Bias of the mixed-effects HCB model in the validation data.

<p>The random effects were estimated using the measured HCB of four randomly selected trees per sample plot [solid line: spatially inexplicit HCB model; dotted line: spatially explicit HCB model].</p

Modelling individual tree height to crown base of Norway spruce (<i>Picea abies</i> (L.) Karst.) and European beech (<i>Fagus sylvatica</i> L.)

<div><p>Height to crown base (HCB) of a tree is an important variable often included as a predictor in various forest models that serve as the fundamental tools for decision-making in forestry. We developed spatially explicit and spatially inexplicit mixed-effects HCB models using measurements from a total 19,404 trees of Norway spruce (<i>Picea abies</i> (L.) Karst.) and European beech (<i>Fagus sylvatica</i> L.) on the permanent sample plots that are located across the Czech Republic. Variables describing site quality, stand density or competition, and species mixing effects were included into the HCB model with use of dominant height (HDOM), basal area of trees larger in diameters than a subject tree (BAL- spatially inexplicit measure) or Hegyi’s competition index (HCI—spatially explicit measure), and basal area proportion of a species of interest (BAPOR), respectively. The parameters describing sample plot-level random effects were included into the HCB model by applying the mixed-effects modelling approach. Among several functional forms evaluated, the logistic function was found most suited to our data. The HCB model for Norway spruce was tested against the data originated from different inventory designs, but model for European beech was tested using partitioned dataset (a part of the main dataset). The variance heteroscedasticity in the residuals was substantially reduced through inclusion of a power variance function into the HCB model. The results showed that spatially explicit model described significantly a larger part of the HCB variations [R²_adj = 0.86 (spruce), 0.85 (beech)] than its spatially inexplicit counterpart [R²_adj = 0.84 (spruce), 0.83 (beech)]. The HCB increased with increasing competitive interactions described by tree-centered competition measure: BAL or HCI, and species mixing effects described by BAPOR. A test of the mixed-effects HCB model with the random effects estimated using at least four trees per sample plot in the validation data confirmed that the model was precise enough for the prediction of HCB for a range of site quality, tree size, stand density, and stand structure. We therefore recommend measuring of HCB on four randomly selected trees of a species of interest on each sample plot for localizing the mixed-effects model and predicting HCB of the remaining trees on the plot. Growth simulations can be made from the data that lack the values for either crown ratio or HCB using the HCB models.</p></div

Parameter estimates, variance-covariance components, and fit statistics of the mixed-effects model (Eq 5).

<p>[<i>R</i>^<i>2</i>_<i>adj</i>: adjusted coefficient determination; <i>RMSE</i>: root mean squared errors; <i>AIC</i>: Akaike’s information criterion; <i>b</i>_<i>1</i>, <i>b</i>_<i>2</i>,<i>…</i>,<i>b</i>_<i>6</i>: fixed parameters; <i>u</i>_<i>j1</i>, <i>u</i>_<i>j2</i> = random effects parameters; <i>σ</i>^<i>2</i>_<i>uj1</i>: variance of <i>u</i>_<i>j1</i>; <i>σ</i>^<i>2</i>_<i>uj2</i>: variance of <i>u</i>_<i>j2</i>; <i>σ</i>^<i>2</i>: variance according to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0186394#pone.0186394.e010" target="_blank">Eq 7</a>; <i>φ</i> = parameter of a power variance function (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0186394#pone.0186394.e011" target="_blank">Eq 8</a>); standard errors are given in the parenthesis].</p

Standardized residuals of spatially inexplicit mixed-effects HCB model with a power variance function (Eq 8) included.

<p>Standardized residuals of spatially inexplicit mixed-effects HCB model with a power variance function (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0186394#pone.0186394.e011" target="_blank">Eq 8</a>) included.</p

Location of sample plots.

<p>(<b>a</b>) training data [purely Norway spruce or Norway spruce-dominated sample plots (red dots), purely European beech or European beech-dominated sample plots (black triangles), light green dots showing forest cover, and grey lines separating Natural Forest Area], (<b>b</b>) model validation data collected on the sample plots in the Krkonoše national park.</p

Data summary.

<p>Data summary.</p