Allometric modelling of crown width for white spruce by fixed- and mixed-effects models

Abstract

Crown width is an important predictor for tree growth, crown surface area, forest canopy cover, tree-crown profiles and wildlife habitat indices. This paper developed crown width models for white spruce (Picea glauca (Moench) Voss) in Alberta using allometric fixed and mixed models with varying degrees of model complexity. Diameter at breast height was the most important predictor and was used in the base model. Crown ratio, height-diameter ratio and two competition indices (CIs) were additional predictors added to the base model to form four expanded models. At each level of complexity, a fixed model and a mixed model were fitted. Improved fits were achieved for both model types as model complexity increased, and all mixed models provided much better fits than their fixed model counterparts. Population-averaged (PA) predictions by fixed models, and typical mean (TM), PA and plot-specific (PS) predictions by mixed models were compared on both model fitting and validation data. TM and PA predictions by each mixed model were almost identical, and they were less accurate than PA predictions by the fixed model counterpart, especially for simpler models. Much better PS predictions by mixed models were observed on both datasets. Although the distancedependent CI was slightly better than the distance-independent CI, both were not recommended due to their marginal contributions to crown width predictions.Keywords: crown width, mixed model, competition index, model prediction, white spruce RÉSUMÉ La largeur de cime est une variable importante pour estimer la croissance, la surface de cime, le couvert forestier, le profil de cime et les indices d'habitat pour la faune. Cet article propose un modèle de largeur de cime élaboré pour l' épinette blanche (Picea glauca (Moench) Voss) de l' Alberta à l'aide de modèles allométriques fixes et mixtes de divers niveaux de complexité. Le diamètre à hauteur de poitrine s' est avéré être la variable la plus importante et celle qui a été retenue dans le modèle de base. Le coefficient de forme, le rapport hauteur-diamètre et deux indices de compétition (CIs) ont été ajoutés au modèle de base afin de créer quatre modèles élargis. On a ajusté un modèle fixe et un modèle mixte pour chaque niveau de complexité. Dans les deux cas, l'ajustement du modèle s'améliorait à mesure qu'augmentait sa complexité, et les modèles mixtes donnaient un meilleur ajustement que les modèles fixes. On a comparé les prévisions moyennes avec celles au niveau des populations (Population-averaged -PA) obtenues avec les modèles fixes et les moyennes-types (TM), les prévisions PA et celles par placette (PS) obtenues avec les modèles mixtes à la fois sur leur ajustement et avec des ensembles de données de validation. Les prévisions TM et PA pour chacun des modèles mixtes ont été identiques et se sont avérées moins précises que les prévisions PA à l'aide des modèles fixes, surtout avec les modèles les plus simples. On a toutefois obtenu de bien meilleures prévisions PS avec les modèles mixtes lors de la validat...

A Crown Width-Diameter Model for Natural Even-Aged Black Pine Forest Management

“…Diameter at breast height (DBH) is the most commonly used independent variable in crown width modeling, since the relationship between DBH and crown width (CW) has been well established in the literature. The mathematical form that links DBH and CW could be either linear [12,14,16,42] or nonlinear [7,23,30,34]. For the needs of the current research, a number of candidate simple linear and nonlinear models were fitted and the best ones, in terms of their fitting ability, were selected for further analysis.…”

“…The incorporation of DBH as the only predictive variable assumes that trees of the same stem diameter also have the same mean crown dimensions, regardless of the competition levels inside the stand; an assumption that does not hold. In order to relax this assumption, previous research efforts have instead used tree variables, such as the crown length or ratio, the tree height and the distance dependent or independent stand-level predictors, which reflect the competition status [7,15,23,34,45]. The inclusion of additional variables leads to the creation of generalized models, which usually improve the crown width predictability in large areas under different stand competition regimes.…”

“…The Φi vector consisted of one fixed effect part which was the same for the entire population and a random effect part which was unique for each plot [20]. The vector of random effects and the error vector were assumed to be uncorrelated and normally distributed with zero mean and variance-covariance matrices D and Ri, respectively [7,27]. The Ri variance-covariance matrix was expressed as [50]…”

“…Therefore, crown size is a critical feature of any tree growing at stand or open conditions, particularly in mountain areas, where water limitations rarely occur. In forestry practice, crown size estimation is considered of vital importance because it is associated with many different factors of forest management concern, such as the photosynthetic capacity [3], stem volume [4], increment efficiency [5], tree competition [6,7], tree health and vigor [8], as well as carbon, water and energy exchange [9]. From an ecological point of view, crown size directly affects many of the understory flora and fauna components by regulating the amount of the penetrated sunlight and precipitation to the lowest layers, thus retaining the forest moisture [10].…”

“…Mixed-effect models of forest growth and yield have been recommended by several authors in the literature as an alternative solution to independency issues emanating from ordinary least squares application (e.g., [19,[21][22][23][24][25][26][27][28]). As far as crown width modeling is concerned, application of the mixed-effect modeling approach has been more limited as yet (Sánchez-González et al [23], Ducey [29], Fu et al [30,31], Hao et al [32], Yang and Huang [7] and Sharma et al [33,34]). In most cases, the mixed-effect models were reported to have generated more accurate predictions than the fixed effect models, since the structural heterogeneity, at tree or at stand level, was reflected by the random-effect parameters.…”