Can random components explain differences in the height–diameter relationship in mixed uneven-aged stands?

Abstract

• Context

Tree height prediction is an important issue in forest management since tree heights are usually measured only in a sample of trees. Although numerous model approaches have been used for this purpose, no agreement on which one is more appropriate has been achieved.

• Aims

To analyse the random effects of basic and generalised height–diameter (h–d) models fitted to multi-species uneven-aged forest stands, and to establish their ability to explain differences between ecoregions, plots and species.

• Methods

Height and diameter measurements for 29,084 trees from 187 sample plots located in the state of Durango (Mexico) were used. Basic and generalised h–d models were fitted in a mixed-models framework. The variability between ecoregions, plots and species was considered in the random effects definition. Model calibration for different height sampling designs and sampling sizes was also analysed.

• Results

Random components performed well in explaining the differences in the h–d relationship between the different plots and species; however, no significant variance for the random effects was found for the different ecoregions. A calibrated basic h–d model produced similar results to a fixed-effects generalised h–d model when a sufficiently large number of trees was used in the calibration process.

• Conclusion

From a practical point of view, if no calibration is carried out, different models should be used for the different species, so that at least the variation among species is captured.

Appendix. Height estimation—an example

This Appendix shows an example of the height estimation using the calibrated Eqs. 9 and 11. Equations 10 and 12 should be calibrated in the same way as Eqs. 9 and 11, respectively.

We suppose that the following data belong to three new plots in which several species are present, but we are interested in calibrating the models for species 1, 2 and 3, which are the commercial ones. Thus, we measure one tree of these species in each one of the plots, resulting in:

Plot	Species	d (cm)	h (m)
1	1	15	12
1	2	17	12
1	3	14	12
2	1	20	13
2	2	19	13
2	3	16	13
3	1	17	11
3	2	15	11
3	3	18	11

1.1 Case 1: use of Eq. 9

In this case, calibration should be done for the whole dataset for which we want to calibrate Eq. 9, using the expression shown in Eq. 6.

Note that subscript i in Eq. 6 is finally not necessary because there were no differences between ecoregions, so hereafter it was omitted.

The estimated variances and the covariance of the random effects (Table 3) are the elements of the variance–covariance matrix $\widehat{\mathbf{D}}$, which should be arranged according to the dataset in the following way: three species, with two random parameters by species and three plots with three random parameters by plot, resulting in a 15 × 15 diagonal matrix:

$$ \widehat{\mathbf{D}}=\left[\begin{array}{ccccccccccccccc}\hfill 0.01090\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0.01090\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0.01090\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0.07487\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0.07487\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0.07487\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0.1206\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0.1206\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0.1206\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0.06554\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0.06554\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0.06554\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0.7793\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0.7793\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0.7793\hfill \end{array}\right] $$

The variance–covariance matrix for the random error term is determined by assuming that all estimations have constant variance (σ ², Table 3) and that the errors are not correlated:

$$ {\widehat{\mathbf{R}}}_{jk}={\sigma}^2\times {\mathbf{I}}_9=\left[\begin{array}{ccccccccc}\hfill 5.678\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 5.678\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 5.678\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 5.678\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 5.678\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 5.678\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 5.678\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 5.678\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 5.678\hfill \end{array}\right] $$

where I ₉ is the identity matrix with dimension (9 × 9) equal to the number of data used for calibration.

The partial derivatives matrix ${\widehat{\mathbf{Z}}}_{jk}$ with respect to the random effects is calculated with the partial derivatives of Eq. 9. Using $z=\left[1- \exp \left(-\frac{a_2}{50}\cdot {d}_{jkl}\right)\right]/\left[1- \exp \left(-\frac{a_2}{50}\cdot 100\right)\right]$ for simplification, the partial derivatives are:

$$ \begin{array}{l}\frac{\partial {h}_{jkl}}{\partial {w}_{0k}}={\left(\frac{1}{a_0}\right)}^2{\left[\left({\left(20{a}_1\right)}^{a_0}-{1.3}^{a_0}\right)z+{1.3}^{a_0}\right]}^{\frac{1}{a_0}}\cdot \\ {}\kern3em \left[\frac{a_0\left(\left( \ln 20+ \ln {a}_1\right){\left(20{a}_1\right)}^{a_0}z-{1.3}^{a_0}z \ln 1.3+{1.3}^{a_0} \ln 1.3\right)}{\left({\left(20{a}_1\right)}^{a_0}-{1.3}^{a_0}\right)z+{1.3}^{a_0}}- \ln \left(\left({\left(20{a}_1\right)}^{a_0}-{1.3}^{a_0}\right)z+{1.3}^{a_0}\right)\right]\end{array} $$

$$ \frac{\partial {h}_{jkl}}{\partial {w}_{1k}}={20}^{a_0}{a}_1{}^{a_0-1}z{\left[\left({\left(20{a}_1\right)}^{a_0}-{1.3}^{a_0}\right)z+{1.3}^{a_0}\right]}^{\left(\frac{1}{a_0}-1\right)} $$

$$ \frac{\partial {h}_{jkl}}{\partial {v}_{0j}}=\frac{\partial {h}_{jkl}}{\partial {w}_{0k}} $$

$$ \frac{\partial {h}_{jkl}}{\partial {v}_{1j}}=\frac{\partial {h}_{jkl}}{\partial {w}_{1k}} $$

$$ \begin{array}{l}\frac{\partial {h}_{jkl}}{\partial {v}_{2j}}=\frac{\left({1.3}^{a_0}-{\left(20{a}_1\right)}^{a_0}\right) \exp \left[\left(2-\frac{d_{jkl}}{50}\right){a}_2\right]}{\left(50{a}_0{\left( \exp \left(2{a}_2\right)-1\right)}^2\right)}.\\ {}\kern3em \left({d}_{jkl}\left(- \exp \left(2{a}_2\right)\right)+100\left( \exp \left(\frac{a_2}{50}{d}_{jkl}\right)-1\right)+{d}_{jkl}\right)\cdot {\left[\left({\left(20{a}_1\right)}^{a_0}-{1.3}^{a_0}\right)z+{1.3}^{a_0}\right]}^{\left(\frac{1}{a_0}-1\right)}\end{array} $$

The values for the partial derivatives are:

Plot	Species	d (cm)	h (m)	$\frac{\partial {h}_{jkl}}{\partial {w}_{0k}}$	$\frac{\partial {h}_{jkl}}{\partial {w}_{1k}}$	$\frac{\partial {h}_{jkl}}{\partial {v}_{0j}}$	$\frac{\partial {h}_{jkl}}{\partial {v}_{1j}}$	$\frac{\partial {h}_{jkl}}{\partial {v}_{2j}}$
1	1	15	12	6.0907	8.2505	6.0907	8.2505	3.3777
1	2	17	12	6.0354	9.0770	6.0354	9.0770	3.5535
1	3	14	12	6.0887	7.8158	6.0887	7.8158	3.2712
2	1	20	13	5.8419	10.2163	5.8419	10.2163	3.7346
2	2	19	13	5.9183	9.8494	5.9183	9.8494	3.6844
2	3	16	13	6.0720	8.6707	6.0720	8.6707	3.4716
3	1	17	11	6.0354	9.0770	6.0354	9.0770	3.5535
3	2	15	11	6.0907	8.2505	6.0907	8.2505	3.3777
3	3	18	11	5.9835	9.4697	5.9835	9.4697	3.6243

The partial derivatives should be arranged in the matrix into different columns, to reflect how the data is arranged in the dataset, to obtain a 9 × 15 matrix (9 observations used in calibration and 15 random parameters) as follows:

$$ {\widehat{\mathbf{Z}}}_{jk}=\left[\begin{array}{ccccccccccccccc}\hfill \frac{\partial {h}_{jk l}}{\partial {w}_{0k}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {w}_{1k}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{0j}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{1j}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{2j}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {w}_{0k}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {w}_{1k}}\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{0j}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{1j}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{2j}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {w}_{0k}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {w}_{1k}}\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{0j}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{1j}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{2j}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill \frac{\partial {h}_{jk l}}{\partial {w}_{0k}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {w}_{1k}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{0j}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{1j}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{2j}}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {w}_{0k}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {w}_{1k}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{0j}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{1j}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{2j}}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {w}_{0k}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {w}_{1k}}\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{0j}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{1j}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{2j}}\hfill & \hfill 0\hfill \\ {}\hfill \frac{\partial {h}_{jk l}}{\partial {w}_{0k}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {w}_{1k}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{0j}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{1j}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{2j}}\hfill \\ {}\hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {w}_{0k}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {w}_{1k}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{0j}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{1j}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{2j}}\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {w}_{0k}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {w}_{1k}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{0j}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{1j}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\partial {h}_{jk l}}{\partial {v}_{2j}}\hfill \end{array}\right] $$

resulting in:

$$ {\widehat{\mathbf{Z}}}_{jk}=\left[\begin{array}{ccccccccccccccc}\hfill 6.0907\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 8.2505\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 6.0907\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 8.2505\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 3.3777\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 6.0354\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 9.0770\hfill & \hfill 0\hfill & \hfill 6.0354\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 9.0770\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 3.5535\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 6.0887\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 7.8158\hfill & \hfill 6.0887\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 7.8158\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 3.2712\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 5.8419\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 10.2163\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 5.8419\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 10.2163\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 3.7346\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 5.9183\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 9.8494\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 5.9183\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 9.9494\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 3.6844\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 6.0720\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 8.6707\hfill & \hfill 0\hfill & \hfill 6.0720\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 8.6707\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 3.4716\hfill & \hfill 0\hfill \\ {}\hfill 6.0354\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 9.0770\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 6.0354\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 9.0770\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 3.5535\hfill \\ {}\hfill 0\hfill & \hfill 6.0907\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 8.2505\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 6.0907\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 8.2505\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 3.3777\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 5.9835\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 9.4697\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 5.9835\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 9.4697\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 3.6243\hfill \end{array}\right] $$

The matrix ${\widehat{\mathbf{e}}}_{jkl}$ containing the errors obtained with the fixed-effects model (Eq. 9 with w _0k , w _1k , v _0j , v _1j and v _2j equal 0) is:

$$ {\widehat{\mathbf{e}}}_{jkl}=\left[\begin{array}{c}\hfill 1.7949\hfill \\ {}\hfill 0.9066\hfill \\ {}\hfill 2.2621\hfill \\ {}\hfill 0.6825\hfill \\ {}\hfill 1.0767\hfill \\ {}\hfill 2.3432\hfill \\ {}\hfill -0.09339\hfill \\ {}\hfill 0.7949\hfill \\ {}\hfill -0.5154\hfill \end{array}\right] $$

Therefore, the random parameters calculated using Eq. 6 were:

$$ {\widehat{\mathbf{b}}}_{jk}=\left[\begin{array}{c}\hfill {w}_{01}\hfill \\ {}\hfill {w}_{02}\hfill \\ {}\hfill {w}_{03}\hfill \\ {}\hfill {w}_{11}\hfill \\ {}\hfill {w}_{12}\hfill \\ {}\hfill {w}_{13}\hfill \\ {}\hfill {v}_{01}\hfill \\ {}\hfill {v}_{02}\hfill \\ {}\hfill {v}_{03}\hfill \\ {}\hfill {v}_{11}\hfill \\ {}\hfill {v}_{12}\hfill \\ {}\hfill {v}_{13}\hfill \\ {}\hfill {v}_{21}\hfill \\ {}\hfill {v}_{22}\hfill \\ {}\hfill {v}_{23}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill 0.01561\hfill \\ {}\hfill 0.02045\hfill \\ {}\hfill 0.05229\hfill \\ {}\hfill 0.002747\hfill \\ {}\hfill 0.009959\hfill \\ {}\hfill 0.03455\hfill \\ {}\hfill 0.05549\hfill \\ {}\hfill 0.04252\hfill \\ {}\hfill -0.0002459\hfill \\ {}\hfill 0.03248\hfill \\ {}\hfill 0.02262\hfill \\ {}\hfill -0.01373\hfill \\ {}\hfill 0.1756\hfill \\ {}\hfill 0.1310\hfill \\ {}\hfill -0.03699\hfill \end{array}\right] $$

This estimation of the random effects can be used to calibrate the model to obtain a specific response for the plots and species of this example, using Eq. 9.

The estimated heights and associated errors using the fixed effects Eq. 9 and the calibrated Eq. 9 are, therefore:

Plot	Species	d (cm)	h (m)	h jkl fixed	e jkl fixed	h jkl cal.	e jkl cal.
1	1	15	12	10.21	1.79	11.54	0.46
1	2	17	12	11.09	0.91	12.58	−0.58
1	3	14	12	9.74	2.26	11.55	0.45
2	1	20	13	12.32	0.68	13.41	−0.41
2	2	19	13	11.92	1.08	13.11	−0.11
2	3	16	13	10.66	2.34	12.22	0.78
3	1	17	11	11.09	−0.09	10.95	0.05
3	2	15	11	10.21	0.79	10.17	0.83
3	3	18	11	11.52	−0.52	11.89	−0.89
				${\displaystyle \sum \left|{\widehat{\mathbf{e}}}_{jkl}\right|}$	10.47		4.54

It can be observed how the prediction errors are reduced from using a fixed-effects prediction to using a calibrated prediction.

1.2 Case 2: use of Eq. 11

This is the usual way of calibrating a mixed-effects model with only one sampling level (i.e. the plot level), as done in many previous studies (e.g. Calama and Montero 2004; Castedo-Dorado et al. 2006; Sharma and Parton 2007). In this case, the calibration process has to be done for each species independently, because each species has a different model (Eq. 11 with the parameters shown in Table 3).

Hereafter, we include an example for species 1 on plot 1. The other species and plots would be calibrated in the same way.

Not that for Eq. 11 and species 1, only the variance for the random effect v ₁ was significant, therefore only one random effect has to be calculated.

The estimated variances and the covariance of the random effects (Table 3) are the elements of the variance–covariance matrix $\widehat{\mathbf{D}}$, conforming a simple (1 × 1) diagonal matrix (1 random parameter):

$$ \widehat{\mathbf{D}}=\left[0.05110\right] $$

The variance–covariance matrix for the random error term is determined by assuming that all estimations have constant variance (σ ², Table 3) and that the errors are not correlated. In this case, it only contains one observation:

$$ {\widehat{\mathbf{R}}}_{jk}={\sigma}^2\times {\mathbf{I}}_1=\left[5.308\right] $$

where I ₁ is the identity matrix with dimension (1 × 1) equal to the number of data points used in calibration.

The partial derivatives matrix ${\widehat{\mathbf{Z}}}_{jk}$ with respect to the random effects is calculated with the partial derivatives of Eq. 11, which have the same expression as those from Eq. 9.

The values for the partial derivatives are:

Plot	Species	d (cm)	h (m)	$\frac{\partial {h}_{jkl}}{\partial {v}_{1j}}$
1	1	15	12	7.7013

Therefore the corresponding matrix is:

$$ {\widehat{\mathbf{Z}}}_{jk}=\left[\frac{\partial {h}_{jk l}}{\partial {v}_{1j}}\right]=\left[7.7013\right] $$

The matrix ${\widehat{\mathbf{e}}}_{jkl}$ containing the errors obtained with the fixed-effects model (Eq. 11 with v _0jk , v _1jk and v _2jk equal 0) is:

$$ {\widehat{\mathbf{e}}}_{jkl}=\left[1.2840\right] $$

Therefore, the random effect estimated using Eq. 6 was:

$$ {\widehat{\mathbf{b}}}_{jk}=\left[{v}_{11}\right]=\left[0.06060\right] $$

This estimation of the random effect can be used to calibrate the model to obtain a specific response for the plot and species of this example, using Eq. 11.

The estimated heights and associated errors using the fixed effects of Eq. 11 and the calibrated Eq. 11 are, therefore:

Plot	Species	d (cm)	h (m)	h jkl fixed	e jkl fixed	h jkl cal.	e jkl cal.
1	1	15	12	10.71	1.28	11.18	0.82