# Stem Taper Approximation by Artificial Neural Network and a Regression Set Models

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

- (1)
- to compare the modelling techniques with respect to their performance to estimate stem profile and tree volume;
- (2)
- to rank the modelling techniques according to predictive performance for various tree species;
- (3)
- to find a modelling technique that combines estimating of stem taper shape for many tree species into one model.

## 2. Data and data Preprocessing

## 3. Methods

#### 3.1. Models

#### 3.2. Methods Used in Models

- root mean squared error, $\mathit{RMSE}=\sqrt{\frac{\sum {\left(\right)open="("\; close=")">{y}_{i}-{{y}^{\prime}}_{i}}^{}2}{}n}$,
- mean error/bias, $\mathit{ME}=\frac{\sum \left(\right)open="("\; close=")">{y}_{i}-{{y}^{\prime}}_{i}}{}n$
- model efficiency, $\mathit{EF}=1-\frac{\sum {\left(\right)open="("\; close=")">{y}_{i}-{{y}^{\prime}}_{i}}^{}2}{}\sum {\left(\right)open="("\; close=")">{{y}^{\prime}}_{i}-{\overline{{y}_{i}}}^{\prime}}^{}2$,

## 4. Results

#### 4.1. Diameter Estimates

#### 4.2. Stem Volume Estimates

#### 4.3. The Models Ranking

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Map of the locations of sample plots. On each sample plot 8 trees have been cut for sectional measurement.

**Figure 2.**From left to right: (

**A**) the stem taper equation (STE) model, (

**B**) the regression tree (REG) model, and (

**C**) the artificial neural network (ANN) model.

Species | Total | In a Training Set | In a Testing Set |
---|---|---|---|

Silver birch | 234 | 139 | 95 |

Common beech | 430 | 257 | 173 |

Common oak | 479 | 286 | 193 |

Silver fir | 219 | 130 | 89 |

European larch | 262 | 156 | 106 |

Black alder | 270 | 161 | 109 |

Scots pine | 504 | 301 | 203 |

Norway spruce | 458 | 274 | 184 |

Total | 2856 |

**Table 2.**Summary statistics of diameter at breast height (dbh) and total tree height (H) for different tree species.

Species | dbh (cm) | H (m) | ||||||
---|---|---|---|---|---|---|---|---|

min | max | avg | std | min | max | avg | std | |

Silver birch | 2.10 | 51.00 | 20.81 | 10.50 | 5.13 | 38.30 | 20.90 | 6.69 |

Common beech | 0.50 | 66.25 | 26.36 | 16.10 | 2.68 | 42.05 | 23.31 | 10.33 |

Common oak | 16.00 | 79.20 | 28.77 | 14.68 | 2.75 | 36.40 | 22.21 | 7.38 |

Silver fir | 0.70 | 66.85 | 26.08 | 15.26 | 1.58 | 37.50 | 20.96 | 9.30 |

European larch | 0.50 | 68.60 | 31.40 | 13.44 | 2.73 | 40.70 | 25.85 | 7.99 |

Black alder | 3.15 | 46.00 | 20.75 | 9.64 | 6.66 | 31.27 | 20.59 | 6.36 |

Scots pine | 0.30 | 65.15 | 25.96 | 11.75 | 1.35 | 35.21 | 22.43 | 7.42 |

Norway spruce | 0.80 | 76.40 | 26.45 | 13.85 | 1.75 | 39.54 | 22.55 | 9.03 |

**Table 3.**Summary statistics of stem diameters (d) and volume (V) estimation using three methods for different tree species.

d | V | |||||
---|---|---|---|---|---|---|

RMSE | ME | EF | RMSE | ME | EF | |

Artificial neural network model (ANN) | ||||||

Birch | 1.5819 | 0.0115 | 0.9696 | 0.1062 | 0.0083 | 0.9517 |

Beech | 2.6475 | −0.2433 | 0.9643 | 0.2817 | 0.0551 | 0.9498 |

Oak | 1.8229 | 0.1431 | 0.9812 | 0.1245 | −0.0150 | 0.9861 |

Fir | 1.7000 | 0.3868 | 0.9819 | 0.1687 | −0.0355 | 0.9761 |

Larch | 1.3573 | 0.1446 | 0.9887 | 0.1137 | −0.0054 | 0.9904 |

Alder | 1.5848 | −0.6322 | 0.9707 | 0.1111 | 0.0481 | 0.9488 |

Pine | 1.1928 | 0.2328 | 0.9865 | 0.0970 | −0.0224 | 0.9799 |

Spruce | 1.5930 | 0.1525 | 0.9787 | 0.1631 | −0.0195 | 0.9649 |

Average | 1.6850 | 0.0245 | 0.9777 | 0.1457 | 0.0017 | 0.9685 |

Regression set model (REG) | ||||||

Birch | 1.6394 | 0.0253 | 0.9674 | 0.1084 | 0.0127 | 0.9497 |

Beech | 2.6596 | 0.1650 | 0.9640 | 0.2638 | 0.0431 | 0.9560 |

Oak | 1.8651 | −0.2855 | 0.9803 | 0.1279 | −0.0274 | 0.9853 |

Fir | 1.7405 | −0.1368 | 0.9810 | 0.1696 | −0.0072 | 0.9759 |

Larch | 1.4105 | −0.1825 | 0.9878 | 0.1395 | −0.0159 | 0.9856 |

Alder | 1.4277 | 0.0025 | 0.9762 | 0.0796 | 0.0051 | 0.9737 |

Pine | 1.2170 | −0.1370 | 0.9859 | 0.0957 | −0.0104 | 0.9804 |

Spruce | 1.8815 | −0.1215 | 0.9703 | 0.2121 | −0.0162 | 0.9407 |

Average | 1.7302 | −0.0838 | 0.9766 | 0.1496 | −0.0020 | 0.9684 |

Variable exponent taper equation model Kozak (2004) | ||||||

Birch | 2.9160 | 0.0773 | 0.9646 | 0.1097 | 0.0157 | 0.9485 |

Beech | 8.0456 | 0.3021 | 0.9590 | 0.2706 | 0.0424 | 0.9537 |

Oak | 3.9225 | −0.1914 | 0.9778 | 0.1221 | −0.0224 | 0.9866 |

Fir | 3.3972 | −0.1200 | 0.9787 | 0.1784 | −0.0135 | 0.9733 |

Larch | 2.5084 | −0.1402 | 0.9846 | 0.1271 | −0.0087 | 0.9880 |

Alder | 2.4097 | 0.0847 | 0.9719 | 0.0814 | 0.0073 | 0.9725 |

Pine | 1.8411 | −0.1064 | 0.9825 | 0.0925 | −0.0114 | 0.9817 |

Spruce | 3.6950 | −0.0484 | 0.9690 | 0.2045 | −0.0137 | 0.9449 |

Average | 3.5919 | −0.0178 | 0.9735 | 0.1483 | −0.0005 | 0.9686 |

**Table 4.**Ranks of estimation errors for the diameters and volume for the three models on testing sets.

Model | RMSE | ME | EF |
---|---|---|---|

Diameters at relative heights | |||

ANN | 9 | 18 | 10 |

Kozak (2004) | 24 | 14 | 23 |

REG | 15 | 16 | 15 |

Tree volume | |||

ANN | 15 | 18 | 15 |

Kozak (2004) | 16 | 15 | 16 |

REG | 17 | 15 | 17 |

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**MDPI and ACS Style**

Socha, J.; Netzel, P.; Cywicka, D.
Stem Taper Approximation by Artificial Neural Network and a Regression Set Models. *Forests* **2020**, *11*, 79.
https://doi.org/10.3390/f11010079

**AMA Style**

Socha J, Netzel P, Cywicka D.
Stem Taper Approximation by Artificial Neural Network and a Regression Set Models. *Forests*. 2020; 11(1):79.
https://doi.org/10.3390/f11010079

**Chicago/Turabian Style**

Socha, Jaroslaw, Pawel Netzel, and Dominika Cywicka.
2020. "Stem Taper Approximation by Artificial Neural Network and a Regression Set Models" *Forests* 11, no. 1: 79.
https://doi.org/10.3390/f11010079