Table 1. Derivation of mean volume growth sub-model. | |||

Model | Invariant | Differential equation | Transition function |

E1 | |||

E2 | |||

E3 | |||

Abbreviations: v − mean stem volume (m^{3}) 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 (m^{3}) | 0.260 | 0.218 | 0.011 | 1.257 |

y (m^{3} 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 |

Table 3. Goodness - of - fit statistics for the whole-stand dynamic model. | |||||||

Global model parameters | Regression estimation ^{a} | ||||||

Parameter: Estimate | : 0.4740 p | Dependent variable | Adj. R^{2} | RMSE | Bias ^{b} | Relative bias (%) ^{b} | |

ASE HCCME | 0.0207 0.0556 | ||||||

Parameter: Estimate | : 37.934 q | 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 | : –0.0945 r | ||||||

ASE HCCME | 0.0067 0.0402 | Mean stem volume (predicted) | 0.740 | 0.0893 | 0.0034 | –10.68 | |

Parameter: Estimate | : 69.279 s | ||||||

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. R^{2} – adjusted coefficient of determination; RMSE – Root Mean Square Error. Total number of observations N = 239. ^{a}The absolute and relative biases for all estimated stand variables are not significantly different from zero. ^{b} |

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 |