Variable-Exponent Taper Equation Based on Multilevel Nonlinear Mixed Effect for Chinese Fir in China
Abstract
:1. Introduction
2. Materials and Methods
2.1. Data
2.2. Basic Model Selection
2.3. Establishment of Mixed Effect Model
2.3.1. Determination of Random Parameters
2.3.2. Variance-Covariance Structure of Stochastic Effect
2.3.3. Intra-Group Variance-Covariance Structure
2.4. Model Evaluation
3. Results and Analysis
3.1. Selection of Basic Model
3.2. Construction of Mixed Effect Model
3.3. Predicted Diameters and Model Calibration
4. Discussion
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Number of Trees | Mean Height/m | S.D | Mean Diameter at Breast Height /cm | S.D | |
---|---|---|---|---|---|
Fit data | |||||
B: 3333 (2 × 1.5 m) | 15 | 13.56 | 2.34 | 12.41 | 2.67 |
C: 5000 (2 × 1 m) | 33 | 12.77 | 2.32 | 11.06 | 2.56 |
D: 6667 (1 × 1.5 m) | 39 | 12.19 | 1.76 | 10.28 | 2.06 |
E: 10,000 (1 × 1 m) | 36 | 11.78 | 2.15 | 9.48 | 2.16 |
Validation data | |||||
B: 3333 (2 × 1.5 m) | 15 | 13.50 | 2.29 | 12.63 | 2.45 |
C: 5000 (2 × 1 m) | 15 | 13.00 | 2.17 | 10.85 | 2.31 |
D: 6667 (1 × 1.5 m) | 15 | 11.60 | 2.03 | 10.21 | 2.15 |
E: 10,000 (1 × 1 m) | 15 | 12.00 | 1.90 | 9.97 | 2.16 |
Models | Parameters | Variables | |
---|---|---|---|
Munro [23] | b1,b2 | D,H,h | |
Kozak et al. (a) [2] | b1 b2 | D,T | |
Kozak et al. (b) [2] | b1 | D,T | |
Bennett and Swindel [24] | b1,b2,b3,b4 | D,X, | |
Cervera [26] | b1,b2,b3,b4,b5 | D,X | |
Demaerschalk [25] | b1,b2,b3,b4 | D,H,h | |
Demaerschalk (a) [27] | b1,b2,b3,b4 | D,H,h | |
Demaerschalk (b) [27] | b1,b2,b3,b4 | D,H,h | |
Ormerod [3] | b1 | D,X | |
Coffre [28] | b1,b2,b3 | D,X | |
Biging [29] | b1,b2 | D,T | |
Reed and Green [30] | b1,b2 | D,T | |
Newberry and Burkhart (a) [31] | b1,b2 | D,H,h | |
Newberry and Burkhart (b) [31] | b1,b2 | D,H,h | |
Real and Moore [32] | b1,b2,b3 | D,X | |
Forslund [33] | b1,b2 | D,T | |
Thomas and Parresol [34] | b1,b2,b3,b4 | D,T | |
Jiménez et al. [35] | b1,b2,b3,b4,b5,b6 | D,T | |
Max and Burkhart [6] | b1,b2,b3,b4,b5,b6 | D,T | |
Valenti and Cao [36] | b1,b2,b3,b4,b5,b6 | D,Z | |
Riemer et al [38] | b1,b2,b3 | D,H,h | |
Zeng and Liao [37] | b1,b2,b3,b4 | D,H,T,X | |
Muhairwe (a) [14] | b1,b2,b3,b4,b5,b6,b7,b8 | D,T,D/H | |
Muhairwe (b) [14] | b1,b2,b3,b4,b5,b6,b7,b8 | D,T,D/H | |
Bi [12] | b1,b2,b3,b4,b5,b6,b7 | D,H,T | |
Lee et al. [39] | b1,b2,b3,b4,b5 | D,T | |
Kozak [1] | b1,b2,b3,b4,b5,b6,b7,b8,b9 | D,T,H | |
Sharma and Zhang [13] | b1,b2,b3,b4 | D,T,H | |
Berhe and Arnoldsson [40] | b1,b2,b3,b4,b5 | D,T | |
Sharma and Parton [41] | b1,b2,b3,b4 | D,T,H |
Model | b1 | b2 | b3 | b4 | b5 | b6 | b7 | b8 | b9 | Bias/cm | RMSE | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Cervera [26] | 0.1089 | 1.6039 | −0.4004 | −1.5954 | 1.3061 | 0.9794 | 0.0163 | 0.5075 | ||||
Max and Burkhar [6] | −4.0941 | 1.7120 | −1.6214 | 57.8924 | 0.8872 | 0.0951 | 0.9806 | 0.0434 | 0.4926 | |||
Zeng and Liao [37] | 4.3567 | −9.0650 | 5.0886 | 0.2407 | 0.9830 | 0.0311 | 0.4613 | |||||
Bi [12] | 1.1652 | −0.4681 | −0.0659 | −0.4987 | −0.0343 | 0.0992 | −0.0628 | 0.9835 | 0.0151 | 0.4540 | ||
Kozak [1] | 1.0097 | 0.9800 | 0.0205 | 0.2983 | −0.3391 | 0.4754 | 0.1523 | 0.03195 | −0.0621 | 0.9829 | 0.0036 | 0.4631 |
Sharma and Zhang [13] | 1.0093 | 2.1547 | −0.4664 | 0.4720 | 0.9824 | 0.0158 | 0.4688 |
Models | Basic Model | Mixed Parameter | Plot-Level Effects | Tree-Level Effects | Mixed Parameter | Nested Effects of Plot and Tree | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
AIC | BIC | -2LL | AIC | BIC | -2LL | AIC | BIC | -2LL | AIC | BIC | -2LL | |||
Zeng and Liao [37] | 3277.3 | 3306.5 | 3267.3 | b1, b4 | 3100.8 | 3147.5 | 3084.8 | 1944.3 | 1991.0 | 1928.3 | b1, b2 | 1947.4 | 2011.6 | 1925.4 |
Bi [12] | 3199.5 | 3246.2 | 3183.5 | b6, b7 | 3071.4 | 3135.6 | 3049.4 | 1839.6 | 1903.8 | 1817.6 | b1, b2 | 1839.8 | 1921.5 | 1811.8 |
Kozak [1] | 3302.5 | 3360.9 | 3282.5 | b5, b6 | 3192.9 | 3268.7 | 3166.9 | 1791.7 | 1867.5 | 1765.7 | b5, b8 | 1789.1 | 1882.5 | 1757.1 |
Sharma and Zhang [13] | 3364.8 | 3411.5 | 3348.8 | b2, b3 | 3170.6 | 3217.3 | 3154.6 | 1798.0 | 1844.7 | 1782.0 | b2, b4 | 1798.5 | 1862.7 | 1776.5 |
Parameters | Zeng and Liao [37] | Bi [12] | Kozak [1] | Sharma and Zhang [13] |
---|---|---|---|---|
Random Effects | b1, b2 | b1, b2 | b5, b8 | b2, b4 |
b1 | 4.3679(0.0745) | 1.1023(0.0472) | 0.9976(0.0136) | 1.0077(0.0021) |
b2 | −9.2297(0.1323) | −0.5123(0.0311) | 1.0049(0.00695) | 2.1588(0.0039) |
b3 | 5.1885(0.0785) | −0.0689(0.0040) | −0.0011(0.0096) | −0.4747(0.0153) |
b4 | 0.3033(0.0489) | −0.4432(0.0336) | 0.2480(0.0118) | 0.4549(0.0221) |
b5 | −0.0056(0.0013) | −0.4742(0.0961) | ||
b6 | 0.1266(0.0159) | 0.4870(0.0373) | ||
b7 | −0.0659(0.0160) | 0.4451(0.1977) | ||
b8 | 0.0915(0.0061) | |||
b9 | −0.2784(0.0228) | |||
0.2086 | 0.0026 | 0.0297 | 0.0019 | |
0.2300 | 0.0033 | 0.0012 | 0.0344 | |
−0.9900 | −0.8640 | −0.757 | −0.2470 | |
0.0922 | 0.0864 | 0.0797 | 0.0841 |
Basic Model | Plot-Level Effects | Tree-Level Effects | Nested Effects of Plot and Tree | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Models | Bias/cm | RMSE | Bias/cm | RMSE | Bias/cm | RMSE | Bias/cm | RMSE | ||||
Zeng and Liao [37] | 0.9830 | 0.0311 | 0.4613 | 0.9848 | 0.0304 | 0.4363 | 0.9935 | 0.0242 | 0.2849 | 0.9935 | 0.0191 | 0.2855 |
Bi [12] | 0.9835 | 0.0151 | 0.4540 | 0.9852 | 0.0153 | 0.4299 | 0.9939 | 0.0136 | 0.2759 | 0.9939 | 0.0139 | 0.2757 |
Kozak [1] | 0.9829 | 0.0036 | 0.4631 | 0.9845 | 0.0030 | 0.4398 | 0.9945 | 0.0002 | 0.2637 | 0.9945 | 0.0002 | 0.2636 |
Sharma and Zhang [13] | 0.9824 | 0.0158 | 0.4688 | 0.9844 | 0.0086 | 0.4425 | 0.9941 | −0.0004 | 0.2716 | 0.9941 | −0.0014 | 0.2716 |
Relative Height | Number | Bias/cm | SD |
---|---|---|---|
0.0 ≤ h/H ≤ 0.1 | 93 | −0.8069 | 1.5047 |
0.1 < h/H ≤ 0.2 | 54 | 0.3019 | 0.4118 |
0.2 < h/H ≤ 0.3 | 49 | 0.2496 | 0.3981 |
0.3 < h/H ≤ 0.4 | 52 | 0.0448 | 0.2955 |
0.4 < h/H ≤ 0.5 | 44 | −0.1583 | 0.2983 |
0.5 < h/H ≤ 0.6 | 56 | −0.4223 | 0.4723 |
0.6 < h/H ≤ 0.7 | 47 | −0.6521 | 0.6854 |
0.7 < h/H ≤ 0.8 | 50 | −0.8677 | 0.8922 |
0.8 < h/H ≤ 0.9 | 51 | −1.0119 | 1.0778 |
0.9 < h/H ≤ 1.0 | 43 | −0.8750 | 1.0249 |
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Zhang, S.; Sun, J.; Duan, A.; Zhang, J. Variable-Exponent Taper Equation Based on Multilevel Nonlinear Mixed Effect for Chinese Fir in China. Forests 2021, 12, 126. https://doi.org/10.3390/f12020126
Zhang S, Sun J, Duan A, Zhang J. Variable-Exponent Taper Equation Based on Multilevel Nonlinear Mixed Effect for Chinese Fir in China. Forests. 2021; 12(2):126. https://doi.org/10.3390/f12020126
Chicago/Turabian StyleZhang, Sensen, Jianjun Sun, Aiguo Duan, and Jianguo Zhang. 2021. "Variable-Exponent Taper Equation Based on Multilevel Nonlinear Mixed Effect for Chinese Fir in China" Forests 12, no. 2: 126. https://doi.org/10.3390/f12020126
APA StyleZhang, S., Sun, J., Duan, A., & Zhang, J. (2021). Variable-Exponent Taper Equation Based on Multilevel Nonlinear Mixed Effect for Chinese Fir in China. Forests, 12(2), 126. https://doi.org/10.3390/f12020126