Construction of Reducible Stochastic Differential Equation Systems for Tree Height–Diameter Connections
Abstract
:1. Introduction
2. Materials and Methods
2.1. Height–Diameter Regression Models
2.2. SDE Models
2.3. Maximum Likelihood Procedure
2.3.1. Regression Equation Models
2.3.2. SDE Models
2.4. Standard Errors of Parameter Estimates
2.5. Random Effects Calibration
2.6. Data
3. Results and Discussion
3.1. Estimates of Parameters
3.2. Bivariate Diameter and Height Distributions
3.3. Comparison of Diameter, Height, Height–Diameter, and Diameter–Height Models
3.4. Mean Tree Basal Area Models
3.5. Mean Tree Volume Models
4. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
- Zea-Camaño, J.D.; Soto, J.R.; Arce, J.E.; Pelissari, A.L.; Behling, A.; Orso, G.A.; Guachambala, M.S.; Eisfeld, R.L. Improving the Modeling of the Height–Diameter Relationship of Tree Species with High Growth Variability: Robust Regression Analysis of Ochroma pyramidale (Balsa-Tree). Forests 2020, 11, 313. [Google Scholar] [CrossRef] [Green Version]
- Shen, J.; Hu, Z.; Sharma, R.P.; Wang, G.; Meng, X.; Wang, M.; Wang, Q.; Fu, L. Modeling Height–Diameter Relationship for Poplar Plantations Using Combined-Optimization Multiple Hidden Layer Back Propagation Neural Network. Forests 2020, 11, 442. [Google Scholar] [CrossRef] [Green Version]
- Tavares Júnior, I.S.; Rocha, J.E.C.; Ebling, Â.A.; Chaves, A.S.; Zanuncio, J.C.; Farias, A.A.; Leite, H.G. Artificial Neural Networks and Linear Regression Reduce Sample Intensity to Predict the Commercial Volume of Eucalyptus Clones. Forests 2019, 10, 268. [Google Scholar] [CrossRef] [Green Version]
- Rupšys, P. New insights into tree height distribution based on mixed-effects univariate diffusion processes. PLoS ONE 2016, 11, e0168507. [Google Scholar] [CrossRef]
- Rupšys, P.; Petrauskas, E. A New Paradigm in Modelling the Evolution of a Stand via the Distribution of Tree Sizes. Sci. Rep. 2017, 7, 15875. [Google Scholar] [CrossRef] [Green Version]
- Bronisz, K.; Mehtätalo, L. Mixed-Effects Generalized Height–Diameter Model for Young Silver Birch Stands on Post-Agricultural Lands. For. Ecol. Manag. 2020, 460, 117901. [Google Scholar] [CrossRef]
- Xie, L.; Widagdo, F.R.A.; Dong, L.; Li, F. Modeling Height–Diameter Relationships for Mixed-Species Plantations of Fraxinus mandshurica Rupr. and Larix olgensis Henry in Northeastern China. Forests 2020, 11, 610. [Google Scholar] [CrossRef]
- Stankova, T.V.; Diéguez-Aranda, U. Height–diameter relationships for Scots pine plantation in Bulgaria: Optimal combination of model type and application. Ann. For. Res. 2013, 56, 149–163. [Google Scholar]
- Temesgen, H.; Gadow, K.V. Generalized height–diameter models–an application for major tree species in complex stands of interior British Columbia. Eur. J. For. Res. 2004, 123, 45–51. [Google Scholar] [CrossRef]
- Zhang, B.; Sajjad, S.; Chen, K.; Zhou, L.; Zhang, Y.; Yong, K.K.; Sun, Y. Predicting Tree Height–Diameter Relationship from Relative Competition Levels Using Quantile Regression Models for Chinese Fir (Cunninghamia lanceolata) in Fujian Province, China. Forests 2020, 11, 183. [Google Scholar] [CrossRef] [Green Version]
- Rupšys, P. Stochastic Mixed-Effects Parameters Bertalanffy Process, with Applications to Tree Crown Width Modeling. Math. Probl. Eng. 2015, 2015, 375270. [Google Scholar] [CrossRef] [Green Version]
- Rupšys, P.; Petrauskas, E. Evolution of Bivariate Tree Diameter and Height Distribution via Stand Age: Von Bertalanffy Bivariate Diffusion Process Approach. J. Forest Res.-Jap. 2019, 24, 16–26. [Google Scholar] [CrossRef]
- Rupšys, P. Modeling Dynamics of Structural Components of Forest Stands Based on Trivariate Stochastic Differential Equation. Forests 2019, 10, 506. [Google Scholar] [CrossRef] [Green Version]
- Rupšys, P. The Use of Copulas to Practical Estimation of Multivariate Stochastic Differential Equation Mixed-Effects Models. AIP Conf. Proc. 2015, 1684, 080011. [Google Scholar]
- Rupšys, P.; Petrauskas, E. A Linkage among Tree Diameter, Height, Crown Base Height, and Crown Width 4-variate Distribution and Their Growth Models: A 4-variate Diffusion Process Approach. Forests 2017, 8, 479. [Google Scholar] [CrossRef] [Green Version]
- Rupšys, P. Understanding the Evolution of Tree Size Diversity within the Multivariate Nonsymmetrical Diffusion Process and Information Measures. Mathematics 2019, 7, 761. [Google Scholar] [CrossRef] [Green Version]
- Rupšys, P. Modeling Perspectives of Forest Growth and Yield: Framework of Multivariate Diffusion Process. AIP Conf. Proc. 2019, 2164, 060017. [Google Scholar]
- Petrauskas, E.; Rupšys, P.; Narmontas, M.; Aleinikovas, M.; Beniušienė, L.; Šilinskas, B. Stochastic Models to Qualify Stem Tapers. Algorithms 2020, 13, 94. [Google Scholar] [CrossRef] [Green Version]
- Deng, C.; Zhang, S.; Lu, Y.; Froese, R.E.; Ming, A.; Li, Q. Thinning Effects on the Tree Height–Diameter Allometry of Masson Pine (Pinus massoniana Lamb.). Forests 2019, 10, 1129. [Google Scholar] [CrossRef] [Green Version]
- Liu, G.; Wang, J.; Dong, P.; Chen, Y.; Liu, Z. Estimating Individual Tree Height and Diameter at Breast Height (DBH) from Terrestrial Laser Scanning (TLS) Data at Plot Level. Forests 2018, 9, 398. [Google Scholar] [CrossRef] [Green Version]
- Mensah, S.; Pienaar, O.L.; Kunneke, A.; Dutoit, B.; Seydack, A.; Uhl, E.; Pretzsch, H.; Seifert, T. Height Diameter allometry in South Africa’s indigenous high forests: Assessing generic models performance and function forms. For. Ecol. Manag. 2018, 410, 1–11. [Google Scholar] [CrossRef]
- Dean, W.C.; Lee, Y.J. A Mixed-Effects Height–Diameter Model for Individual Loblolly and Slash Pine Trees in East Texas. South. J. Appl. For. 2011, 35, 12–17. [Google Scholar]
- Huxley, A. Problems of Relative Growth; The Dial Press: New York, NY, USA, 1932. [Google Scholar]
- Richards, F.J. A flexible growth function for empirical use. J. Exp. Bot. 1959, 10, 290–300. [Google Scholar] [CrossRef]
- Petrauskas, E.; Rupšys, P.; Memgaudas, R. Q-Exponential Variable-form of a Steam Taper and Volume Model for Scots Pine (Pinus sylvesteris L.) in Lithuania. Baltic For. 2011, 17, 118–127. [Google Scholar]
- Xingji, J.; Fengri, L.; Weiwei, J.; Lianjun, Z. Modeling and Predicting Bivariate Distributions of Tree Diameter and Height. Sci. Silvae Sin. 2013, 49, 74–82. [Google Scholar]
- Ogana, F.N.; Osho, J.S.A.; Gorgoso-Varela, J.J. An approach to modeling the joint distribution of tree diameter and height data. J. Sustain. Forest. 2018, 37, 475–488. [Google Scholar] [CrossRef]
- Pogoda, P.; Ochał, W.; Orzeł, S. Performance of Kernel Estimator and Johnson SB Function for Modeling Diameter Distribution of Black Alder (Alnus glutinosa (L.) Gaertn.) Stands. Forests 2020, 11, 634. [Google Scholar] [CrossRef]
- Mønness, E. The bivariate power-normal distribution and the bivariate Johnson system bounded distribution in forestry, including height curves. Can. J. For. Res. 2015, 45, 307–313. [Google Scholar] [CrossRef]
- Rupšys, P. Univariate and Bivariate Diffusion Models: Computational Aspects and Applications to Forestry. In Stochastic Differential Equations: Basics and Applications; Tony, G., Deangelo, T.G., Eds.; Nova Science: New York, NY, USA, 2018; pp. 1–77. [Google Scholar]
- Davidian, M.; Giltinan, D.M. Some simple methods for estimating intra-individual variability in nonlinear mixed-effects model. Biometrics 1993, 49, 59–73. [Google Scholar] [CrossRef]
- Joe, H. Accuracy of Laplace Approximation for Discrete Response Mixed Models. Comput. Stat. Data An. 2008, 52, 5066–5074. [Google Scholar] [CrossRef]
- Fisher, R.A. On the Mathematical Foundations of Theoretical Statistics. Philos. T. Roy. Soc. A 1922, 222, 309–368. [Google Scholar]
- MacPhee, C.; Kershaw, J.A.; Weiskittel, A.R.; Golding, J.; Lavigne, M.B. Comparison of Approaches for Estimating Individual Tree Height–Diameter Relationships in the Acadian Forest Region. Forestry 2018, 91, 132–146. [Google Scholar] [CrossRef]
- Ishihara, M.I.; Konno, Y.; Umeki, K.; Ohno, Y.; Kikuzawa, K. A New Model for Size-Dependent Tree Growth in Forests. PLoS ONE 2016, 11, e0152219. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Ramos-Ábalos, E.M.; Gutiérrez-Sánchez, R.; Nafidi, A. Powers of the Stochastic Gompertz and Lognormal Diffusion Processes, Statistical Inference and Simulation. Mathematics 2020, 8, 588. [Google Scholar] [CrossRef] [Green Version]
- Narmontas, M.; Rupšys, P.; Petrauskas, E. Models for Tree Taper Form: The Gompertz and Vasicek Diffusion Processes Framework. Symmetry 2020, 12, 80. [Google Scholar] [CrossRef] [Green Version]
- Burkhart, H.E.; Tomé, M. Modelling Forest Trees and Stands; Springer Sience + Business Media: Dordrecht, The Netherlands, 2012. [Google Scholar]
SDE 1 | ||
---|---|---|
Vasicek | ||
Gompertz | ||
Bertalanffy | ||
Gamma |
SDE | Trajectory Type | Equation |
---|---|---|
Vasicek | Mean, median and mode | |
Quantile (0 < p < 1) | 1 | |
Variance | ||
Gompertz Bertalanffy Gamma | Mean | |
Median | ||
Mode | ||
Quantile (0 < p < 1) | ||
Variance |
SDE | ||
---|---|---|
Vasicek | ||
Gompertz Bertalanffy Gamma |
SDE | Trajectory Type | Equation |
---|---|---|
Vasicek | Mean, median, and mode | |
Quantile (0 < p < 1) | ||
Variance | ||
Gompertz Bertalanffy Gamma | Mean | |
Median | ||
Mode | ||
Quantile (0 < p < 1) | ||
Variance |
Data | Number of Plots | Min | Max | Mean | SD | Number of Plots | Min | Max | Mean | SD |
---|---|---|---|---|---|---|---|---|---|---|
Estimation | Validation | |||||||||
d (cm) | 23 | 1.20 | 13.40 | 3.96 | 1.61 | 8 | 1.80 | 8.00 | 4.15 | 1.25 |
h (m) | 23 | 1.50 | 7.90 | 3.55 | 1.11 | 8 | 2.07 | 6.56 | 3.84 | 0.91 |
t (year) | 23 | 53.00 | 123.00 | 92.21 | 20.82 | 8 | 53.00 | 103.00 | 80.60 | 17.50 |
Model | Parameters of SDE Models | ||||||||
---|---|---|---|---|---|---|---|---|---|
αd | βd | σdd | αh | βh | σhh | σdh | σd | σh | |
4 fixed effect | 4.0760 (0.0037) | 0.0501 (0.0003) | 0.2661 (0.0016) | 3.5518 (0.0021) | 0.1861 (0.0056) | 0.4596 (0.0139) | 0.3400 (0.0081) | - | - |
4 mixed effect | 4.2783 (0.0023) | 0.0841 (0.0009) | 0.2165 (0.0024) | 3.7127 (0.0011) | 0.2348 (0.0640) | 0.1453 (0.0396) | 0.01285 (0.0258) | 1.5029 (0.0097) | 1.1790 (0.0076) |
5 fixed effect | 0.1136 (0.0004) | 0.0772 (0.0003) | 0.0224 (0.0001) | 0.3599 (0.0442) | 0.2754 (0.0338) | 0.0457 (0.0056) | 0.0303 (0.0029) | - | - |
5 mixed effect | 0.1414 (0.0003) | 0.0946 (0.0002) | 0.0156 (0.0001) | 0.3639 (0.0004) | 0.2749 (0.0003) | 0.0147 (4.6 × 10−5) | 0.0112 (4.4 × 10−5) | 0.0274 (0.0002) | 0.0750 (0.0005) |
6 fixed effect | 5.2039 (0.0173) | 0.0668 (0.0002) | 0.0021 (5.8 × 10−6) | 75.0842 (37.1990) | 0.4296 (0.0492) | 0.0013 (3.4 × 10−6) | 0.0013 (4.0 × 10−6) | - | - |
6 mixed effect | 7.2647 (0.0166) | 0.0907 (0.0002) | 0.0012 (3.3 × 10-6) | 67.3215 (0.0788) | 0.4159 (0.0001) | 0.0004 (1.1 × 10−6) | 0.0004 (1.6 × 10−6) | 0.5511 (0.0035) | 17.1946 (0.1104) |
7 fixed effect | 3.1269 (0.0024) | 0.0395 (6.3 × 10−5) | 0.0023 (6.1 × 10−6) | 1.0109 (0.0018) | 0.0153 (4.8 × 10−5) | 0.0013 (3.4 × 10−6) | 0.0014 (4.1 × 10−6) | - | - |
7 mixed effect | 3.1690 (0.0017) | 0.0389 (4.6 × 10−5) | 0.0012 (3.3 × 10−6) | 1.0551 (0.0010) | 0.0155 (2.6 × 10−5) | 0.0004 (1.1 × 10−6) | 0.0004 1.6 × 10−6 | 0.1480 (0.0010) | 0.1368 (0.0009) |
Model | Parameters of Regression Models | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
β1 | β2 | β3 | Σ | σϕ | ||||||
Diameter | Height | |||||||||
1 fixed effect | 1.0772 (0.0022) | 1.0248 (0.0014) | - | 0.9898 (0.0027) | - | 1.4949 (0.0021) | 0.6405 (0.0009) | - | 0.4830 (0.0013) | - |
1 mixed effect | 0.9214 (0.0017) | 1.1529 (0.0013) | - | 0.7511 (0.0021) | 0.1130 (0.0007) | 2.2444 (0.0012) | 0.3685 (0.0004) | - | 0.1724 (0.0005) | 0.4405 (0.0028) |
2 fixed effect | 198.4369 (21.2221) | 0.0157 (0.0013) | 1.4884 (0.0050) | 0.9846 (0.0026) | - | 23.7733 (1.2223) | 0.0277 (0.0018) | 1.0383 (0.0055) | 0.4759 (0.0013) | - |
2 mixed effect | 73.9614 (0.2618) | 0.0499 (0.0002) | 1.8394 (0.0029) | 0.7466 (0.0020) | 12.3094 (0.0802) | 25.1334 (0.0386) | 0.0037 (2.3 × 10−5) | 0.5584 (0.0010) | 0.1745 (0.0005) | 7.3270 (0.0471) |
3 fixed effect | 0.0018 (1.9 × 10−5) | −61.6512 (0.4348) | 0.3074 (0.0018) | 0.9843 (0.0078) | - | 0.0186 (0.0048) | −34.9128 (9.6510) | −0.0357 (0.0022) | 0.4764 (0.0013) | - |
3 mixed effect | 0.1861 (0.0006) | −1.3312 (0.0036) | 0.5858 (0.0008) | 0.7405 (0.0020) | 0.0314 (0.0002) | 0.2888 (0.0004) | −6.1644 (0.0472) | −0.7894 (0.0034) | 0.1750 (0.0005) | 0.0834 (0.0005) |
Model (Predictors) | Estimation Dataset 1 | Validation Dataset | ||||||
---|---|---|---|---|---|---|---|---|
B (%B) | AB (%AB) | RMSE (%RMSE) | R2 | B (%B) | AB (%AB) | RMSE (%RMSE) | R2 | |
Diameter | ||||||||
Vasicek (t) | 0.0068 (−8.47) | 0.8278 (23.70) | 1.1114 (28.10) | 0.5213 | −2.5 × 10−10 (−6.91) | 0.8675 (22.65) | 1.0674 (25.74) | 0.2768 |
Vasicek (t,h) | 0.0025 (−4.60) | 0.6462 (17.72) | 0.8548 (21.62) | 0.7168 | −1.9 × 10−10 (−4.56) | 0.7243 (18.62) | 0.9406 (22.69) | 0.4384 |
Gompertz (t) | 0.0076 (−8.49) | 0.8302 (23.61) | 1.1207 (28.36) | 0.5132 | 0.1339 (−3.43) | 0.8615 (21.70) | 1.0687 (25.77) | 0.2750 |
Gompertz (t,h) | 0.0030 (−4.91) | 0.6418 (17.55) | 0.8567 (21.66) | 0.7156 | 0.0951 (−2.30) | 0.7177 (17.96) | 0.9266 (22.35) | 0.4549 |
Bertalanffy (t) | −0.0226 (−9.38) | 0.8348 (23.90) | 1.1256 (28.46) | 0.5090 | 0.1442 (−3.45) | 0.8621 (21.76) | 1.0698 (27.86) | 0.2741 |
Bertalanffy (t,h) | −0.0127 (−5.33) | 0.6448 (17.68) | 0.8587 (21.71) | 0.7142 | 0.0950 (−2.28) | 0.7184 (17.98) | 0.9272 (22.36) | 0.4543 |
Gamma (t) | −0.0216 (−9.28) | 0.8334 (23.85) | 1.1222 (28.37) | 0.5121 | 0.1338 (−3.44) | 0.8623 (21.75) | 1.0689 (25.78) | 0.2751 |
Gamma (t,h) | −0.0135 (−5.30) | 0.6442 (17.67) | 0.8574 (21.68) | 0.7151 | 0.0950 (−2.28) | 0.7185 (17.99) | 0.9273 (22.37) | 0.4543 |
Power (h) | 0.0064 (−4.50) | 0.6398 (17.51) | 0.8507 (21.51) | 0.7195 | 0.0026 (−4.59) | 0.7199 (18.57) | 0.9269 (22.35) | 0.4546 |
Richards (h) | 0.0006 (−5.07) | 0.6397 (17.74) | 0.8485 (21.45) | 0.7209 | 0.0056 (−4.63) | 0.7167 (18.50) | 0.9269 (22.36) | 0.4546 |
Q-exp (h) | −0.0004 (−5.50) | 0.6387 (17.97) | 0.8482 (21.45) | 0.7211 | 0.0147 (−4.53) | 0.7183 (18.55) | 0.9326 (22.49) | 0.4479 |
Height | ||||||||
Vasicek (t) | 0.0026 (−2.69) | 0.4049 (12.45) | 0.5448 (15.35) | 0.7609 | −4.0 × 10−11 (−1.92) | 0.3912 (10.80) | 0.5160 (13.44) | 0.6822 |
Vasicek (t,h) | 0.0034 (−1.55) | 0.3219 (9.75) | 0.4191 (11.81) | 0.8585 | 0.0106 (−1.01) | 0.3363 (9.05) | 0.4566 (11.89) | 0.7512 |
Gompertz (t) | 0.0049 (−2.69) | 0.4063 (12.44) | 0.5476 (15.42) | 0.7585 | 0.0352 (−0.99) | 0.3941 (10.76) | 0.5162 (13.44) | 0.6821 |
Gompertz (t,h) | −0.0004 (−1.54) | 0.3171 (9.55) | 0.4084 (11.50) | 0.8657 | −0.0189 (−1.68) | 0.3435 (9.22) | 0.4657 (12.13) | 0.7412 |
Bertalanffy (t) | −0.0048 (−2.98) | 0.4069 (12.48) | 0.5484 (15.45) | 0.7578 | 0.0352 (−1.01) | 0.3943 (10.77) | 0.5164 (13.48) | 0.6819 |
Bertalanffy (t,h) | −0.0060 (−1.69) | 0.3174 (9.57) | 0.4087 (11.51) | 0.8654 | −0.0196 (−1.71) | 0.3443 (9.25) | 0.4668 (12.16) | 0.7400 |
Gamma (t) | −0.0031 (−2.92) | 0.4065 (12.47) | 0.5477 (15.43) | 0.7583 | 0.0352 (−1.03) | 0.3941 (10.77) | 0.5162 (13.44) | 0.6821 |
Gamma (t,h) | −0.0050 (−1.66) | 0.3175 (9.57) | 0.4087 (11.51) | 0.8654 | −0.0195 (−1.71) | 0.3444 (9.25) | 0.4669 (12.17) | 0.7398 |
Power (h) | 0.0010 (−1.51) | 0.3161 (9.56) | 0.4066 (11.45) | 0.8668 | 0.0118 (−0.93) | 0.3422 (9.15) | 0.4652 (12.11) | 0.7418 |
Richards (h) | −0.0002 (−1.77) | 0.3197 (9.73) | 0.4091 (11.52) | 0.8652 | 0.0169 (−0.86) | 0.3450 (9.20) | 0.4709 (12.26) | 0.7354 |
Q-exp (h) | −0.0032 (−1.82) | 0.3202 (9.76) | 0.4112 (11.58) | 0.8638 | 0.0156 (−0.90) | 0.3442 (9.18) | 0.4699 (12.24) | 0.7365 |
Model (Predictors) | Estimation Dataset | Validation Dataset | ||||||
---|---|---|---|---|---|---|---|---|
B (%B) | AB (%AB) | RMSE (%RMSE) | R2 | B (%B) | AB (%AB) | RMSE (%RMSE) | R2 | |
Diameter | ||||||||
Vasicek (t) | −0.0025 (−15.31) | 1.1752 (33.58) | 1.6318 (41.26) | 0 | 0.2470 (−2.78) | 1.0200 (25.08) | 1.2912 (31.14) | 0 |
Vasicek (t,h) | −0.0006 (−5.60) | 0.7237 (19.53) | 0.9833 (24.86) | 0.6253 | −0.0888 (−6.90) | 0.7815 (20.54) | 0.9960 (24.02) | 0.3702 |
Gompertz (t) | 0.2425 (−2.94) | 1.0273 (25.29) | 1.2999 (31.35) | 0 | 0.2425 (−2.94) | 1.0273 (25.29) | 1.2999 (31.35) | 0 |
Gompertz (t,h) | 0.0018 (−5.78) | 0.7172 (19.43) | 0.9757 (24.67) | 0.6310 | −0.0732 (−6.67) | 0.7714 (20.27) | 0.9771 (23.56) | 0.3940 |
Bertalanffy (t) | −0,1293 (−19.04) | 1,2091 (35.47) | 1.6420 (41.52) | 0 | 0.1323 (−5.87) | 1.0373 (26.22) | 1.3092 (31.58) | 0 |
Bertalanffy (t,h) | −0.0069 (−6.24) | 0.7185 (19.55) | 0.9785 (24.74) | 0.6290 | −0.0662 (−6.61) | 0.7710 (20.24) | 0.9751 (23.52) | 0.3964 |
Gamma (t) | −0,0982 (−18.76) | 1.2670 (37.10) | 1.7045 (43.10) | 0 | −0.0485 (−10.95) | 1.1219 (29.41) | 1.3827 (33.25) | 0 |
Gamma (t,h) | −0.0207 (−6.49) | 0.7323 (19.97) | 0.9885 (24.99) | 0.6213 | −0.1528 (−8.83) | 0.7935 (21.17) | 1.0064 (24.27) | 0.3571 |
Power (h) | 0.0043 (−5.57) | 0.7357 (19.78) | 0.9977 (25.23) | 0.6142 | −0.1330 (−7.97) | 0.7862 (20.78) | 1.0147 (24.47) | 0.3465 |
Richards (h) | −0.0017 (−6.21) | 0.7423 (20.11) | 0.9932 (25.11) | 0.6177 | −0.0961 (−7.15) | 0.7847 (20.58) | 1.0127 (24.42) | 0.3490 |
Q-exp (h) | −0.0024 (−6.26) | 0.7409 (20.15) | 0.9930 (25.11) | 0.6178 | −0.0926 (−7.08) | 0.7843 (20.56) | 1.0125 (24.42) | 0.3493 |
Height | ||||||||
Vasicek (t) | −0.0016 (−8.73) | 0.8132 (23.77) | 1.1142 (31.39) | 0 | 0.2881 (2.68) | 0.6839 (16.67) | 0.9155 (23.84) | 0 |
Vasicek (t,h) | −0.0010 (−3.33) | 0.4980 (14.42) | 0.6708 (18.89) | 0.6376 | 0.1526 (1.69) | 0.5297 (13.27) | 0.6963 (18.13) | 0.4214 |
Gompertz (t) | 0.0062 (−8.49) | 0.8114 (23.67) | 1.1142 (31.38) | 0 | 0.2957 (2.89) | 0.6848 (16.66) | 0.9154 (23.84) | 0 |
Gompertz (t,h) | 0.0045 (−3.28) | 0.5006 (14.50) | 0.6711 (18.90) | 0.6372 | 0.1340 (1.15) | 0.5311 (13.35) | 0.6923 (18.03) | 0.4281 |
Bertalanffy (t) | −0.0921 (−11.50) | 0.8369 (25.05) | 1.1142 (31.38) | 0 | 0.1975 (0.19) | 0.6787 (16.95) | 0.9154 (23.84) | 0 |
Bertalanffy (t,h) | −0.0385 (−4.34) | 0.5029 (14.75) | 0.6686 (18.83) | 0.6399 | 0.0886 (0.04) | 0.5369 (13.64) | 0.6963 (18.13) | 0.4214 |
Gamma (t) | −0.0389 (−9.95) | 0.8328 (24.84) | 1.1207 (31.57) | 0 | 0.1214 (−1.95) | 0.6948 (17.74) | 0.9296 (24.21) | 0 |
Gamma (t,h) | −0.0144 (−3.57) | 0.4940 (14.46) | 0.6547 (18.44) | 0.6547 | 0.0960 (0.37) | 0.5331 (13.58) | 0.6901 (17.97) | 0.4316 |
Power (h) | 0.0037 (−3.07) | 0.5162 (14.87) | 0.6970 (19.63) | 0.6087 | 0.1598 (1.81) | 0.5534 (13.82) | 0.7325 (19.08) | 0.3596 |
Richards (h) | 0.0014 (−3.31) | 0.5095 (14.66) | 0.6905 (19.45) | 0.6159 | 0.1783 (2.26) | 0.5439 (13.51) | 0.7224 (18.81) | 0.3772 |
Q-exp (h) | −0.0006 (−3.44) | 0.5096 (14.67) | 0.6908 (19.46) | 06156 | 0.1826 (2.33) | 0.5395 (13.38) | 0.7196 (18.74) | 0.3820 |
Type | Mean | Variance |
---|---|---|
Marginal | ||
Conditional |
Type | Trajectory Type | Equation |
---|---|---|
Marginal | Mean | |
Median | ||
Mode | ||
Quantile (0 < p < 1) | 1 | |
Variance | ||
Conditional | Mean | |
Median | ||
Mode | ||
Quantile (0 < p <1) | ||
Variance |
Estimation Dataset | Validation Dataset | |||||||
---|---|---|---|---|---|---|---|---|
B (%B) | AB (%AB) | RMSE (%RMSE) | R2 | B (%B) | AB (%AB) | RMSE (%RMSE) | R2 | |
Mixed-effect scenario | ||||||||
0.0078 (0.23) | 0.0187 (8.69) | 0.0300 (15.94) | 0.9602 | 0.0066 (4.08) | 0.0080 (5.08) | 0.0079 (4.84) | 0.9886 | |
0.0040 (0.27) | 0.104 (5.23) | 0.0151 (8.02) | 0.9899 | 0.0062 (3.72) | 0.0072 (4.45) | 0.0075 (4.56) | 0.9899 | |
0.0038 (−1.67) | 0.0171 (8.51) | 0.0286 (15.19) | 0.9639 | 0.0066 (4.08) | 0.0080 (5.09) | 0.0079 (4.84) | 0.9886 | |
−0.0001 (−1.69) | 0.0090 (4.99) | 0.0135 (7.19) | 0.9919 | 0.0062 (3.72) | 0.0072 (4.45) | 0.0075 (4.56) | 0.9899 | |
Fixed-effect scenario | ||||||||
0.0119 (−0.37) | 0.0394 (18.67) | 0.0684 (36.35) | 0.7930 | −0.0066 (−3.00) | 0.0224 (14.27) | 0.0260 (15.80) | 0.8787 |
Estimation Dataset | Validation Dataset | ||||||
---|---|---|---|---|---|---|---|
B (%B) | AB (%AB) | RMSE (%RMSE) | R2 | B (%B) | AB (%AB) | RMSE (%RMSE) | R2 |
0.0003 (−0.35) | 0.0009 (10.27) | 0.0017 (24.64) | 0.9500 | −3.5 ×10−5 (1.41) | 0.0003 (5.07) | 0.0005 (9.34) | 0.9765 |
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Narmontas, M.; Rupšys, P.; Petrauskas, E. Construction of Reducible Stochastic Differential Equation Systems for Tree Height–Diameter Connections. Mathematics 2020, 8, 1363. https://doi.org/10.3390/math8081363
Narmontas M, Rupšys P, Petrauskas E. Construction of Reducible Stochastic Differential Equation Systems for Tree Height–Diameter Connections. Mathematics. 2020; 8(8):1363. https://doi.org/10.3390/math8081363
Chicago/Turabian StyleNarmontas, Martynas, Petras Rupšys, and Edmundas Petrauskas. 2020. "Construction of Reducible Stochastic Differential Equation Systems for Tree Height–Diameter Connections" Mathematics 8, no. 8: 1363. https://doi.org/10.3390/math8081363