This work aimed to model the growth and yield of
Individual tree models are constituted by a set of submodels that estimate diameter and height growth as well as mortality probability through tree- and stand-related variables and through competition data [
Since Newnham [
Typically, estimates of equation parameters for the models are derived from linear and nonlinear regressions [
Artificial neural networks (ANNs) are a type of artificial intelligence system similar to human brain, having a computational capability which is acquired through learning [
Artificial neurons are processing units composed of an activation function which is also known as transfer function. This function is applied to a linear combination among the input variables and weights that reach a given neuron and then returns an output value [
The architecture of an artificial neural network refers to how neurons are organized. They comprise three layers: an input layer where variables are introduced to the network, a hidden or intermediate layer where most of the signal processing occurs, and an output layer where the end result is completed and presented.
MLP (Multilayer Perceptron) is the most widely used artificial neural network model for predicting continuous variables. In MLP training, network functioning is as follows: each neuron is connected to every neuron in the subsequent layer; there are no connections between neurons in the same layer; it has at least one processing hidden layer and a high connectivity level between neurons, which is defined by synaptic weights.
The input layer distributes the inputs to subsequent layers. Input nodes have liner activation functions and no thresholds. Each hidden unit node and each output node have thresholds associated with them in addition to the weights. The hidden unit nodes have nonlinear activation functions and the outputs have linear activation functions. Hence, each signal feeding into anode in a subsequent layer has the original input multiplied by a weight with a threshold added and then is passed through an activation function that may be linear or nonlinear (hidden units) [
According to this architecture, an approximation was obtained of an unknown function
The ANN method was successfully used for modeling regular mortality of individual trees of
In Brazil, despite there being more than 4 million hectares planted with
Data were obtained from 63 permanent plots, approximately 500 m² in area, containing clonal stands of
Characteristics of
Age (months) | 24–72 |
Diameter at breast height: |
4.0–29.4 |
Average diameter: |
7.3–18.4 |
Total height: Ht (m) | 8.5–34.1 |
Dominant height: Hd (m) | 13.1–34.8 |
Basal area (m² ha−1) | 4.7–27.2 |
Volume (m³ ha−1) | 23.8–353.9 |
Density (trees ha−1) | 760–1180 |
Equations to estimate individual volume with bark.
Area code | Equation |
---|---|
Stratum 001 |
|
Stratum 016 |
|
Stratum 026 |
|
Stratum 041 |
|
Stratum 042 |
|
Stratum 077 |
|
Study site. Source: adapted from Silva [
In every permanent plot, measurements were taken of the diameter at breast height (d.b.h.) using a measuring tape, and total height (Ht) of each tree, using a digital hypsometer, for five annual measurements (24, 36, 48, 60, and 72 months). The last measurement is equivalent to the age of cutting forest. The volume with bark (
The set of permanent plots was randomly divided into two groups. The first group was composed of 33 plots (11 in each productive capacity class) and was used for artificial neural network training. In five measurements, this group totaled 8,735 cases (measurements of individual trees), as each plot had 53 trees on average throughout this evaluation. The second group was composed of the remaining 30 plots (10 in each productive capacity class) and was used for model validation. This group totaled 7,756 cases.
The annual mortality probability (
Distance-independent competition indices (
Five hundred artificial neural networks were trained for the following output variables: annual mortality probability (
Output and input variables for neural network training.
Output variable | Input variables | Number of trained networks |
---|---|---|
|
|
100 |
|
100 | |
|
100 | |
|
100 | |
|
100 | |
| ||
Total | 500 | |
| ||
|
|
100 |
|
100 | |
|
100 | |
|
100 | |
|
100 | |
| ||
Total | 500 | |
| ||
|
|
100 |
|
100 | |
|
100 | |
|
100 | |
|
100 | |
| ||
Total | 500 |
For the training of artificial neural networks, software Statistica 8.0 [
MLP networks are feedforward multilayer networks having one or more layers of neurons between the input and output layers, known as hidden layer [
The feedforward type of training was used, by the supervised method. In this procedure, the data flow algorithm moves in only one, noncyclic direction, to initially define the synapse weights, excluding the input variables with low-weight synapses, while the supervised method indicates the input and output variables [
The training stages, such as preprocessing, actual training, with selection of architectures and stopping methods, and postprocessing, were performed by the optimization tool Intelligent Problem Solver (IPS), from software Statistica. 500 networks were initially trained in order to estimate each variable. Without precise and accurate networks, this number would be increased.
This software normalizes data in the range 0-1 and tests various architectures and network models. In the supervised method, input and output variables are set by the user. In the feedforward procedure, the data flow algorithm moves in only one, noncyclic direction, to initially determine the synapse weights. The back-propagation algorithm corrects the initial synapse weights so as to minimize prediction error. Therefore, in this process, initial input variables can be excluded during training for not helping (low synapse weight) minimize prediction error [
The definition of network architecture, that is, number of neurons per layer, number of layers, and parameterization was optimized by the tool Intelligent Problem Solver, from software Statistica.
The selection of best network for each output variable was based on the following criteria [
The validation of selected networks was done by annually projecting the tree mortality, the height and diameter of living trees, and the volume per hectare of plots until age of 72 months, according to the flowchart of basic steps and decisions for an individual tree model [
To verify network behavior under different growing conditions, the plots were divided into three productivity classes (high, medium, and low) based on site indices (
The tree mortality rule was the one that is used by Pretzsch et al., [
In order to evaluate the accuracy of mortality estimates, a graphical analysis was performed of the estimated number of surviving trees in relation to observations. For the variables height and diameter, the
The network with the best performance to estimate annual mortality probability
Architecture of artificial neural networks trained to obtain the mortality probability, height and diameter at future ages, and respective statistics.
ANN | Input variables | Number of neurons (Layers) |
|
CV % | RMSE | Bias | Bias % | AMD | ||
---|---|---|---|---|---|---|---|---|---|---|
Input | Hidden | Output | ||||||||
Mortality probability |
||||||||||
| ||||||||||
|
|
4 | 9 | 1 | 0.744 | 43.1 | 0.034 | 0.0004 | −16.91 | 0.0255 |
|
IID2 | 1 | 3 | 1 | 0.579 | 52.3 | 0.041 | 0.0051 | −15.84 | 0.0305 |
|
|
4 | 8 | 1 | 0.799 | 38.6 | 0.030 | 0.0009 | −12.73 | 0.0235 |
|
|
4 | 5 | 1 | 0.711 | 45.2 | 0.037 | 0.0112 | 4.01 | 0.0286 |
|
|
3 | 4 | 1 | 0.790 | 39.3 | 0.031 | 0.0000 | −16.34 | 0.0244 |
| ||||||||||
Height (Ht2) | ||||||||||
| ||||||||||
|
|
3 | 6 | 1 | 0.993 | 3.0 | 0.626 | −0.0011 | −0.12 | 0.4370 |
|
|
5 | 6 | 1 | 0.993 | 2.9 | 0.592 | −0.0089 | −0.17 | 0.4069 |
|
|
5 | 7 | 1 | 0.994 | 2.6 | 0.532 | 0.0001 | −0.07 | 0.4048 |
|
|
5 | 7 | 1 | 0.995 | 2.5 | 0.526 | −0.0028 | −0.10 | 0.3979 |
|
|
5 | 9 | 1 | 0.993 | 2.9 | 0.608 | 0.0028 | −0.14 | 0.4222 |
| ||||||||||
Diameter ( |
||||||||||
| ||||||||||
|
|
4 | 5 | 1 | 0.990 | 4.6 | 0.618 | −0.0048 | −0.31 | 0.4089 |
|
|
5 | 7 | 1 | 0.990 | 4.4 | 0.601 | −0.0118 | −0.42 | 0.3986 |
|
|
3 | 4 | 1 | 0.993 | 3.8 | 0.523 | 0.0008 | −0.31 | 0.3913 |
|
|
5 | 5 | 1 | 0.993 | 3.7 | 0.508 | −0.0021 | −0.25 | 0.3821 |
|
|
5 | 6 | 1 | 0.988 | 4.9 | 0.667 | 0.0013 | −0.28 | 0.5049 |
Mortality probability, height and diameter, as observed and estimated by artificial neural network training.
The best networks for the variables height and diameter were
The projection of the number of surviving trees per hectare indicates a slight underestimation bias in the validation plots (Figure
Histogram of residuals by the artificial neural network model for independent data (generalization).
The neural networks selected for the variables diameter and height provided accurate estimates, regardless of the productivity class (Table
Statistics of height (Ht) and diameter (
Productive capacity class | Age (months) | Variable | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Ht (m) |
| ||||||||||||
|
CV % (±) | RMSE | Bias | Bias % | AMD |
|
CV % (±) | RMSE | Bias | Bias % | AMD | ||
High ( |
36 | 0.979 | 3.5 | 0.707 | −0.080 | −0.489 | 0.456 | 0.976 | 5.5 | 0.767 | −0.081 | −0.856 | 0.516 |
48 | 0.965 | 4.6 | 1.040 | 0.027 | 0.003 | 0.741 | 0.961 | 7.1 | 1.065 | −0.016 | −0.443 | 0.752 | |
60 | 0.953 | 5.1 | 1.272 | −0.223 | −1.067 | 0.930 | 0.945 | 8.5 | 1.379 | −0.184 | −1.642 | 0.961 | |
72 | 0.934 | 5.9 | 1.540 | −0.153 | −0.784 | 1.159 | 0.925 | 10.3 | 1.748 | −0.211 | −1.890 | 1.248 | |
| |||||||||||||
Medium ( |
36 | 0.981 | 3.4 | 0.621 | 0.090 | 0.494 | 0.453 | 0.981 | 4.8 | 0.624 | 0.147 | 1.246 | 0.461 |
48 | 0.957 | 5.1 | 1.091 | 0.219 | 1.017 | 0.848 | 0.962 | 7.0 | 0.971 | 0.135 | 1.174 | 0.754 | |
60 | 0.938 | 6.3 | 1.437 | 0.170 | 0.635 | 1.154 | 0.940 | 9.0 | 1.345 | 0.137 | 0.964 | 1.067 | |
72 | 0.921 | 7.1 | 1.747 | 0.149 | 0.369 | 1.425 | 0.922 | 10.7 | 1.685 | 0.162 | 0.965 | 1.351 | |
| |||||||||||||
Low ( |
36 | 0.966 | 4.1 | 0.635 | 0.025 | 0.101 | 0.486 | 0.975 | 5.4 | 0.565 | 0.061 | 0.559 | 0.416 |
48 | 0.947 | 5.7 | 1.031 | 0.216 | 1.015 | 0.811 | 0.953 | 7.7 | 0.924 | 0.265 | 2.163 | 0.735 | |
60 | 0.921 | 7.0 | 1.440 | 0.442 | 2.017 | 1.138 | 0.929 | 9.6 | 1.266 | 0.414 | 3.138 | 1.012 | |
72 | 0.903 | 7.8 | 1.684 | 0.420 | 1.642 | 1.327 | 0.909 | 11.0 | 1.507 | 0.427 | 2.806 | 1.201 |
Heights and diameters, as observed and estimated per age and productivity class (high, medium, and low) by the artificial neural network model for independent data (generalization).
Estimates of volume outside bark per hectare, as projected until age of 72 months (Figure
Estimates of volume outside bark (m3 ha−1) per age (36–72) and productivity class (high, medium, and low). Values on the bars indicate the difference.
In recent decades, a major concern in the field of forest mensuration has been to develop growth and yield models using individual trees [
A widely used resource in this type of modeling is regression analysis through linear and nonlinear functions [
ANNs are increasingly becoming a popular tool in forest mensuration [
A large number of authors have discussed the structure, technique, and operation of ANNs [
As far as results found in this study are concerned, the competition index of the network selected as best for mortality probability (
Studies involving individual tree mortality through regression models usually estimate mortality probability only as a function of the competition index as independent variable [
Network input variables for estimating height and diameter include the
Other than the competition index, tree- and stand-related variables such as age and site index are also widely used to express height and diameter growth at the individual tree level, using linear and nonlinear regression [
Martins [
In the validation at the stand level, the percentage errors using artificial neural networks were in the order of 0.5%, while those using regression models were in the order of 6% for total volume per hectare.
Since the same data were used, with the same methodology, it can be said that the tool artificial neural networks is effective in individual tree modeling and provides superior results if compared to regression models, particularly in the model generalization or validation stage.
Considering the results found, artificial neural networks should be studied for describing the structure and dynamics of natural tropical forests, with all of their complexity resulting from size and species diversity [
Results in this study confirm the use potential of MLP artificial neural networks, through the supervised learning method, for individual tree modeling of commercial
The authors wish to thank Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPQ), Brazil, for granting the scholarship and Monica Castellani for English revision.