Compatible taper and volume equations for pure and natural Scots pine stands in the northeastern part of Turkey (Ardahan Province) were developed using a nonlinear mixedeffects modeling approach. Experimental data were obtained from 137 felled sample trees in different diameter and height classes. The most successful model (Jiang et al. 2005) explained 98.3% of the variance in stem diameter estimation and the RMSE, ME, MAE, AIC and BIC value obtained using this model were 1.955 cm, 0.043 cm, 1.300 cm, 9783.8 and 9812.6, respectively. Considering the criterion values of AIC, BIC and 2LnL, the model with randomeffects in two parameters (b_{1} and b_{3}) was the most successful for Scots pine. While the mixed model including random parameters did not completely solved the problem of the autocorrelation of errors in this study, the use of the autoregressive error structure AR(1) eliminated the autocorrelation in the residuals. In addition, the best estimation results among 20 different calibration options were obtained using the option of measuring two tree diameters at d_{1.30} and d_{5.30 }with validation data. The most successful model explained 99.18% of the total variance in stem volume estimation in Scots pine.
Scots pine (
While the volume of trees can be practically calculated using various tree volume equations, the volume or proportions of derived wood assortments such as mine poles, industrial wood, pulp and paper, and firewood cannot be so easily determined. For this reason, taper models that provide detailed estimates of both tree volume and wood assortments are needed (
Since the 1900s, many stem profile models in different forms have been developed. According to
Diameters measured at different heights on a single tree are used as an important data source for developing stem profile models. Consequently, sequential measurements along the tree stem are related with each other (
Stem diameters at different heights and stem volume predictions of commercial trees such as Scots pine are critical for developing thorough forest management plans. This study focuses on developing compatible taper and volume models, which allow comprehensive volume and diameter estimations for pure and natural Scots pine stands distributed in the Ardahan SFE of Erzurum Regional Directorate of Forestry, using the nonlinear mixedeffect modeling approach.
Pure Scots pine stands are naturally distributed in the Ardahan province, Northeastern Turkey (40° 45′ 24″  41° 36′ 13″ N, 42° 25′ 43″  43° 29′ 17″ E), where the monthly average temperature ranges from 11.1 to +16.4 °C (annual average of 3.9°C), the lowest temperature is between 39.8 °C and 2.2 °C, and the highest temperature reaches 35 °C. The average total annual precipitation is 573.9 mm and the average annual relative humidity varies between 65% and 71% (
The area of the Ardahan State Forest Enterprise is approximately 547,671 ha, of which 30,757.4 ha (5.6%) are covered by forests and 516,913.6 ha (94.4%) are bare lands. Of the forested area, 24,343.3 ha (79%) are productive forests, while 6,414.1 ha (21%) are unproductive or degraded forests.
The case study area covers 24.106 ha of pure Scots pine forests in the Ardahan region, which are managed under five different management units; 3819 ha are assigned to the Ardahan forest management unit, 4525 ha are assigned to the Köroglu forest management unit, 1582 ha are assigned to the Posof forest management unit, 7865 ha are assigned to the Ugurlu forest management unit and 6314 ha are assigned to the Yalnizçam forest management unit (
Data from 137 sample trees felled in pure and natural Scots pine stands in Ardahan SFE, Erzurum Regional Directorate of Forestry, were used as source material for this case study. In the selection of the sample trees, due care was taken to distribute them as equally and evenly as possible in different diameter and height classes and to best reflect the variability in volume development.
The sample trees were cut from the base diameter of the stem (0.3 m above the ground), and then the bottom diameter of he log (0.3 m), the diameter at breast height (1.3 m), and other diameters at 1 meter regular intervals (2.3 m, 3.3 m, 4.3 m etc.) were measured. All measures were taken using calipers with an accuracy of ± 1.0 mm. In addition, the heights of the trees were measured with a steel tape measure to an accuracy of ± 1.0 cm. The diameters of noncircular sections on the stem were measured in two directions perpendicular to the stem section and their average was calculated.
A total of 2939 diameter measurements were taken from the 137 sample tree stems felled within the scope of this study. The data used in the study were randomly divided into two groups: (i) the data used to estimate the parameters of the stem profile models (Group I data: approximately 80% of the total data  2340 stem diameter records from 110 trees); and (ii) the data used to validate the suitability of these models for the stand (Group II data: approximately 20% of the total data  599 record from 27 trees).
We used four segmented polynomial stem profile models which have been employed in other similar studies: Model 1 by
The NLIN procedure of the software package SAS/STAT^{®} v. 9.3 was used to estimate the parameters of the four stem taper and volume models (
where
After determining the bestfit of the above four segmented stem profile models, the mixedeffects modeling approach was used to estimate the bestfit stem diameter model.
Different diameter values measured along each log were used to develop the taper models. Diameters taken from the same log are interdependent (serial correlation or “autocorrelation”), thus one of the basic assumptions of regression analysis is neglected (
In the mixedeffects model, parameters are divided into two groups: fixed and random effects. The fixed effect parameter expresses the general relationships that apply to the entire population. The random effect parameter describes the variability between different sampling units (sample trees). The structure of the nonlinear mixedeffects model is presented below in the form of a matrix (
where
The parameters of
The variance components and constant parameters of the bestfit stem profile model were estimated using the PROC NLMIXED procedure in the SAS/ETS® v. 9.3 statistical package (
Another important issue to be evaluated and solved in mixedeffects modeling is the identification of the “calibration responses” of the model. Calibrated models offer the possibility of more accurate, consistent, and reliable estimates (
BLUP requires the measurement of a certain number of new data in a site or sample area to be calibrated, especially when estimating the randomeffect parameter (
where
In determining the calibration response of the mixed effects models with validation data, the Sum of Squared Errors (SSE,
Goodnessoffit statistics of the stem profile models successfully fitted to the data set are given in
When evaluating the best fit quality criteria of the four segmented stem profile models used in the study, Model 1 (
Our results are in agreement with the findings obtained using the same model in Caucasian fir/Oriental spruce (
To use the model equation of
In this model equation, all parameters were found to be significant (p<0.001). The corrected coefficient of determination of the model was 0.976, the standard error of the estimate was 2.451 cm, the mean error was 0.115 cm, and the mean absolute error was 2.006 cm.
The AR (1) equation structure based on
where
The parameters associated with the
The AIC, BIC and 2LnL values were used to compare the nonlinear mixedeffects regression models. The model with the lowest numerical value for these criteria is considered the best performing model (
The parameters and fit statistics for the
In this study, the
Random effective parameter values (
Calibrated models offer the possibility of obtaining more accurate, consistent and reliable estimates (
The error values of the stem diameter models developed for Scots pine trees in Ardahan region were also examined in terms of standard error, mean error and mean absolute error values of the estimates of fixedeffects and mixedeffects stem taper and AR(1) for relative height values (
The lowest error value was obtained at 0.250.35, while the highest error value was 0.9 for relative length for the nonlinear fixedeffect model. On the other hand, for the mixed effects models and AR(1) models of the same stem diameter model, the lowest error value for relative length was 0.3, while the highest was 0.9 (
When examining the stem forms of Scots pine in the Ardahan region, we recorded that branching begins at about 7080% of the tree height. As a consequence, the reliability of diameter estimates above these heights may decrease due to stem swelling.
The distribution of errors in relative length estimates for the fixed and mixedeffect and AR(1) error structure of the
The autocorrelation of errors for the first three logs of the stem diameter model constructed for the Scots pine trees in Ardahan region is shown in
The boxplots of residuals in each diameter class for the four stem volume models are shown in
The suitability of the stem volume equations for the stand from which the samples were taken was tested using an independent data set (
A compatible segmented taper model for pure Scots pine stands in the Ardahan region of Turkey was developed using the mixedeffects modeling approach and the AR(1) structure. Four segmented polynomial taper models were used to predict the variation of tree stem diameters. The stem taper equation developed by
Autocorrelation and heterogeneous distribution of error variance are common problems in models with hierarchical data structure. These problems cannot be eliminated in traditional regression models, but can be solved by the mixedeffects modeling approach and the autoregressive error structure AR(1), which represents the most important advantage of mixedeffects models over traditional regression models. In this study, the inclusion of random effects parameters to the model reduced the problem of errorrelated autocorrelation, resulting in a more homogeneous error variance structure in almost all relative height values and in a reduction of the RMSE values for both the AR(1) and mixedeffects models, compared with the nonlinear regression model. Finally, the autocorrelation problem was solved using the autoregressive error structure AR(1), as proven by the homogeneous error variance distribution.
Accurate and reliable assessments of tree or stand volume is highly dependent on the accuracy of stem taper estimation. In this study we showed that the estimation of stem taper in pure Scots pine stands of the Ardahan region can be successfully performed using nonlinear mixedeffects modeling technique, especially the AR(1) model, which provided accurate predictions within a wide range of diameter at breast height (6.0 to 75.0 cm) in Scots pine. However, when it comes to choose the best of various models with similar prediction success for a tree species in any region, the practical application of the method and the preferences of forest managers and practitioners should not be overlooked.
This study is part of the master’s thesis prepared by BS and supervised by AK for the Institute of Natural and Applied Sciences, Artvin Çoruh University, Turkey. The authors thank the other researchers involved in this study and the staff of Ardahan SFE.
BS and AK planned the experiments, contributed to data collection, data analysis and wrote the first draft. AK contributed to statistical analysis of the data, reviewed and edited the manuscript.
Plots of the relative height versus relative diameter outside bark for (a) the fitting data points, and (b) the validation data points.
RMSE values of fixed and mixed effect and AR(1) models by relative size classes.
Residual plots for the fixed (left) and mixed (center) effects and AR(1) models (right).
Residuals plotted against lagged residuals for fixed effects models (left column), mixed effects models (center column) and AR(1) models (right column).
Box plots of total volume residuals against diameter classes for Scots pine.
Descriptive statistics for the fitting and the validation data points. (D): diameter at breast height (cm); (H): total height (m); (t): tree age (year); (v): tree volume (m^{3}); (d): diameter outside bark at specific height; (h): height at specific diameter; (n): number of observations; (SD): standard deviation.
Datatype  Variable  n  Minimum  Mean  Maximum  SD 

Fitting data  D (cm)  110  6.0  35.3  74.5  16.2 
H (m)  110  6.6  21.0  32.0  5.5  
t (yil)  110  19.0  99.2  181.0  37.3  
v (m^{3})  110  0.008  1.317  5.920  1.315  
d (cm)  2340  0.2  24.0  86.0  15.2  
h (m)  2340  0.3  11.1  31.3  7.1  
Validation data  D (cm)  27  7.5  36.5  66.0  15.0 
H (m)  27  8.7  21.9  30.7  4.6  
t (yil)  27  17.0  103.3  171.0  32.2  
v (m^{3})  27  0.022  1.337  4.465  1.123  
d (cm)  599  0.5  24.2  72.0  14.2  
h (m)  599  0.3  11.3  30.3  7.1 
Goodnessoffit statistics of the taper models used.
Model  Parameter  Estimates  R^{2}  RMSE  ME  MAE  AIC  BIC 

Model 1 
b_{1}  76.47923  0.9834  1.955  0.043  1.300  9783.8  9812.6 
b_{2}  7.49191  
b_{3}  0.823433  
b_{4}  3.475003  
Model 2 
b_{1}  6.31331  0.9774  2.282  0.111  1.674  10509.6  10549.9 
b_{2}  3.114493  
b_{3}  35.87637  
b_{4}  3.26965  
a_{1}  0.128333  
a_{2}  0.843195  
Model 3 
b_{1}  4.845493  0.9726  2.514  0.291  1.881  10960.1  10994.6 
b_{2}  6.807834  
b_{3}  11.1853  
b_{4}  11.10006  
a_{1}  0.308675  
Model 4 
a_{1}  0.000027  0.9782  2.234  0.134  1.663  10411.7  10463.5 
a_{2}  1.834183  
a_{3}  1.269305  
b_{1}  0.000011  
b_{2}  0.000038  
b_{3}  0.00003  
p_{1}  0.085951  
p_{2}  0.698639 
Parameter estimates and fit statistics for mixedeffects models and after autoregressive modeling.
Model  Components  Parameter  Estimate  Std. Error  p  

Mixedeffects model  Fixed Parameters 

85.8733  5.2503  16.36  <0.0001 

7.5179  0.4927  15.26  <0.0110  

0.8822  0.0037  239.07  <0.0001  

3.4950  0.0772  45.29  <0.0001  
Variance component  σ^{2}_{u}( 
7931.7000  0.0164  483861.00  <0.0001  
σ^{2}_{v}( 
0.0078  0.0015  5.36  <0.0001  
Covariance  σ^{2}_{uv}( 
2.8341  1.1269  2.51  0.0134  
Model error  σ^{2}  1.5132  0.0466  32.47  <0.0001  
AR(1) model  Parameters 

80.4428  1.4594  55.12  <0.0001 

7.9281  0.5688  13.94  <0.0001  

0.8120  0.0067  120.42  <0.0001  

3.1730  0.0909  34.89  <0.0001 
Variation of various error values according to relative height values for fixed and mixedeffects and AR(1) models.
Relativeheight  n  Fixedeffects models  Mixedeffects models  AR(1) model  

RMSE  ME  MAE  RMSE  ME  MAE  RMSE  ME  MAE  
0.00.1  249  0.755  0.021  0.147  0.333  0.045  0.065  0.391  0.020  0.217 
0.10.2  238  0.326  0.015  0.066  0.339  0.017  0.069  0.276  0.004  0.068 
0.20.3  228  0.231  0.011  0.048  0.234  0.008  0.047  0.219  0.006  0.052 
0.30.4  229  0.353  0.001  0.076  0.315  0.003  0.066  0.238  0.004  0.053 
0.40.5  235  0.546  0.002  0.125  0.357  0.003  0.079  0.300  0.002  0.072 
0.50.6  229  0.645  0.002  0.149  0.299  0.026  0.068  0.288  0.013  0.068 
0.60.7  234  0.758  0.004  0.176  0.284  0.034  0.068  0.315  0.000  0.076 
0.70.8  234  0.836  0.019  0.199  0.426  0.031  0.101  0.344  0.002  0.085 
0.80.9  231  0.807  0.017  0.183  0.674  0.031  0.153  0.357  0.015  0.085 
0.91.0  233  0.573  0.057  0.130  0.573  0.056  0.130  0.347  0.036  0.084 
Total  2340  1.955  0.043  1.300  1.282  0.130  0.846  1.249  0.054  0.859 
Fit statistics of the four stem volume equations used in this study.
Model  Parameter 





Fit data  ME  0.01  0.01  0.08  0.07 
MAE  0.08  0.10  0.12  0.11  
RMSE  0.12  0.18  0.20  0.18  
R^{2}_{adj}  0.99  0.98  0.98  0.98  
Validationdata  ME  0.03  0.02  0.06  0.04 
MAE  0.07  0.11  0.10  0.09  
RMSE  0.11  0.17  0.16  0.14  
R^{2}_{adj}  0.99  0.98  0.98  0.99 
Fig. S1  The location of the case study area.
Tab. S1  The taper functions of the four models evaluated based on fitting data.