Heightdiameter relationships are of critical importance in tree and stand volume estimation. Stand description, site quality determination and appropriate forest management decisions originate from reliable stem height predictions. In this work, the predictive performances of heightdiameter models developed for Taurus cedar (Cedrus libani A. Rich.) plantations in the Western Mediterranean Region of Turkey were investigated. Parametric modeling methods such as fixedeffects, calibrated fixedeffects, and calibrated mixedeffects were evaluated. Furthermore, in an effort to come up with more reliable stemheight prediction models, artificial neural networks were employed using two different modeling algorithms: the LevenbergMarquardt and the resilient backpropagation. Considering the prediction behavior of each respective modeling strategy, while using a new validation data set, the mixedeffects model with calibration using 3 trees for each plot appeared to be a reliable alternative to other standard modeling approaches based on evaluation statistics regarding the predictions of tree heights. Regarding the results for the remaining models, the resilient propagation algorithm provided more accurate predictions of tree stem height and thus it is proposed as a reliable alternative to preexisting modeling methodologies.
Tree Height Model PredictionGeneralized ModelsMixedEffects ModelsLevenbergMarquardt AlgorithmResilient PropagationForest InventoryIntroduction
Natural forest areas have continued to decline in the last 30 years while plantations have increased in the same timeframe, now comprising some 7% of the global forest area. Plantations are of great importance in economic and ecological terms. Thus, robust and reliable information on the growth and development characteristics of these species are essential in order to sustainably manage plantations and determine the success of afforestation efforts. Diameter at breast height (d) of trees and total treebole height (h) are two basic commonly used parameters in forest inventory, planning and management of plantations. These parameters are fundamental variables in several forest practices such as prediction of standing yield, growth projection, carbon accounting, identification of stand structural diversity, determination of the extent of damage in forest or stands, and to estimate missing tree heights (Curtis 1967, Adame et al. 2008).
In forest management applications or field inventories, d in trees can be measured rapidly and precisely, while h is costly, more time consuming, and challenging to measure (Meng & Huang 2009, Sharma & Parton 2007, Ozçelik et al. 2018). It is also known that there is a strong correlation between d and h in the stem of trees. Because of this, models can be created to discover the relationship between d and h based on the tree diameter at breast height and the total stem height of trees that are measured. Ultimately, these models are used to estimate the height of any tree in the stand with unknown height (Robinson & Wykoff 2004, Lei et al. 2009).
There are many studies describing tree height estimation and prediction models, mainly using wellknown classical modeling techniques, such as nonlinear regression models (Colbert et al. 2002, Kearsley et al. 2017) and Bayesian modeling (Zhang et al. 2014), using in most cases allometric models. Further research evaluating more recent and promising modeling methods such as techniques based on the artificial intelligence are warranted to improve efficacy. Previous research on heightdiameter (hd) modeling of cedar plantations in Turkey was conducted by Catal (2012), although it was based on a limited number of traditional hd models, without further investigation of the potential efficiency of other modeling approaches.
Generally, hd models may be simple or generalized. Simple models utilize d to calculate h, as the relationship between the two variables is wellknown (Bronisz & Mehtätalo 2020). The limitation with simple models, however, is the large sampling effort required (GómezGarcía et al. 2014). In contrast, the correlation between d and h (total height) varies among species and can be influenced by standspecific variables such as stand structure and site productivity (Lhotka 2012). Models that include measures of stand structure, site productivity and tree social position are commonly referred to as generalized hd (GM) models. Mehtätalo et al. (2015) used fixedeffects (FE) and generalized mixedeffects (ME) modeling techniques to show differences between marginal and plotspecific hd relationships, in a range of different geographical/ecological regions and tree species. The use of additional standlevel variables was previously shown to improve tree height estimates (VargasLarreta et al. 2009, Lhotka 2012). For example, Newton & Amponsah (2007) used dominant height (H_{0}) within a stand to estimate standlevel competition.
Since the estimation and prediction of tree heights are crucial in tree growth and tree volume prediction, alternative and accepted modeling techniques exist such as the generalized, FE and ME models, and artificial neural network modeling techniques, that may be useful in modeling the relationship between hd and reducing the measurement efforts on the ground. The most basic (Huang et al. 1992, Peng et al. 2001, Soares & Tomé 2002) and generalized hd models (GómezGarcía et al. 2015, Adame et al. 2008, VargasLarreta et al. 2009, CrecenteCampo et al. 2010, Lhotka 2012, Zang et al. 2016, Ozçelik et al. 2018) are available for many tree species. Most research has focused on mixedmodels to understand hd relationships (Calama & Montero 2004, Sharma & Parton 2007, Lhotka 2012, Ozçelik et al. 2013, Ozçelik et al. 2018, GómezGarcía et al. 2014, GómezGarcía et al. 2015, Mehtätalo et al. 2015, Adamec & Drápela 2016, Bronisz & Mehtätalo 2020). In Turkey, hd equations were developed for some tree species at regional scale (Diamantopoulou & Ozçelik 2012, Ozçelik et al. 2014, Catal & Carus 2018).
Taurus cedar (Cedrus libani A. Rich.) natural range includes the Taurus Mountains of Turkey, where it is an ecologically and economically important component of the forest ecosystems (Boydak 2003). The most recent inventory data show that cedar covers an area of approximately 463,521 ha in Turkey, with a total growing stock of 27.4 million m^{3} (GDF 2015). It is not only an important resource of raw material for the forest product industry, but it also fulfils critical ecological tasks including the reduction of soil erosion and the conservation of water resources, the mitigation of the adverse impacts of climate change, and is essential in the maintenance of biological diversity in Turkey. It has moderate soil requirements and is tolerant to extremes of temperature and drought in summer and to cold temperatures in winter (Saatçioglu 1979); therefore, C. libani has great potential for afforestation, especially in areas with semiarid climates. Given that 35% of Turkey’s territory has semiarid climate, cedar is a critical tree species for use in afforestation. As of 2000, an area of approximately 110,000 ha was reafforested with cedar (Konukçu 2001). However, there is still relatively little information concerning the growth of Taurus cedar plantation compared with the quantity of information available for natural cedar stands. Hd relationship is one of the most important components of growth and yield models for sustainable management of Taurus cedar plantations. Moreover, to our best knowledge, hd generalized mixedeffects model has not been developed for Taurus cedar plantations.
Until now, the use of artificial neural networks (ANNs) in forest modeling has led to the conclusion that they can be considered as significant alternative techniques for many characteristics of trees growth compared to classical modeling methods, both in classification tasks (Schmoldt et al. 1997, Liu et al. 2003, Cosenza et al. 2017), and for estimation and prediction problems (Leite et al. 2011, Soares et al. 2013, Reis et al. 2016, Ozçelik et al. 2017, Ercanli et al. 2018, Monteiro Da Silva et al. 2018, Vacchiano et al. 2018, Wu et al. 2019, Sun et al. 2021). Specifically, artificial intelligence was successfully implemented to the development of total tree height models as well (Li & Jiang 2010, Diamantopoulou & Ozçelik 2012, Ozçelik et al. 2013, Vieira et al. 2018, Thanh et al. 2019, Ercanli 2020). The majority of the ANN modeling studies employed multilayer perceptron architecture combined by the standard backpropagation algorithm.
The incentive for developing an artificial neural network modeling approach in the present work was that ANNs can discover and thus automatically model the relationships underlying input and output variables, a property reflected in the connection weights of the network. Instead, traditional approaches, such as the widely accepted generalized, fixed and mixedeffects modeling methods, require certain assumptions about the form of a fitting function which must be specified in advance, introducing limitations that mainly impact model applicability to different scenarios. In contrast, an ANN model is trained to find this relationship, thus fitting complex nonlinear models. In addition, given that ANNs provide resilience to outliers, the techniques generally perform well where there are missing or inaccurate data; this problem is typically found in forestdata measurements which require substantial effort inthefield, increasing proneness to errors. Furthermore, this generalization ability enables ANN models to generate reliable predictions with new data sets. ANN modeling techniques, therefore, can overcome problems in basic data gathered in the forest, including nonlinear relationships, multicollinearity, heteroscedasticity, outliers and noise. Although ANNs may suffer from overfitting of data, this problem can be avoided by selecting a suitable training architecture followed by testing and validation.
The objective of this work was to compare the effectiveness of different modeling approaches in providing accurate predictions of total stem height of C. libani plantations of Turkey. After developing the ANN models, validation was carried out using simple measurements from the forest environment, such as d. Variability between sampling plots was included in the models by recording dominant heights (H_{0}) and diameters (D_{0}) of each sample plot. The adjusted fixedand mixedeffects models, after being localized using calibration data from one to three sample cedar trees were compared against the prediction abilities of the novel ANN models. For this purpose, two different learning algorithms, that have shown significant ability to cope the known disadvantages and limitations of the most used standard backpropagation learning algorithm (Wu & Ji 2015), i.e., the resilient propagation (RPANN) and the LevenbergMarquardt (LMANN), were applied as innovative modeling strategies in forest model building, with the aim of optimizing the learning process and better exploting the information in real data measurements originating from the forests.
Materials and methodsSite description
Data used for development of models were gathered in cedar plantations. Sample plots were in the Isparta Regional Directorate of Forestry, which includes the Isparta and Burdur provinces (southwestern Turkey), at an elevation between 1000 to 1600 m a.s.l. Climate in this region is characterized as transitional from the Mediterranean to continental climate. Annual precipitation varies between 426 to 814 mm, while annual mean temperature ranges from 11.9 to 13.2 °C (MGM 2013). The bedrock of the study site is composed of sedimentary rock including limestone, sandstone, and claystone.
Data collection
To determine diameterheight associations, data were collected from 30 sampling plots randomly distributed among Taurus cedar plantations in Isparta and Burdur Forest Regions. Sample plots were selected to represent a range of age classes, site conditions, and densities for the species in the Mediterranean Region of Turkey. Plot size varied between 270 to 540 m^{2}, depending on density, with a minimum of 30 trees per plot. The age of the sampling plots varied from 17 to 35. Two perpendicular overbark diameters (di) were measured with a precision of 0.1 cm and averaged to find actual diameter (d, cm) for each tree. A BlumeLeiss hypsometer was used to obtain total heights with precision of 0.5 m. Depending on plot size, dominant heights (H_{0}) and diameters (D_{0}) were obtained as the average heights and diameters of the 100 trees with the thickest diameter at breast height (d) per hectare.
The available Cedar data set was randomly split into two groups; some 21 sample plots (70% of all plots) were randomly selected for the model fitting while the remaining 9 sample plots (30% of all plots) were used for evaluating the models’ performance on this new data set (Tab. 1). The basic descriptive statistics for the dataset used has been given in Tab. 1and illustrated in Fig. 1.
Generalized heightdiameter models (GM)
Twentyone generalized heightdiameter models selected from earlier studies were evaluated using nonlinear least squares (NLS  CrecenteCampo et al. 2010, CorralRivas et al. 2014). These models included different standlevel variables such as H_{0}, D_{0}, and quadratic mean diameter, to represent the variation between forest stands.
Fixedeffects model (FE)
A large number of nonlinear model forms with two and three parameters were evaluated for cedar plantations, including those noticed by Huang et al. (1992) and Lei et al. (2009) as Curtis, Weibull, Exponential, ChapmanRichards, Gompertz, Schnute, and KorfLundgvist. The NLS were fitted to the datasets to determine the most appropriate model for cedar plantations. To evaluate the prediction accuracy of the models, statistical and graphical evaluations were used. Of the models assessed, predictive capability of the ChapmanRichards appeared best to model heightdiameter relationships in cedar plantations (eqn. 1):
where h_{ij} is the total tree height (m) of the jth tree in the ith plot, d_{ij} is the diameter (cm), and β_{1} β_{3} are model parameters.
To calibrate the FE model, a correction factor k^{*} (eqn. 2) suggested by Temesgen et al. (2008) was used. Temesgen et al. (2008) indicated that when the heights of a subsample of n_{im} trees from the ith stand is known, the predicted heights of the remaining trees from the same stand can be calibrated (eqn. 2):
The influence of the correction factor on the prediction accuracy of a nonlinear FE model was also investigated.
Nonlinear mixedeffects model (ME)
In the ME model structure, parameters of eqn. 1 comprise the plotspecific randomeffect parameters and the population level fixedeffect parameters that are common to all trees. In matrix notation, eqn. 1 can be represented as (eqn. 4):
where y_{i} = [h_{i1}, h_{i2}, …, h_{in}]^{T}, d_{i} = [d_{i1}, d_{i2}, …, d_{in}]^{T}, ε_{i} = [ε_{i1}, ε_{i2}, …, ε_{in}]^{T}, n_{i} is the sample size for tree height values in plot i, and u_{i} and b and are column vectors representing random and fixedeffects parameters, respectively; ε_{i} ~N(0,R) and u_{i} ~N(0,D), where R and D are diagonal matrices. Additionally, it is assumed that u_{i} and ε_{i} are independent. SAS procedure NLMIXED (SAS Institute 2010) was used to obtain the parameters of nonlinear ME models (eqn. 4).
The firstorder Taylor series expansion can be used to obtain the randomeffect parameters u_{i }for the ith plot (Hall & Clutter 2004, Meng & Huang 2009  eqn. 5):
where û_{i}^{k }is an estimate of the randomeffect parameters for plot i at the kth iteration, hat{D} is an estimate of D, the variancecovariance matrix for u_{i}, and Z_{i} is (eqn. 6):
hat{R} is an estimate of R, the variancecovariance matrix for ε_{i}, y_{i} is the m × 1 vector of observed tree heights, and m is the number of tree height measurements used to localize the height growth model.
Estimating u_{i} requires an iterative process. In eqn. 5, a null starting value (û_{i}^{0}=0) was used and repeatedly updated until the absolute difference between û_{i}^{k} and û_{i}^{k+1} was smaller than a predetermined tolerance limit. For the random effects, the outcome is the approximated empirical best linear unbiased predictor (EBLUP).
ANN models
The most significant advantage of an ANN model is that it is trained from measured data without the need to add further information. That is, after the fitting of the system with the available groundtruth data, the ANN model automatically discovers the existing dependencies, and finally produces a trained model. On the other hand, the net topology, the learning algorithms used, and the learning constrains are specified by the modeler.
LevenbergMarquardt artificial neural network model (LMANN)
The LevenbergMarquardt (LM) algorithm was selected for application in the multilayer perceptron learning of ANN (Levenberg 1944, Marquardt 1963). Past studies showed that the LM algorithm has the ability, on the one hand, to cope with known disadvantages and limitations of the standard backpropagation learning algorithm, such as slow convergence, much offline training requirements, instability, trapped at local minima (Adeloye & De Munari 2006, Wilamowski & Yu 2010, Scanlan & Mulvaney 2013, Wu & Ji 2015); on the other hand, LM algorithm have showed significant ability on the prediction of environmental variables (Wu 2014, Diamantopoulou et al. 2015, Zhou et al. 2018, Ozçelik et al. 2019).
A detailed description of how the LevenbergMarquardt algorithm is embedded to neural network training can be found in Hagan & Menhaj (1994). Concisely, the Hessian matrix (Hm) approximation is introduced by the LM algorithm as (Ozçelik et al. 2019  eqn. 7):
Hm = J^T J+ \mu I
where I is the identity matrix, J is the Jacobian matrix, and μ is the combination coefficient.
The update rule of the LM algorithm is w_{t+1 }= w_{t } (H_{m})^{1} J^{T}e, where w are the weights, and e are the biases. The efficiency and convergence of the LMANN models are very sensitive to the configuration of the coefficient value (μ) of eqn. 7. As the value becomes larger, the more weight is given to gradient descent learning with a small step size. The smaller it is, the more weight is given to large step sizes. For μ equal to zero, the algorithm turns to the GaussNewton method.
Resilient propagation artificial neural network model (RPANN)
Resilient propagation, known as Rprob, is a neural network algorithm which was fully described by Riedmiller & Braun (1993) and successfully used (Saputra et al. 2017, Florescu & Igel 2018), so that known disadvantages of the standard backpropagation algorithm can be overcome. The backpropagation algorithm has known drawbacks, due to elementary learning processes. The main problems of the learning phase include slow convergence resulting from small changes in weights, and the neural network weights trapping around local optima.
Rprob mainly refers to the gradient direction. The algorithm calculates a delta value (Δ_{ij}) for each weight that increases when the gradient does not change sign under the prerequisite that the system works on the right direction, or decreases when the gradient does change sign. The learning rule used to calculate delta value which improves the change of the network weights is the following (Riedmiller & Braun 1993  eqn. 8):
Finally, the new weight value w^{t}_{ij} between i and j neurons in two consecutive layers on the (t1) is given by (eqn. 9):
w_{ij}^t = w_{ij} + \Delta w_{ij}^{t 1}Aspects of ANN model training
In geotechnical engineering, the selection of the proper input variables for ANN model building is usually based on a priori knowledge of the physical problem (Maier & Dandy 2000). In order to construct a reliable model for total stem height prediction, the sensitivity analysis was used to choose the proper input layer nodes of the ANNs. The analysis was performed on the available variables measured in the field (Tab. 1), that were selected in advance due to the physical problem (Cariboni et al. 2007).
To avoid overfitting and undertraining of the network in reaching the best possible training efficiency of the model, the training process was initiated with the smallest possible network and no hidden nodes. Then the construction of the model continued by adding any necessary hidden nodes in order the network learning to be completed. Finally, the architecture is reached to an end, when the agreement obtained between ANN model predictions and targets can be considered as satisfactory.
Overfitting avoidance of the ANN models led to acceptable generalization ability of the models. For this reason, the kfold cross validation resampling method with k=10 was applied (Olson & Delen 2008, May et al. 2010) to both training and testing data sets (Leahy 1994), due to its efficiency and easiness to apply. This procedure was not applied in regression model building, where there is little to be gained by separating data into parts (Hursch 1991). As described above, the whole data set was randomly divided into two parts. The fitting data set was used for the construction of all different types of models. The validation data set was used only for the evaluation of the predictive capability of all constructed models.
Due to the fact that the ANN model error value is strongly affected by the number of the weights, the credibility and the generalization capability of the model was determined through the mean error rate on the crossvalidation examples, and the validation data set, respectively (Fig. 2).
As mentioned above, the efficiency and convergence of training of the LMANN models significantly depends on the proper value selection for the combination coefficient (μ in eqn. 7). The initial value of μ was set to 0.001 (Diamantopoulou et al. 2015) and the final value that led to the lowest sum of square errors model value was found equal to 1.00e07.
Similar to the LMANN models, training of RPANN models is very sensitive to the value of delta rule Δ_{ij}^{(t)} (eqn. 8), which is responsible for the sign (positive or negative) of the gradient to show the direction of the adjustment weight. Through this process the iteration of learning stops when the error target is reached. The initial delta value was 0.07, with increment steps of 1.2 and decrease steps of 0.5.
The architecture of both the LMANN and RPANN algorithms finalized after the detection of the correct number of nodes in the hidden layer. This final number was obtained during a trialanderror process: the process began with one hidden node and was repeated until the desired target error value was reached. During the trialanderror process, selection of the correct combination of transfer functions was made. The combination of functions that provided the optimal behavior of both the ANN models was: (i) the hyperbolic tangent transfer function (tansig) to transfer knowledge from the input layer to the hidden layer; and (ii) the linear activation function (purelin), transferring information from the hidden layer to the output layer (Fausett 1994). The ANN models were constructed using the neural network toolbox of the MATLAB software ver. R2017a (Beale et al. 2017) developed by The Math Works™ Inc.
Evaluation
The evaluation of the different modeling approaches (i) Generalized Model (GM), (ii) noncalibrated fixedeffects model [FE(0)], (iii) calibrated fixedeffect model [FE(13)], (iv) calibrated mixedeffects model [ME (13)], (v) LMANN, and (vi) RPANN techniques, was made using the mean difference (MD, m), the mean absolute difference (MAD, m), the fit index (FI) and the root square mean error (RSME), which were calculated as follows (eqn. 10 to eqn. 13):
where n is number of plots, n_{i} is number of tree height values for plot i, h_{ij} and hat{h}_{ij} are measured and predicted heights, respectively, and bar{h}_{i} is mean height of h_{ij} for plot i.
It is worth noting, that the adjusted FE and the calibrated ME models’ parameters were “localized” by the use of a number of sampled heights in each plot. Trincado et al. (2007) recommended that the number of tree heights measured in each plot ranged from one to three. Finally, to compare and evaluate the generalization ability of the constructed models by the different modeling approaches, the above metrics were computed for the validation data set as well (Ozçelik et al. 2018, Bronisz & Mehtätalo 2020).
Twentyone GM models were fitted using the NLS method with the fitting data set, while the predictive capability of the models was evaluated using the validation data set. The final GM model (Krumland & Wensel 1988) including D_{0} and H_{0} as additional independent variables, was (eqn. 14):
The model given in eqn. 14 and its modified forms was successfully used by many authors (e.g., CrecenteCampo et al. 2010). H_{0 }was the most important contributor among stand level variables since it can allow a plotlevel adaptation related to many factors, such as genetics, topography, silvicultural regime, and environment (Huang et al. 2009). Further, the accuracy of hd models can be improved by the incorporation of D_{0} and H_{0} as covariates (CrecenteCampo et al. 2010, Raptis et al. 2021).
A series of parameters from eqn. 1 was implemented (Tab. 2). Model (1) with random components β_{1} and β_{3} resulted in the lowest values for both Akaike’s (AIC) and Bayesian (BIC) information criteria. The final ME model was (eqn. 15):
where u_{1} and u_{2} are random parameters. Parameter estimates were obtained using the fitting data set for the best models (GM, FE and ME models) and are given in Tab. 3.
According to the predictions, the marginal noncalibrated fixedeffects (FE) response model gave more unsuccessful outcomes, when random parameters of the mixed models were zero and prior height information was not available for model calibration for both datasets (Tab. 4).
Artificial neural network modeling approaches
ANN models used a threelayer architecture that included: (i) one input layer with three variables (diameter at breast height, d; dominant height, H_{0}; and dominant diameter, D_{0}). Sensitivity analysis was performed using the available variables listed in Tab. 1. The three variables given above had significant entry on the target variable configuration, with a value of error ratio higher than one; (ii) one output layer with one target variable (total tree stem height, h) specified by the physical problem; and (iii) one hidden layer, comprising the proper number of hidden nodes specified by the network learning.
Using the LM algorithm for ANN model learning, the LMANN showing the best adaptation to the fitting data set was 321/0.5512, i.e., the specific model included a single input layer with the input nodes d, H_{0} and D_{0}, one hidden layer with two hidden nodes and an output layer with one node (h), with RMSE equal to 0.5512. Using the RP algorithm for ANN model learning, the RPANN showing the best adaptation to the fitting data set was the 321/0.5387 model, i.e., the specific model comprised one input layer with input variables d, H_{o} and D_{o}, a single output layer with output node h as in LMANN models building and, between these layers, a single hidden layer with two hidden nodes resulted from the trialanderror process. The RMSE was 0.5387. For both ANN algorithms, the optimum number of hidden nodes was determined following examination of different numbers of nodes. The optimum values were selected as those leading to the best learning of the ANN models, with the smallest RMSE value (Fig. 3).
The k=10 cross validation method applied during the learning using the fitting data set for both ANN algorithms for training and testing in the construction phase, ensured the absence of overfitting and increased the ability of the ANNs to generalize. Both networks were trained for as many epochs as required for the models to reach the minimum MSE value (Fig. 4). As shown in Fig. 4, the estimated errors for the training and the test data sets were not significantly different for either ANN model, which indicated the absence of overfitting. The LMANN was stopped after 200 training interactions (epochs) as no influential reduction of the MSE value occurred after 28 epochs, while the RPANN was trained for 500 interactions as there was no influential reduction in MSE after 167 epochs (Fig. 4).
Evaluation of the different modeling approaches
A synopsis of the evaluation statistics calculated with the fitting and validation data sets using the different modeling approaches is given in Tab. 4. The best outcomes among the GM models are reported in Tab. 4. Comparing the MD, MAD, RMSE, and FI values for the fitting data set, both the RPANN and the LMANN models gave the best results (Tab. 4). Further, in order to enrich the conclusions derived from the results for the constructed models using the fitting data set, where the total height approximately normally distributed, the tpaired test was applied. The twotailed pvalues of the test were <0.05 for both the LMANN and the RPANN constructed models, suggesting that there is no mean difference between the observed and the estimated height values. The same conclusion can be derived for both the fixedeffects (FE) ChapmanRichards model (eqn. 1) and the generalized Krumland & Wensel (eqn. 14) model, while the mean error of mixedeffects model (eqn. 14) was found significant (p<0.05) for the fitting data set.
The accuracy of the estimated results from the ANN models was examined further by analyzing the correlation between the observed h values and their predictions from the models (Freedman et al. 2007). The Pearson’s correlation coefficient (r) was computed as the ratio between the covariance of the observed heights and the height predictions by the constructed models. The LMANN model’s estimations gave r = 0.9223, while for the RPANN model gave r = 0.9259, indicating a high accordance between observed and predicted values. The variability of the estimation errors (CV% = (RMSE/hat{h} · 100) in relation to the mean of the total tree height for the fitting data set was 9.00% and 8.80% of the mean height for the LMANN and the RPANN models, respectively. The reliability and the predictive capabilities of both ANN models was also examined based on the validation data set taken from 270 trees in 9 plots in the same forest areas. The LMANN model predicted an r value of 0.9318 with a CV% of 10.27%, while the r value for the RPANN model was 0.9336, with a CV% of 10.02%. According to the validation data set, the ChapmanRichards mixedcalibrated model using three trees for calibration showed the next best predictive capabilities, with CV% equals to 10.17% and r = 0.9309, while the noncalibrated generalized model of Krumland & Wensel (1988) gave similar results as compared to ChapmanRichards mixedcalibrated model, with CV% equals to 10.83% and r = 0.9269. Finally, the ChapmanRichards fixedmodel produced the poorest results for the validation data set with CV% equals to 17.13 and r = 0.8140.
Discussion
Important ecosystem services including usable forest products and biodiversity conservation can be obtained through sustainably managed forests. The development of a reliable model that could estimate the total height of trees using easily measurable characteristics of trees, can contribute efficiently to the above goal. Our approach attempts to compare, suggest and finally provide an alternative new methodology to the field of optimal forest management design, which would be able to give reliable predictions of tree heights, as this tree characteristic has great influence on the stand description, site quality determination, and tree and stand wood volume estimation.
Parametric modeling techniques assessment
The adjusted fixedeffects model developed for Taurus cedar stem heights in the present work had a much lower mean deviation (MD) for all calibration alternatives compared to the fixedeffects model. With all calibration parameters tested, the fit index increased at rates varying from 27% to 39%, with a mean increase of 35%. Similar findings were also reported in the literature (Lhotka 2012, VanderSchaaf 2013, Ozçelik et al. 2018). The most effective improvement in predictions, among the tested calibration alternatives, was obtained through the use of 3 trees for calibration in each sampling plot. Different sampling alternatives tested for both adjusted fixedeffects model and ME model, corresponding to number of trees measured in each plot. These tree heights were used to estimate the random parameters of the ME model, to calibrate heights and to calculate the assessment statistics for the validation data set (Tab. 4).
According to the ME model, lower mean deviations were obtained following calibration with 3 trees, when compared with the adjusted FE model. Comparing all calibration alternatives, ME model gave a decrease of 32% and 39% in the mean deviation and mean absolute difference, respectively. Previous studies (Trincado et al. 2007, Lhotka 2012, Huang et al. 2009, GómezGarcía et al. 2014, VanderSchaaf 2013) also demonstrated that calibration can significantly improve height predictions. There were no large differences between the calibrations performed with 2 and 3 trees for cedar plantations (Tab. 4) through testing different calibration alternatives with the use of the mixedeffects model, while there were similar trends between different calibration alternatives for MAD and FI values. When the calibrated fixed effect model and the calibrated mixed effect models were compared, for all calibration alternatives, the ME model was more successful than the calibrated FE model in accurately estimating heights in cedar plantations.
The results obtained using the Krumland & Wensel (1988) model, compared with the outcomes of the FE model, suggest that the GM model gave more accurate results than the FE model. The generalized hd model gave an average reduction of 17% in MAD values, when compared to the FE model. The GM models exhibited better results than the adjusted FE models for all evaluation statistics, except for MD values. The model of Krumland & Wensel (1988) includes H_{0} and D_{0} as fixed effects at plot level. H_{0} is preferred to other parameters, as measurement of mean height across plots involves more sampling effort than H_{0} (Huang et al. 2009). However, Bronisz & Mehtätalo (2020) did not suggest H_{0} to be included in the model as a fixed effect predictor when a small sample of H_{0} values based on a certain number of dominant height measurements is used in the model fitting process. In this case, the inclusion of a similar number of height values could improve model accuracy; otherwise, model outcomes may suffer from large and unpredictable measurement errors impacts on the estimates (Bronisz & Mehtätalo 2020). The calibrated basic mixed hd model produced more accurate tree height predictions than the generalized hd model for Turkish cedar plantations without the inclusion of other measurements as covariates. This is an important issue with practical implications in forest economics, such as the reduction of time, effort, and cost for collecting additional covariates. Similar conclusions have been reported by Huang et al. (2009).
The sample size used in model calibration had an impact on prediction accuracy, as height predictions improved depending on the number of trees used in the calibration process (Trincado et al. 2007, Huang et al. 2009, VanderSchaaf 2013, Ozçelik et al. 2018). The accuracy of prediction, however, decreased substantially beyond a certain number of trees. Therefore, the number of trees to be used for calibration should be chosen so as to strike a balance between the costs of inventory and the prediction accuracy. Calama & Montero (2004) suggested that calibration should be performed with additional trees, thus increasing the success of height predictions. Trincado et al. (2007) also argued that height prediction accuracy increases with increasing the numbers of trees used for calibration. According to these authors, the prediction accuracy reached to maximum using three calibration trees. Similar outcomes were also reported by Temesgen et al. (2008). Adame et al. (2008) tested different calibration alternatives and suggested that the use 2 or 3 trees was necessary and adequate for calibration. Huang et al. (2009) used between 1 to 9 trees for calibration, obtaining the most accurate results in height predictions when the calibration was performed with a single tree. The results of the present study suggested that there was a minimal improvement in prediction capability of the model when the number of calibration trees was increased above three.
Assessment of artificial neural network modeling techniques
We examined the performances of the LM and RP algorithms in tree height prediction, with the disadvantages and restrictions of the standard backpropagation algorithm (Fausett 1994, Patterson 1996, Wilamowski & Yu 2010). The low dispersion of predicted values around the 1:1 line throughout the observed range of values (Fig. 5a, Fig. 5b) indicates that both ANN models had similar adjustments to the data used in model construction. Moreover, both ANN models showed better adaptation to the fitting data set than the fixed (Fig. 5c), mixed (Fig. 5d) and generalized (Fig. 5e) models. Furthermore, the statistical evaluation of the null hypothesis (H_{0}: intercept = 0 and H_{0}:slope = 1) for the linear relatonship h_{obs} = intercept + slope · h_{pred} was performed for all models (Fig. 5). The null hypotheses were not rejected at α=0.05 for either ANN models, indicating no significant differences between the 45 degree line and the linear trendline derived by the observed and the estimated height values (Fig. 5a, Fig. 5b). Finally, the RPANN model (Fig. 5b) appeared to provide slightly more accurate fitting to the observed data as compared to the LMANN model (Fig. 5a). The distribution of prediction errors from ANN models (Fig. 6a, Fig. 6b) revealed a frequency peak at zero and a rapid decline of the frequency at larger error values, thus corroborating that both ANN models are welltrained networks. Nonetheless, it is worth noting that the most accurate predictions were obtained by the best RPANN model, with a RMSE value of 2.25% lower than the error obtained from the best LMANN model.
The prediction errors histograms and the relative errors box plots (Fig. 7, Fig. 8), along with the validation statistics presented in Tab. 4, clearly show that the RPANN model provided the most accurate and reliable prediction results as compared to the other models tested. The variability of the prediction errors (CV%) of the RPANN model was approximately 2.43%, which was lower than that of the LMANN model, and 1.47% lower than the variability derived from the ChapmanRichards mixed effect calibrated model [ME(3)]. The distribution analysis of prediction errors for all tested models suggested that the performances of the RPANN model are superior to those of all the other uncalibrated and calibrated models. Therefore, we conclude that the RPANN model provides the best performances in the estimation and prediction of tree height, as compared with fixed, generalized, fixedcalibrated and mixed calibrated models.
Conclusions
In this study, equations for height prediction were developed for Cedrus libani plantations, which involved the evaluation of five alternative modeling methods: (i) the Generalized (GM); (ii) the noncalibrated fixedeffects [FE(0)]; (iii) the calibrated fixedeffect, using 1 to 3 calibration trees per plot [FE(13)]; (iv) the calibrated mixedeffects using 1 to 3 calibration trees per plot [ME (13)]; and (v) the ANNs using two different learning algorithms, the RP and the LM. As compared to the FE, all the constructed models (generalized, adjusted FE, ME, LMANN and BPANN) gave better performance, providing accurate and reliable tree height predictions. As expected, the predictive ability of the adjusted and ME models improved with increasing the calibration sample size. However, increasing the number of trees used for calibration had a minimal effect on the accuracy of the models.
The LMANN and RPANN models constructed in the present work showed reliable abilities to generalize from the input data and gave accurate predictions of the total tree height variable. Both alternatives exhibited similar performance in the construction phases, in addition to the ability to generalize. Based on the cumulative results from all approaches, the resilient propagation algorithm was more effective in predicting total stem height of C. libani growing in Turkey. The effectiveness and efficiency of ANN modeling in general and, specifically of the resilient propagation algorithm, along with the fit to the current dataset, the potential of this modeling approach to forest management, are clear. This direction is being followed in future research.
We thank the Turkish General Directorate of Forestry for its contribution to field work. We also thank Prof. Steve Woodward from University of Aberdeen for his valuable comments and suggestions for revising the English grammar of the text.
Funding
This work was financially supported by the Scientific Research Project Coordination Unit of the Suleyman Demirel University, Isparta, Turkey (project no. 2262YL10).
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Scatter plot of total height (h) against diameter at breast height (d) for cedar pine trees for both fitting and validation data sets.
The 10fold cross validation division for the ANN models construction.
Optimum number of nodes in the hidden layer for the LMANN and RPANN models.
LMANN (a) and RPANN (b) models’ training and test performance for the best case of two hidden nodes in the hidden layer, using the fitting data set.
Fortyfivedegree plots with the linear trendline of height estimations obtained by (a) LMANN, (b) RPANN, (c) fixed, (d) mixed, and (e) generalized models.
Errors histogram for height estimations obtained by the (a) LMANN, (b) RPANN models, and (c) errors box plots for both ANN models using the training data set.
Errors histograms for height predictions obtained by (a) LMANN, (b) RPANN, (c) ChapmanRichards [eqn. 1  FE(0)], (d) Krumland & Wensel (eqn. 13  GM) models, and (e) errors’ box plots for all models using the validation data set..
Errors histograms for height predictions obtained by the ChapmanRichards (a) FE(1), (b) FE(2), (c) FE(3), (d) ME(1), (e) ME(2), and (f) ME(3) models, and (g) errors box plots for all models using the validation data set.
Summary statistics for the fitting and the validation data sets. (d): diameter at breast height (1.3 m above the ground); (h): total tree height; (H_{0}): dominant height; (D_{0}): dominant diameter; (G): basal area per hectare; (d_{m}): mean diameter; (h_{m}): mean height; (D_{g}): quadratic mean diameter; (SD): standard deviation.
Variable
Fitting data (70%)(630 trees in 21 plots)
Validation data (30%)(270 trees in 9 plots)
Mean
Min
Max
SD
Mean
Min
Max
SD
d (cm)
10.31
3.20
20.00
3.12
11.37
5.40
20.00
3.50
h (m)
6.12
3.00
11.75
1.42
6.66
3.20
11.00
1.81
Age (years)
26.00
17.00
35.00
6.30
31.00
23.00
35.00
4.70
D_{0 }(cm)
13.96
9.72
18.30
2.38
14.86
7.82
19.24
3.49
H_{0}(m)
7.51
5.44
10.02
1.21
8.30
5.66
10.65
1.67
G (m^{2 }ha^{1})
30.30
12.75
50.70
10.86
36.96
12.01
58.25
15.02
d_{m}(cm)
10.29
6.68
13.53
2.02
11.35
6.74
14.64
2.51
h_{m}(m)
6.10
4.31
7.93
1.03
6.66
4.50
8.65
1.37
D_{g}(cm)
10.57
6.98
13.92
2.08
11.60
6.77
14.92
2.59
Fit statistics for eqn. 1 for different combinations of random parameters.
Random Parameters
AIC (smaller is better)
BIC (smaller is better)
None (Fixedeffects)
2503
2522
β_{1}
1617
1624
β_{2}
1821
1828
β_{3}
1605
1612
β_{1} and β_{2}
1540
1550
β_{1} and β_{3}
1539
1548
β_{2} and β_{3}
1714
1724
β_{1}, β_{2} and β_{3}
1656
1670
Parameter estimates and standard errors (in parentheses) for the fixed and mixedeffects regression models. (σ^{2}): residual variance of the model; (σ^{2}_{u1}): variance of the randomeffect u_{1}; (σ^{2}_{u2}): variance of the randomeffect u_{2;} (σ^{2}_{u1 u2}): covariance of the randomeffects u_{1 }and u_{2}.
Parameter estimates
Generalized Model
Fixedeffects
Mixedeffects
β_{1}
6.9087 (0.5670)
7.8237 (0.6187)
24.7062 (11.3829)
β_{2}
0.0294 (0.0045)
0.1348 (0.0315)
0.0159 (0.0113)
β_{3}

1.5604 (0.2998)
0.8643 (0.0750)
σ^{2}

0.8311 (0.0468)
0.2216 (0.0129)
σ^{2}_{u1}


73.9740 (113.22)
σ^{2}_{u2}


0.0448 (0.0176)
σ^{2}_{u1 u2}


1.6671 (1.3800)
Fit statistics for the constructed basic hd mixedmodel, generalized hd model and artificial neural network models (LMANN and RPANN) for (a) the fitting dataset, (b) the validation data sets, and (c) the calibrated adjusted fixed and mixed models, using 1, 2, and 3 sampled trees for calibration. (§): denote the best method for cedar plantations; (*): for pvalue <0.05, the null hypothesis (H_{o}: μ_{1}=μ_{2}) was rejected.
Dataset
Model
Type
RMSE
FI
MD/two tailed pvalue*
MAD
(a) Fitting data set(630 trees in 21 plots)
LMANN: 321
ANN
0.5512
0.8241
0.0407494
0.4306
RPANN: 321
ANN
0.5387^{ §}
0.8353
0.0074/0.729
0.4155^{ §}
ChapmanRichards (eqn. 1)
Fixed
0.9116
0.5912
0.0019833
0.7088
ChapmanRichards (eqn. 15)
Mixed
0.6316
0.8059
0.1905/0.000
0.4684
Krumland & Wensel (eqn. 14)
Generalized
0.5677
0.8422
0.7526316
0.4299
(b) Validation data set(270 trees in 9 plots)
LMANN: 321
ANN
0.6839
0.8
0.1138
0.5673
RPANN: 321
ANN
0.6671^{ §}
0.8176
0.0995^{ §}
0.55^{ §}
ChapmanRichards (eqn. 1)
Fixed [F(0)]
1.1401
0.6017
0.2101
0.8943
Krumland & Wensel (eqn. 14)
Generalized (GM)
0.721
0.7825
0.2272
0.5985
(c) Validation data(270 trees in 9 plots)after using localization