Nonlinear fixed and random generalized height–diameter models for Portuguese cork oak stands - Paulo, Joana Amaral; Tomé, José; Tomé, Margarida (Annals of Forest Science, 2011)
• Objective The objective of the research was to develop a generalized height–diameter model for Quercus suber L. in Portugal, which can be applied both to undebarked and debarked trees, with diameter at breast height over cork larger than 2.5 cm. • Methods A nonlinear fixed effects model (NLFEM) and a nonlinear mixed effects model (NLMEM) approaches were used. Parameters estimates were obtained using the SAS macro NLINMIX, which uses a linear approximation to the marginal likelihood function by expanding it with a first-order Taylor series on the random effects. The option of expanding on the random effects at their current empirical best linear unbiased predictors (EBLUP) was used. The fitted models were evaluated using an independent data set, together with an existing model specific for undebarked trees. To obtain subject specific predictions with the NLMEM, a conventional and an improved calibration procedures were applied, considering four different tree sub-sampling designs. Both proposed models included dominant height and stand density as covariates to explain plot variability. • Conclusions Validation indicated that, even in the situations where the NLMEM calibration is not possible, this model should be preferred. The differences between the validated models, which were more evident for young stands, were considered. No large differences in predictive accuracy were found between the calibrated NLMEM using the conventional or the improved calibration procedures, for all the considered sub-sampling designs.
Keywords: height-diameter relationship; model calibration; nonlinear fixed effects model; nonlinear mixed effects model; Quercus Suber L

Generalized height-diameter and crown diameter prediction models for cork oak forests in Spain - Sanchez-Gonzalez, M; Cañellas, I; Montero, G (Investigación Agraria: Sistemas y Recursos Forestales, 2007)
A generalized height-diameter equation, along with a crown diameter prediction equation for cork oak forests in Spain were developed based on data from the Second Spanish Forest Inventory. Nine generalised height-diameter equations were selected as candidate functions to model the height-diameter under cork relationship, while for the crown diameter prediction model five linear and non-linear equations were tested. The equations were fitted using the mixedeffects model approach. The Stoffels & Van Soest power equation, constrained to pass through the point of dominant diameter and dominant height, was selected as the generalised height-diameter model. Regarding the crown diameter prediction model, the parable function without the intercept and with quadratic mean diameter incorporated as a fixed effect into the b parameter, proved to be the model with best prediction capabilities. The models were validated by characterising the model error using the PRESS (Prediction Sum of Squares) statistic. Both equations will be submodels of the ALCORNOQUE v1.0, a management oriented growth and yield model for cork oak forests in Spain.
Keywords: crown width; forest growth modelling; height-diameter relationship; mixed effects models; Quercus suber

Generalized height-diameter and crown diameter prediction models for cork oak forests in Spain - Sánchez-González, M.; Cañellas, I.; Montero, G. (Investigación Agraria: Sistemas y Recursos Forestales, 2007)
A generalized height-diameter equation, along with a crown diameter prediction equation for cork oak forests in Spain were developed based on data from the Second Spanish Forest Inventory. Nine generalised height-diameter equations were selected as candidate functions to model the height-diameter under cork relationship, while for the crown diameter prediction model five linear and non-linear equations were tested. The equations were fitted using the mixedeffects model approach. The Stoffels & Van Soest power equation, constrained to pass through the point of dominant diameter and dominant height, was selected as the generalised height-diameter model. Regarding the crown diameter prediction model, the parable function without the intercept and with quadratic mean diameter incorporated as a fixed effect into the b parameter, proved to be the model with best prediction capabilities. The models were validated by characterising the model error using the PRESS (Prediction Sum of Squares) statistic. Both equations will be submodels of the ALCORNOQUE v1.0, a management oriented growth and yield model for cork oak forests in Spain.
Keywords: crown width; forest growth modelling; height-diameter relationship; mixed effects models; Quercus suber