^{1}

^{2}

^{*}

^{3}

Linear and nonlinear crown variable functions for 173 Brutian pine (

Crown characteristics are an important component of growth and yield models. Tree crown research contributes to several key forest ecosystem attributes: biodiversity, productivity, forest management, forest environment, and wildlife (

Tree stem shape has been commonly modeled using taper models (

For stem taper and volume predictions using regression analysis, an appropriate nonlinear function must first be identified, which is a very difficult task. The main reason that artificial neural network (ANN) applications have received attention is that the methodology is comparable to statistical modeling and ANNs can be seen as a complementary effort (without the restrictive assumption of a particular statistical model) or as an alternative approach to fitting nonlinear models to data. Due to the fact that Neural Networks (NNs) attempt to find the best nonlinear function based on the network’s complexity, without the constraint of pre-specified nonlinearity, we investigated their applicability in over-bark diameter and stem volume predictions using the same crown variables previously described.

ANNs have been successfully applied in the field of forest modeling. Among others, ANNs have been used for: (a) prediction of diameter distribution (

The objective of this study was to investigate the level of improvement in diameter over bark (

One hundred seventy three sample trees of Brutian pine were selected from natural even-aged managed stands in Bucak Forest Enterprise, southern Turkey, on lands owned by the Forest Service. Trees were felled through the clear-cutting of areas of the Bucak Forest Enterprise and were systematically sampled to cover the range of diameters within a stand, with emphasis on dominant and codominant individuals. Trees possessing multiple stems, broken tops, obvious cankers or crooked boles were not included in the sample. Total height was measured to the nearest 0.05 m. Diameter over-bark (D) at breast height (1.3 m) was measured and recorded to the nearest 0.1 cm using digital calipers. Diameter over-bark was measured at 0.3, 1.3, 2.3 m and then at intervals of 1 m along the remainder of the stem. In each section, two perpendicular diameters over-bark were measured and then arithmetically averaged. The height to base of the live crown was determined by identifying that point along the bole where the lowest live branch or branch whorl was attached to the main bole as indicated by

A scatter plot of relative diameter against relative height was examined visually to detect possible anomalies in data. Extreme data points were observed; therefore the systematic approach proposed by

All trees with total height less than 5.3 m were eliminated, as they could not be used to fit the modified

The modified form of the segmented polynomial model published by

Two crown variables, CL and CR were included into the best fitted model identified for Brutian pine.

where λ_{i} are the parameters to be estimated from data. _{1}, _{2}, _{3}, and _{4} to ascertain the effects of incorporating crown variables into the existing model forms for Brutian pine. The parameters (_{1}_{2}_{3} and _{4}) in the taper and volume equations were replaced with all combinations of CR, CL and CR with CL functions and were evaluated for model improvement.

In addition to the evaluation of the entire stem, model performance was examined using sectional relative height classes from 10% to 90% of total height. For the taper and volume model forms used in this study, the upper stem diameter at 5.30 m is a required input variable. Diameters at 5.30 m were obtained through actual field measurements.

Known advantages of ANNs over traditional approaches (

Furthermore, in the multilayer-perceptron learning step two different optimization algorithms were used: (1) the back-propagation (BP), which produces the back-propagation artificial neural network (BPANN) models (

The generalized regression neural network (GRNN) is one type of NN which was devised by

Appropriate input variables to the NN models can be selected in advance based on

For the development of the BPANN models, the effectiveness and convergence of training depends significantly on the values of learning rate (LR) and momentum factor (M). The numbers of neurons in the hidden layer of the ANNs were finalized after a trial and error procedure using different combinations of learning rates and momentum factors. Each combination of LR and M was tested for different numbers of hidden neurons. For the LMANN models development, the effectiveness and convergence of training depends significantly on the adjustment of the damping factor (

The statistics used to compare the models included the average bias (

where _{i} is the observed value for the _{i} is the predicted value for the _{i}_{i},

To concurrently minimize taper and volume errors, both equations were fitted simultaneously using SAS PROC MODEL (

Crown ratio (CR) and crown length (CL) were incorporated into the existing taper and volume equations. The parameters (_{1}, _{2}, _{3} and _{4}) for the existing taper and volume equations (_{4} with the linear

Model OM represents the original model forms without the addition of the crown variable functions, while models MCR and MCRCL represent the modified model after incorporating _{4} parameter, respectively. Based on the fit statistics for the OM, MCR, and MCRCL model forms (

For

For stem volume prediction, significant improvements were observed in the modified

The BPANN, LMANN and GRNN model fit statistics for

Results indicate that the inclusion of both crown variables (CR and CL) had positive effects in

Similarly to what noted with the model fitting data, the inclusion of both crown variables as input variables resulted in the most accurate

The SEE for predicting

As seen in

Validation results for all the models are shown in

The results obtained for the validation data set for all models were in agreement with their performance for the model fitting data set (

One of the underlying goals for efficient timber resources management is that of optimizing the prediction accuracy of constructed forest-data models. The performance of nonlinear regression models and NN models (BPANN, LMANN and GRNN models) were compared for the estimation of diameter over-bark and cubic meter volume based on Brutian pine data from southeast Turkey.

The incorporation of crown variables to regression and NN modeling procedures showed improvements in the accuracy for diameter and stem volume predictions in Brutian pine. Slightly better results were obtained for estimating stem form than stem volume, when employing taper and volume equations. As indicated by _{4} parameter was changed.

For environmental issues, such as forest modeling where the complexity of the natural problem is faced, it is very difficult to suggest a specific approach for a given problem. As pointed out by

In practical forestry, the application of NN models by practitioners can be achieved through the use of trained models that have been constructed by experts for this purpose. However, their use by practitioners requires computational skills but not

Accurate estimation of over-bark diameter and stem volume is crucial for the efficient management of forest resources. The inclusion of linear CR and linear CR with CL functions in existing segmented taper and cubic meter volume equations for Brutian pine in Turkey resulted in significant reduction of model sum of squared error. Prediction improvements for upper stem diameter and volume were greater for model forms with CR and CL than model forms with CR alone, though overall improvements were small. Similar results were obtained using the back-propagation, Levenberg-Marquardt and generalized regression neural network models. The incorporation of the crown variables to these models also exhibited improved performance.

Our results indicate that the nonlinear regression model had larger SEE and smaller FI values than the three types of NN models tested, when evaluating both

Implementation of the NN approaches does offer a number of advantages over the more traditional regression method of forest-data modeling and should be viewed as a useful alternative to this technique (

This study was supported by the Scientific and Technological Research Council of Turkey (TUBITAK-BIDEB).

Plot of relative height (h/H)

(a) The multilayer-perceptron (MLP) and (b) the generalized regression neural network (GRNN) architectures.

The standard errors of estimate (SEE) for estimating diameter over-bark for the taper compatible volume system (a) and for the Levenberg-Marquardt models (c) and volume over-bark along the stem for the taper compatible volume system (b) and for the Levenberg-Marquardt models (d), by relative height classes, using the fitting data.

The standard errors of estimate (SEE) and fit indexes (FI) for predicting diameter over-bark and volume over-bark along the stem for the taper and compatible volume system (OM) and for the back-propagation (BPANN_OM, BPANN_MCR and BPANN_MCRCL), Levenberg-Marquardt (LMANN_OM, LMANN_MCR and LMANN_MCRCL), and the generalized regression (GRNN_OM, GRNN_MCR and GRNN_MCRCL) models, using the validation data set.

Summary of Brutian pine tree attributes for model fitting and for the validation data. (SD): Standard deviation.

Data set | Parameter | Unit | Mean | SD | Min | Max |
---|---|---|---|---|---|---|

Fitting data (measurements from 131 trees, n = 2173) | Over-bark dbh (D) | cm | 39.85 | 12.16 | 9 | 64 |

Total height (H) | m | 18.77 | 3.9 | 8.8 | 26.8 | |

Disk diameter ( |
cm | 26.34 | 13.27 | 2 | 73 | |

Disk height (h) | m | 8.63 | 5.57 | 0.3 | 24.3 | |

Stem volume (V) | m^{3} |
1.06 | 0.77 | 0.02 | 3.31 | |

Validation data (measurements from 42 trees, n = 729 ) | Over-bark dbh (D) | cm | 42.39 | 13.8 | 11 | 72 |

Total height (H) | m | 19.35 | 4.13 | 9.5 | 26.6 | |

Disk diameter ( |
cm | 27.72 | 14.13 | 2 | 80 | |

Disk height (h) | m | 8.95 | 5.72 | 0.3 | 24.3 | |

Stem volume (V) | m^{3} |
1.23 | 0.98 | 0.03 | 4.14 |

The best combinations of all parameters that conduct to the best learning of the BPANN, LMANN and GRNN models for the prediction of the diameter over-bark (dob) to a measurement point, and the prediction of the stem volume (V) over-bark. (OM): the model without incorporation of the crown variables; (MCR): the model with the CR variable inclusion; (MCRCL): the model with CR and CL variables inclusion; (D): diameter at breast height over-bark (cm); (dob): diameter over-bark (cm) to measurement point at height h; (H): total tree height (m): (h): height above the ground to the measurement point (m); (F): diameter over-bark (cm) at 5.3 m above ground; (V): stem volume over-bark from stump (m^{3}); (CL): crown length; (CR): crown ratio.

BPANN models that resulted to the best learning | ||||||
---|---|---|---|---|---|---|

Model | Number of nodes | Number of Epochs | Learning rate | Momentum factor | ||

Input layer | Hidden layer | Output layer | ||||

OM | 4 : (D, H, h, F) | 8 | 1 : ( |
1000 | 0.10 | 0.30 |

MCR | 5 : (D, H, h, F, CR) | 10 | 1 : ( |
1000 | 0.10 | 0.30 |

MCRCL | 6 : (D, H, h, F, CR, CL) | 13 | 1 : ( |
1000 | 0.10 | 0.30 |

OM | 5 : (D, H, |
6 | 1 : (V) | 1000 | 0.09 | 0.20 |

MCR | 6 : (D, H, |
10 | 1 : (V) | 1000 | 0.07 | 0.30 |

MCRCL | 7 : (D, H, |
11 | 1 : (V) | 1000 | 0.05 | 0.30 |

LMANN models that resulted to the best learning | ||||||

Model | Number of nodes | Number ofEpochs | Initial (μ)value | Adjustment factor (v) | ||

Input layer | Hidden layer | Output layer | ||||

OM | 4 : (D, H, h, F) | 8 | 1 : ( |
3000 | 0.1 | 10 |

MCR | 5 : (D, H, h, F, CR) | 10 | 1 : ( |
1000 | 0.1 | 10 |

MCRCL | 6 : (D, H, h, F, CR, CL) | 13 | 1 : ( |
1000 | 0.1 | 10 |

OM | 5 : (D, H, |
6 | 1 : (V) | 2000 | 0.1 | 10 |

MCR | 6 : (D, H, |
10 | 1 : (V) | 2000 | 0.1 | 10 |

MCRCL | 7 : (D, H, |
11 | 1 : (V) | 2000 | 0.1 | 10 |

GRNN models that resulted to the best learning | ||||||

Model | Number of nodes | Smoothing coefficient (σ) | ||||

Input layer | 1st Hidden layer | 2nd Hidden layer | Output layer | |||

OM | 4 : (D, H, h, F) | 1956 | 2 | 1 : ( |
0.041 | |

MCR | 5 : (D, H, h, F, CR) | 1956 | 2 | 1 : ( |
0.049 | |

MCRCL | 6 : (D, H, h, F, CR, CL) | 1956 | 2 | 1 : ( |
0.049 | |

OM | 5 : (D, H, |
1956 | 2 | 1 : (V) | 0.041 | |

MCR | 6 : (D, H, |
1956 | 2 | 1 : (V) | 0.041 | |

MCRCL | 7 : (D, H, |
1956 | 2 | 1 : (V) | 0.039 |

Parameter estimates for the compatible taper and volume equations based on the model fitting data. (OM): the original model forms (

Model | _{1} |
_{2} |
_{3} |
_{4} |
_{1} |
_{2} |
_{3} |
---|---|---|---|---|---|---|---|

OM | 85.9076 | 6.7407 | 0.6977 | 2.3021 | - | - | - |

MCR | 84.8311 | 6.7349 | 0.6973 | - | 2.9278 | -1.4063 | - |

MCRCL | 85.90755 | 6.7201 | 0.7034 | - | 2.913 | 0.0988 | -3.3829 |

Stem fit statistics for the compatible volume and taper equation systems for Brutian pine based on the model fitting data. (OM): the original model forms (

Model | Taper (cm) | Volume (m^{3}) |
||||||
---|---|---|---|---|---|---|---|---|

Bias | SEE | MAE | FI | Bias | SEE | MAE | FI | |

OM | 0.045 | 1.7535 | 1.1907 | 0.9825 | 0.001 | 0.0072 | 0.0043 | 0.9846 |

MCR | 0.0163 | 1.7328 | 1.1815 | 0.983 | 0.0009 | 0.0071 | 0.0043 | 0.9848 |

MCRCL | 0.0872 | 1.696 | 1.1602 | 0.9837 | 0.0009 | 0.007 | 0.0042 | 0.985 |

Fit statistics for the BPANN, LMANN and GRNN models for diameter over-bark (

Model | V (m^{3}) |
|||||||
---|---|---|---|---|---|---|---|---|

Bias | SEE | MAE | FI | Bias | SEE | MAE | FI | |

BPANN_OM | -0.0089 | 1.588 | 1.1418 | 0.9857 | 0.00079 | 0.00279 | 0.002 | 0.9977 |

BPANN_MCR | -0.0072 | 1.534 | 1.1252 | 0.9865 | 0.00073 | 0.00259 | 0.0018 | 0.998 |

BPANN_MCRCL | -0.005 | 1.5045 | 1.0867 | 0.9871 | -0.00015 | 0.00247 | 0.0017 | 0.9981 |

LMANN_OM | -0.0154 | 1.5196 | 1.0859 | 0.9869 | -1.0·10^{-6} |
0.00227 | 0.0016 | 0.9985 |

LMANN_MCR | -0.0184 | 1.496 | 1.08 | 0.9872 | 1.5·10^{-7} |
0.00212 | 0.0015 | 0.9987 |

LMANN_MCRCL | 0.0025 | 1.444 | 1.0336 | 0.9881 | -4.8·10^{-6} |
0.00204 | 0.0014 | 0.9988 |

GRNN_OM | -0.0221 | 1.5254 | 1.0712 | 0.9868 | 5.7·10^{-5} |
0.0028 | 0.0018 | 0.9977 |

GRNN_MCR | -0.0269 | 1.496 | 1.012 | 0.9873 | 7.5·10^{-5} |
0.00268 | 0.0015 | 0.9979 |

GRNN_MCRCL | -0.0211 | 1.4661 | 0.9833 | 0.9878 | 8.7·10^{-5} |
0.0026 | 0.0014 | 0.9979 |

Fit statistics for the compatible volume and taper equation systems and for the BPANN, LMANN and GRNN models for diameter over-bark (

Model | V (m^{3}) |
|||||||
---|---|---|---|---|---|---|---|---|

Bias | SEE | MAE | FI | Bias | SEE | MAE | FI | |

OM | 0.0751 | 1.9673 | 1.3238 | 0.9806 | 0.0006 | 0.0075 | 0.0048 | 0.9883 |

MCR | 0.0686 | 1.9236 | 1.2885 | 0.9813 | 0.0007 | 0.0071 | 0.0046 | 0.989 |

MCRCL | 0.0841 | 1.9635 | 1.3307 | 0.9814 | 0.0004 | 0.0076 | 0.005 | 0.989 |

BPANN_OM | 0.0935 | 1.871 | 1.3066 | 0.9824 | 0.0043 | 0.0055 | 0.0044 | 0.9933 |

BPANN_MCR | 0.0707 | 1.863 | 1.3291 | 0.9826 | 0.0043 | 0.0054 | 0.0043 | 0.9936 |

BPANN_MCRCL | -0.0178 | 1.857 | 1.3188 | 0.9827 | 0.0038 | 0.005 | 0.0038 | 0.9945 |

LMANN_OM | 0.0788 | 1.8093 | 1.2721 | 0.9836 | 0.0035 | 0.0047 | 0.0036 | 0.995 |

LMANN_MCR | -0.0787 | 1.7908 | 1.2976 | 0.9839 | 0.0035 | 0.0046 | 0.0036 | 0.9952 |

LMANN_MCRCL | 0.0552 | 1.7873 | 1.2717 | 0.984 | 0.0033 | 0.0044 | 0.0034 | 0.9956 |

GRNN_OM | 0.5644 | 2.7244 | 1.9538 | 0.9628 | 0.0032 | 0.0058 | 0.0041 | 0.9926 |

GRNN_MCR | 0.4131 | 2.7221 | 2.0848 | 0.9628 | 0.0026 | 0.0056 | 0.0034 | 0.9931 |

GRNN_MCRCL | 0.3122 | 2.7211 | 2.113 | 0.9629 | 0.002 | 0.0054 | 0.0034 | 0.9934 |