# Do AI Models Improve Taper Estimation? A Comparative Approach for Teak

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}), Root Mean Square of Error (RMSE), Mean Error of Bias (MBE) and Mean Absolute Error (MAE) statistical indices were used to evaluate the models’ performance. Goodness of fit criterion of all the models suggests that Kozak’s model shows the best results, closely followed by the ANN model. However, PG, PGR and CatBoost outperformed the Fang model. Artificial intelligence methods can be an effective alternative to describe the shape of the stem in Tectona grandis trees with an excellent accuracy, particularly the ANN and CatBoost models.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Data and Data Preprocessing

#### 2.3. Artificial Intelligence Models

#### 2.3.1. Genetic Programming

#### 2.3.2. Gaussian Process Regression (GPR)

#### 2.3.3. Category Boosting (CatBoost)

_{i}} i = 1,…, n, where X

_{i}= (x

_{i}, 1, …, x

_{i}, m) is a vector of m characteristics, the category of the k-th training example can be replaced by a numerical characteristic expressed in Equation (5) according to the requested TS. The substitution of a given categorical example $x{\sigma}_{p,k}$, k can be obtained by calculating its average value with the same category value placed before in a random permutation of data set σ = (σ

_{1}, …, σ

_{n}). In addition, CatBoost can combine various categorical features into a new one in a greedy way by establishing a new tree split.

#### 2.3.4. Artificial Neural Networks (ANN)

_{ij}and W

_{jk}, and each unit adds its entries, adding a bias or threshold term to the sum and the non-linearity, transforming the sum to produce an exit. This non-linear transformation is called a node activation function. The nodes of the exit layer tend to have linear activations. In MLP, the logistic sigmoid function (Equation (7)) and the linear function (Equation (8)) are generally used in the hidden and exit layer, respectively.

^{®}software version R2019a (Mathworks Inc, Natik MA, USA).

#### 2.4. Non-Linear Regression Models

#### 2.5. Goodness of Fit of the Models

^{2}; Equation (12)), Root-Mean-Square Error (RMSE; Equation (13)), Mean Bias Error (MBE; Equation (14)) and Mean Absolute Error (MAE; Equation (15)).

^{2}is closest to unity and the other criteria are closest to zero.

## 3. Results and Discussion

^{2}= 0.985, RMSE = 1.070, MAE = 0.746 and MBE = −0.063). Regarding the artificial intelligence models for estimating stem diameters, these were capable of describing the diametric profile with accuracy. Particularly, the ANN model obtained the best performance with relation to other models evaluated, followed by the CatBoost model. The GP model performed the lowest in comparison to the other artificial intelligence models. The results obtained using the ANN model were close enough to those obtained using Kozak’s model; however, in terms of the RMSE and MAE statistics the differences are negligible (Table 4).

_{1}to a height h

_{2}, which can be executed without difficulty in a spreadsheet (numerical integrating this function).

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Perez, D. Growth and Volume Equations Developed from Stem Analysis for Tectona Grandis in Costa Rica. J. Trop. For. Sci.
**2008**, 20, 66–75. [Google Scholar] - Moret, A.; Jerez, M.; Mora, A. Determinación de Ecuaciones de Volumen Para Plantaciones de Teca (Tectona Grandis L.) En La Unidad Experimental de La Reserva Forestal Caparo, Estado Barinas–Venezuela. Rev. For. Venez.
**1998**, 42, 41–50. [Google Scholar] - CONAFOR www.gob.mx/conafor/. Available online: https://www.gob.mx/conafor/documentos/plantaciones-forestales-comerciales-27940/ (accessed on 1 July 2020).
- Kozak, A. My Last Words on Taper Equations. For. Chron.
**2004**, 80, 507–515. [Google Scholar] [CrossRef] - Fang, Z.; Borders, B.E.; Bailey, R.L. Compatible Volume-Taper Models for Loblolly and Slash Pine Based on a System with Segmented-Stem Form Factors. For. Sci.
**2000**, 46, 1–12. [Google Scholar] - Quiñonez-Barraza, G.; los Santos-Posadas, D.; Héctor, M.; Álvarez-González, J.G.; Velázquez-Martínez, A. Sistema Compatible de Ahusamiento y Volumen Comercial Para Las Principales Especies de Pinus En Durango, México. Agrociencia
**2014**, 48, 553–567. [Google Scholar] - Pompa-García, M.; Corral-Rivas, J.J.; Ciro Hernández-Díaz, J.; Alvarez-González, J.G. A System for Calculating the Merchantable Volume of Oak Trees in the Northwest of the State of Chihuahua, Mexico. J. For. Res.
**2009**, 20, 293–300. [Google Scholar] [CrossRef] - Cruz-Cobos, F.; los Santos-Posadas, D.; Héctor, M.; Valdez-Lazalde, J.R. Sistema Compatible de Ahusamiento-Volumen Para Pinus Cooperi Blanco En Durango, México. Agrociencia
**2008**, 42, 473–485. [Google Scholar] - Tamarit, U.J.C.; De los Santos Posadas, H.M.; Aldrete, A.; Valdez Lazalde, J.R.; Ramírez Maldonado, H.; Guerra De la Cruz, V. Sistema de Cubicación Para Árboles Individuales de Tectona Grandis L. f. Mediante Funciones Compatibles de Ahusamiento-Volumen. Rev. Mex. Cienc. For.
**2014**, 5, 58–74. [Google Scholar] - Schikowski, A.B.; Corte, A.P.; Ruza, M.S.; Sanquetta, C.R.; Montano, R.A. Modeling of Stem Form and Volume through Machine Learning. An. Acad. Bras. Cienc.
**2018**, 90, 3389–3401. [Google Scholar] [CrossRef] [PubMed] - Nunes, M.H.; Görgens, E.B. Artificial Intelligence Procedures for Tree Taper Estimation within a Complex Vegetation Mosaic in Brazil. PLoS ONE
**2016**, 11, e0154738. [Google Scholar] [CrossRef] - Sakici, O.; Ozdemir, G. Stem Taper Estimations with Artificial Neural Networks for Mixed Oriental Beech and Kazdaği Fir Stands in Karabük Region, Turkey. Cerne
**2018**, 24, 439–451. [Google Scholar] [CrossRef] - Socha, J.; Netzel, P.; Cywicka, D. Stem Taper Approximation by Artificial Neural Network and a Regression Set Models. Forest
**2020**, 11, 79. [Google Scholar] [CrossRef] - Koza, J.R. Introduction to Genetic Programming. In Proceedings of the 9th Annual Conference Companion on Genetic and Evolutionary Computation, London, UK, 7–11 July 2007; pp. 3323–3365. [Google Scholar]
- Rasmussen, C.E. Gaussian Processes for Machine Learning. In Summer School Machine Learning; Springer: Berlin/Heidelberg, Germany, 2003; pp. 63–71. [Google Scholar]
- Jamei, M.; Ahmadianfar, I.; Olumegbon, I.A.; Karbasi, M.; Asadi, A. On the Assessment of Specific Heat Capacity of Nanofluids for Solar Energy Applications: Application of Gaussian Process Regression (GPR) Approach. J. Energy Storage
**2021**, 33, 102067. [Google Scholar] [CrossRef] - Samarasinghe, M.; Al-Hawani, W. Short-Term Forecasting of Electricity Consumption Using Gaussian Processes. Master’s Thesis, University of Agder, West Agdelshire, Norway, 2012. [Google Scholar]
- Williams, C.K.; Rasmussen, C.E. Gaussian Processes for Machine Learning; MIT Press: Cambridge, MA, USA, 2006; Volume 2, No. 3; p. 4. [Google Scholar]
- Prokhorenkova, L.; Gusev, G.; Vorobev, A.; Dorogush, A.V.; Gulin, A. CatBoost: Unbiased Boosting with Categorical Features. Adv. Neural Inf. Process. Syst.
**2018**, 31, 6638–6648. [Google Scholar] - R Foundation for Statistical Computing. R Core Team: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2022. [Google Scholar]
- Haykin, S.; Lippmann, R. Neural Networks, A Comprehensive Foundation. Int. J. Neural Syst.
**1994**, 5, 363–364. [Google Scholar] - Basheer, I.A.; Hajmeer, M. Artificial Neural Networks: Fundamentals, Computing, Design, and Application. J. Microbiol. Methods
**2000**, 43, 3–31. [Google Scholar] [CrossRef] - Esmaeelzadeh, S.R.; Adib, A.; Alahdin, S. Long-Term Streamflow Forecasts by Adaptive Neuro-Fuzzy Inference System Using Satellite Images and K-Fold Cross-Validation (Case Study: Dez, Iran). KSCE J. Civ. Eng.
**2015**, 19, 2298–2306. [Google Scholar] [CrossRef] - Borders, B.E. Systems of Equations in Forest Stand Modeling. For. Sci.
**1989**, 35, 548–556. [Google Scholar] - Durbin, J.; Watson, G.S. Testing for Serial Correlation in Least Squares Regression. I; Oxford University Press: Oxford, UK, 1992; pp. 237–259. [Google Scholar]

**Figure 4.**Residual distributions of the models (

**a**) Kozak 2004, (

**b**) Fang 2000, (

**c**) ANN, (

**d**) GPR, (

**e**) CatBoost, (

**f**) GP.

Variable | Maximum | Mean | Minimum | Standard deviation |
---|---|---|---|---|

Normal diameter D with bark (cm) | 45.00 | 26.89 | 8.50 | 6.81 |

Total height H of the tree (m) | 27.00 | 18.96 | 9.03 | 3.39 |

Commercial height (Hc) of the tree (m) | 18.15 | 10.82 | 2.62 | 2.60 |

Age (years) | 22.00 | 15.99 | 7.5 | 4.62 |

Parameter | Characteristic |
---|---|

Size of the population | 500 individuals |

Criterion of finishing | 100 generations |

Maximum size of the tree | 150 nodes, 12 levels |

Elites | 1 individual |

Parent selection | Selection per tournament |

Cross | Sub-tree, 90% of probability |

Mutation | 15% of mutation rate |

Function of evaluation | Coefficient of determination R^{2} |

Symbolic functions | (+, −, ×, ÷, exp, log) |

Symbolic terminals | Constant, weight × variable |

Model | Expression | Number of Equation |
---|---|---|

Fang 2000 | $d={C}_{1}\sqrt{{H}^{\frac{k-{b}_{1}}{{b}_{1}}}{\left(1-z\right)}^{\frac{\left(k-\beta \right)}{\beta}}{\alpha}_{1}{}^{{I}_{1}+{I}_{2}}{\alpha}_{2}{}^{{I}_{2}}}$ I _{1} = 1 if p_{1} ≤ z ≤ p_{2} otherwise I_{1} = 0I _{2} = 1 if p_{2} ≤ z ≤ 1 otheriwse I_{2} = 0${C}_{1}=\sqrt{\frac{{a}_{0}{D}^{{a}_{1}}{H}^{{a}_{2}-\frac{k}{{b}_{1}}}}{{b}_{1}\left({t}_{0}-{t}_{1}\right)+{b}_{2}\left({t}_{1}-{\alpha}_{1}{t}_{2}\right)+{b}_{3}{\alpha}_{1}{t}_{2}}}$ ${t}_{0}={\left(1-{p}_{0}\right)}^{\frac{k}{{b}_{1}}}$ ${p}_{0}=\frac{{h}_{0}}{H}$ ${t}_{1}={\left(1-{p}_{1}\right)}^{\frac{k}{{b}_{1}}}$ ${t}_{2}={\left(1-{p}_{2}\right)}^{\frac{k}{{b}_{2}}}$ ${\alpha}_{1}={\left(1-{p}_{1}\right)}^{\frac{\left({b}_{2}-{b}_{1}\right)k}{{b}_{1}{b}_{2}}}$ ${\alpha}_{2}={\left(1-{p}_{2}\right)}^{\frac{\left({b}_{3}-{b}_{2}\right)k}{{b}_{2}{b}_{3}}}$ $\beta ={b}_{1}{}^{1-\left({I}_{1}+{I}_{2}\right)}{b}_{2}{}^{{I}_{1}}{b}_{3}{}^{{I}_{2}}$ $z=\frac{h}{H}$ | (9) |

Kozak 2004 | $d={a}_{0}{D}^{{a}_{1}}{H}^{{a}_{2}}{\left(\frac{1-{z}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{1-{b}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\right)}^{\left[{b}_{1}{z}^{4}+{b}_{2}\left(\frac{1}{{e}^{\raisebox{1ex}{$D$}\!\left/ \!\raisebox{-1ex}{$H$}\right.}}\right)+{b}_{3}{\left(\frac{1-{z}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{1-{b}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\right)}^{0.1}+{b}_{4}\left(\frac{1}{D}\right)+{b}_{5}{H}^{\left(1-{z}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)}+{b}_{6}\left(\frac{1-{z}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{1-{b}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\right)\right]}$ b = 1.3/H | (10) |

**Table 4.**Summary statistics for diameter estimate along the stem (d) and DW parameter obtained by autocorrelation correction of the conventional models.

Model | R^{2} | RMSE (cm) | MBE (cm) | MAE (cm) | DW |
---|---|---|---|---|---|

Kozak2004 | 0.985 | 1.070 | −0.063 | 0.746 | 2.055 |

Fang2000 | 0.974 | 1.405 | −0.125 | 1.120 | 2.053 |

CatBoost | 0.978 | 1.299 | −0.038 | 0.920 | - |

GPR | 0.978 | 1.314 | −0.010 | 0.952 | - |

ANN | 0.985 | 1.085 | −0.082 | 0.751 | - |

PG | 0.977 | 1.343 | −0.098 | 0.964 | - |

Fang 2000 | Kozak 2004 | |||
---|---|---|---|---|

Parameter | Estimation | Standard Error | Estimation | Standard Error |

a_{0} | 0.000068 | 2.181 × 10^{−8} | 1.223695 | 0.0385 |

a_{1} | 1.928423 | 2.507 × 10^{−7} | 0.990858 | 0.0063 |

a_{2} | 0.854570 | 0.08590 | −0.05868 | 0.0132 |

b_{1} | 2.259 × 10^{−6} | 2.181 × 10^{−8} | 0.124234 | 0.0546 |

b_{2} | 9.93 × 10^{−6} | 2.507 × 10^{−7} | −1.10823 | 0.0765 |

b_{3} | 0.000034 | 2.264 × 10^{−7} | 0.406955 | 0.0151 |

b_{4} | - | - | 7.265247 | 0.5388 |

b_{5} | - | - | 0.113903 | 0.00364 |

b_{6} | - | - | −0.44487 | 0.0393 |

p_{1} | 0.016437 | 0.000183 | - | - |

p_{2} | 0.082406 | 0.00205 | - | - |

${\gamma}_{1}$ | 0.507385 | 0.0173 | 0.413999 | 0.0158 |

${\gamma}_{2}$ | 0.159728 | 0.0109 | 0.136901 | 0.0106 |

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**MDPI and ACS Style**

Fernández-Carrillo, V.H.; Quej-Chi, V.H.; De los Santos-Posadas, H.M.; Carrillo-Ávila, E.
Do AI Models Improve Taper Estimation? A Comparative Approach for Teak. *Forests* **2022**, *13*, 1465.
https://doi.org/10.3390/f13091465

**AMA Style**

Fernández-Carrillo VH, Quej-Chi VH, De los Santos-Posadas HM, Carrillo-Ávila E.
Do AI Models Improve Taper Estimation? A Comparative Approach for Teak. *Forests*. 2022; 13(9):1465.
https://doi.org/10.3390/f13091465

**Chicago/Turabian Style**

Fernández-Carrillo, Víctor Hugo, Víctor Hugo Quej-Chi, Hector Manuel De los Santos-Posadas, and Eugenio Carrillo-Ávila.
2022. "Do AI Models Improve Taper Estimation? A Comparative Approach for Teak" *Forests* 13, no. 9: 1465.
https://doi.org/10.3390/f13091465