# Stem Taper Estimation Using Artificial Neural Networks for Nothofagus Trees in Natural Forest

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

^{2}were randomly established. In each plot, the diameter at breast height of all trees (D ≥ 10 cm) was measured, ruling out trees of other species. The three plots were processed to generate an average stand table, which was used as the basis for the selection of the sample trees in each stand. In each mixed stand, five trees of each species (N. obliqua, N. alpina, N. dombeyi) were selected. In monospecific stands, 10 trees of each species present were selected. All trees were destructively sampled.

_{i}is the volume without bark of each section of the stem (m

^{3}); L is the length of the stem section (m); A

_{1}and A

_{2}are the areas at the base and up of each stem section without bark (m

^{2}), respectively. The total volume of each tree was obtained from the sum of the volumes of each stem section from a height of 0.3 m to the top of the tree.

#### 2.2. Taper Models

_{i}is the diameter without bark measured in variable height h

_{i}, D is the diameter and H is the total height of the tree. Where $H{r}_{i}=\frac{{h}_{i}}{H}$, ${X}_{i}=\frac{H-{h}_{i}}{H-1.3}$, ${Z}_{i}=\frac{{\left(1-H{r}_{i}\right)}^{\frac{1}{2}}}{{\left(1-\frac{1.3}{H}\right)}^{\frac{1}{3}}}$, and ${\beta}_{1,2,\dots 8}$ are the parameters to be estimated from the models.

#### 2.3. Modeling with Artificial Neural Networks

#### 2.4. Artificial Neural Networks Structure

#### 2.5. Performance Metrics and Cross-Validation

#### 2.6. Generation of Estimated Volume

#### 2.7. Proof of Methods Performance

## 3. Results

#### 3.1. Stem Diameter Modeling

#### 3.2. Modeling for the Accumulated Volume

^{3}in the accumulated volume at the individual tree level. The bias in the testing phase was −0.0001, with this value being the one that also produced less bias in relation to the other two configurations. The resulting configuration used three hidden layers of 40, 20 and 10 neurons in each of them, using the ReLU activation function. This configuration generated 1441 weight parameters, originating from 2000 epoch training cycles. Yet, as in the ANN trained for the stem profile, the stabilization of the cost function and the MSE was achieved before 500 cycles (Figure 3). The ANN training curves show that the cost function and MSE stabilize after 500 cycles, and after that, the ANN does not show significant improvements in the testing phase. In this estimation phase, the ANN shows consistent estimates in relation to the measured volumes and the normal distribution of errors (Figure 4).

#### 3.3. Individual Volume Estimation

#### 3.4. Proof of Methods Performance

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

^{o}025/2012 “Desarrollo de herramientas de cuantificación biométrica generalizadas para el manejo y uso integral sustentable de renovales de Nothofagus spp.”. The authors would like to acknowledge the technical support of Jorge Cancino and Carlos Valenzuela.

## Conflicts of Interest

## Appendix A

- # Setting callbacks and creating routes
- name <- "model_name"
- root0 <- "directory_path/Callbacks_Weights/"
- dir.create(paste0(root0,name,"/"), showWarnings = FALSE)
- root <- file.path(paste0(root0,name),"weights.{epoch:02d}-{val_loss:.2f}.hdf5")
- # Creating Artificial Neural Network (ANN)
- model <- keras_model_sequential()
- # Create architecture for diameter estimates using eight predictor variables
- model %>% layer_dense(units = 30, activation = "relu", input_shape = c(8)) %>%
- layer_dense(units = 25, activation = "relu", input_shape = c(8)) %>%
- layer_dropout(0.1) %>%
- layer_dense(units = 1)
- # Compile ANN
- modelo %>% compile(loss='mse',
- optimizer = 'adam',
- metrics = 'mae')
- # Create Callback
- cp_callback <- callback_model_checkpoint(filepath = root,
- save_best_only = TRUE,
- save_weights_only = TRUE,
- save_freq = "epoch",
- monitor = "val_loss")
- # Training ANN
- history <- model %>% fit(data_train,
- y_train,
- epochs=2000,
- batch_size=32,
- validation_split=0.2,
- callbacks=list(cp_callback))
- # Summary of ANN architecture
- summary(model)
- # Evaluation performance ANN using testing data
- model %>% evaluate(data_test, y_test)
- # Generating performance indices
- loss <- history[["metrics"]][["loss"]]
- val_loss <- history[["metrics"]][["val_loss"]]
- mae <- history[["metrics"]][["mae"]]
- val_mae <- history[["metrics"]][["val_mae"]]
- epoch <- seq(1:length(loss))
- # Setting a dataframe with performance indices
- data_epochs <- as.data.frame(cbind(epoch,loss,val_loss,mae,val_mae))
- # Writting a csv file with performance índices
- write.csv(data_epochs, paste0("directory_path/Models_save/epochs_my_",name,".csv"),
- row.names = FALSE)
- # Save ANN architecture
- model %>% save_model_hdf5(paste0("directory_path/Models_save/my_",name,".h5"))
- # Load ANN architecture and best weights
- best_file <- "weights.1860-1.95.hdf5"
- root2 <- paste0("directory_path/Callbacks_Weights/",name,"/",best_file)
- model <- keras_model_sequential()
- model %>% layer_dense(units = 30, activation = "relu", input_shape = c(8)) %>%
- layer_dense(units = 25, activation = "relu", input_shape = c(8)) %>%
- layer_dense(units = 1)
- model_load <- load_model_weights_hdf5(object=model, filepath = root2)
- # Predictions and performance índices in new dataset testing
- y_est <- model_load %>% predict(data_test_new)
- error <- (y_obs - y_est)
- sesgo <- mean(error)
- RMSE_test <- sqrt(mean(error^2))
- RMSE_test_p <- (RMSE_test/mean(y_test))*100

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**Figure 1.**ANN training for the diameter without bark. Black denotes the training dataset and blue the validation dataset.

**Figure 2.**Estimation of the diameter without bark in the testing dataset using the general ANN and the taper function by Kozak [5]. For the ANN, the prediction that is made jointly for the species is shown, while the taper models were adjusted independently for each species.

**Figure 3.**ANN training for accumulated volume without bark. Black denotes the training dataset and blue the validation dataset.

**Figure 4.**Estimation of the volume without bark accumulated in the base-up direction in the first k-fold of the dataset testing, using the ANN structure selected with the lowest RMSE.

**Figure 5.**Proof for the estimation of the diameter without bark at different heights for the three selected trees of each species (dots). The blue line represents the estimate generated by Kozak’s (2004) taper function and the red line represents the estimate generated by the one-phase ANN.

**Figure 6.**Proof for the prediction of the accumulated volume at different heights for the three selected trees of each species (dots) in the two ANN training phases. The black line represents the volume estimate generated by Kozak’s (2004) taper function, the blue line represents the volume estimate generated by the one-phase ANN (diameter) and the red line represents the volume estimate generated by the two-phase ANN (diameter-volume).

Species | n | Diameter (cm) | Height (m) | Volume (m^{3} tree^{−1}) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Av | SD | Min | Max | Av | SD | Min | Max | Av | SD | Min | Max | ||

N. obliqua | 635 | 24.3 | 9.4 | 4.9 | 54.7 | 20.5 | 5.6 | 6.0 | 36.8 | 0.512 | 0.444 | 0.006 | 3.138 |

N. alpina | 459 | 24.5 | 8.9 | 4.6 | 52.0 | 21.7 | 4.6 | 7.0 | 33.4 | 0.529 | 0.383 | 0.007 | 2.368 |

N. dombeyi | 286 | 22.7 | 10.0 | 4.9 | 53.2 | 19.7 | 6.0 | 5.6 | 33.7 | 0.481 | 0.500 | 0.007 | 2.839 |

Model | Model Structure | Cite |
---|---|---|

1 | ${d}_{i}={\left[{D}^{2}({\beta}_{1}{X}_{i}^{1.5}+{\beta}_{2}{X}_{i}^{3}+{\beta}_{3}{X}_{i}^{32}\right]}^{0.5}$ | Bruce et al. [33] |

2 | ${d}_{i}={\left[{D}^{2}({\beta}_{1}{X}_{i}^{1.5}+{\beta}_{2}{X}_{i}^{32}+{\beta}_{3}{X}_{i}^{40}\right]}^{0.5}$ | Bruce et al. [33] |

3 | ${d}_{i}={D}^{2}{\left[{10}^{2{\beta}_{1}}{D}^{2{\beta}_{2}-2}{H}^{2{\beta}_{3}}{\left(H-{h}_{i}\right)}^{2{\beta}_{4}}\right]}^{0.5}$ | Demaerschalk [34] |

4 | ${d}_{i}=D\left\{{\beta}_{0}+{\beta}_{1}\mathrm{ln}\left[1-H{r}_{i}^{\frac{1}{3}}\left(1-{e}^{-\frac{{\beta}_{1}}{{\beta}_{2}}}\right)\right]\right\}$ | Biging [35] |

5 | ${d}_{i}={\beta}_{1}{D}^{{\beta}_{2}}{\left(1-Hr\right)}^{{\beta}_{3}H{r}_{i}+{\beta}_{4}H{r}_{i}+{\beta}_{5}}$ | Lee et al. [36] |

6 | ${d}_{i}={\beta}_{1}{D}^{{\beta}_{2}}+{\beta}_{3}^{D}{Z}_{i}^{\lambda}$ $\begin{array}{c}\lambda ={\beta}_{4}H{r}_{i}^{2}+{\beta}_{5}\mathrm{ln}\left(H{r}_{i}+0.001\right)+{\beta}_{6}H{r}_{i}^{1/2}+{\beta}_{7}\mathrm{exp}\left(H{r}_{i}\right)\\ \hfill +{\beta}_{8}\left(D/H\right)\hfill \end{array}$ | Kozak [5] |

**Table 3.**Estimated parameters and precision fit results obtained in the traditional taper functions for the diameter without bark in the three species evaluated using the testing dataset.

Species | Model | b1 | b2 | b3 | b4 | b5 | b6 | b7 | b8 | Bias (cm) | AIC | BIC | RMSE | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

(cm) | (%) | |||||||||||||

N. obliqua | 1 | 0.9793 * | −0.0079 * | 0.0013 * | 0.082 | 6.51 | 1.61 | 1.66 | 10.8 | |||||

2 | 0.9541 * | 0.0157 * | −0.0035 * | 0.179 | 6.49 | 1.59 | 1.64 | 10.7 | ||||||

3 | 0.1240 * | 0.9806 * | −0.7845 * | 0.7204 * | 0.029 | 8.50 | 1.89 | 1.65 | 10.7 | |||||

4 | 1.1601 * | 0.3885 * | 0.119 | 4.48 | 1.18 | 1.62 | 10.5 | |||||||

5 | 1.2676 * | 0.9473 * | 1.3716 * | −1.8155 * | 1.3268 * | 0.036 | 10.48 | 2.09 | 1.61 | 10.4 | ||||

6 | 0.9850 * | 0.9917 * | 1.0001 * | 0.4059 * | −0.1256 * | 0.5402 * | −0.1274 * | 0.0208 * | 0.018 | 16.43 | 2.51 | 1.54 | 10.0 | |

N. alpina | 1 | 1.0015 * | −0.0195 * | 0.0015 ns | 0.025 | 6.47 | 1.57 | 1.60 | 10.5 | |||||

2 | 0.9634 * | 0.0260 * | −0.0078 * | 0.141 | 6.45 | 1.55 | 1.57 | 10.2 | ||||||

3 | 0.1072 * | 0.9497 * | −0.7479 * | 0.7318 * | 0.023 | 8.46 | 1.85 | 1.59 | 10.4 | |||||

4 | 1.1710 * | 0.3985 * | 0.107 | 4.43 | 1.12 | 1.53 | 10.0 | |||||||

5 | 1.3108 * | 0.9414 * | 1.9184 * | −2.4435 * | 1.5119 * | 0.032 | 10.41 | 2.01 | 1.50 | 9.8 | ||||

6 | 0.8925 * | 1.0267 * | 0.9994 * | 1.0690 * | −0.2439 * | 1.3983 * | −0.7274 * | 0.1219 * | 0.002 | 16.32 | 2.40 | 1.38 | 9.0 | |

N. dombeyi | 1 | 0.9891 * | 0.0015 * | 0.0000 ns | −0.044 | 6.41 | 1.51 | 1.51 | 10.5 | |||||

2 | 0.9821 * | 0.0047 * | −0.0010 * | −0.009 | 6.41 | 1.50 | 1.50 | 10.5 | ||||||

3 | 0.0957 * | 0.9604 * | −0.7890 * | 0.7760 * | 0.017 | 8.40 | 1.79 | 1.49 | 10.4 | |||||

4 | 1.1989 * | 0.4342 * | 0.121 | 4.42 | 1.11 | 1.52 | 10.6 | |||||||

5 | 1.2683 * | 0.9504 * | 1.5661 * | −1.8257 * | 1.2954 * | 0.027 | 10.36 | 1.97 | 1.43 | 10.0 | ||||

6 | 0.8236 * | 1.0669 * | 0.9982 * | 0.5046 * | −0.0820 * | 0.3046 * | −0.0852 * | 0.1123 * | 0.008 | 16.30 | 2.38 | 1.35 | 9.5 |

**Table 4.**Hyper-parameters obtained from the best ANN structures for the diameter without bark and the accumulated volume in the training and testing phase, 2000 epochs and activation function ReLU.

Variable | Optimizer | Batch Size | Layer | Units | n pars. | Bias Test | RMSE Train | AIC | BIC | RMSE Test | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|

(cm) | (%) | (cm) | (%) | |||||||||

Dsc | Adam | 32 | 2 | 30–25 | 1071 | −0.0812 | 1.1503 | 7.5 | 16.14 | 2.22 | 1.1643 | 7.7 |

Adam | 32 | 2 | 30–20 | 911 | 0.0324 | 1.2508 | 8.1 | 16.22 | 2.30 | 1.2596 | 8.3 | |

Adam | 32 | 3 | 30–25–10 | 1316 | 0.1282 | 1.2037 | 7.8 | 16.19 | 2.26 | 1.2963 | 8.5 | |

(m^{3}) | (%) | (m^{3}) | (%) | |||||||||

Volume | Adam | 32 | 3 | 40–20–10 | 1441 | −0.0001 | 0.0266 | 8.2 | 14.37 | −1.43 | 0.0296 | 9.0 |

Adam | 32 | 3 | 40–30–20 | 2271 | 0.0008 | 0.0269 | 8.3 | 14.38 | −1.42 | 0.0318 | 9.7 | |

Adam | 32 | 2 | 30–25 | 1101 | 0.0012 | 0.0284 | 8.8 | 14.44 | −1.36 | 0.0327 | 10.1 |

**Table 5.**Summary of total tree volume estimation in the test basis using Kozak’s taper functions, one-phase ANN and two-phase ANN of prediction. The best ranked ANNs according to the architecture shown in Table 4 were used for the calculation.

Species | Model | RMSE | |
---|---|---|---|

(m^{3} tree^{−1}) | (%) | ||

N. obliqua | Kozak | 0.0378 | 11.6 |

One-phase ANN | 0.0336 | 10.2 | |

Two-phase ANN | 0.0332 | 9.8 | |

N. alpina | Kozak | 0.0266 | 9.9 |

One-phase ANN | 0.0245 | 8.9 | |

Two-phase ANN | 0.0236 | 8.3 | |

N. dombeyi | Kozak | 0.0301 | 10.3 |

One-phase ANN | 0.0294 | 9.3 | |

Two-phase ANN | 0.0278 | 8.6 |

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## Share and Cite

**MDPI and ACS Style**

Sandoval, S.; Acuña, E.
Stem Taper Estimation Using Artificial Neural Networks for Nothofagus Trees in Natural Forest. *Forests* **2022**, *13*, 2143.
https://doi.org/10.3390/f13122143

**AMA Style**

Sandoval S, Acuña E.
Stem Taper Estimation Using Artificial Neural Networks for Nothofagus Trees in Natural Forest. *Forests*. 2022; 13(12):2143.
https://doi.org/10.3390/f13122143

**Chicago/Turabian Style**

Sandoval, Simón, and Eduardo Acuña.
2022. "Stem Taper Estimation Using Artificial Neural Networks for Nothofagus Trees in Natural Forest" *Forests* 13, no. 12: 2143.
https://doi.org/10.3390/f13122143