A spatially explicit height–diameter model for Scots pine in Estonia
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Abstract
This contribution presents an approach to model individual tree height–diameter relationships for Scots pine (Pinus sylvestris) in multisize and mixedspecies stands in Estonia using the Estonian Permanent Forest Research Plot Network. The dataset includes 22,347 trees. The main focus of the study was to use an approach that is spatially explicit allowing for high accuracy prediction from a minimum set of predictor variables that can be easily derived. Consequently, the height–diameter relationship is modeled as a function of only the stand quadratic mean diameter (dg) and the plot geographical coordinates. A specific generalized additive model gam is employed that allows for the integration of a varying coefficient term and 2dimensional surface estimators representing a spatial trend and a spatially varying coefficient term. The high flexibility of the model is needed due to the very few predictor variables that subsume a variety of potential influential factors. Subsequently, a linear mixed model is used that quantifies the random variation between plots and between measurement occasions within plots, respectively. Hence, our model is based on the theory of structured additive regression models (Fahrmeir et al. 2007) and separates a structured (correlated) spatial effect from an unstructured (uncorrelated) spatial effect. Additionally, the linear mixed model allows for calibration of the model using height measurements as preinformation. Model bias is small, despite the somewhat irregular distribution of experimental areas within the country. The overall model shows some similarity with earlier applications in Finland. However, there are important differences involving the model form, the predictors and the method of parameter estimation.
Keywords
Estonia Scots pine Height–diameter model Generalized additive model Varying coefficient model VCM Mixed model Structured additive regression model STARIntroduction
Measurements of tree heights and diameters are essential in forest assessment and modeling. Tree heights are used for estimating timber volume, site index and other important variables related to forest growth and yield, succession and carbon budget models (Peng 2001). Considering that the diameter at breast height (dbh) can be more accurately obtained, and at lower cost than total tree height, only a subsample of heights is usually measured in the field. Height–diameter equations are then used to predict the heights of the remaining trees, thus reducing the cost of data acquisition. For these reasons, developing suitable height–diameter models may be considered one of the most important elements in forest design and monitoring.
The height–diameter relationship varies from stand to stand, and even within the same stand this relation is not constant over time (Curtis 1967). Therefore, a single height–diameter function without further predictor variables is not able to correctly describe all the possible relationships that may be found within a given forest. To reduce the level of variance, some basic relationships can be improved by taking into account stand variables that account for the dynamics of each stand. A generalized height–diameter function estimates the specific relationship between individual tree heights and diameters using stand variables such as basal area per hectare and quadratic mean diameter (see for example, Larsen and Hann 1987; López Sánchez et al. 2003; Temesgen and von Gadow 2004). The reason for using this approach is to avoid having to establish individual height–diameter relationships for every stand. In contrast in Germany socalled Einheitshöhenkurven have been routinely used for many decades (Lang 1938; Kramer 1964; von Laer 1964; Kennel 1972; Nagel 1991). In this approach, a height–diameter pair of a mean or dominant tree is used as predictor for the general height–diameter relationship. Hence, generalized height–diameter functions are used to estimate individual height–diameter relationships using additional predictor variables, while “Einheitshöhenkurven” are employed by applying additional measurements of a representative height–diameter pair.
In forest field inventories, height and diameter data are generally taken from trees growing in plots that are located in different stands. Often these plots are permanently marked, and the same trees are remeasured over time. Such clustered and longitudinal data are characterized by a lack of independence between observations, since data coming from the same sampling cluster and measurement occasion tend to resemble each other more than the average (Fox et al. 2001). This type of data can be also characterized as having a unit, cluster and occasionspecific heterogeneity (Brezger and Lang 2006), whereas in forest management a unit may refer to an ecoregion or forest district, and a cluster may refer to a sample plot. The lack of independence between observations results in biased estimates of the confidence intervals of the parameters if ordinary least squares regression techniques are used (Searle et al. 1992). To deal with this problem, the extension of ordinary regression to mixed model theory has been proposed in forest research (Gregoire 1987; Lappi 1991; Calama and Montero 2004; Mehtätalo 2004; Nothdurft et al. 2006). In mixed models, both fixed and random parameters are estimated simultaneously, providing consistent estimates of the fixed parameters and their standard errors. Furthermore, the inclusion of random parameters allows to model the variability detected for a given phenomenon among different locations, clusters and measurement occasions within a given population, after defining a common fixed functional structure (CastedoDorado et al. 2006). Mixed models also improve the predictive ability if it is possible to predict the value of the random parameters for a location and measurement occasion, which is not part of the dataset by using additional measurements as preinformation (Lappi 1997; Mehtätalo 2004). This particular method of predicting random effects by BLUPS (best linear unbiased predictors) is usually described as model calibration (Lappi 1997). Hence, applying mixed model methodology to generalized height–diameter functions allows for a prediction using both information in sync: predictors as mean quadratic diameter and additionally measured diameter–height pairs. The use of an arbitrary number of diameter–height pairs for calibration is an additional advantage. These measurements need not be from representative mean or dominant trees.
However, in many applications a theoretical problem remains since the random effects are usually assumed to be independently distributed. This assumption is violated if the data originating, for example, from largescale inventories show spatial correlation patterns. Up to now, this topic has been rarely discussed in a forestry context (Brezger and Lang 2006; Augustin et al. 2009). An approach to model the spatial correlation of the random effects related to only one parameter of a height–diameter model via a geostatistical approach is presented by Nanos et al. (2004).
Spatial effects are usually a surrogate of the effects of other unobserved influential factors. But in many applications, the spatial effect is not only the result of factors that exist only locally, i.e. on cluster level but also the result of factors that show a strong spatial structure. Hence, the overall spatial trend can be separated into a spatially correlated (structured) and an uncorrelated (unstructured) effect (Brezger and Lang 2006). Subsequently, only the unstructured effect is modeled by (uncorrelated) random effects on cluster level. For the structured effect, two main approaches may be distinguished: (I) the structured spatial effect is modeled in the framework of socalled geoadditive models via a Gaussian Markov random field, i.e. spatially correlated random effects are estimated for the spatial units of the observations (Kammann and Wand 2003). (II) the structured spatial effect is modeled via 2dimensional surface fitting by applying specific generalized additive models based on e.g. penalized regression splines with thin plate basis (Wahba 1990; Wood 2006). The second approach is particularly suited if the data locations are described by exact coordinates. The first approach can also be employed if the observations are assigned to adjacent geographic units like forest districts. Both approaches can be combined with the estimation of (uncorrelated) random effects to account for the unstructured spatial effect simultaneously. The resulting model types are still called either geoadditive models concerning approach I or generalized additive mixed models gamm concerning approach II (Lin and Zhang 1999; Fahrmeir and Lang 2001). However, the separation of both (structured and unstructured) spatial effects might easily lead to numerical problems using approach II, especially if the model for the 2dimensional surface is rather complex (Wood 2006). If not only cross sectional but also longitudinal data should be modeled, the theory could be extended by modeling additionally a (nonlinear) time trend or even 3dimensional space–time trends (Wood 2006; Augustin et al. 2009).
Further extensions are the implementation of nonlinear effects for additional covariates leading to the wellknown generalized additive models gam (Hastie and Tibshirani 1990). If not only main effects but also interactions between nonlinear effects and categorical or continuous covariates are to be modeled, this specific type of a gam is called a varying coefficient model VCM (Hastie and Tibshirani 1993). In this model type the nonlinear effects are effect modifiers of the categorical or continuous covariates. If the interaction covariate is categorical, then this results in separate nonlinear effects for each level of the covariate. If the covariate is continuous, a special case of an interaction is modeled because one of the covariates enters the model still linearly. The natural extension to allow for an interaction term that is nonlinear with respect to several covariates is again based on 2 or multidimensional surface fitting (Wood 2006). In contrast to 2dimensional surface fitting for describing structured spatial effects, usually tensor product splines are employed that allow for different degrees of smoothing (anisotropy) for the different dimensions (Wood 2006).
Both approaches (I) and (II) for describing structured spatial effects could be also combined with continuous or categorical covariates. In this case, a special type of a VCM with a spatially varying effect modifier results (Fotheringham et al. 2002). Hence, in this case not only the intercept is allowed to vary in space but also the coefficients of one or more covariates. All listed model classes are subsumed by Fahrmeir et al. (2007) under the class of structured additive regression models STAR.
The main objective of this paper is to predict tree height for a certain dbh with high accuracy using only a minimum set of information, i.e. only mean quadratic diameter and geographic coordinates that can be easily derived. No additional information should be used to allow for a very simple practical application, and no predictors should be used that are themselves resulting from extensive modeling processes and that include a prediction error. Hence, the main objective is prediction rather than quantification of stand and site effects. The effect of dg that changes with age and as a result of silviculture should account for the longitudinal development of the height–diameter relationship. The effect of the geographic coordinates subsumes the effects of all potential factors that are spatially correlated. This structured spatial effect can be employed in predictions even if no additional measurements are available. Finally, uncorrelated random effects subsume the effects of all spatially uncorrelated factors that appear only locally on plot level.
For this purpose, the wellknown Näslund height–diameter function is parameterized within the generalized additive regression gam frame work (Hastie and Tibshirani 1990) applying the specific methodology of Wood (2006). A high flexibility of the model is needed, because only few predictor variables are used. The approach should additionally allow for a calibration of a mean population model if additional height–dbh measurements are available. As a result, we present a height–diameter model approach that is parameterized in the framework of methodologically wellfounded STAR models (Fahrmeir et al. 2007) employing several of the initially stated model components.
Materials and methods
This section introduces the Estonian Permanent Forest Research Network that provides an extensive dataset for the study and explains the methods which were applied.
The Estonian Permanent Forest Research Network
During the past two decades, a permanent Forest Research Network database has been established in Estonia including measurements from different original data sources. The database was developed with the understanding that a longterm series of remeasured plots would provide a useful basis for forest growth modeling. All basic forest types, stand ages and stand densities should be represented. Tree coordinates should be assessed, and plot areas should be large enough to characterize stand structures (Kiviste et al. 2007).
Choice of model
For ensuring centered random effects, a coefficient α _{0} was also estimated. Hence, the estimated conditional expectation value \( \hat{y}_{ijk} \) enters the model as a predictor not just as an offset. To take heteroscedasticity into account, the variance can be modeled through a function involving either the predicted value or some explanatory variables (Davidian and Giltinan 1995; Pinheiro and Bates 2000). We used the powerofthemean function which is a wellknown example of a variance function based on predicted values, where σ ^{2} is the residual variance, and θ is a parameter to be estimated (Eq. 2b2). The linear mixed model (Eq. 2b2) was parameterized employing the Rlibrary nlme (Pinheiro et al. 2006).
Calibration
Results
Generalized additive model
Statistical characteristics of the generalized additive model (Eq. 2b1)
Estimate  Std. error  tValue  Pr(>t)  

Coefficients of parametric terms  
α  0.9799  0.0084  117.1  <2e16*** 
β  0.3418  0.0004  771.6  <2e16*** 
edf  F  Pvalue  

Approximate significance of smooth terms  
f _{1} (dg)  2.915  41.67  <2e16***  
f _{2} (north, east)  28.204  29.92  <2e16***  
f _{3} (dg)  8.995  495.83  <2e16***  
f _{4} (north, east)  29.000  105.48  <2e16***  
R ^{2} (adjusted) = 0.986  Deviance explained = 98.6%  
GCV score = 0.089  n = 22,347  
Significance codes: ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 
Effects of squared mean diameter (dg)
The overall effect of the dg on the height–diameter curve is illustrated in Fig. 4. For a constant geographic location (northing 58.5, easting 25), the increase in dg results in larger heights for a predefined dbh. However, from a dg of 34 cm the height–diameter curve is assumed to be constant. This constraint was set during model parametrization since the derived (unconstraint) nonlinear effects resulted in an unfeasible set of curves for plots exceeding a dg of 34 cm up to the maximum dg within the database of 41.4 cm. Parameterizing unconstrained effects of dg resulted in a decrease in tree height for a given dbh for the entire diameter range. The crossing of height–diameter curves over time or with increasing dg is a natural consequence of different growth rates in different diameter and height classes that should not be constrained (Lappi 1997). However, in our investigation the entire curve moved toward lower heights, and this did not seem to be feasible. Only about 3% of the observed pine stands in our database have a dg of more than 34 cm. Thus, the decrease in the height–diameter curve with increasing dg in the area of large dg’s might be due to confounding effects and an unbalanced data structure. For example, site index is often correlated with stand age, and thus patterns like “old stands grow mostly on poor sites” confound the original correlation. The results show that the database needs to be extended for large dg’s that cover the whole range of sites in Estonia.
Geographical effects (structured spatial effects)

both parameters show a considerable variability within Estonia, which underlines the need for two 2dimensional surface estimators that vary α and β simultaneously.

the parameters are negatively correlated as the spatial distributions are similar but inverted;

the variable width of the confidence intervals in Fig. 5 reflects the different sample densities; as expected, it is rather narrow in the southcentral part where the highest concentration of research plots is found.
Mixed model
Statistical characteristics of the linear mixed model (Eq. 2b2)
Value  Std. error  df  tValue  PValue  

Fixed effects  
α _{0}  1.003051  0.001832160  21,557  547.469  0 
Value  Lower 95% confidence bound  Upper 95% confidence bound  

Random effects  
sd (a _{ i })  0.2629  0.2382  0.2902 
sd (b _{ i })  0.01804  0.01649  0.01974 
cor (a _{ i }, b _{ i })  −0.6731  −0.7360  −0.59867 
sd (a _{ ij })  0.08218  0.07165  0.09426 
sd (b _{ ij })  0.002854  0.002138  0.003809 
sd (ε _{ ijk })  0.1415  0.1347  0.1487 
With  \( D_{1} = \left[ {\begin{array}{*{20}c} {\text{var} (a_{i} )} \hfill & {\text{cov} (a_{i} ,b_{i} )} \hfill \\ {\text{cov} (a_{i} ,b_{i} )} \hfill & {\text{var} (b_{i} )} \hfill \\ \end{array} } \right] \)  \( D_{2} = \left[ {\begin{array}{*{20}c} {{\text{var}}(a_{ij} )} \hfill & 0 \hfill \\ 0 \hfill & {{\text{var}}(b_{ij} )} \hfill \\ \end{array} } \right] \)  
Variance function  
θ  0.2052  0.1807  0.2299 
Discussion and conclusions
The presented model for Scots pine provides a first comprehensive basis for tree height estimation from measured diameters for the whole area of Estonia, based on the extensive database of the Estonian forest growth experiments. Like Mehtätalo (2004), we use dg instead of age (Lappi 1997; Eerikäinen 2003) to describe the longitudinal development of the height–diameter curves. Since the height–diameter relationship is modeled as a function of dg, the model can be used as a surrogate for a real height growth model also. This would require a combination with a diameter growth model. The advantage of this oftenused approach is that tree height increments are not required for model parameterization. The linear mixed model used in our analysis shows some similarity with the approaches presented by Lappi (1997) and Mehtätalo (2004). However, there some differences. Firstly, Lappi and Mehtätalo use a reparameterized logarithmic height–diameter relationship (KorfFunction) where the parameters have some biological meaning and a somewhat lower correlation than the Näslund function. However, the Näslund function was already applied successfully in linear mixed models (Kangas and Maltamo 2002; Kinnunen et al. 2007). Lappi and Mehtätalo present generalized models that include different sets of predictor variables to describe the trends of the original parameters. They also include a mean tree diameter, as we have done, but additionally basal area. Mehtätalo (2004) also uses the north coordinate and a categorical variable to differentiate the site nutrient supply. The different sets of predictors should account for differences in the availability of information. However, their model variants that are predictor parsimonious are considerably less (spatially) flexible compared with our approach since the model effects are simple linear relationships. It can be assumed that their variants with several predictors are more flexible especially if the predictors are itself predictions from regionalization processes. But our model combines a high flexibility with parsimonious input information. In the future, additional predictors might be used in our model also, especially in view of improving the biological interpretability. However, it can be assumed due to unobserved influencing factors and limited regionalization processes that spatial trends still will be present in many databases. Hence, our methodology of estimating 2dimensional surfaces for structured spatial trends will still be required for optimum estimation.
The approach by Nanos et al. (2004) shows a high spatial flexibility also. But their procedure does not separate between a structured (correlated) spatial effect and an unstructured (uncorrelated) spatial effect like we do. Our 2step approach is a result of numerical problems, but theoretically the procedure employs a simultaneous estimation of 2dimensional surfaces and (uncorrelated) random effects. In contrast, the 2step approach employed by Nanos et al. (2004) is the result of the underlying methodology: The random effects of a nonlinear mixed model are predicted in a second step by applying geostatistical methods. Hence, several methodological aspects have to be discussed: (I) Is the geostatistical model flexible enough to account for the unstructured spatial trend also? In our model, this unstructured spatial trend is quantified by uncorrelated random effects. (II) In the geostatistical approach, the preceding (first step) estimation of random effects is employed without considering spatial correlation. Additionally, only one parameter is assumed to vary randomly, and it is not clear in which way the geostatistical approach could be extended to models with random variation of several parameters since the random effects are usually correlated. In our approach, this correlation is accounted for, because the two 2dimensional surfaces are estimated simultaneously. (III) In practical applications, the approach of Nanos et al. (2004) employs two different models depending on whether height–diameter measurements are available or not. If prior information is available, the random effects are predicted via the mixed nonlinear model; otherwise, the random effects are estimated via the geostatistical approach. In our case, the fixed part of the model (Eq. 2b1) is already spatially explicit and is employed even if prior information is not available. If prior information is available, the calibration is based on the same model but the random part is employed also.
Because of the more complex specification of the 2dimensional trend function within a gam, we had to apply a 2step procedure by first fitting the gam and then using the prediction as ‘a priori information’ in a linear mixed model. This procedure might not be ideal, and further approaches like geoadditive models that estimate the structured spatial trend via a Gaussian Markov random field should be tested. However, the quantification of the spatial pattern of the height–diameter curve is a considerable advantage of the model and accounts already for the structured spatial effects. Additionally, further studies are required concerning the temporal autocorrelation of the random effects of different measurement occasions from the same plot. Until now, only a subsample of the database includes remeasurements, and the observed time series are rather short.
We recommend that the already excellent Forest Research Network database will be extended in the future or complemented with the dataset of the Estonian National Forest Inventory, especially with the aim of capturing areas with large dg’s and regions of Estonia that are sparsely covered with sample plots. A such extended database would possibly improve our model, e.g. removal of the unconstraint decrease of the height–diameter curve for dg larger 34 cm. As an alternative monotonicity constraints could be explicitly integrated in the model (Brezger and Steiner 2003).
If the spatial distribution of sample plots will be more regular, further investigation into the choice of the basis dimension of the 2dimensional trend function could be conducted. Wood (2006) recommends to test informally by increasing the maximum basis dimension, if the chosen dimension is sufficient to represent the underlying data structure reasonably well. An indication for inadequate model flexibility would be an estimated basis dimension near the defined maximum. Given sufficient flexibility (i.e. the basis dimension is large enough), the degree of smoothing is almost exclusively governed by a “penalizing term”, which is controlled by a smoothing parameter. To determine the optimal smoothing parameter, extended generalized crossvalidation techniques are applied which result in minimizing a generalized crossvalidation score (GCV score; Wood 2006).
First attempts of increasing the basis dimension of the 2dimensional smooth function resulted in a reduced bias. Increasing the dimension considerably from 29 to 225 resulted in a decrease of the overall height bias from 0.156 to 0.106 m. Further enlargement of the dimension had almost no effect on the bias reduction. However, this part of model selection needs to be supplemented by extensive crossvalidation, especially due to the unbalanced spatial distribution of the experimental plots.
Notes
Acknowledgments
We thank two anonymous reviewers for valuable comments that improved significantly the quality of the manuscript.
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