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Six generalized algebraic difference equations (GADAs) derived from the base models of log-logistic, Bertalanffy-Richards, and Lundqvist-Korf were used to develop site index model for

In Tunisia,

Although

Site index models based on the height growth of dominant trees are the classical way of indirectly estimating site quality (mostly a combination of soil fertility and climate) in forestry management (e.g., [

The study concerned all the

Data come from 62 temporary sample plots located in the planted

Map showing the studied area of

Summary statistics of the modelling data can be seen in Table

Characteristics of the stem analyses data coming from the 62 fallen trees used for modelling height growth.

Variable | Average | Minimum | Maximum | Standard deviation |
---|---|---|---|---|

Tree age (years) | 33.06 | 22 | 55 | 7.370 |

Height (m) | 11.02 | 6.3 | 21.4 | 3.034 |

Number of cross-sectional cuts | 9.87 | 5 | 20 | 3.070 |

Six dynamic equations with two site-specific parameters were tested using a dummy variable procedure [

Table

Base models and GADA formulations considered.

Base equation | Parameter related to site | Solution for | Dynamic equation | Model |
---|---|---|---|---|

Log-logistic (log-transformed): | (M1) | |||

(M2) | ||||

Bertalanffy-Richards: | (M3) | |||

(M4) | ||||

(M5) | ||||

Lundqvist-Korf: | (M6) |

The methodology employed in this work is based on the generalized algebraic differences equations (GADAs) suggested by Cieszewski and Bailey [

For the GADA, it is generally needed that two parameters (the parameter related to the asymptote and one among the shape parameters) are functions of site productivity to allow simultaneously polymorphism and variable asymptotes for growth curves. Productivity is assumed to be dependent on an unobservable variable

To illustrate this method, consider the model M3 (Table

The use of stem analysis data implies the autocorrelation among observations within the same tree (correlation between the residuals within the same tree), which invalidates the standard hypothesis testing [

To evaluate the presence of autocorrelation and the order of the CAR(

To determine the order of the function CAR(

Height-lag1-residuals and height-lag2-residuals versus height-residuals for model M6 fitted without considering the autocorrelation parameters (a), and using continuous-time autoregressive error structures of first and second order (b and c, resp.).

The comparison of the estimates for the different models was based on numerical and graphical analyses of the residuals. Also, four statistical criteria obtained from the residuals were examined: root mean square error (RMSE), mean residual (Bias), adjusted coefficient of determination

Residual mean square error:

Mean residual (Bias):

Adjusted coefficient of determination:

Akaike’s information criterion differences (AICd), which is an index to select the best model based on minimising the Kullback-Leibler distance, were used in order to compare models with a different number of parameters:

Graphical analysis of the residuals and the biological realism of the fitted curves as well were an important step in comparing the different models. This is essential because curve profiles may differ drastically, even though the statistics and residuals of the fit are similar (e.g., [

The practical use of the model to estimate site quality from any given pair of height and age data requires the selection of a base age to which site index will be referenced. Inversely, site index and its associated base age may be used to estimate dominant height at any desired age. Therefore, the selection of a base age becomes an important issue when only one observation of a new individual is available [

According to Goelz and Burk [

It is important to note that the statistic described in this section is only meaningful if the site-specific parameters are discarded because of the lack of repeated data within the same individual [

Figure

The parameter estimates for each model and their corresponding goodness-of-fit statistics are shown in Table

Parameter estimates and goodness-of-fit statistics.

Model | Par. | Estim. | S.E. | RMSE | Bias | AICd | ||||
---|---|---|---|---|---|---|---|---|---|---|

M1 | _{1} | −5.095 | 11.107 | −0.46 | 0.6466 | |||||

_{2} | 6021.461 | 2533.2 | 2.38 | 0.0178 | ||||||

_{3} | 1.611 | 0.045 | 35.92 | <0.0001 | 0.4883 | 0.072 | 0.9826 | 8.17 | 26.1 | |

0.842 | 0.024 | 34.90 | <0.0001 | |||||||

0.796 | 0.024 | 33.21 | <0.0001 | |||||||

M2 | _{1} | 47.812 | 10.281 | 4.65 | <0.0001 | |||||

_{2} | −8.719 | 3.563 | −2.45 | 0.0147 | ||||||

_{3} | 1.609 | 0.045 | 35.83 | <0.0001 | 0.4881 | 0.072 | 0.9826 | 7.62 | 25.9 | |

0.842 | 0.024 | 34.90 | <0.0001 | |||||||

0.796 | 0.024 | 33.24 | <0.0001 | |||||||

M3 | _{1} | 0.049 | 0.003 | 17.10 | <0.0001 | |||||

_{2} | −0.052 | 0.457 | −0.11 | 0.9093 | ||||||

_{3} | 4.999 | 1.385 | 3.61 | 0.0003 | 0.4864 | 0.072 | 0.9846 | 3.42 | 23.7 | |

0.837 | 0.025 | 34.16 | <0.0001 | |||||||

0.791 | 0.025 | 31.95 | <0.0001 | |||||||

M4 | _{2} | 0.046 | 0.003 | 16.06 | <0.0001 | |||||

_{3} | 4.957 | 0.144 | 34.38 | <0.0001 | 0.4974 | 0.075 | 0.9821 | 23.26 | 23.6 | |

0.848 | 0.024 | 35.77 | <0.0001 | |||||||

0.796 | 0.024 | 33.08 | <0.0001 | |||||||

M5 | _{2} | 0.048 | 0.003 | 16.37 | <0.0001 | |||||

_{3} | 1.393 | 0.063 | 22.16 | <0.0001 | 0.4932 | 0.071 | 0.9823 | 19.49 | 23.8 | |

0.848 | 0.024 | 35.87 | <0.0001 | |||||||

0.804 | 0.023 | 34.83 | <0.0001 | |||||||

M6 | _{2} | 35.574 | 0.728 | 48.78 | <0.0001 | |||||

_{3} | 0.558 | 0.034 | 16.30 | <0.0001 | 0.4832 | 0.067 | 0.9830 | 0 | 37.6 | |

0.829 | 0.025 | 32.83 | <0.0001 | |||||||

0.794 | 0.024 | 32.68 | <0.0001 |

Par

Among the 4 models for which all the coefficients are significant, M2 and M6 were the best models in terms of RMSE,

Height growth curves for site index values of 6, 9, 12 and 15m at reference age of 30 years for the dynamic models M2, M4–M6, which was developed considering two site-specific parameters in their corresponding base equations.

To compare M2 and M6, we used graphs showing

bias and root mean square error in height estimation by 5-year age classes (Figure

fitted height growth curves for different site index values overlaying the trajectories of observed height and current annual height growth curves observed for both models (Figure

Bias and root mean square error (RMSE) in height estimation for models M2 and M6 by 5-year age classes.

Height growth curves for site index values of 6, 9, 12 and 15 m at reference age of 30 years overlaying the trajectories of the observed heights over time for models M2 and M6 (left) and current annual height growth curves derived for both models (right).

Considering the reduced number of observations (3 observations) whose age exceeds 47 years, the last observations were grouped in the 45-year old age class. Concerning the bias and RMSE in height predictions by age classes, Figure

Residuals versus predicted heights (a) and site index at reference age of 30 years (b) for model M6 (the GADA formulation of the Lundqvist-Korf base equation that considers two site-specific parameters) fitted with a second-order continuous-time autoregressive error structure (CAR(2)).

Site index predictions against total age using model M6 and the stem analysis data.

Model M6 (the GADA formulation derived from the Lundqvist-Korf base function by considering two parameters as related to site productivity) was selected for height growth prediction of

We were aware of the restrictions of our data, having only two plots with measurements over the age of fifty years. The lack of permanent sample plots measured at young and old ages was also an important restriction. Therefore, model selection was viewed as a compromise between biological and statistical considerations, rather than as a pure exercise in statistical inference. The model selection procedure indicated that (

To determine the best reference age to define the site index of stands, we used model M6. Figure

Relative error in height prediction (RE) related to the choice of a reference age for Model (M6) adjusted with CAR(2): RE% by 5-year age classes and corresponding number of observations (

Site index

Distribution of sampled plots in fertility classes and descriptive statistics of site index

Fertility classes | Limits of fertility classes | Midvalue of fertility classes | Number of plots | Site index | |||

Average | Minimum | Maximum | Standard deviation | ||||

1 | 15 | 4 | 14.22 | 13.52 | 14.79 | 0.648 | |

2 | 12 | 27 | 11.75 | 10.56 | 13.41 | 0.854 | |

3 | 9 | 25 | 9.18 | 7.79 | 10.47 | 0.790 | |

4 | 6 | 6 | 6.34 | 5.26 | 7.39 | 0.839 | |

Total | — | — | 62 | 10.35 | 5.26 | 14.79 | 2.154 |

Finally, Figure

Height growth curves obtained with model M6 for site index values of 6, 9, 12, and 15 m at the reference age of 30 years, overlaid on the trajectories of the observed heights of sampled trees over time.

In the absence of other existing models for

Site index curves (site index values of 6, 9, 12, and 15 m at reference age of 30 years) (Lundqvist-Korf: GADA) overlaid on the trajectories of the observed dominant heights over time for

At the older ages when the two Spanish models are quite similar, the Tunisian model presents on the contrary higher asymptotes. Even though we do not have any information about size of old

A site index model that describes the dominant height growth of

The elaborated curves define 4 site quality classes (Figure

This study (Project no. A/4996/06) was carried out within the framework of the bilateral cooperation between Tunisia (INRGREF) and Spain (University of Lleida) and was financed by the Ministry for Scientific Research and Technology of Tunisia (MSRT) and the Spanish Agency of the International Cooperation and Development (AECID). The authors acknowledge the support received from EFIMED. In addition, the framework provided by AGORA project has facilitated the transfer of scientific methods from Portugal and Spain to Tunisia that has made it possible to conduct this study.