The experience of using an artificial neural network for approximating the average height and average diameter of 187 pine stand of various ages (from 7 to 120 years) and density (from 0.4 to 10.7 thousand pieces / ha) is described in the article. As an object of research, there are pure pine stands growing in the ribbon burs of the Altai Krai territory and the Republic of Kazakhstan. All considered stands grow in dry forest growing conditions and have a different origin. Approximation of the data was carried out using the Neural Network Toolbox, which is part of the MATLAB software package. A two-layer neural network with a direct connection, a hidden layer of sigmoid-type neurons and linear output neurons was used in the course of the work. The number of neurons in the hidden layer of the network was chosen experimentally and was chosen equal to five. The aim of the work was to create a mathematical model that allows to determine the average height and average diameter of pine stand of a certain age and density. The article provides a table of the approximated values of the above taxation indices. A comparison of the approximating ability of an artificial neural network and the Mitcherlich function is made, based on the data of absolute and average approximation errors. The conclusion is drawn that the artificial neural network coped with the approximation of the taxation indices better than it was possible to do with the help of the Mitcherlich function. However, the model obtained does not describe the initial data, since the allowable limit of the mean error of approximation was exceeded.
stands under consideration; felling of various (from 9 to 30%) intensity was carried out in 37 stands; there are no data on logging in 22 stands.
The taxation characteristics of 93 pine stands were obtained by the authors of the article using the method of trial
plots and common methods [
12000 10000
The data was approximated using the Neural Network Toolbox, which is part of the MATLAB software package. In the course of the work, a two-layer network with a direct link, a hidden layer of sigmoidal type neurons and linear output neurons was applied. This type of network is suitable for multidimensional mapping tasks, when specifying consistent data and a sufficient number of neurons in the hidden layer. Bayesian regularization is the used algorithm training of ANN. This algorithm usually requires more time, but can lead to a good generalization for complex, small or "noisy" data sets. The training of the ANN is stopped in accordance with the adaptive minimization of weight (regularization) [19].
The choice of the optimal number of neurons in the hidden layer is an important stage of the work, since too few of them will reduce the accuracy of the ANN, and too many can lead to an imaginary increase in the accuracy and deterioration of the network's generalizing ability [20]. In accordance with the Kolmogorov theorem [10], when creating an ANN with one hidden layer, 2N + 1 neurons are sufficient, where N is the number of input neurons. In practice, networks with one hidden layer with the number of neurons from N to 3N are more often used [12]. In this research work, N = 2, since the mean diameter and average height are approximated depending on 2 factors: age and density of the stands.
Selection of the optimal value of neurons in the hidden layer was made by repeated repetitions of the ANN learning process with different numbers of neurons. At the end of the training process, the following indices were evaluated: mean square error, the correlation coefficient, and the model's conformity to the biological features of the object of study. As a result, it was decided to use ANN with 5 neurons in the hidden layer. The root-mean-square error of the ANN for the training and test data sets was 4.5 and 4.3, respectively. The correlation coefficient (R) for both samples was 0.9. Figure 4 shows the scheme of the used neural network.
As a result of training ANN, a mathematical model was obtained, which allows to determine the average height and average diameter of pine stand of a certain age and density. With the help of the latter, Table 1 was compiled. 10,0 2,0 1,2 4,5 3,7 6,6 5,3 8,2 6,0 9,4 6,0 _ _ _ _ _ _ _ 0,5 2,0 3,8 5,5 7,8 8,7 11,2 11,3 13,7 13,4 15,6 15,0 16,8 16,1 17,8 16,9 18,7 17,3 19,6 17,6 20,6 17,8 21,7 17,8 23,0 1,0 2,0 3,3 5,4 7,3 8,5 10,5 11,1 13,0 13,1 14,6 14,6 15,8 15,7 16,7 16,4 17,5 16,9 18,3 17,1 19,3 17,3 20,3 17,4 21,5 where is the maximum value of height (diameter), m (cm); e is the mathematical constant; a and b are the coefficients of the equation; T is the age of the stand, years.
Grades of absolute error values modulo, m (cm)
The data in Table 2 indicate that the ANN has better coped with the problem of approximating the mean height and average diameter of the stand by 2.5 and 2.8%, respectively. However, the permissible limit of the mean error of approximation (10-15%) was still exceeded, which indicates that the model for describing the initial data is not good enough.
To increase the accuracy of the mathematical model of ANN, it is necessary to increase the amount of data and the number of input neurons. For example, variables such as the origin of the stand, its position relative to the terrain, the slope exposition, the presence and intensity of cutting, the area of growth and other factors affecting the average diameter and height of the stand might be added.
Conclusions 1. The artificial neural network coped with the approximation of taxation indices better than it was possible to do with the help of the Mitcherlich function.
2. The model obtained does not describe the initial data sufficiently well, since the permissible limit of the mean error of approximation (10-15%) was exceeded.
3. To increase the accuracy of the mathematical model of the artificial neural network, to approximate the average height and average diameter of the stands, it is necessary to increase the amount of data and the number of input neurons.
4. There is a need to compile a database containing as accurate and detailed a description of pine stands of belt forest.
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