Table 1. Derivation of mean volume growth sub-model.
Model Invariant Differential equation Transition function

E1

t1e1a t1e1b t1e1c

E2

t1e2a t1e2b t1e2c

E3

t1e3a t1e3b t1e3c
Abbreviations: v − mean stem volume (m3) and H − dominant stand height (m); p, q – global model parameters
Table 2. Description of the data set.
Stand variable Mean Standard deviation Minimum Maximum
DBH (cm) 17.7 6.0 5.6 34.7
N (trees ha–1) 2270 1862 393 9305
H (m) 16.3 5.0 6.3 32.5
Age (years) 38 14 12 80
v (m3) 0.260 0.218 0.011 1.257
y (m3 ha–1) 340.7 144.6 61.4 723.4
Abbreviations: DBH – diameter at breast height; H − dominant stand height; N – stand density; v – mean stem volume; y – total stand volume
1

Fig. 1. Data examination charts. a – Stand age-site index data range of this study. Basal area values below (×) and above (▲) average are indicated. Solid lines connect measurements of the same plot, for site index averaged per plot. Dotted gridlines delimit 4m-site index classes. b – Dominant stand height-stand density data range. Site index values below (●) and above (□) average are indicated. Thinning trajectories for some of the managed plots are represented by solid lines; c – Site index-stand age data comparison between the permanent sample plots of this study (◌) and temporary sample plot data (♦). Site index model by Stankova and Diéguez-Aranda (2012) for reference age 50 years and temporary sample plots data from Stankova and Shibuya (2007) are used. View larger in new window/tab.

2

Fig. 2. Plots of residuals vs lag−residuals. a1 to a3 – mean stem volume, m3; b1 to b3 – stand density, ha-1; c1 to c3 – total stand volume m3/ha. View larger in new window/tab.

Table 3. Goodness - of - fit statistics for the whole-stand dynamic model.
Global model parameters   Regression estimation a
Parameter: Estimate p: 0.4740 Dependent variable Adj. R2 RMSE Bias b Relative
bias (%) b
ASE HCCME 0.0207 0.0556
Parameter: Estimate q: 37.934 Stand density 0.948 403 –11 –0.37
ASE HCCME 2.9410 9.1053
Mean stem volume (projected) 0.931 0.0460 –0.0002 –2.42
Parameter: Estimate r: –0.0945
ASE HCCME 0.0067 0.0402 Mean stem volume (predicted) 0.740 0.0893 0.0034 –10.68
Parameter: Estimate s: 69.279
ASE HCCME 8.6749 27.317 Total stand volume 0.919 41.81 –0.480 –1.89
Abbreviations: ASE – Asymptotic Standard Error; HCCME – Heteroscedasticity - Consistent Covariance Matrix Estimator; Adj. R2 – adjusted coefficient of determination; RMSE – Root Mean Square Error.
a Total number of observations N = 239. b The absolute and relative biases for all estimated stand variables are not significantly different from zero.
3

Fig. 3. Goodness-of-fit and prediction charts. a, b − density decrease (a) and volume growth (b) models fitted to the experimental data. Fitted curves for five data series comprising the data range are shown; c, d actual vs. projected values: stand density (c) and mean stem volume (d); e − actual vs. predicted mean stem volume; f − actual vs. estimated total stand volume. Linear regressions of observed against estimated variable values are fitted and the results from simultaneous F-test for line slope equals 1 and zero intercept (Gadow and Hui 1999) are shown in the plots. View larger in new window/tab.

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Fig. 4. Plots of residuals vs. predicted values. a – projected mean stem volume; b – predicted mean stem volume; c – stand density; d – total stand volume. The lines connect the residuals (◌) by data series, the symbol ● denoting values of just-thinned plots. View larger in new window/tab.

Table 4. Validation test statistics of the whole-stand dynamic model.

Dependent variable 		Tolerance intervals (TI) for the relative errors

1 – α = 99% probability for the future errors
1 – γ = 50% 1 – γ = 75% 1 – γ = 95%
of future observations
Stand density –19.6 17.6 –32.7 30.7 –54.9 53.0
Projected mean stem volume –17.4 16.4 –29.4 28.4 –49.7 48.7
Predicted mean stem volume –33.4 12.9 –49.8 29.2 –77.5 57.0
Total stand volume –18.8 19.2 –32.2 32.6 –54.9 55.4