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ŠUMARSKI LIST 1-2/2021 str. 36     <-- 36 -->        PDF

Another criterion for the selection of the best fit model was the characteristics of the residual homogeneity. In addition, the significance and predicting accuracy of the model parameters and some biological presumptions (presence of a typical S growth shape, polymorphism and asymptote) were also used to examine the best site index prediction model.
To verify the model, we used the predicted residual error sum of squares (PRESS). The procedure consisted of model fitting, refitting with one omitted observation and calculation of the predicted value of the omitted observation that was not used in the model estimation. The predicted value was calculated for each omitted observation, and PRESS statistics was calculated as follows (Eq. 9):
where: n is the number of observations, yi is ith observation, yi,-1 is the predicted omitted observation.
As a rule, this method is used on an independent data set. In the absence of an independent data set, the basic data set was used for model verification. This may be partially justified by the fact that the parameters of the presented model (eq.1-3) were not fitted directly but indirectly through the SI value at the age of 100. So, in the PRESS procedure, the SI value was parameterized and with it, the whole model was reparametrized with the omitted observations (eq.1-3). The starting values of the SI variables were the observed values at the age of 100. The calculation was performed by site classes for each tree in the site class. The values of the PRESS statistics were summarized and presented for all trees in one class. Besides the PRESS statistics, the residual sum of square (RSS) was calculated. All calculations were conducted in R language (R core team 2008).
The first results obtained showed that due to the existence of one or more shorter or longer periods of highly suppressed tree growth and the growth rates that deviated significantly from the expected rates (the shapes of their height growth curve were qualitatively different from the remaining trees), some felled dominant trees should be rejected. After this procedure, the definitive sample was compiled from 53 (ŽA) and 54 (RU) trees to be analyzed, and their height growth per location is shown in Figure 1.
The next step in the development of polymorphic site index curves represents a parameterization of the growth functions per site class. The obtained model parameters and the statistics for the site classes are presented in Table 2.
The CHR model has the best overall features (the highest R2, parameters of all models are statistically significant), except for slightly higher values of RSE compared to the other models). The values of the asymptote parameters are highest for the Korf, followed by the Korsun and the CHR model.
The obtained statistics of the residuals of growth functions (not presented) showed that the arithmetic mean of the residuals is very close to zero, and the standard deviation of the residuals does not exceed a value greater than half the site class width of 4 m. The values of the skewness coefficients for the CHR function are in a range between -0.207 and 0.636, and they are the smallest compared to the values for the remaining two applied functions (from -1.234 to 0.593, on average). The values of the kurtosis coefficients are from 2.011 to 3.423 (the CHR), from 2.017 to 5.065 (The Korsun) and from 2.037 to 5.182 (the Korf model).