Generalized Nonlinear Mixed-Effects Individual Tree Crown Ratio Models for Norway Spruce and European Beech
Abstract
:1. Introduction
2. Materials and Methods
2.1. Sampling and Measurements
2.2. Data Analysis
2.2.1. Tree and Stand Measures
- canopy height class I, CC1: height greater than 66% of the tallest tree;
- canopy height class II, CC2: height between 33% and 66% of the tallest tree; and
- canopy height class III, CC3: height less than 33% of the tallest tree.
2.2.2. Model Development
2.3. Estimation of Model Parameters and Evaluation
2.4. The Localizing Mixed-Effects Model and Subject-Specific CR Prediction
3. Results
4. Discussion
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Variables | Mean ± Std. (Range) | |
---|---|---|
Norway Spruce | European Beech | |
Number of sample plots | 90 (18 monospecific + 72 mixed species) | 88 (18 monospecific + 70 mixed species) |
Number of sample trees | 6736 | 7933 |
Number of trees per sample plot | 149 ± 115 (3–430) | 142 ± 124 (3–515) |
Number of trees per hectare (N ha−1) | 1005 ± 676 (92–2568) | 827 ± 745 (41–2568) |
Basal area (BA, m2 ha−1) | 47.2 ± 19.4 (7.2–91.4) | 43.3 ± 14 (0.7–89.1) |
BA proportion of a species (BAPRO) | 0.72 ± 0.31 (0.0005–1) | 0.72 ± 0.24 (0.0004–1) |
BA of trees larger in diameter than a subject tree (BAL, m2 ha−1) | 36.7 ± 21.2 (0–88.8) | 33.9 ± 16.7 (0–86.3) |
Quadratic mean DBH (QMD, cm) | 28.9 ± 9.6 (11.9–60.4) | 32.9 ± 11.8 (15.4–87.4) |
DBH-to-QMD ratio (dq) | 1.2 ± 0.5 (0.1–6.8) | 1.1 ± 0.64 (0.12–5.7) |
Arithmetic mean DBH (cm) | 24.8 ± 10.5 (9.4–53.9) | 28.6 ± 12.8 (9.4–84.4) |
DBH sum (DBHSUM, cm) | 4845 ± 1860 (813–9333) | 4172 ± 1911 (675–9333) |
Dominant diameter (DDOM, cm) | 49.9 ± 13.3 (18.4–77.1) | 56.7 ± 11.2 (24.7–84.3) |
Dominant height (HDOM, m) | 27.1 ± 8.2 (8–42) | 30.6 ± 6.3 (13.6–42.8) |
Mean height (m) | 16.2 ± 6.7 (4.6–37.4) | 19.2 ± 7.9 (6.6–42.8) |
Relative spacing index (RSI) | 0.15 ± 0.07 (0.05–0.44) | 0.15 ± 0.06 (0.06–0.41) |
Total height (m) | 16.9 ± 9.9 (1.5–48.7) | 19.4 ± 10.3 (1.5–50.6) |
Diameter at breast height (DBH, cm) | 26.7 ± 18.5 (2.5–119) | 28.3 ± 20.1 (2.9–118.7) |
Height-to-DBH ratio (HDR, m cm−1) | 0.6 ± 0.3 (0.03–6.6) | 0.8 ± 0.3 (0.1–6.1) |
Height to crown base (HCB, m) | 6.26 ± 5.7 (0.01–37) | 8.5 ± 6.4 (0.1–35) |
Crown diameter (CW, m) | 3.7 ± 1.6 (0.5–13.9) | 6.1 ± 3.2 (0.8–24.1) |
Crown depth (CL, m) | 10.6 ± 6.7 (0.2–35.3) | 10.9 ± 6.4 (0.5–45.2) |
Crown ratio (CR) | 0.65 ± 0.19 (0.05–0.99) | 0.58 ± 0.18 (0.05–0.99) |
Model Components | Parameter Estimates | |
---|---|---|
Norway Spruce | European Beech | |
Fixed | ||
α1 | 0.186249 (0.0372) | 0.343277 (0.0233) |
α2 | −0.10861 (0.0362) | 0.073573 (0.0233) |
α3 | −0.2362 (0.0379) | −0.05535 (0.0245) |
β1 | 0.025698 (0.000945) | 0.026731 (0.000564) |
β2 | 0.002476 (0.000246) | −0.00229 (0.000218) |
β3 | 1.106038 (0.0972) | 0.912028 (0.0576) |
β4 | 0.048479 (0.0149) | −0.19403 (0.0156) |
β5 | −0.09135 (0.00115) | −0.07912 (0.000764) |
b2 | 0.194021 (0.009281) | 0.201351 (0.015073) |
Variance | ||
σ2ui1 | 1.9746 | 1.8083 |
σui1ui2 | −0.19082 | −0.08725 |
σ2ui2 | 0.01296 | 0.00957 |
σ2 | 0.00998 | 0.00986 |
Fit statistics | ||
R2adj | 0.6334 | 0.7203 |
RMSE | 0.0999 | 0.0993 |
AIC | −28612 | −36931 |
© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Sharma, R.P.; Vacek, Z.; Vacek, S. Generalized Nonlinear Mixed-Effects Individual Tree Crown Ratio Models for Norway Spruce and European Beech. Forests 2018, 9, 555. https://doi.org/10.3390/f9090555
Sharma RP, Vacek Z, Vacek S. Generalized Nonlinear Mixed-Effects Individual Tree Crown Ratio Models for Norway Spruce and European Beech. Forests. 2018; 9(9):555. https://doi.org/10.3390/f9090555
Chicago/Turabian StyleSharma, Ram P., Zdeněk Vacek, and Stanislav Vacek. 2018. "Generalized Nonlinear Mixed-Effects Individual Tree Crown Ratio Models for Norway Spruce and European Beech" Forests 9, no. 9: 555. https://doi.org/10.3390/f9090555