 0.13 | 1.42 | 27,465 | 27,407 | 0.32 | 0.10 | 1.49 | 27,762 | 27,740 |

Data Set | Type | MLP | ANFIS | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

r | R^{2} | RMSE | BIC | AIC | r | R^{2} | RMSE | BIC | AIC | ||

A | Training | 0.65 | 0.42 | 1.58 | - | - | 0.68 | 0.46 | 1.33 | - | - |

Validation | 0.34 | 0.12 | 2.24 | - | - | 0.52 | 0.27 | 1.7 | - | - | |

Total | 0.59 | 0.35 | 1.76 | 2413 | 2375 | 0.63 | 0.40 | 1.52 | 2264 | 2260 | |

B | Training | 0.60 | 0.36 | 1.63 | - | - | 0.65 | 0.42 | 1.42 | - | - |

Validation | 0.29 | 0.08 | 1.56 | - | - | 0.57 | 0.33 | 1.62 | - | - | |

Total | 0.56 | 0.31 | 1.59 | 2345 | 2306 | 0.61 | 0.37 | 1.48 | 2259 | 2255 | |

C | Training | 0.29 | 0.08 | 1.93 | - | - | 0.62 | 0.38 | 1.42 | - | - |

Validation | 0.31 | 0.10 | 1.82 | - | - | 0.45 | 0.20 | 1.75 | - | - | |

Total | 0.29 | 0.08 | 1.84 | 2442 | 2403 | 0.56 | 0.32 | 1.51 | 2275 | 2271 | |

D | Training | 0.51 | 0.26 | 1.65 | - | - | 0.62 | 0.38 | 1.41 | - | - |

Validation | 0.18 | 0.03 | 2.23 | - | - | 0.45 | 0.20 | 1.74 | - | - | |

Total | 0.47 | 0.22 | 1.94 | 2479 | 2440 | 0.57 | 0.33 | 1.5 | 2254 | 2250 | |

Final model | 0.53 | 0.28 | 1.55 | 2328 | 2289 | 0.62 | 0.39 | 1.47 | 2251 | 2247 |

Data Set | Type | MLP | ANFIS | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

r | R^{2} | RMSE | BIC | AIC | r | R^{2} | RMSE | BIC | AIC | ||

A | Training | 0.52 | 0.27 | 1.53 | - | - | 0.50 | 0.25 | 1.52 | - | - |

Validation | 0.27 | 0.07 | 1.87 | - | - | 0.33 | 0.11 | 1.64 | - | - | |

Total | 0.50 | 0.25 | 1.55 | 6555 | 6536 | 0.46 | 0.21 | 1.54 | 6557 | 6538 | |

B | Training | 0.65 | 0.42 | 1.42 | - | - | 0.53 | 0.28 | 1.43 | - | - |

Validation | 0.46 | 0.22 | 1.51 | - | - | 0.32 | 0.10 | 1.85 | - | - | |

Total | 0.63 | 0.40 | 1.48 | 6474 | 6455 | 0.47 | 0.22 | 1.54 | 6507 | 6488 | |

C | Training | 0.59 | 0.35 | 1.48 | - | - | 0.54 | 0.29 | 1.57 | - | - |

Validation | 0.40 | 0.16 | 1.52 | - | - | 0.35 | 0.12 | 1.49 | - | - | |

Total | 0.56 | 0.31 | 1.48 | 6482 | 6463 | 0.5 | 0.25 | 1.51 | 6499 | 6480 | |

D | Training | 0.59 | 0.35 | 1.48 | - | - | 0.53 | 0.28 | 1.49 | - | - |

Validation | 0.52 | 0.27 | 1.49 | - | - | 0.37 | 0.14 | 1.64 | - | - | |

Total | 0.58 | 0.34 | 1.48 | 6466 | 6466 | 0.48 | 0.23 | 1.53 | 6511 | 6492 | |

Final model | 0.57 | 0.32 | 1.44 | 6432 | 6413 | 0.49 | 0.24 | 1.5 | 6498 | 6479 |

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## Share and Cite

**MDPI and ACS Style**

Salehnasab, A.; Bayat, M.; Namiranian, M.; Khaleghi, B.; Omid, M.; Masood Awan, H.U.; Al-Ansari, N.; Jaafari, A.
Machine Learning for the Estimation of Diameter Increment in Mixed and Uneven-Aged Forests. *Sustainability* **2022**, *14*, 3386.
https://doi.org/10.3390/su14063386

**AMA Style**

Salehnasab A, Bayat M, Namiranian M, Khaleghi B, Omid M, Masood Awan HU, Al-Ansari N, Jaafari A.
Machine Learning for the Estimation of Diameter Increment in Mixed and Uneven-Aged Forests. *Sustainability*. 2022; 14(6):3386.
https://doi.org/10.3390/su14063386

**Chicago/Turabian Style**

Salehnasab, Abotaleb, Mahmoud Bayat, Manouchehr Namiranian, Bagher Khaleghi, Mahmoud Omid, Hafiz Umair Masood Awan, Nadir Al-Ansari, and Abolfazl Jaafari.
2022. "Machine Learning for the Estimation of Diameter Increment in Mixed and Uneven-Aged Forests" *Sustainability* 14, no. 6: 3386.
https://doi.org/10.3390/su14063386