NONLINEAR MIXED-EFFECT HEIGHT-DIAMETER MODEL FOR Pinus pinaster AIT. AND Pinus radiata D. DON

ABSTRACT

Tree height-diameter (H-D) relationships are important for routine forest assessment. Several H-D relationships have been developed for different species and more are still evolving. This study introduces new H-D model developed for Pinus pinaster and Pinus radiata in Spain, based on data from 184 and 96 permanent sample plots, respectively, collected in the northwest region of the country. Nonlinear mixed-effect modelling technique was used to fit the generalized H-D model. The mixed-effect H-D model was calibrated using the random effects predicted from one to three randomly selected trees per sample plot. Different indices including root mean square error (RMSE) and adjusted coefficient of determination () were used to assess the predictive performance of the model. The results showed that the new model had and RMSE of 0.906 and 1.156 m and 0.814 and 1.703 m for P. pinaster and P. radiata, respectively. The calibration response involved the selection of one tree per sample plot and resulted in a reduction of RMSE by 6.5% and 13.5% for pinaster and P. radiata, respectively.

Keywords:
Hossfeld IV; Fixed-effect; Model calibration; Maritime pine; Monterrey pine

INTRODUCTION

Total height and diameter at breast height (D at 1.3 m above the ground) are fundamental tree variables that are routinely measured in forest inventory. They are required for the assessment of non-spatial structure of forest stands and estimation of volume (Adame et al., 2008; Gómez-García et al., 2014), basal area and determination of the competitive position of a tree in forest stand (West, 2015; Ogana, 2019a) and assessment of site productivity (Jayaraman and Lappi, 2001; West, 2015). Measurement of tree diameter at breast height is relatively simple, accurate and with low cost (Ferraz-Filho et al., 2018; Corral-Rivas et al., 2019). Conversely, tree height measurement is difficult, time consuming and expensive (Mehtätalo et al., 2015; Ozcelik et al., 2018). Owing to the associated problem with tree height measurement, only sub-sample of trees is measured. Thus, height-diameter (H-D) models are often used to estimate the height of trees for which the diameters have been measured (Kalbi et al., 2017).

Several nonlinear H-D models with single-variable (D) have been developed for different species (e.g. Yuancai and Parresol, 2001; Calama and Montero, 2004; Sharma, 2009; Ferraz-Filho et al., 2018). These models have been developed using either fixed-effect or mixed-effect techniques. In fixed-effect H-D models, the assumption of independence is violated (Ozcelik et al., 2018) and sufficient number of measurements is required for unbiased estimate of tree height (Arcangeli et al., 2014; Kalbi et al., 2017). On the other hand, mixed-effect models “allow the prediction of a response when using only the fixed-effect, and a calibrated response where random effects are predicted and included in the model using measurements of height from a sample trees” (Burkhart and Tome, 2012). Single-variable mixed-effect H-D models have been consistently used in the recent times (e.g. Sharma and Parton, 2007; Budhathoki et al., 2008; Zhang et al., 2016; Kalbi et al., 2017; Ogana, 2019a).

Corral-Rivas et al. (2019), asserted that modelling H-D relationships with single-variable (one predictor) [D] may not be adaptable to different stand dynamics and silvicultural conditions; and as such, may not possibly estimate all H-D relationships in the stands. This has led to the introduction of generalized H-D model so that different stand variations and conditions can be accounted for (Krisnawati et al., 2010). However, developing generalized H-D models often requires additional inventory costs, especially, model with mean height as one of the predictor variables (López-Sánchez et al., 2003). Some of the stand variables that have been used to develop generalized H-D models include number of trees per ha, basal area per ha, quadratic mean diameter, dominant and mean height, dominant diameter and age, among others. López-Sánchez et al. (2003) compared twenty-six fixed-effect generalized H-D models for P. radiata D. Don in Galicia, Spain using different stand variables. Recently, Corral-Rivas et al. (2019) compared ten generalized H-D models for seven pine species in Durango, Mexico.

Monterrey pine (Pinus radiata) and Maritime pine (Pinus pinaster) stands are important natural resources in northwest Spain. These species are considered as fast growing mainly occur in pure stands but sometimes Pinus pinaster also occurs in mixed stands. Pinus spp. and Eucalyptus spp. are the most commonly used species in productive stands in this area of Spain where the timber harvest represent more than 50% of the total country (Gorgoso-Varela et al., 2015). Pure stands of maritime pine are mainly derived from natural regeneration, although they are occasionally established as plantations. Exotic Monterrey pine stands are derived from plantations.

Therefore, the main objective of this study was to fit a new H-D function based on the variation of the Hossfeld IV model using fixed and mixed-effect models, and to compare it with other classical models. The Pinus pinaster and Pinus radiata stands in northwest Spain were used as case study.

MATHERIAL AND METHODS

Data

The data used for this study were obtained from two species of pine - Maritime pine (Pinus pinaster Ait) and Monterrey pine (Pinus radiata D. Don) in northwest Spain. The plantations and natural regeneration stands of P. pinaster cover 217,281 ha and 22,523 ha in the regions of Galicia and Asturias, respectively. The pure plantations stand of P. radiata occupy 96,177 ha and 25,385 ha in Galicia and Asturias, respectively (MMAMRM, 2011). The map of the study area is presented in Fig 1. A total of 184 permanent sample plots (PSPs) from P. pinaster stands and 96 PSPs from P. radiata stands were used for this study. The plot sizes ranged from 375 to 900 m²; to achieve a minimum of 30 trees per plot. Square plots were used in this study. Diameter at breast height (D at 1.3 m above the ground) and total tree height (H) were measured with calliper and hypsometer to an accuracy of 0.1 cm and 0.1 m, respectively. A total of 17,845 and 12,722 trees were measured from P. pinaster and P. radiata, respectively. The data from the two species were randomly split into two groups: 60% (fitting data) and 40% (validation data). The descriptive statistics of the quadratic mean diameter (dg, cm), basal area per ha (G, m² ha^-1), number of trees per ha (N trees ha^-1), dominant height (Ho, m), dominant diameter (i.e., average diameter of the 100 largest trees per ha; Do, cm), mean height (Hm, m) and age (t, years) for the two species are presented in Table 1. The scatterplots showing the relationship between tree height and diameter for the species are shown in Fig. 2a and b.

FIGURE 1
Distribution of Pinus pinaster and Pinus radiata stands in northwest Spain (regions of Galicia and Asturias).

FIGURE 2
Scatterplots of the relationship between tree height and diameter for (a) Pinus pinaster and (b) Pinus radiata.

TABLE 1
Descriptive statistics of the data groups.

Species	Variables	Fitting data (N trees = 10776)				Validation (N trees = 7184)
		Mean	SD	Min	Max	Mean	SD	Min	Max
P. pinaster (184 plots)	D (cm)	17.4	7.0	1.8	56.7	17.4	6.8	1.6	63.2
	H (m)	11.5	3.8	1.4	32.7	11.4	3.7	2.3	36.0
	G (m2.ha-1)	34.4	12.2	7.8	115.1	34.3	12.0	7.8	115.1
	dg (cm)	18.0	5.4	10.4	41.5	17.9	5.2	10.4	41.5
	Hm (m)	11.4	3.4	4.7	24.3	11.3	3.3	4.7	24.3
	Ho (m)	13.1	3.7	5.5	30.6	13.0	3.6	5.5	30.6
	t (years)	19.8	8.1	8.0	61.5	19.6	7.9	8.0	61.5
	N (trees.ha-1)	1466	510	375	2480	1465	507	375	2480
	Do (m)	26.1	6.9	13.8	55.8	26.0	6.6	13.8	55.8
		Fitting data (N trees = 7737)			Validation (N trees = 5158)
P. radiata (96 plots)	D (cm)	16.3	6.1	1.3	45.5	16.4	6.2	1.8	47.1
	H (m)	14.6	3.9	2.6	31.7	14.7	4.0	2.8	31.5
	G (m2.ha-1)	30.7	7.1	14.3	46.6	30.7	7.1	14.3	46.6
	dg (cm)	17.1	2.7	11.9	27.1	17.2	2.8	11.9	27.1
	Hm (m)	14.5	2.4	9.7	22.6	14.5	2.4	9.7	22.6
	Ho (m)	17.8	3.0	12.9	28.7	17.9	3.1	12.9	28.7
	t (years)	17.1	3.8	12.0	32.0	17.1	3.6	12.0	32.0
	N (trees ha-1)	1373	397	530	2610	1363	399	530	2610
	Do (m)	26.4	3.9	19.3	38.3	26.6	3.9	19.3	38.3

• Max = maximum, min = minimum, SD = standard deviation, N trees = number of trees measured

Single variable Height-Diameter models

We derived a new H-D model from the Hossfeld IV growth function (Zeide 1993). The generic Hossfeld IV function is given by [1], where H = total tree height (m), D = diameter at breast height (1.3 m above the ground, cm), and a, b, c = model parameters. Preliminary application of the above function for height prediction in P. pinaster and P. radiata performed poorly - extremely large estimates for the parameters and larger standard errors, especially for a parameter. The inclusion of the point of measurement did not improve the function.

$H=\frac{D^c}{\raisebox{1ex}{$\left(b+D^c\right)$}\!\left/ \!\raisebox{-1ex}{$a$}\right.}$ (1)

To improve the Hossfeld IV function, the third parameter in the denominator was replaced with power 3. The rationale for this was because only the adjustment of parameter c achieved convergence and raised to power 3 since it was the third parameter that was replaced. The point of measurement of D was included. We termed this new growth function as modified Hossfeld IV model and expressed as [2]:

$H=1.3+\frac{D^c}{\raisebox{1ex}{$\left(b+D^3\right)$}\!\left/ \!\raisebox{-1ex}{$a$}\right.}$ (2)

The variables and the parameters in the model are previously defined in equation (1). The new model was compared with fifteen single-variable H-D models, where eight have 2-parameters and seven have 3-parameters. These models include: Bertalanffy, Curtis, Meyer, Michailoff, Michaelis-Menten (MM), Naslund, Power, Wykoff, Chapman-Richards (Richards), Gompertz, Korf, Logistic, Prodan, Ratkowsky and Weibull models (Table 2). These models have been consistently used in forestry including recent work by Corral-Rivas et al. (2019) and Ogana (2019a). The models were first fitted to the height-diameter data (fitting and validation) of P. pinaster and P. radiata using ordinary non-linear least square (ONLS) method, implemented in the ‘nls’ function in R (R Core Team, 2017). Here all parameters in the models were considered as fixed.

TABLE 2
Single H-D models.

Model name	Form	Reference	Eq.
Bertalanffy	$H=1.3+a{\left(1-e^{\left(-bD\right)}\right)}^3$ (3)	Von Bertalanffy (1957)	(3)
Curtis	$H=1.3+a{\left(D/\left(1+D\right)\right)}^b$ (4)	Curtis (1967)	(4)
Meyer	$H=1.3+a\left(1-e^{-bD}\right)$ (5)	Meyer (1940)	(5)
Michailoff	$H=1.3+a{e}^{\left(-b{D}^{-1}\right)}$ (6)	MIchailoff (1943)	(6)
Michaelis-Menten (MM)	$H=1.3+\frac{aD}{\left(b+D\right)}$ (7)	Michaeli and Menten (1913)	(7)
Naslund	$H=1.3+\frac{D^2}{{\left(a+bD\right)}^2}$ (8)	Näslund (1936)	(8)
Power	$H=1.3+a{D}^b$ (9)	Stoffels and van Soest (1953)	(9)
Wykoff	$H=1.3+e^{a+b/\left(D+1\right)}$ (10)	Wykoff et al. (1982)	(10)
Chapman-Richards	$H=1.3+a{\left(1-e^{-bD}\right)}^c$ (ll)	Richards (1959)	(11)
Gompertz	$H=1.3+{ae}^{\left(-b{e}^{\left(-cD\right)}\right)}$ (12)	Gompertz (1825), Huang et al. (1992)	(12)
Korf	$H=1.3+a\left(-e^{-b{D}^{-c}}\right)$ (13)	Lundqvist (1957)	(13)
Logistic	$H=1.3+\frac{a}{1+b{e}^{-cD}}$ (14)	Pearl and Reed (1920)	(14)
Prodan	$H=1.3+\frac{D^2}{a+bD+c{D}^2}$ (15)	Strand (1959)	(15)
Ratkowsky	$H=1.3+a{e}^{-b/\left(D+c\right)}$ (16)	Ratkowsky (1990)	(16)
Weibull	$H=1.3+a\left(1-e^{b{D}^c}\right)$ (17)	Yang et al. (1978)	(17)

Generalized H-D models

The new function, that is, equation (2) was generalized to account for the different stand conditions. At first, different combinations of stand variables (as shown in Table 1 above) were evaluated. And it was observed that the inclusion of the quadratic mean diameter (dg), dominant height (Ho) and number of trees per ha (N) improved tree height prediction than other alternatives. The new model was termed generalized modified Hossfeld H-D model. It is expressed as [18], where H = total tree height, D = diameter at breast height, dg = quadratic mean diameter, Ho = dominant height, N = number of trees per ha, ln = natural logarithm and a, b, c, d, e = model parameters.

$H=1.3+\frac{D^c}{\raisebox{1ex}{${exp}^{\left(\frac{-e}{ln\left(H_o\right)}\right)}\left(b+D^3\right)$}\!\left/ \!\raisebox{-1ex}{$a$}\right.}{exp}^{\left(\raisebox{1ex}{$-d$}\!\left/ \!\raisebox{-1ex}{$\sqrt{\raisebox{1ex}{$N$}\!\left/ \!\raisebox{-1ex}{$d_g$}\right.}$}\right.\right)}$ (18)

The new generalized modified Hossfeld model was compared with eleven established generalized H-D models. Ten of the models have been used for P. radiata in Spain (López-Sánchez et al., 2003). While the generalized Michaelis-Menten (Gen.MM) was developed by Ogana (2019b). Different categories of generalized models were used including those ranging from 1 to 5 parameters. The parameters of the generalized models were considered as fixed. The models are presented in Table 3. The models were fitted to the fitting data sets from P. pinaster and P. radiata using ONLS, implemented in the ‘nls’ function in R (R Core Team, 2017). The models were also validated.

TABLE 3
Generalized H-D models.

Model name	Form	Reference	Eq.
Canadas IV	$H=1.3+{\left[b_0\left(\frac{1}{D}-\frac{1}{D_o}\right)+{\left(\frac{1}{H_o-1.3}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right]}^{-2}$ (19)	Cañadas et al. (1999)	(19)
Mønness	$H=1.3+{\left[b_0\left(\frac{1}{D}-\frac{1}{D_o}\right)+{\left(\frac{1}{H_o-1.3}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right]}^{-3}$ (20)	Mønness (1982)	(20)
Gaffrey	$H=1.3+\left(H_o-1.3\right)e^{b_0\left(1-\frac{d_g}{D}\right)+b_1\left(\frac{1}{d_g}-\frac{1}{D}\right)}$ (2l)	Gaffrey (1988)	(21)
Sloboda	$H=1.3+\left(H_o-1.3\right)e^{\left(b_0\left(1-\frac{D}{d_g}\right)\right)}{e}^{b_1\left(\frac{D}{d_g}-\frac{1}{D}\right)}$ (22)	Sloboda et al. (1993)	(22)
Gen.MM	$H=1.3+\frac{b_{0}D}{\left(b_1+D\right)}{e}^{\frac{-b_2}{\sqrt{H_o}}}$ (23)	Ogana (2019b)	(23)
Pienaar	$H=b_0{H}_o{\left(1-e^{\frac{-b_{1}D}{d_g}}\right)}^{b_2}$ (24)	Pienaar (1991)	(24)
Mirkovich	$H=1.3+\left(b_0+b_1{H}_o-b_2{d}_g\right)e^{\raisebox{1ex}{$-b_3$}\!\left/ \!\raisebox{-1ex}{$D$}\right.}$ (25)	Mirkovich (1958)	(25)
S.A I	$H=1.3+\left(b_0+b_1{H}_o-b_2{d}_g\right)e^{\raisebox{1ex}{$-b_3$}\!\left/ \!\raisebox{-1ex}{$\sqrt{D}$}\right.}$ (26)	Schroder, Álvarez-González (2001)	(26)
Tome	$H=H_o{e}^{\left(b_0+b_1{H}_o+b_2\frac{N}{1000}+b_{3}t\right)\left(\frac{1}{D}-\frac{1}{D_o}\right)}$ (27)	Tomé (1989)	(27)
Cox III	$H=H_m\left[b_0+b_1{H}_m+b_2\frac{H_m}{d_g}+b_{3}D+b_4\frac{N}{\frac{d_g\left(H_m{d}_g\right)}{D}}\right]$ (28)	Cox (1994); López-Sánchez et al. (2003)	(28)
S.A II	$H=1.3+\left(b_0+b_1{H}_o-b_2{d}_g+b_{3}G\right)e^{\raisebox{1ex}{$-b_4$}\!\left/ \!\raisebox{-1ex}{$\sqrt{D}$}\right.}$ (29)	Schröder, Álvarez-González (2001)	(29)

Model Evaluation and Assessment

Model assessment was based on residual graphs and numerical comparisons of the root mean square error (RMSE), adjusted coefficient of determination (), Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). The BIC evaluates a model based on its simplicity (Corral-Rivas et al., 2019). Models with low RMSE, AIC, and BIC and high were regarded as good models. Based on these indices, the H-D models were ranked (Tewari and Singh, 2018). A value of 1 was assigned to the best model and the largest value was assigned to the worst model with respect to each fit index. The ranks were summed for each model; this was used as the indicator of the performance of the individual model with respect to the four fit indices. The smaller the rank sum, the better the model. The same indices for the H-D models were computed for the fitting and validation data sets from P. pinaster and P. radiata, Where: RSS = residual sum of square, n = sample size, p = number of parameters; = average total tree height; Yi is the observed tree height and is the theoretical value predicted by the model.

$RMSE=\sqrt{\frac{{\sum }_{i=1}^n{\left(Y_i-{\widehat{Y}}_i\right)}^2}{n}}$ (30)

${\stackrel{-}{R}}^2=1-\frac{\left(n-1\right)}{\left(n-p\right)}\frac{\sum _{i=1}^n{\left(Y_i-{\widehat{Y}}_i\right)}^2}{\sum _{i=1}^n{\left(Y_i-{\stackrel{-}{Y}}_i\right)}^2}$ (31)

$AIC=n\mathrm{l}\mathrm{n}\left(\frac{RSS}{n}\right)+2p$ (32)

$BIC=n\mathrm{l}\mathrm{n}\left(\frac{RSS}{n}\right)+p\ln n$ (33)

Mixed-effects model

The new generalized modified Hossfeld model was refitted using mixed-effect models. In this technique both the within and between plot-height variability was considered by the introduction of random parameters. This technique also helps to overcome the issue of lack of independence between observations (Corral-Rivas et al., 2019). The fixed and random parameters of a mixed-effects model are estimated simultaneously (Pinheiro and Bates, 1998). Following the methodology of Pinheiro and Bates (1998) and recently used by Mehtätalo et al. (2015) and Corral-Rivas et al. (2019), the non-linear mixed-effect was defined as [34], where hij is the total tree height of tree j on plot i and corresponding diameter dij; f is the nonlinear model; is the parameter vector r x 1 where r represents the whole parameters in the model. The lambda λ is a p x 1 vector for the fixed parameters (a, b and c) and p is the number of parameters. The bi represent the q x 1 vector for the random parameters (q = number of random parameters) - it is the plot effect that shows the variation in the parameters of plot i from the typical plot. Аi=r x p and Вi=r x q, these are the dimensional matrix for both the fixed and random effects, respectively, unique to plot i (Corral-Rivas et al., 2019). The underlying assumption is that the “plot effects have a common multivariate normal distribution with mean 0 and variance-covariance matrix var(bi)=D for all values of i” (Mehtätalo et al., 2015). The epsilon is the vector form of the random error and is assumed to be normal and independent with zero mean and constant variance var(εij) = σ² . To decide which fixed parameter should be considered as random, different combinations of fixed and random parameters were evaluated using RMSE, , AIC, and BIC.

$h_{ij}=f\left(d_{ij};{\mathit{\varphi }}_i\right)+{\epsilon }_{ij}$ (34)

${\mathit{\varphi }}_i={\mathit{{\rm A}}}_{\mathit{i}}\mathit{\lambda }+{{\mathit{B}}_{\mathit{i}}\mathit{b}}_i$ (35)

The parameters of the mixed-effects models were estimated with the method of restricted maximum likelihood (REML) to diminish biases through the ‘nlme’ function in R (R Core Team, 2017). The empirical best linear unbiased predictor (EBLUP) approximation was used for the maximization of the marginal likelihood function as recommended by Beal and Sheiner (1982). The whole data sets were used for the mixed model.

Calibration of the mixed models

Predicting the response variable (total tree height) with the random effects estimated from its prior information is termed as calibration of the mixed-effects H-D model (Sharma et al. 2019). Prediction of the random effect for a given stand and the adjustment of the fixed part of the mixed-effects H-D model requires the measured height of one or more trees in each sample plot to be used to predict the specific random parameters for that stand. We calibrated the mixed-effects H-D model using the random effects predicted from one to three randomly selected trees per sample plot in the validation data set. The following procedures were used in selecting the sample trees:

One sample tree: the tree nearest to the 25th quantile;

Two sample trees: each one nearest to the 50th and 75th quantiles

Three sample trees: each one nearest to the quadratic mean diameter, minimum and maximum diameter

To make predictions for the random components using a sub-sample of heights the empirical best linear unbiased predictor (EBLUP) theory was applied following the expression 36 (Vonesh and Chinchilli, 1997), where is a random effect vector that describes plot-level H-D variations for the i^th plot; is the mj × mj variance-covariance matrix for within-plot variability; Zi is m × q matrix of the partial derivatives of the valued random parameters from and m × 1 residuals vector, whose components result from the difference between the observed height of each tree and the predicted value from the model, considering only fixed parameters. The values were estimated iteratively, for that purpose, we developed an R script using matrix functions in R (R Core Team, 2017). The calibration alternatives were evaluated in terms of the previously defined statistics (RMSE and ) and compared with the RMSE estimations obtained with ONLS in the individual fit of the selected model to each of the calibration plots.

${\widehat{b}}_i=\widehat{D}{\widehat{Z}}_i^T{\left({\widehat{R}}_i+{\widehat{Z}}_i\widehat{D}{\widehat{Z}}_i^T\right)}^{-1}{\widehat{\epsilon }}_i$ (36)

RESULTS AND DISCUSSION

Single variable fixed-effect models

In the fitting data set of P. pinaster, the RMSE, , AIC and BIC of the models ranged from 2.291 - 2.649 m, 0.513 - 0.636, 48453 - 51589 and 48482 - 51611, respectively (Table 4). When the models were validated, the RMSE, , AIC and BIC values of the models ranged from 2.249 - 2.591 m, 0.514 - 0.634, 11649 - 13685 and 11669 - 13698, respectively. The assessment of the models based on their rank positions with respect to the fit indices showed that modified Hossfeld function ranked 4^th behind Gompertz, Ratkowsky and Logistic functions for the fitting data. It also ranked 6^th for the validation data set. Michailoff and Bertalanffy had highest rank sum of 60 (15^th) and 64 (16^th), respectively.

TABLE 4
Parameter estimates and fit indices for both fitting and validation data set of Pinus pinaster.

Model	Parameter estimates			Fitting data				Validation data
	a	b	c	RMSE		AIC	BIC	∑Rank	RMSE		AIC	BIC	∑Rank
Bertalanffy	16.45(0.095)	0.117(0.001)		2.649	0.513	51589	51611	6416	2.591	0.514	13685	13698	6416
Curtis	23.885(0.170)	14.306(0.131)		2.504	0.565	50370	50392	5614	2.450	0.566	12878	12892	5614
Meyer	34.36(0.814)	0.021(0.001)		2.335	0.622	48864	48886	328	2.287	0.621	11892	11906	328
Michailoff	23.262(0.163)	13.408(0.125)		2.521	0.559	50517	50539	6015	2.466	0.56	12974	12988	6015
MM	56.677(1.526)	78.169(2.667)		2.33	0.623	48822	48844	287	2.283	0.623	11864	11878	287
Naslund	2.229(0.019)	0.18(0.001)		2.409	0.597	49542	49563	4812	2.359	0.598	12333	12347	4812
Power	1.145(0.019)	0.768(0.005)		2.305	0.632	48580	48602	195	2.259	0.631	11714	11728	164
Wykoff	3.20(0.007)	-15.26(0.137)		2.487	0.571	50227	50248	5213	2.434	0.572	12784	12798	5213
Chapman-Richards	25.00(0.666)	0.04(0.002)	1.30(0.035)	2.396	0.602	49420	49449	4411	2.343	0.603	12240	12261	4411
Gompertz	36.087(1.306)	2.322(0.023)	0.03(0.001)	2.291	0.636	48453	48482	41	2.249	0.634	11649	11669	41
Korf	70.00(9.374)	6.00(0.088)	0.40(0.029)	2.375	0.609	49228	49257	4010	2.322	0.610	12110	12131	4010
Logistic	27.647(0.548)	5.869(0.094)	0.069(0.001)	2.291	0.636	48459	48489	73	2.250	0.634	11658	11679	93
Prodan	-3.455(0.074)	1.746(0.014)	0.009(0.001)	2.305	0.632	48587	48615	216	2.259	0.631	11716	11736	185
Ratkowsky	96.23(6.848)	116.69(8.033)	34.39(1.923)	2.291	0.636	48459	48488	62	2.249	0.634	11653	11673	62
Weibull	85.00(23.107)	0.01(0.002)	0.900(0.023)	2.337	0.621	48885	48914	369	2.295	0.619	11940	11960	369
*Hossfeld	1.134(0.019)	-3.555(0.773)	3.771(0.006)	2.304	0.632	48573	48602	164	2.261	0.630	11727	11748	246

• *= modified Hossfeld IV; a, b, c = model parameters; values in parenthesis are the standard errors; subscripts represent relative positions of the rank sum

In the case of P. radiata, the results from the fitting data set showed that the RMSE, , AIC and BIC values of the models ranged from 2.452 - 2.704 m, 0.523 - 0.607, 35843 - 37350 and 35871 - 37371, respectively (Table 5). The fit indices of the models for the validation data were in the range of 2.513 - 2.788 m, 0.516 - 0.607, 9510 - 10582 and 9530 - 10596, respectively. Evaluation of the models based on their relative position showed that modified Hossfeld IV was the best performed model with the lowest ranks sum of 4 (1^st) for both fitting and validation data sets. This was followed by Power and Ratkowsky with rank sum of 8 (3^rd) and 12 (4^th), respectively. Weibull and Bertalanffy had highest rank sum of 60 (15^th) and 64 (16^th), respectively. Other H-D models including Chapman-Richards, Curtis, Korf, Meyer, Michaelis-Menten (MM), Nalsund and Wykoff performed relatively. The fit indices provided identical ranks, especially the RMSE and in both species.

TABLE 5
Parameter estimates and fit indices for both fitting and validation data set of Pinus radiata.

Model	Parameter estimates			Fitting data				Validation data
	a	b	c	RMSE	R 2	AIC	BIC	∑Rank	RMSE	R2	AIC	BIC	∑Rank
Bertalanffy	17.105(0.070)	0.178(0.001)		2.704	0.523	37350	37371	6416	2.788	0.516	10582	10596	6416
Curtis	24.001(0.149)	8.96(0.102)		2.537	0.58	36368	36388	5213	2.623	0.575	9914	9927	5313
Meyer	22.748(0.244)	0.058(0.001)		2.484	0.597	36042	36063	369	2.552	0.595	9668	9681	369
Michailoff	23.434(0.142)	8.272(0.095)		2.55	0.575	36447	36468	5614	2.627	0.570	9969	9982	5614
MM	33.362(0.481)	23.117(0.587)		2.47	0.602	35952	35973	195	2.534	0.600	9603	9616	267
Naslund	1.218(0.014)	0.193(0.001)		2.500	0.592	36141	36162	4411	2.572	0.588	9748	9761	4411
Power	2.642(0.044)	0.587(0.006)		2.455	0.606	35860	35880	82	2.515	0.606	9518	9531	82
Wykoff	3.203(0.006)	-9.696(0.108)		2.525	0.584	36294	36315	4812	2.600	0.579	9863	9876	4812
Chapman-Richards	25.322(0.156)	0.040(0.007)	0.832(0.008)	2.468	0.602	35939	35960	164	2.534	0.600	9597	9610	246
Gompertz	24.911(0.573)	1.699(0.017)	0.063(0.003)	2.470	0.602	35953	35981	216	2.531	0.601	9585	9605	215
Korf	35.00(2.196)	5.00(0.195)	0.60(0.037)	2.496	0.593	36117	36145	4010	2.566	0.590	9727	9746	4010
Logistic	22.808(0.378)	3.318(0.056)	0.097(0.003)	2.478	0.599	36009	36037	328	2.539	0.599	9620	9639	328
Prodan	-1.243(0.019)	0.875(0.010)	0.024(0.001)	2.473	0.601	35974	36002	287	2.527	0.603	9567	9587	154
Ratkowsky	39.943(1.547)	31.196(2.221)	12.71(0.996)	2.462	0.604	35904	35931	123	2.524	0.603	9556	9575	123
Weibull	25.00(1.807)	0.110(0.004)	0.070(0.033)	2.553	0.574	36465	36493	6015	2.632	0.569	9990	10010	6015
*Hossfeld	2.819(0.065)	12.995(3.530)	3.566(0.008)	2.452	0.607	35843	35871	41	2.513	0.607	9510	9530	41

• *= modified Hossfeld IV; values in parenthesis are the standard errors; subscripts represent relative positions of the rank sum

To further assess the performance of the single-variable H-D models, the average residuals of the predicted height from the validation data set were computed for different diameter classes and plotted for the two species. A diameter class of 5 cm interval was used and the mean residual in height prediction was assessed. Only the residual graphs of the two best single-variable H-D models and the modified Hossfeld function for P. pinaster and P. radiata were presented (Fig 3a and b). The graphs showed that modified Hossfeld IV function had the same behavior as Gompertz and Logistic models. These models both overestimated and underestimated tree height in the lower (< 5 cm) and larger (> 55 cm) diameter classes, respectively in P. pinaster (Fig 3a). In the case of P. radiata, the mean residuals plot of the modified Hossfeld function was more stable than Ratkowsky and Power H-D models. The modified Hossfeld was consistent across the diameter classes except in the large classes (> 40 cm). In contrast, Ratkowsky and Power models both overestimated and underestimated tree height in the lower (< 5 cm) and larger (≥ 40 cm) diameter classes.

FIGURE 3
Mean residual plot against diameter classes for the single-variable H-D model in (a) P. pinaster and (b) P. radiata (validation data).

The performance of the new modified Hossfeld in P. pinaster and P. radiata was more stable than most of the functions evaluated in this study. The parameter estimates are significant with smaller standard errors. This is a new single variable H-D model. The Logistic and Gompertz functions with the least bias in P. pinaster, did not perform well in P. radiata. This implies that the nature of data could affect the performance of a function. Mehtätalo et al. (2015) selected the logistic function for modelling H-D relationships for the pure Scots pine. They reported a better performance with the logistic function compared to other models. Similar observation was reported in Ogana (2018) for Gmelina arborea Roxb stands. Another function which seems to be relatively stable is the Ratkowsky. Its ranked 2^nd and 3^rd in P. pinaster and P. radiata, respectively. This model was selected by Liu et al. (2018) as the most suitable non-linear model between 32 H-D models evaluated for Metasequoia in China.

When considering model simplicity and the ease of fitting the functions, the new modified Hossfeld HD model could be adopted for estimating tree height especially, in the P. radiata stand.

Generalized h-d models

The results of the generalized H-D models are presented in Table 6 and 7. In P. pinaster, the new generalized Hossfeld model had RMSE, , AIC and BIC of 1.290, 0.885, 36076 and 36120, respectively for the fitting data set; and 1.271, 0.883, 3454 and 3489, respectively for the validation data. The model ranked 3^rd behind Sloboda (1^st) and Tomé (2^nd) models both for the fitting and validation data sets. Mirkovich and Gaffrey had the highest rank sum of 44 (11^th) and 48 (12^th), respectively. In the case of P. radiata, the new generalized Hossfeld model had the lowest rank sum of 4 for both the fitting and validation data sets and as such, was the best H-D model. Its RMSE, , AIC and BIC were 1.775, 0.794, 30848 and 30890, respectively for the fitting data set; and 1.841, 0.789, 6307 and 6340, respectively for the validation data. This was followed by Schroder and Alvarez (S.A II) and S.A I. Sloboda and Gaffrey had the poorest results. The inclusion of stand variables (dg, Ho and N) improved the prediction of the modified Hossfeld model. The RMSE decreased from 2.261 to 1.271 (44%) and from 2.513 to 1.841 (27%) in P. pinaster and P. radiata, respectively. This is expected because the introduction of stand variables in a model usually decrease the variability within sample unit. Several researchers have recommended the inclusion of stand variables in H-D relationships (e.g., López-Sánchez et al., 2003; Canga-Libano et al., 2009; Crecente-Campo et al., 2010; Uzoh, 2017). Corral-Rivas et al. (2019) compared ten generalized H-D models for seven pine species in Durango, Mexico. They asserted that modelling H-D relationship with D only may not be adaptable to different stand dynamics and silvicultural conditions; and as such, may not possibly estimate all H-D relationships in the stands.

TABLE 6
Parameter estimates and fit indices for the generalized H-D models of P. pinaster and P. radiata.

Model	Parameter estimates					Fitting data
	b0	b1	b2	b3	b4	RMSE		AIC	BIC	∑Rank
P. pinaster
Canadas	0.992(0.008)					1.292	0.884	36102	36117	164
Monness	0.982(0.007)					1.301	0.883	36255	36269	236
Gaffrey	0.278(0.019)	2.447(0.340)				1.797	0.776	43221	43243	4812
Sloboda	-0.411(0.005)	0.014(0.001)				1.271	0.888	35759	35781	41
Gen.MM	95.899(0.818)	9.976(0.177)	6.352(0.042)			1.309	0.881	36390	36419	369
Piennaar	1.109(0.012)	1.498(0.086)	0.805(0.036)			1.310	0.881	36407	36437	4010
Mirkovich	2.844(0.096)	1.145(0.011)	0.158(0.007)	6.273(0.007)		1.313	0.881	36456	36492	4411
S.A I	5.665(0.179)	1.741(0.021)	0.302(0.012)	3.346(0.037)		1.305	0.882	36328	36364	287
Tome	-2.508(0.254)	-0.277(0.163)	0.770(0.083)	-0.026(0.008)		1.284	0.886	35976	36013	82
Cox III	0.718(0.010)	-0.029(0.001)	0.559(0.014)	0.017(0.001)	-0.42(0.024)	1.305	0.882	36336	36380	308
SA II	5.654(0.178)	1.663(0.022)	0.311(0.012)	0.035(0.004)	3.349(0.037)	1.299	0.883	36238	36282	205
*Hossfeld	40.728(1.104)	34.663(3.674)	3.368(0.006)	0.573(0.035)	-5.988(0.043)	1.290	0.885	36076	36120	123
P. radiata
Canadas	0.972(0.007)					1.812	0.786	31158	31172	267
Monness	1.005(0.007)					1.817	0.784	31205	31219	349
Gaffrey	0.424(0.037)	0.579(0.638)				2.833	0.476	38078	38099	4812
Sloboda	-0.486(0.006)	0.028(0.002)				1.941	0.754	32227	32249	4411
Gen.MM	122.169(2.221)	14.567(0.264)	6.492(0.081)			1.817	0.784	31203	31231	338
Piennaar	1.121(0.012)	1.466(0.067)	0.964(0.033)			1.801	0.788	31069	31097	164
Mirkovich	6.502(0.241)	1.101(0.015)	0.255(0.016)	7.200(0.068)		1.811	0.787	31159	31193	256
S.A I	12.959(0.441)	1.825(0.029)	0.512(0.028)	3.956(0.035)		1.787	0.791	30948	30983	123
Tome	-4.676(0.377)	-0.154(0.016)	1.554(0.124)	-0.061(0.012)		1.807	0.787	31121	31156	205
Cox III	0.699(0.016)	-0.042(0.001)	0.659(0.017)	0.025(0.001)	-2.032(0.092)	1.827	0.782	31291	31333	4010
S.A II	12.175(0.447)	1.803(0.028)	0.565(0.029)	0.065(0.009)	3.949(0.035)	1.780	0.793	30895	30937	82
*Hossfeld	59.376(2.467)	49.397(3.976)	3.436(0.007)	1.413(0.068)	-7.197(0.089)	1.775	0.794	30848	30890	41

• *= modified Hossfeld IV; b0, b1, b2, b3, b4 = estimated parameter; values in parenthesis are the standard errors; subscripts represent relative positions of the rank sum.

TABLE 7
Fit indices of the generalized H-D models for validation data of Pinus pinaster and Pinus radiata.

	Pinus pinaster					Pinus radiata
Model	RMSE		AIC	BIC	∑Rank	Model	RMSE		AIC	BIC	∑Rank
Canadas	1.277	0.882	3511	3518	195	Canadas	1.881	0.780	6517	6524	287
Monness	1.287	0.880	3625	3632	246	Monness	1.885	0.779	6543	6550	4010
Gaffrey	1.780	0.771	8290	8304	4812	Gaffrey	2.893	0.479	10964	10977	4812
Sloboda	1.255	0.886	3271	3285	41	Sloboda	1.982	0.755	7059	7072	4411
Gen.MM	1.293	0.879	3703	3724	297	Gen.MM	1.877	0.781	6501	6521	216
Piennaar	1.302	0.877	3795	3815	4010	Piennaar	1.881	0.780	6523	6543	329
Mirkovich	1.312	0.876	3909	3937	4411	Mirkovich	1.881	0.780	6521	6541	308
S.A I	1.299	0.878	3771	3799	369	S.A I	1.853	0.786	6371	6397	133
Tome	1.268	0.884	3416	3443	82	Tome	1.879	0.789	6512	6538	195
Cox III	1.275	0.882	3504	3539	174	Cox III	1.870	0.782	6468	6501	174
S.A II	1.293	0.879	3701	3735	297	S.A II	1.846	0.788	6336	6369	92
*Hossfeld	1.271	0.883	3454	3489	123	*Hossfeld	1.841	0.789	6307	6340	41

• S.A = Schroder and Alvarez; Gen.MM = generalized Michaelis-Menten

Also, the mean residuals graph of the generalized Hossfeld showed similar behavior with Sloboda and Tome H-D models for P. pinaster, and SA I and SA II in P. radiata (Fig 4a and b). The models both overestimated and underestimated tree height in the lower (> 5 cm) and larger (> 55 cm) diameter classes, respectively in P. pinaster (Fig 4a).

FIGURE 4
Mean residual plot against diameter classes for the generalized H-D in (a) P. pinaster and (b) P. radiata (validation data).

Generalized Hossfeld with Mixed-effect

The result from the adjustment of the new generalized modified Hossfeld model with mixed effects is presented in Table 8. Of the different combination of fixed and random parameter tried, the best fit was found by relating the parameter b4 with a random parameter uj in an additive form (b4 + uj). The model is represented by equation (37). The estimated values and signs of all parameters are biologically plausible and interpretable. The predicted trajectories with new generalized height-diameter model showed appropriate trends, logical asymptotes, and adaptation at observed values; therefore, predictions out of the range of the used dataset could be done within reasonable limits, as the biological behaviour of the transition functions was adequate. The result showed that the inclusion of the random parameter improved the generalized Hossfeld function with RMSE and of 1.156 and 0.906, respectively for P. pinaster; 1.703 and 0.814, respectively for P. radiata. Also, the graph of residual against predicted height showed homogeneous variance (homoscedastic) in both species (Fig. 5a and b), where uj ~ N [0, σu] is the random parameter which is assumed to be normal with a zero mean and constant variance due to random effect (σu²). Other parameters in the model were previously defined in equation (18).

FIGURE 5
Residual plot against predicted height for the Hossfeld generalized model with mixed effect in (a) P. pinaster and (b) P. radiata.

TABLE 8
Parameters estimated, variance components and fit indices of the mixed-effect H-D model (equation 37) for both two species.

Parameters	P. pinaster	P. radiata
	Fixed parameters
b0	34.759	57.021
b1	41.344	46.516
b2	3.367	3.441
b3	0.358	1.211
b4	-5.651	-7.199
	Variance components
σ2u	0.022	0.013
σ2	1.439	2.92
	Fit indices
RMSE	1.156	1.703
	0.906	0.814
AIC	56171	50329
BIC	56218	50373

The mixed-effects H-D model developed (equation 37) was calibrated with the random effect predicted from measured heights of one to three randomly selected trees on each sample plot. The calibrated response described 89.9% and 70.6% of H-D variations in P. pinaster and P. radiata, respectively with one tree selected randomly (Table 9). It was observed that when one tree was selected randomly to calibrate the model, the RMSE value was reduced by 6.5% with respect to the estimated value from equation (18) by ONLS in P. Pinaster. On the other hand, in P. radiata the RMSE value was reduced by 13.5% with one randomly tree with respect to the ONLS fitting. The Hossfeld generalized H-D model with mixed-effect was precise enough for predicting the total height tree for the species studied. The importance of calibrating mixed-effect models to the forest owners cannot be overemphasized because only few sample trees are required to provide information on the height of all trees in the stand (Corral-Rivas et al., 2019). Different sample trees for model calibration have been reported. For example, Castedo-Dorado et al. (2006) recommended three sampled trees; Kalbi et al. (2017) used four sampled trees for the H-D model developed for Oriental beech stand in Iran. In this study, the selection of 1, 2 and sampled trees had RMSE of 1.199, 1.229 and 1.227, respectively in P. pinaster; and 2.122, 2.468 and 2.273 in P. radiata. The inclusion of more trees would require additional inventory cost. Thus, the generalized H-D model developed in this study would be valuable to the forest owners as only one sample tree is needed to obtain information of the height of trees in the pine stands.

TABLE 9
Comparison of fit indices calibration options of the generalized mixed-effect H-D model (equation 37).

Species	Calibration options	RMSE		AIC	BIC
P. pinaster	ONLS	1.282	0.884	59908	59954
	Mixed model	1.156	0.906	56171	56218
	One sample tree	1.199	0.899	57506	57553
	Two sample trees	1.229	0.894	58380	58426
	Three sample trees	1.227	0.894	58326	58373
P. radiata	ONLS	2.452	0.607	35843	35871
	Mixed	1.690	0.814	30081	30108
	One sample tree	2.122	0.706	33601	33629
	Two sample trees	2.468	0.602	35940	35968
	Three sample trees	2.273	0.663	34669	34696

CONCLUSION

This study has introduced new single-variable and a generalized height-diameter model with mixed-effect for P. pinaster and P. radiata. Fixed-parameters were expanded with random-parameters in a non-linear manner using dominant height, quadratic mean diameter and number of trees per ha as the stand-specific covariates. Its selection is justified for these species because it is the only model of those analyzed that is exempt from the presence of collinearity between its variables, together with the fact that the estimate of the height corresponds to the dominant height when the normal diameter of the tree It is equal to the dominant diameter of the mass. The calibrated response allows accurate results to be obtained with a very small sampling effort, making this approach highly effective and useful in traditional Spanish forest inventories and dynamic forest growth models. Thus, with one sample tree it will be possible to determine the height of all trees in the P. pinaster and P. radiata stands.

ACKNOWLEDGEMENT

The authors are thankful for the helpful comments of four anonymous reviewers which improved the quality of the manuscript.

Conflict of Interest

The authors declare that they have no conflict of interest

FUNDING

This work was supported by the Government of Spain, Department of Economy, Industry and Competitiveness under the Torres Quevedo Contract PTQ-16-08445. The study also was financially supported by the Gobierno del Principado de Asturias through the project entitled “Estudio del crecimiento y producción de Pinus pinaster Ait. en Asturias” (CN-07-094); by the Ministerio de Ciencia e Innovación through the project entitled “Influencia de los tratamientos selvícolas de claras en la producción, estabilidad mecánica y riesgo de incendios forestales en masas de Pinus radiata D. Don y Pinus pinaster Ait. en el Noroeste de España” (AGL2008-02259).

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HIGHLIGHTS

• 1
  New generalized height-diameter model was developed for Pinus pinaster and P. radiata.
• 2
  The properties and predictions of the models are biological reasonable.
• 3
  The model compares well with other established height-diameter models used in quantitative forestry.
• 4
  One tree selected at random was sufficient to calibrate the generalized model.

Publication Dates

• Publication in this collection
  17 June 2020
• Date of issue
  Jan-Mar 2020

History

• Received
  20 Nov 2019
• Accepted
  20 Feb 2020