# Artificial Neural Networks and Linear Regression Reduce Sample Intensity to Predict the Commercial Volume of Eucalyptus Clones

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data

^{3}and the maximum was 0.6822 m

^{3}.

#### Methods for Estimation of the Tree Attributes

#### 2.2. Clone Cluster Analysis

#### 2.3. Settings to Select the Best Model

^{0.5}× MS

^{res}] to correct the logarithmic discrepancy, where: F = Meyer correction factor; e = base of natural logarithms; and MS

^{res}= mean square of the residuals. The variance inflation factor (VIF) was calculated for models 2, 4, and 5, to verify the presence of multicollinearity in the predictor variables.

^{2}

_{adj}), residual standard error (S

_{y.x}), variation coefficient (VC%), Student’s t-test for the estimated parameters (5% significance), Akaike’s Information Criterion (AIC), graphical analysis of percentage relative errors (E%) vs. real volumes, and histogram of frequency of E%.

#### 2.4. ANN Training

^{3}) was the output variable. The activation functions used in training were exponential, identity, logistic, and hyperbolic tangent. The Broyden–Fletcher–Goldfarb–Shanno (BFGS) training algorithm was used. The minimum learning rate was 0.0001, while the maximum was 0.001.

_{yŷ}): r

_{yŷ}= cov(y, ŷ)/[s

^{2}(y) × s

^{2}(ŷ)]

^{1/2}, where: s

^{2}= variance; cov = covariance; y = real commercial volume with bark v (m

^{3}), and ŷ = predicted commercial volume with bark v (m

^{3}).

#### Selection of the Best ANN

_{yŷ}, root mean square error in percentage, RMSE% = 100/ӯ × [1/n × ∑(ŷ

_{i}− y

_{i})

^{2}]

^{0.5}, bias = 1/n × ∑[ŷ

_{i}− y

_{i}] and error variance = 1/n × ∑[bias − (ŷ

_{i}− y

_{i})]

^{2}. The best network, per approach, was selected using the weighted value method [33]: WV = ∑Nr

_{i}× W

_{i}, where: WV = weighted value; y

_{i}= real commercial volume with bark v (m

^{3}); ŷ

_{i}= predicted commercial volume with bark v (m

^{3}); ӯ = average of real volumes (m

^{3}); n = total number of observations; Nr

_{i}= number of records with the ith placement; and W

_{i}= weight of the ith placement.

#### 2.5. Experimental Scenarios

#### 2.6. Methods Analysis

_{yŷ}, RMSE%, bias and error variance. Scatter plots of the standardized residuals as a function of the predicted v were plotted using the following equation: sr = ŷ

_{i}− y

_{i}/(∑e

_{i}− ē/n − 1)

^{0.5}, where sr = standardized residuals; y

_{i}= real commercial volume with bark v (m

^{3}); ŷ

_{i}= predicted commercial volume with bark v (m

^{3}); e

_{i}= difference between the real and predicted volume of the ith tree (m

^{3}); ē = mean of the difference between the real and predicted volume (m

^{3}); and n = total number of observations.

#### 2.7. Methods Validation

## 3. Results

#### 3.1. Clone Grouping

^{3}, respectively. The horizontal line intercepted the dendrogram between the two and four Euclidean distances, configuring the following groups of clones: group A formed by clones C2, C3, C4, C5, C8, C9, and C11 (n = 222); B by C1, C7, and C13 (n = 373); and C by C6, C10, C12, and C14 (n = 71) (Figure 4). The union of clones by clustering, with high, medium, and low values of v in groups A, B, and C, respectively, reduced the variability of v within the groups. Due to the greater number of clones and a sufficient number of trees to reduce the sample intensity, clone group A was chosen for the model fit and ANN training in scenario (b) (Table A2).

#### 3.2. Best Model Selection

^{2}

_{adj}value and smaller S

_{y.x}, VC%, and AIC (Table 3). The distribution frequency of E% was more concentrated in the zero value for Model 5. The variation in errors of Model 5 was also smaller than that in the other models, and therefore, this model was selected for evaluation with ANN in the later stages (Figure 5). Models 2 and 5 presented VIF > 10, indicating the presence of multicollinearity.

#### 3.3. ANN Retained in Scenarios (a) and (b)

_{yŷ}> 0.9860 (Table 4). The exponential activation function was the most frequent and the identity was the least frequent in the selected ANNs. The neuron number in the hidden layer varied from 3–25 in scenario (a) and from 1–30 in (b). In scenario (a), the number of input variables of ANNs with the general data was higher (22–48) than the other approaches of this same scenario and, in scenario (b), the number of input variables in the intensity of the one tree per diameter class was lower (13–19).

#### 3.4. Predictions Assessment in Scenario (a)

_{yŷ}≥ 0.9940 (Table 5). The ANN was modeled more efficiently in the approach with the general data, with an RMSE smaller than 8%. The error variations of Model 5 in clone groups A and B were smaller than those of ANN. The accuracy in the v predictions of upper classes was lower in both methods (Figure 6).

#### 3.5. Predictions Assessment in Scenario (b)

_{yŷ}values in the two methods were relatively high (≥0.9843) (Table 6). The RMSE% values of ANN with five, four, and two trees per class, and of Model 5 with six, three, and one tree per class were smaller. The dispersion of the residuals, for both methods, was more accurate along the zero axis in the smaller and intermediate values of v and less accurate in the larger values (Figure 7).

#### 3.6. Variance Analysis for Means in Scenario (b)

_{cal}= 0.25 and p-value = 0.98 (Table A4).

#### 3.7. Predictions Validation in Both Scenarios

## 4. Discussion

_{yŷ}of ANN retained in both scenarios was due to their ability to model complex relationships between qualitative and quantitative variables [47,48,49]. The largest neuron number in the hidden layer for the general data was due to the modeling complexity with qualitative variables (region, farm, and groups of clones) [50]. The input number reduction in ANN up to the one tree per class intensity was due to data randomization without the inclusion of some categories of qualitative variables in the training (region and farm). According to Silva et al. [51], obtaining the appropriate architecture of the neural network is optimized by successive attempts that produce satisfactory results. Görgens et al. [15] argue that architecture is directly related to the learning power of the network, and, in more complex data, the demand for neurons and even for layers increases. In the present study, the non-linear behavior of the data required the use of more complex variables, and, therefore, influenced the number of neurons in the input and hidden layers. Thus, it is important to test as many architectures as possible to find the best fit for the data distribution.

_{yŷ}value and lowest RMSE% of the ANN, with the general data, indicate that the precision increases with the inclusion of the clone group as a qualitative variable [22,23,52]. The smaller error variations in the equations with Model 5 in the predictions per clone group are necessary to reduce the data variability by stratification, generating an equation per clone group. Similar results were obtained by Ribeiro et al. [23] using the Schumacher-Hall model and ANNs for estimation of the bole volume for different species in the Tapajós National Forest. After stratifying the data for the regression, they observed an improvement in the parameter estimates in comparison to the use of a single equation. The need to generate an equation per approach requires more time for data collection and fit [23] and, therefore, v can be predicted with an ANN for the general data. The smaller precision in the upper classes of v for the approach of both methods in scenario (a) was due to the greater data variation in these classes.

_{yŷ}value in scenario (b) indicates a good fit of the methods in predicting the data. The highest r

_{yŷ}value and lowest RMSE% of the equations with Model 5 compared to ANN in the intensities of six and three trees per class suggested that a greater number of trees decreased the quadratic error and increased the accuracy of the predictions. For Model 5, at least three trees per class are better for predicting v, while for ANN, at least four trees per class are better for prediction, based on the increase in RMSE% in the intensities of two and three trees per class, respectively. This result was similar to that found by David et al. [53], in which they affirmed that it is possible to predict stem volume of Eucalyptus grandis W. Hill ex Maiden × Eucalyptus urophylla ST Blake by measuring at least two trees per class. However, these authors used sampling proportional to the frequency of trees by diameter class; the present study used the same number of trees per class, except for the intensity with only one tree in the class.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

Clone | Species/Hybrid | n |
---|---|---|

C1 | Eucalyptus grandis W. Hill ex Maiden × Eucalyptus urophylla S.T. Blake | 339 |

C2 | Eucalyptus grandis W. Hill ex Maiden × Eucalyptus urophylla S.T. Blake | 42 |

C3 | Eucalyptus grandis W. Hill ex Maiden × Eucalyptus urophylla S.T. Blake | 113 |

C4 | Eucalyptus grandis W. Hill ex Maiden | 43 |

C5 | Eucalyptus platyphylla F. Muell. | 2 |

C6 | Eucalyptus platyphylla F. Muell. | 2 |

C7 | Eucalyptus grandis W. Hill ex Maiden × Eucalyptus urophylla S.T. Blake | 2 |

C8 | Eucalyptus grandis W. Hill ex Maiden × Eucalyptus urophylla S.T. Blake | 2 |

C9 | Eucalyptus grandis W. Hill ex Maiden × Eucalyptus urophylla S.T. Blake | 13 |

C10 | Eucalyptus urophylla S.T. Blake | 11 |

C11 | Eucalyptus urophylla S.T. Blake | 7 |

C12 | Eucalyptus urophylla S.T. Blake | 54 |

C13 | Eucalyptus urophylla S.T. Blake | 32 |

C14 | Eucalyptus camaldulensis Dehnh. × Eucalyptus urophylla S.T. Blake | 4 |

Total | 666 |

**Table A2.**Number of trees (n) per diametric class in clone A group at two and three years of age, simultaneously.

dbh Class (cm) | n |
---|---|

4–6 | 23 |

6–8 | 28 |

8–10 | 28 |

10–12 | 27 |

12–14 | 21 |

14–16 | 13 |

16–18 | 10 |

18–20 | 1 |

Total | 151 |

**Table A3.**Summary of the Lilliefors (normality) and Bartlett’s (homoscedasticity) tests for the real and predicted values in the different sample intensities of scenario (b).

Intensity Sample | Treatment | n | Original Value | Transformed Value | ||||
---|---|---|---|---|---|---|---|---|

Lilliefors | p-Value | Bartlett | p-Value | Lilliefors | p-Value | |||

Real | 108 | 0.13486 * | 0.00005 | 0.07881 ^{ns} | >0.01 | |||

6/class | ANN | 108 | 0.15964 * | 0.00038 | 0.14973 ^{ns} | >0.01 | 0.07341 ^{ns} | >0.01 |

Model 5 | 108 | 0.14441 * | 0.00861 | 0.08355 ^{ns} | >0.01 | |||

5/class | ANN | 108 | 0.16288 * | 0.00019 | 0.21673 ^{ns} | >0.01 | 0.07269 ^{ns} | >0.01 |

Model 5 | 108 | 0.14355 ^{ns} | >0.01 | 0.09035 ^{ns} | >0.01 | |||

4/class | ANN | 108 | 0.14298 ^{ns} | >0.01 | 0.21918 ^{ns} | >0.01 | 0.08726 ^{ns} | >0.01 |

Model 5 | 108 | 0.14409 * | 0.00917 | 0.08969 ^{ns} | >0.01 | |||

3/class | ANN | 108 | 0.16724 * | 0.00007 | 0.40207 ^{ns} | >0.01 | 0.07236 ^{ns} | >0.01 |

Model 5 | 108 | 0.14370 * | 0.00989 | 0.08892 ^{ns} | >0.01 | |||

2/class | ANN | 108 | 0.16676 * | 0.00008 | 0.16563 ^{ns} | >0.01 | 0.06155 ^{ns} | >0.01 |

Model 5 | 108 | 0.14492 * | 0.00781 | 0.09548 ^{ns} | >0.01 | |||

1/class | ANN | 108 | 0.16423 * | 0.00014 | 0.46086 ^{ns} | >0.01 | 0.06843 ^{ns} | >0.01 |

Model 5 | 108 | 0.15562 * | 0.00090 | 0.10664 * | 0.00416 |

^{ns}non-significant value; n = number of observations.

**Table A4.**Analysis of variance (ANOVA) of the means of the real and predicted values by ANN and regressions with Model 5 at sample intensities of six to two trees per diameter class in scenario (b).

Source of Variance | df | SS | MS | F_{cal} |
---|---|---|---|---|

Treatments | 2 | 0.6 | 0.293 | 0.303 ^{ns} |

Intensities | 4 | 1.2 | 0.295 | 0.305 ^{ns} |

Treatments × Intensities | 8 | 2 | 0.246 | 0.254 ^{ns} |

Residuals | 1605 | 1549.8 | 0.966 | |

Total | 1619 | 1553.6 |

^{ns}non-significant value (p > 0.05).

**Table A5.**Results of the Kolmogorov–Smirnov test calculated (D

_{cal}) for validation of the predicted values by ANN and equations with Model 5 in all approaches to scenarios (a) and (b).

Scenario | Approach | n | ANN | Model 5 | ||
---|---|---|---|---|---|---|

D_{cal} | p-Value | D_{cal} | p-Value | |||

A | General | 133 | 0.0526 ^{ns} | >0.01 | 0.0376 ^{ns} | >0.01 |

A | Group A | 44 | 0.1136 ^{ns} | >0.01 | 0.0682 ^{ns} | >0.01 |

A | Group B | 74 | 0.0405 ^{ns} | >0.01 | 0.0541 ^{ns} | >0.01 |

A | Group C | 14 | 0.2857 ^{ns} | >0.01 | 0.1429 ^{ns} | >0.01 |

B | 6/class | 108 | 0.0648 ^{ns} | >0.01 | 0.0556 ^{ns} | >0.01 |

B | 5/class | 108 | 0.1389 ^{ns} | >0.01 | 0.0648 ^{ns} | >0.01 |

B | 4/class | 108 | 0.0556 ^{ns} | >0.01 | 0.0648 ^{ns} | >0.01 |

B | 3/class | 108 | 0.0648 ^{ns} | >0.01 | 0.0463 ^{ns} | >0.01 |

B | 2/class | 108 | 0.1389 ^{ns} | >0.01 | 0.0556 ^{ns} | >0.01 |

B | 1/class | 108 | 0.1111 ^{ns} | >0.01 | 0.0648 ^{ns} | >0.01 |

^{ns}non-significant value; n = number of observations.

## References

- Ferraz Filho, A.C.; Mola-Yudego, B.; Ribeiro, A.; Scolforo, J.R.S.; Loos, R.A.; Scolforo, H.F. Height-diameter models for eucalyptus sp. plantations in Brazil. Cerne
**2018**, 24, 9–17. [Google Scholar] [CrossRef] - Binkley, D.; Campoe, O.C.; Alvares, C.; Carneiro, R.L.; Cegatta, I.; Stape, J.L. The interactions of climate, spacing and genetics on clonal Eucalyptus plantations across Brazil and Uruguay. For. Ecol. Manag.
**2017**, 405, 271–283. [Google Scholar] [CrossRef] - Dasgupta, M.G.; Dharanishanthi, V. Identification of PEG-induced water stress responsive transcripts using co-expression network in Eucalyptus grandis. Gene
**2017**, 627, 393–407. [Google Scholar] [CrossRef] [PubMed] - Rocha, J.E.C.; Brasil Neto, A.B.; Noronha, N.C.; Gama, M.A.P.; Carvalho, E.J.M.; Silva, A.R.; Santos, C.R.C. Organic matter and physical-hydric quality of an oxisol under eucalypt planting and abandoned pasture. Cerne
**2016**, 22, 381–388. [Google Scholar] [CrossRef] - Santana, R.C.; Barros, N.F.; Leite, H.G.; Comerford, N.B.; Novais, R.F. Estimativa de biomassa de plantios de eucalipto no Brasil. Rev. Árvore
**2008**, 32, 697–706. [Google Scholar] [CrossRef] [Green Version] - Matos, G.S.B.; Silva, G.R.; Gama, M.A.P.; Vale, R.S.; Rocha, J.E.C. Desenvolvimento inicial e estado nutricional de clones de eucalipto no nordeste do Pará. Acta Amazon.
**2012**, 42, 491–500. [Google Scholar] [CrossRef] - Ounban, W.; Puangchit, L.; Diloksumpun, S. Development of general biomass allometric equations for Tectona grandis Linn.f. and Eucalyptus camaldulensis Dehnh. plantations in Thailand. Agric. Nat. Resour.
**2016**, 50, 48–53. [Google Scholar] [CrossRef] - Paul, K.I.; Roxburgh, S.H.; Chave, J.; England, J.R.; Zerihun, A.; Specht, A.; Lewis, T.; Bennett, L.T.; Baker, T.G.; Adams, M.A.; et al. Testing the generality of above-ground biomass allometry across plant functional types at the continent scale. Glob. Chang. Biol.
**2016**, 22, 2106–2124. [Google Scholar] [CrossRef] - Forrester, D.I.; Tachauer, I.H.H.; Annighoefer, P.; Barbeito, I.; Pretzsch, H.; Ruiz-Peinado, R.; Stark, H.; Vacchiano, G.; Zlatanov, T.; Chakraborty, T.; et al. Generalized biomass and leaf area allometric equations for European tree species incorporating stand structure, tree age and climate. For. Ecol. Manag.
**2017**, 396, 160–175. [Google Scholar] [CrossRef] - Fortier, J.; Truax, B.; Gagnon, D.; Lambert, F. Allometric equations for estimating compartment biomass and stem volume in mature hybrid poplars: General or site-specific? Forests
**2017**, 8, 309. [Google Scholar] [CrossRef] - García-Espinoza, G.G.; Aguirre-Calderón, O.A.; Quiñonez-Barraza, G.; Alanís-Rodríguez, E.; De Los Santos-Posadas, H.M.; García-Magaña, J.J. Taper and volume systems based on ratio equations for Pinus pseudostrobus Lindl. in Mexico. Forests
**2018**, 9, 344. [Google Scholar] [CrossRef] - IBÁ. INDÚSTRIA BRASILEIRA DE ÁRVORES. Report IBÁ-2017. Indicators of Performance of the National Sector of Planted Trees for the Year 2016. Available online: http://iba.org/images/shared/Biblioteca/IBA_RelatorioAnual2017.pdf (accessed on 10 August 2018).
- Cosenza, D.N.; Leite, H.G.; Marcatti, G.E.; Binoti, D.H.B.; Alcântara, A.E.M.; Rode, R. Classificação da capacidade produtiva de sítios florestais utilizando máquina de vetor de suporte e rede neural artificial. Sci. For.
**2015**, 43, 955–963. [Google Scholar] [CrossRef] - Zhang, D.; Zhang, L.; Ye, Q.; Ruan, H. Robust learning-based prediction for timber-volume of living trees. Comput. Electron. Agric.
**2017**, 136, 97–110. [Google Scholar] [CrossRef] - Görgens, E.B.; Leite, H.G.; Gleriani, J.M.; Soares, C.P.B.; Ceolin, A. Influência da arquitetura na estimativa de volume de árvores individuais por meio de redes neurais artificiais. Rev. Árvore
**2014**, 38, 289–295. [Google Scholar] [CrossRef] [Green Version] - Görgens, E.B.; Leite, H.G.; Santos, H.N.; Gleriani, J.M. Estimação do volume de árvores utilizando redes neurais artificiais. Rev. Árvore
**2009**, 33, 1141–1147. [Google Scholar] [CrossRef] [Green Version] - Binoti, M.L.M.S.; Binoti, D.H.B.; Leite, H.G.; Garcia, S.L.R.; Ferreira, M.Z.; Rode, R.; Silva, A.A.L. Redes neurais artificiais para estimação do volume de árvores. Rev. Árvore
**2014**, 38, 283–288. [Google Scholar] [CrossRef] [Green Version] - Cosenza, D.N.; Soares, A.A.V.; de Alcântara, A.E.M.; da Silva, A.A.L.; Rode, R.; Soares, V.P.; Leite, H.G. Site classification for eucalypt stands using artificial neural network based on environmental and management features. Cerne
**2017**, 23, 310–320. [Google Scholar] [CrossRef] - Özçelik, R.; Diamantopoulou, M.J.; Crecente-Campo, F.; Eler, U. Estimating Crimean juniper tree height using nonlinear regression and artificial neural network models. For. Ecol. Manag.
**2013**, 306, 52–60. [Google Scholar] [CrossRef] - Diamantopoulou, M.J.; Özçelik, R.; Crecente-Campo, F.; Eler, Ü. Estimation of Weibull function parameters for modelling tree diameter distribution using least squares and artificial neural networks methods. Biosyst. Eng.
**2015**, 133, 33–45. [Google Scholar] [CrossRef] - Vahedi, A.A. Artificial neural network application in comparison with modeling allometric equations for predicting above-ground biomass in the Hyrcanian mixed-beech forests of Iran. Biomass Bioenergy
**2016**, 88, 66–76. [Google Scholar] [CrossRef] - Reis, L.P.; Souza, A.L.; Mazzei, L.; Reis, P.C.M.; Leite, H.G.; Soares, C.P.B.; Torres, C.M.M.E.; Silva, L.F.; Ruschel, A.R. Prognosis on the diameter of individual trees on the eastern region of the amazon using artificial neural networks. For. Ecol. Manag.
**2016**, 382, 161–167. [Google Scholar] [CrossRef] - Ribeiro, R.B.S.; Gama, J.R.V.; Souza, A.L.; Leite, H.G.; Soares, C.P.B.; Silva, G.F. Métodos para estimar o volume de fustes e galhos na Floresta Nacional do Tapajós. Rev. Árvore
**2016**, 40, 81–88. [Google Scholar] [CrossRef] - Rocha, S.J.S.S.; Torres, C.M.M.E.; Jacovine, L.A.G.; Leite, H.G.; Gelcer, E.M.; Neves, K.M.; Schettini, B.L.S.; Villanova, P.H.; da Silva, L.F.; Reis, L.P.; et al. Artificial neural networks: Modeling tree survival and mortality in the Atlantic Forest biome in Brazil. Sci. Total Environ.
**2018**, 645, 655–661. [Google Scholar] [CrossRef] [PubMed] - Husch, B.; Beers, T.W.; Kershaw, J.A., Jr. Forest Mensuration, 4th ed.; John Wiley & Sons: Hoboken, NJ, USA, 2003; 443p. [Google Scholar]
- Mesquita, D.P.P.; Gomes, J.P.P.; Souza Junior, A.H.; Nobre, J.S. Euclidean distance estimation in incomplete datasets. Neurocomputing
**2017**, 248, 11–18. [Google Scholar] [CrossRef] - Ward, J.H., Jr. Hierarchical grouping to optimize an objective function. J. Am. Stat. Assoc.
**1963**, 58, 236–244. [Google Scholar] [CrossRef] - Camolesi, J.F.; Scolforo, J.R.S.; Oliveira, A.D.; Acerbi Júnior, F.W.; Rufini, A.L.; Mello, J.M. Ajuste, seleção e teste de identidade de modelo para volume e número de moirões da candeia (Eremanthus erythropappus). Cerne
**2010**, 16, 431–441. [Google Scholar] [CrossRef] - Schneider, P.R.; Tonini, H. Utilização de variáveis dummy em equações de volume para Acacia mearnsii De Wild. Ciênc. Florest.
**2003**, 13, 121–129. [Google Scholar] [CrossRef] - Rolim, S.G.; Couto, H.T.Z.; Jesus, R.M.; França, J.T. Modelos volumétricos para a Floresta Nacional do Tapirapé-Aquirí, Serra dos Carajás (PA). Acta Amazon.
**2006**, 36, 107–114. [Google Scholar] [CrossRef] - Meyer, H.A. A Correction for a Systematic Error Occurring in the Application of the Logarithmic Volume Equation; Forestry School Research: Scranton, PA, USA, 1941; 15p.
- Haykin, S. Neural Networks and Learning Machines; Pearson: Hoboken, NJ, USA, 2009; ISBN 9780131471399. [Google Scholar]
- Thomas, C.; Andrade, C.M.; Schneider, P.R.; Finger, C.A.G. Comparação de equações volumétricas ajustadas com dados de cubagem e análise de tronco. Ciênc. Florest.
**2006**, 16, 319–327. [Google Scholar] [CrossRef] - Lek, S.; Guégan, J.F. Artificial neural networks as a tool in ecological modelling, an introduction. Ecol. Model.
**1999**, 120, 65–73. [Google Scholar] [CrossRef] - Diamantopoulou, M.J. Artificial neural networks as an alternative tool in pine bark volume estimation. Comput. Electron. Agric.
**2005**, 48, 235–244. [Google Scholar] [CrossRef] - Che, S.; Tan, X.; Xiang, C.; Sun, J.; Hu, X.; Zhang, X.; Duan, A.; Zhang, J. Stand basal area modelling for Chinese fir plantations using an artificial neural network model. J. For. Res.
**2018**. [Google Scholar] [CrossRef] - Reis, L.P.; Souza, A.L.; Reis, P.C.M.; Mazzei, L.; Soares, C.P.B.; Torres, C.M.M.E.; Silva, L.F.; Ruschel, A.R.; Rêgo, L.J.S. Estimation of mortality and survival of individual trees after harvesting wood using artificial neural networks in the amazon rain forest. Ecol. Eng.
**2018**, 112, 140–147. [Google Scholar] [CrossRef] - Diamantopoulou, M.J. Assessing a reliable modeling approach of features of trees through neural network models for sustainable forests. Sustain. Comput. Inform. Syst.
**2012**, 2, 190–197. [Google Scholar] [CrossRef] - Soares, F.A.A.M.N.; Flôres, E.L.; Cabacinha, C.D.; Carrijo, G.A.; Veiga, A.C.P. Recursive diameter prediction and volume calculation of eucalyptus trees using Multilayer Perceptron Networks. Comput. Electron. Agric.
**2011**, 78, 19–27. [Google Scholar] [CrossRef] - Sabatia, C.O.; Burkhart, H.E. Predicting site index of plantation loblolly pine from biophysical variables. For. Ecol. Manag.
**2014**, 326, 142–156. [Google Scholar] [CrossRef] - Ebling, Â.A.; Péllico Netto, S. Modelagem de ocorrência de coortes na estrutura diamétrica da Araucaria angustifolia (Bertol.) Kuntze. Cerne
**2015**, 21, 251–257. [Google Scholar] [CrossRef] - Campos, J.C.C.; Leite, H.G. Mensuração Florestal: Perguntas e Respostas, 4rd ed.; Editora UFV: Viçosa, Brazil, 2013; pp. 491–493. ISBN 978-85-7269-465-0. [Google Scholar]
- Cysneiros, V.C.; Pelissari, A.L.; Machado, S.A.; Figueiredo Filho, A.; Souza, L. Modelos genéricos e específicos para estimativa do volume comercial em uma floresta sob concessão na Amazônia. Sci. For.
**2017**, 45, 295–304. [Google Scholar] [CrossRef] - Návar, J.; Ríos-Saucedo, J.; Pérez-Verdín, G.; Rodríguez-Flores, F.J.; Domínguez-Calleros, P.A. Regional aboveground biomass equations for North American arid and semi-arid forests. J. Arid Environ.
**2013**, 97, 127–135. [Google Scholar] [CrossRef] - Sales, F.C.V.; Silva, J.A.A.; Ferreira, R.L.C.; Gadelha, F.H.L. Ajustes de modelos volumétricos para o clone Eucalyptus grandis x E. urophylla cultivados no agreste de Pernambuco. Floresta
**2015**, 45, 663–670. [Google Scholar] [CrossRef] - Gimenez, B.O.; Santos, L.T.; Gebara, J.; Celes, C.H.S.; Durgante, F.M.; Lima, A.J.N.; Santos, J.; Higuchi, N. Tree climbing techniques and volume equations for Eschweilera (Matá-Matá), a hyperdominant genus in the Amazon forest. Forests
**2017**, 8, 154. [Google Scholar] [CrossRef] - Özçelik, R.; Diamantopoulou, M.J.; Brooks, J.R.; Wiant Junior, H. Estimating tree bole volume using artificial neural network models for four species in Turkey. J. Environ. Manag.
**2010**, 91, 742–753. [Google Scholar] [CrossRef] - Nunes, M.H.; Görgens, E.B. Artificial intelligence procedures for tree taper estimation within a complex vegetation mosaic in Brazil. PLoS ONE
**2016**, 11. [Google Scholar] [CrossRef] - Vahedi, A.A. Monitoring soil carbon pool in the Hyrcanian coastal plain forest of Iran: Artificial neural network application in comparison with developing traditional models. Catena
**2017**, 152, 182–189. [Google Scholar] [CrossRef] - Diamantopoulou, M.J.; Milios, E. Modelling total volume of dominant pine trees in reforestations via multivariate analysis and artificial neural network models. Biosyst. Eng.
**2010**, 105, 306–315. [Google Scholar] [CrossRef] - Silva, M.L.M.; Binoti, D.H.B.; Gleriani, J.M.; Leite, H.G. Ajuste do modelo de Schumacher e Hall e aplicação de redes neurais artificiais para estimar volume de árvores de eucalipto. Rev. Árvore
**2009**, 33, 1133–1139. [Google Scholar] [CrossRef] [Green Version] - Bhering, L.L.; Cruz, C.D.; Peixoto, L.A.; Rosado, A.M.; Laviola, B.G.; Nascimento, M. Application of neural networks to predict volume in eucalyptus. Crop Breed. Appl. Biotechnol.
**2015**, 15, 125–131. [Google Scholar] [CrossRef] [Green Version] - David, H.C.; Otávio, R.; Miranda, V.; Welker, J.; Fiorentin, L.D.; Ebling, Â.A.; Henrique, P.; Martins, B.; Silva, D. Strategies for stem measurement sampling: A statistical approach of modelling individual tree volume. Cerne
**2016**, 22, 249–260. [Google Scholar] [CrossRef]

**Figure 1.**Location of the municipalities of Dom Eliseu, Paragominas, and Ulianópolis, southeast mesoregion of the Pará state, Brazil, generated with ArcGIS 10.4.1 software.

**Figure 4.**Dendrogram of the cluster analysis using Ward’s method and the Euclidean distance with interception of the branches by the horizontal dashed line for the formation of clusters with Eucalyptus clones: group A: C3, C4, C11, C5, C2, C8, and C9; group B: C13, C1, and C7; and group C: C6, C12, C10, and C14.

**Figure 5.**Percentage error vs. real commercial volume with bark up to 4 cm in diameter at the tree top (v) and frequency histogram of E% of the fitted models.

**Figure 6.**Dispersion of the standardized residuals as a function of the predicted commercial volume with bark up to 4 cm in diameter at the tree top (v), in the sample intensities of scenario (a).

**Figure 7.**Dispersion of the standardized residuals as a function of the predicted commercial volume with bark up to 4 cm in diameter at the tree top (v), in the sample intensities of scenario (b).

**Table 1.**Number of trees (n) and mean (and standard deviation) of diameter at breast height (dbh), h, and v per diameter class.

dbh Class (cm) | n | dbh (cm) | h (m) | v (m^{3}) |
---|---|---|---|---|

4–6 | 85 | 5.09 (0.50) | 9.23 (1.19) | 0.0069 (0.0028) |

6–8 | 87 | 6.86 (0.48) | 11.46 (1.42) | 0.0185 (0.0047) |

8–10 | 90 | 8.99 (0.54) | 13.35 (2.02) | 0.0381 (0.0090) |

10–12 | 92 | 10.77 (0.54) | 14.86 (2.00) | 0.0617 (0.0126) |

12–14 | 86 | 12.83 (0.54) | 17.04 (2.09) | 0.0995 (0.0199) |

14–16 | 71 | 14.95 (0.56) | 19.02 (1.88) | 0.1492 (0.0236) |

16–18 | 55 | 16.83 (0.53) | 21.56 (2.70) | 0.2155 (0.0432) |

18–20 | 35 | 18.92 (0.61) | 24.92 (1.96) | 0.3203 (0.0451) |

20–22 | 35 | 20.83 (0.68) | 26.23 (2.32) | 0.3804 (0.0570) |

22–24 | 22 | 22.95 (0.63) | 26.20 (2.34) | 0.4671 (0.0864) |

24–26 | 8 | 24.54 (0.58) | 29.53 (1.40) | 0.5929 (0.0257) |

**Table 2.**Volumetric models tested to predict volume with bark up to 4 cm in diameter at the top of Eucalyptus clone trees.

No. | Author * | Model |
---|---|---|

1 | Husch | Ln(v) = β_{0} + β_{1}Ln(dbh) + ε_{i} |

2 | Brenac | Ln(v) = β_{0} + β_{1}Ln(dbh) + β_{2}dbh^{−1} + ε_{i} |

3 | Spurr | Ln(v) = β_{0} + β_{1}Ln(dbh^{2}h) + ε_{i} |

4 | Schumacher-Hall | Ln(v) = β_{0} + β_{1}Ln(dbh) + β_{2}Ln(h) + ε_{i} |

5 | Prodan | Ln(v) = β_{0} + β_{1}Ln(dbh) + β_{2}Ln^{2}(dbh) + β_{3}Ln(h) + β_{4}Ln^{2}(h) + ε_{i} |

_{i}= model parameters, where i: 0, 1, 2, 3, and 4; v = commercial volume with bark up to 4 cm in diameter at the top of the tree; h = total height (m); dbh = diameter at breast height (cm); and ε

_{i}~ N (0, σ

^{2}) = random error; * Models found in: Camolesi et al. [28]; Schneider and Tonini [29]; and Rolim et al. [30].

**Table 3.**Adjusted coefficient of determination (R

^{2}

_{adj}), residual standard error (S

_{y.x}, in m

^{3}), variation coefficient (VC%), Akaike’s Information Criterion (AIC), and parameters estimated (and standard errors) with significance in the t-test for the fitted models.

Model No. | R^{2}_{adj} | Sy.x | VC% | AIC | Estimated Parameters | ||||
---|---|---|---|---|---|---|---|---|---|

β_{0} | β_{1} | β_{2} | β_{3} | β_{4} | |||||

1 | 0.9582 | 0.02773 | 23.39 | −439.13 | −9.606008 * (±0.0361) | 2.855458 * (±0.0148) | |||

2 | 0.9642 | 0.02568 | 21.64 | −509.33 | −7.834555 * (±0.2062) | 2.329743 * (±0.0620) | −5.061533 * (±0.5809) | ||

3 | 0.9777 | 0.02028 | 17.10 | −742.82 | −10.67147 * (±0.0330) | 1.049066 * (±0.0043) | |||

4 | 0.9771 | 0.02055 | 17.32 | −756.89 | −10.50457 * (±0.0527) | 2.227204 * (±0.0332) | 0.875632 * (±0.0433) | ||

5 | 0.9894 | 0.01395 | 11.74 | −1051.54 | −10.39049 * (±0.2743) | 4.890782 * (±0.1599) | −0.598349 * (±0.0352) | −1.508759 * (±0.2963) | 0.471110 * (±0.0551) |

**Table 4.**Configurations, correlation coefficients between real and predicted values (r

_{yŷ}), and weighted value (WV) of the best artificial neural network in the generalization of the approaches in scenarios (a) and (b).

Scenario | Approach | Network | Neurons ^{1} | r_{yŷ} | Activation Function | WV | |
---|---|---|---|---|---|---|---|

HL ^{2} | OL ^{3} | ||||||

A | General | ANN 2 | 48–25–1 | 0.9977 | Exponential | H. tangent ^{4} | 5 |

A | Group A | ANN 5 | 30–3–1 | 0.9940 | Logistic | Logistic | 5 |

A | Group B | ANN 5 | 36–9–1 | 0.9955 | H. tangent | Identity | 4 |

A | Group C | ANN 2 | 22–3–1 | 0.9992 | Exponential | Exponential | 6 |

B | 6/class | ANN 5 | 19–30–1 | 0.9901 | H. tangent | Exponential | 5 |

B | 5/class | ANN 1 | 18–12–1 | 0.9933 | Exponential | Logistic | 8 |

B | 4/class | ANN 3 | 17–1–1 | 0.9912 | Logistic | Identity | 5 |

B | 3/class | ANN 3 | 16–9–1 | 0.9888 | Exponential | Exponential | 7 |

B | 2/class | ANN 2 | 15–4–1 | 0.9880 | Exponential | Exponential | 7 |

B | 1/class | ANN 5 | 13–1–1 | 0.9861 | Exponential | Exponential | 6 |

^{1}Number of neurons per layer;

^{2}Hidden layer;

^{3}Output layer;

^{4}Hyperbolic tangent.

**Table 5.**Correlation coefficient (r

_{yŷ}), root mean square error (RMSE%), bias, and error variance (EV) of the artificial neural networks and equations with Model 5 in scenario (a).

Method | Statistic | Approach | |||
---|---|---|---|---|---|

General | Group A | Group B | Group C | ||

ANN | r_{yŷ} | 0.9977 | 0.9940 | 0.9955 | 0.9992 |

RMSE% | 7.87 | 12.70 | 9.95 | 4.99 | |

Bias | 0.00187 | −0.00046 | −0.00176 | 0.00127 | |

EV | 0.00009 | 0.00031 | 0.00014 | 0.00003 | |

Model 5 | r_{yŷ} | 0.9952 | 0.9947 | 0.9959 | 0.9994 |

RMSE% | 11.08 | 11.83 | 9.59 | 8.32 | |

Bias | 0.00006 | −0.0005 | −0.00168 | −0.00275 | |

EV | 0.00018 | 0.00027 | 0.00013 | 0.00007 |

**Table 6.**Correlation coefficient (r

_{yŷ}), root mean square error (RMSE%), bias, and error variance (EV) of the artificial neural networks, and equations for Model 5 in the sample intensities of scenario (b).

Method | Statistic | Sample Intensity | |||||
---|---|---|---|---|---|---|---|

6/Class | 5/Class | 4/Class | 3/Class | 2/Class | 1/Class | ||

ANN | r_{yŷ} | 0.9901 | 0.9933 | 0.9912 | 0.9888 | 0.9880 | 0.9861 |

RMSE% | 12.12 | 10.33 | 11.73 | 14.46 | 14.31 | 17.09 | |

Bias | −0.00039 | 0.00154 | −0.00099 | −0.00136 | 0.00222 | 0.00339 | |

EV | 0.00005 | 0.00003 | 0.00005 | 0.00007 | 0.00011 | 0.00009 | |

Model 5 | r_{yŷ} | 0.9914 | 0.9913 | 0.9910 | 0.9907 | 0.9871 | 0.9843 |

RMSE% | 11.78 | 12.17 | 12.42 | 12.32 | 14.33 | 16.21 | |

Bias | −0.00001 | 0.00051 | 0.00047 | 0.00009 | 0.00001 | −0.00095 | |

EV | 0.00005 | 0.00005 | 0.00005 | 0.00005 | 0.00007 | 0.00009 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tavares Júnior, I.d.S.; Rocha, J.E.C.d.; Ebling, Â.A.; Chaves, A.d.S.; Zanuncio, J.C.; Farias, A.A.; Leite, H.G.
Artificial Neural Networks and Linear Regression Reduce Sample Intensity to Predict the Commercial Volume of *Eucalyptus* Clones. *Forests* **2019**, *10*, 268.
https://doi.org/10.3390/f10030268

**AMA Style**

Tavares Júnior IdS, Rocha JECd, Ebling ÂA, Chaves AdS, Zanuncio JC, Farias AA, Leite HG.
Artificial Neural Networks and Linear Regression Reduce Sample Intensity to Predict the Commercial Volume of *Eucalyptus* Clones. *Forests*. 2019; 10(3):268.
https://doi.org/10.3390/f10030268

**Chicago/Turabian Style**

Tavares Júnior, Ivaldo da Silva, Jonas Elias Castro da Rocha, Ângelo Augusto Ebling, Antônio de Souza Chaves, José Cola Zanuncio, Aline Araújo Farias, and Helio Garcia Leite.
2019. "Artificial Neural Networks and Linear Regression Reduce Sample Intensity to Predict the Commercial Volume of *Eucalyptus* Clones" *Forests* 10, no. 3: 268.
https://doi.org/10.3390/f10030268