EUCALYPTUS GRANDIS X EUCALYPTUS UROPHYLLA GROWTH CURVE IN DIFFERENT SITE CLASSIFICATIONS, CONSIDERING RESIDUAL AUTOCORRELATION

Brazil is a major producer in the timber sector, mainly with the use of wood from species of the genus Eucalyptus, with 26.1% of planted forests located in Minas Gerais. Researchers and manufacturers have been searching for techniques with the objective of making full use of these forests, with a primary focus on greater growth. A modeling of growth curves is an alternative for the estimation of floral production and an important aid tool for the researcher’s decision making. Growth curves are commonly studied by nonlinear regression models, which have important assumptions that if not met should be added to the model. The present work aims to select among nonlinear Logistic, Gompertz and von Bertalanffy regression models the most suitable to describe the growth in wood volume of Eucalyptus urophylla x Eucalyptus grandis hybrids in three Forest Site categories, including whether assumption deviations are required. Methods were executed by the Gauss-Newton iterative method implemented in nls() and it gnls() functions of the R software. Determination coefficient, Akaike information criterion (AICc) and Residual Standard Deviation (RSD) were used as selection evaluators of the best model. The results demonstrate that for all site categories, the Gompertz model with addition of autoregressive parameters AR (1) is the most appropriate to describe the growth in wood volume of Eucalyptus urophylla x Eucalyptus grandis hybrids. The addition of the first-order autoregressive parameter does not affect the quality of fit, but it is the correct procedure. Site I, which presents the largest trees according to pre-defined variations, recorded 308 m/ha of wood volume, followed by 286 m/ha and 263 m/ha for Sites II and III, respectively. The time for Site III to reach the maximum point of volume growth is between the fourth and sixth year, while the other sites are more precocious, reaching this point between the second and third year. 1Universidade Federal de Lavras UFLA, Instituto de Ciências Exatas e Tecnológicas, Departamento de Estat́ıstica, CEP: 37200-900, Lavras, MG, Brasil. E-mail: walleffsilva@gmail.com; fernandesfelipest@gmail.com; joamuniz@ufla.br; tales.jfernandes@ufla.br 2Escola Superior de Agricultura “Luiz de Queiroz” Esalq-USP, CEP: 13418-900, Piracicaba, SP, Brasil. E-mail: fabiana.muniz@hotmail.com.br 122 Rev. Bras. Biom., Lavras, v.39, n.1, p.122-138, 2021 doi: 10.28951/rbb.v39i1.511


Introduction
Planted wood of the genus Eucalyptus has great potential due to its availability in a short period of time and use in the production of raw material for cellulose, charcoal and poles. According to data from IBGE (2018), Brazil has around 9.9 million hectares of planted forests, of which 7.5 million are eucalyptus, with the southeastern region being responsible for 42.3% of the total planted area in Brazil, mainly in the states of Minas Gerais (26.1%) and São Paulo (12.2%). For maximum use of these hectares, researchers have been looking for more efficient ways to use the forest, mainly in the breeding of trees and forest management.
The growth of a tree consists of the thickening and elongation of roots, trunk and branches and increase in the number of leaves, resulting in changes in tree volume and shape (SCOLFORO, 2006). In a forestry company, production planning is essential and, in this context, forest growth and production modeling is a tool that assists researchers in making decisions. The growth and production of a forest stand depends on age, productive capacity or site, degree of utilization of productive potential and silvicultural treatments. Other strategies to assist growth are better management of fertilizers, the use of high-quality seedlings and the control of weeds, pests and diseases (ELLI et al., 2019).
Among tree characteristics, height and diameter constitute an important variable, which is determined or estimated essentially for the calculation of volume and for the calculation of height and volume increments. Retslaff et al. (2015) point out that, in forest inventories, determining the height of all trees can lead to more errors than estimating it, due to visualization difficulties in addition higher demand of time and costs. Mendonça et al. (2017) reported that the wood volume, both present and future, is the most important item in forest planning.
For Scolforo (2006), the use of data in which height and age are related is the most practical, efficient and consistent method to use data in the classification of forest sites, because height correlates very well with the development potential of most species used in reforestation. Therefore, the greater the height, the better the site quality. According to Bila (2012), the practical importance of the classification of sites in terms of productivity is related to the decrease in costs for forestry companies with permanent installments, with the provision of information that will define management techniques (precision forestry) more adapted to a given situation, considering sustainable aspects and economic profitability.
According to Coutinho et al. (2017), when grouping all trees and analyzing their increments, it was observed that the production curve has a sigmoidal shape, in which the first phase corresponds to the juvenile age, the second to the mature age and the third to the senile age. Thus, the optimum cutting age, from the technical point of view, is when the population cycle allows obtaining the largest wood volume per unit of area per year. This age varies according to the growth curve of the forest and must be evaluated for each plantation using inventory techniques and forest biometry.
The expression 'growth curve' usually refers to sigmoidal curves that represent the behavior of dimension measurements over time. Animal, plant and fruit growth curves are commonly studied using nonlinear regression models, as in , Jane et al. (2019) and Silva et al. (2020), respectively. Specifically in the forestry area, some works in literature have been using nonlinear models to describe the growth curve. Venduscrolo et al. (2016) evaluated modeling using nonlinear regression and artificial neural networks to estimate the height of eucalyptus trees. Coutinho et al. (2017) adjusted nonlinear regression Weibull, Logistic, Gompertz, Schumacher and Chapman-Richards models to assess the growth pattern and describe the probabilistic distribution of the annual increase in the diameter of Cryptomeria japonica. Santos et al. (2017) using nonlinear models, estimated characteristics such as dominant height, average diameter, basal area and volume with bark according to age.
According to Mendonça et al. (2017), when using regression models, important assumptions are added to the problem of establishing a relationship among variables of interest. Prado et al. (2020) emphasize that the assumption of residual independence is almost always not accepted when studying growth curves, as measures are taken repeatedly in the same individual, the residue of an observation can be associated with the residue of adjacent observations, that is, residues are autocorrelated. Studies on growth in volume of forests have not taken this assumption deviation into account.
Thus, the aim of the present work was to select among the nonlinear regression models Logistic, Gompertz and von Bertalanffy, the most adequate to describe the growth in wood volume of Eucalyptus urophylla x Eucalyptus grandis hybrids in three site classifications, considering assumption deviations if necessary.

Material
Data from Gonçalves et al. (2017) referring to the planting of Eucalyptus grandis x Eucalyptus urophylla hybrids that were historic and active since the second cycle, with reference in December 2012, were used. For the estimation of the growth curves of eucalyptus plantations information on age (years) and volumetric production of wood (m 3 /ha) from monoclonal plantations were used.
Data from were obtained from the Forest Inventory of Fibria Celulose S.A., Aracruz branch, located at Aracruz, Espírito Santo, Brazil. The average temperature of the region is 28 • C and the average annual rainfall is 1,200 mm. The predominant soil type in the region is Distrophic Red-Yellow Latosol and Red-Yellow Podzol (GONÇ ALVES et al., 2017).
The forest area was divided into 3 site classes, with large trees (site I), medium trees (site II) and small trees (site III). According to Gonçalves et al. (2017), site classification included three steps: adjustment of a general model, projection of the dominant height and site classification defining the upper and lower limits of each class, creating the 3 site classes. The wood volume (in m 3 /ha) from the first to the 15 th year was then calculated.

Methods
The growth analysis of Eucalyptus grandis x Eucalyptus urophylla plants was performed using nonlinear models. Logistic (1), Gompertz (2) and von Bertalanffy (3) models were adjusted.
where Y i represents the volume of trees in the site in m 3 in the i-th observation; α represents the maximum horizontal asymptote, that is, the maximum volume that a tree of that site can reach; k represents the growth rate (the higher the k, the less time the tree takes to reach α); β is interpreted as the abscissa of the inflection point, from which growth decelerates, x i is the i-th year of measurement. i corresponds to the random error, which is assumed and is independent and identically distributed following normal distribution with mean of zero and constant variance, that is i ∼ N (0; σ 2 )). For the analysis of residues, Shapiro-Wilk tests (SHAPIRO and WILK, 1965) were used to verify normality of residues, in which the null hypothesis is that the residues follow a normal distribution, the Breuch-Pagan test (BREUSCH and PAGAN, 1979) to assess the homogeneity of the variance, whose null hypothesis is that the residues are homoscedastic and the Durbin-Watson test (DURBIN and WATSON, 1950) to check for the existence of residual autocorrelation, the null hypothesis for this test is the independence of waste. If there is violation of the independence assumption, it will be added one parameter autoregressive first-order AR (1) to the model.
Heterogeneity of variances and residual autocorrelation are characteristics inherent to growth data over time that, in most works, are not considered in the modeling, which can lead to inaccurate estimates and results (MAZZINI et al., 2005;MENDES et al ., 2008;PRADO, SAVIAN and MUNIZ, 2013).
When residual dependence was identified, in order to obtain reliable parameter estimates, it was necessary to incorporate this feature in the modeling. Thus the error vector was as described below: where: i = 2, 3, ..., 15 is the number of years in which the volume was estimated; i is the residue estimated in the i-th year; φ 1 is first-order autoregressive parameter; i−1 is the residual estimated for the year immediately preceding the i-th year; and φ p is the autoregressive parameter of order p; i−p is the residual estimated for p years before the i-th year and υ i is the random error of the model that follows normal distribution with mean zero and constant variance. When residues are independent, then parameters φ 1 , ..., φ p are null, hence i = υ i (SILVA et al., 2020).
In order to compare and evaluate model adjustments, the corrected Akaike information criterion (AICc) was used, and the model with the lowest AICc estimation was chosen. Regarding the determination coefficient (R 2 ), the best model is the one with the highest value. In addition, the Residual Standard Deviation (RSD) was used to compare adjustments, with those with the lowest RSD values being considered the best models. To illustrate the estimates of adjusted models, graph analysis was used.
The estimates of parameters and figures presented here were obtained by the iterative Gauss-Newton method implemented in nls () and gnls () functions of the R software. The significance of parameters was verified using the t test at 5% significance level.

Results and Discussion
Initially, the above models were adjusted and analysis of residues was performed ( Table 1).
The assumptions that must be assumed by the error (normality, independence and homogeneity of variances) were evaluated, respectively, by the Shapiro-Wilk (SW), Durbin-Watson (DW) and Breusch-Pagan (BP) tests, considering 5% significance level. It was observed that the assumptions of normality and homogeneities of variance for the random error were validated. However, for all models and for all site classifications, the Durbin-Watson test was significant, violating the assumption of independence, so it was necessary to include a first-order autoregressive parameter AR (1) in the model.
In a study on the growth of pepper cultivar 'Doce', Jane et al. (2019) observed that there was no need for the inclusion of the autoregressive parameter. However, Muniz et al. (2017) added this parameter to their model and found significant improvement in the growth adjustment of cocoa fruits, including significant reduction in the residual standard deviation, making estimates more reliable. Pereira et al. (2016), considering the residual autocorrelation with the addition of the first-order autoregressive parameter AR (1), obtained the lowest values in all densities and irrigation regimes analyzed for the height growth of coffee plants.
Describing the growth of coconut fruits, Prado et al. (2013) showed that adjustment of the logistic model to experimental data of longitudinal external diameter (LED), considering the first-order autoregressive structure for residues, is adequate in the description of data and resulted in estimates of parameters quite consistent with those reported in literature. Prado et al. (2020) reported that in the description of coconut growth, both for longitudinal and cross-sectional diameter of the internal cavity, it was necessary to incorporate the residual dependence modeling, since this assumption was not met. Studying the blackberry growth curve, Silva et al. (2020) pointed out the need to add parameter φ 1 for some cultivars and not for others, thus reinforcing the need to always conduct the analysis of residues, even if for a specific cultivar or variable, it was not necessary to use the autoregressive parameter.
The parameter estimates, obtained by the Gauss-Newton iterative method, for the wood volume of Site I and II are contained in Tables 2 and 3, observing the selection criteria of the models (higher R 2 , lower AICc and SRD values) and comparing the choices of adequate adjustments according to the addition or not of the autoregressive parameter. It could be observed that there is a change in the indication of the model, in these scenarios, without the addition of the first-order autoregressive parameter AR (1). Observing the previously mentioned selection criteria, von Bertalanffy model proved to be satisfactory, while with the inclusion of this parameter, the adjustment quality evaluators of Gompertz model presented better results compared to the others.
For Sites I and II, analyzing models adjusted without adding the autoregressive parameter and observing the evaluators to select the best adjustment with lower AICc, high R 2 and lower RSD, these criteria indicated the von Bertalanffy model as the most appropriate. However, with the violation of the assumption of independence, it was necessary to add the first-order autoregressive parameter AR (1) to the model in both sites. In this scenario, AICc selected the von Bertalanffy model, while R 2 and RSD selected the Gompertz model as the best fit. Table 2 shows that the estimates for the maximum wood volume without adding the autoregressive parameter ranges from 303.2850 m 3 /ha to 313.3403  Pereira et al. (2016), the moment when plants reach maximum growth rate, shifting from a period of accelerated growth to the inhibition period (inflection point), is represented by parameter β. Thus, it was observed that the deceleration in volume growth in Site I is between the second and fourth years. For Site II, according to Table 3, it was found that the maximum estimated wood volume ranges from 277.8772 m 3 /ha to 291.0234 m 3 /ha for models without the addition of the autoregressive parameter. For models without adding the autoregressive parameter, there is a slowdown in the wood volume growth between the third and fourth year. Observing models with the addition of the autoregressive parameter, variation from 272.4648 m 3 /ha to 314.3186m 3 /ha was observed for maximum wood volume for Site II.  Table 4, a different behavior from the other Sites was observed. The selection criteria agree to select the Gompertz model as the best fit, both with the addition of the AR parameter (1) and without the addition. The estimates for the chosen models are close to each other with small variations. For models without the addition of the autoregressive parameter, it was observed that the maximum wood volume growth is between 245.9575 m 3 /ha and 277.3292 m 3 /ha. However, with the addition of the autoregressive parameter, the variation is between 263.8590 m 3 /ha and 270.9575 m 3 /ha. These values are closer to the estimated maximum volume of 218.6599 m 3 /ha for Eucalyptus camaldulensis x Eucalyptus urophylla obtained by Mendonça et al. (2017). Analyzing the estimates for the inflection point (parameter β), which indicates the point of maximum wood volume growth, this point occurs later than in the other Sites, between the fourth and sixth year.
Thus, it could be concluded that the Gompertz model with the addition of the  (2014) showed that for variable plant height, the Gompertz model, sometimes with structure of independent errors, sometimes with autocorrelated errors, presented, in six situations analyzed, values closest to the one for the adjusted determination coefficient and also the smallest Akaike information criterion values, corroborating to the fact that the Gompertz model was the one that best described the height growth of coffee plants. Studying the growth curve of coffee plants, Fernandes et al. (2014) also concluded that the Gompertz model with incorporation of assumption deviations on the residues vector was the one that provided the best adjustments. Pereira et al. (2016) reported that the Gompertz model, considering heterogeneity of variances and residual autocorrelation, presented the lowest values in all analyzed densities and irrigation regimes, indicating that this model is the one that best describes the height growth of coffee plants over time. According to Muniz et al. (2017), the incorporation of the first-order autoregressive parameter improved the quality of fit, with reduction of about 40% in the residual standard deviation for the Gompertz model. Venduscrolo et al. (2017) showed that the estimates presented values statistically equal to those observed by the t-test, indicating that the Gompertz model is efficient for estimating the dependent variable (total height) as a function of the independent variable (diameter at chest height). Lundgren et al. (2017) evaluated the influence of the sampling type in the estimation of the Eucalyptus wood volume and found that the total wood volume that was measured at 7.5 years provided the value of 166.14 cubic meters for an area of 2.4 ha, result similar to the estimates for Site III. For Marangon et al. (2017), in the dynamics of the diametric distribution and eucalyptus production in different ages and spaces, at the age of 7 years the Site reaches its maximum accumulated volumetric productivity, and at the age of 13 years the Site reaches its minimum for the three ages analyzed, possibly due to the fact that trees of this age have higher individual volumes and smaller diametric distribution.
Regarding wood volume production for all sites, verifying estimates obtained by parameter α, it was observed that Site I can be considered the most productive and, according to Gonçalves et al. (2017), Site III shows slower growth and smaller production than the other sites, since its inflection point is estimated to occur later. Coutinho et al. (2017) explained that the growth of trees is not regular throughout their life cycles, and for cryptomeria individuals studied, a decrease in diameter at chest height was observed over the years and that the largest increments are concentrated in the juvenile phase of trees, where naturally there is greater vigor, when compared to maturity and senescence phases. Lima et al. (2017) observed maturity or growth rate estimates (k) and found the occurrence of small values, mainly when the von Bertalanffy model was adjusted, and these small values indicate that plants take longer to reach maturity. Marangon et al. (2017) reported that the growth and productivity level of sites may vary according to the water availability, spacing between plants, soil richness, genetic potential of plants, site capacity, etc. Almeida et al. (2017) analyzed the estimated wood volume for Eucalyptus sp. with high spatial resolution satellite images, observing the generated wood volume maps, and showed that it was possible to verify how much development can be uneven within the same species. Figures 1, 2 and 3 show the graphic adjustment for the description of the Eucalyptus grandis x Eucalyptus urophylla wood volume for Sites I, II and III (respectively), showing the difference in prediction for models with or without the addition of the first-order autoregressive parameter AR (1).  According to results shown in Tables 2 and 3 for Sites I and II, it was observed that, in principle, the von Bertalanffy model was the preferred candidate among the other models based on the adjustment quality evaluators. However, when adding the assumption of independence, this model presented poor estimates, underestimating the asymptote (Figures 1 and 2). This evidences that the analysis of results, frequently ignored in literature, constitutes one of the most important stages when studying regression models. Underestimation and overestimation in asymptote estimates were observed in other studies. Santos et al. (2017) found underestimation and overestimation points in the estimation of Tauari volume in the Tapajós National Forest, with tendency of overestimation for volumes of up to 5 m 3 and underestimation for volumes from 10 m 3 . Gomes et al. (2018) observed tendency of overestimation in commercial height in the lower stratum and underestimation in the upper stratum for volumetric estimates of wood volume in the Tapajós National Forest. Nunes et al. (2017), using Hohenadl-Kren, Husch, Stoate and Schumacher-Hall models for the volumetric estimation of a dense Eucalyptus sp. population, found significant tendency of underestimation for smaller diameter trees, while the Brenac model showed a slight tendency of overestimation for trees with diameter at chest height (DCH) between 6 and 8 cm.
Andrade (2017) used a volumetric model for Eucalyptus urophylla and Eucalyptus grandis trees aged 5-7 years developed from the form factor equation adapted to the Gompertz bio-mathematical model, presenting smaller variation of errors around the zero axis, with less pronounced tendency of underestimation. Volumetric modeling resulted in estimates similar to other studies, like Santos et al. (2017) that found satisfactory results using the Gompertz model for volume, presenting mean growth values similar to those found in the present work. Vendruscolo et al. (2017) found that the equations obtained with the Gompertz model showed adjustment statistics equal to or slightly higher than other models for estimating the height of Tectona grandis L.f. trees.
Despite the violation of the assumption of independence, the adjustment of models without the addition of the autoregressive parameter AR (1) was presented in result tables to enable the comparison between estimates and possible adjustment improvement, mainly due to disagreement in the selection of the best model in Sites I and II. Tables 2, 3 and 4 showed that there was an increase in the standard error of the estimate of all parameters when considering the first-order residual dependence AR (1). This increase in the standard error of the estimate was also observed by Prado et al. (2020), that show that incorporating residual dependence does not necessarily improve the quality of fit, but it is the most coherent, because without adding the autoregressive parameter, it would be accepted that residues are independent, when in fact they are not.

Conclusions
The Gompertz model with the addition of the first-order autoregressive parameter AR (1) was the most adequate to describe the growth in wood volume of Eucalyptus grandis x Eucalyptus urophylla hybrids for the three Site classifications, obtaining satisfactory estimates for wood volume.
Estimates without the autoregressive parameter AR (1) for the von Bertalanffy model, although similar to those obtained with the addition of this parameter in the Gompertz model, have become obsolete. An increase in the standard error of the estimate was also observed when incorporating AR (1) in all models, reinforcing that the addition of the first-order autoregressive parameter does not necessarily improve the quality of fit. However, it is more coherent, since the assumption of residual independence was not met.
The estimate for maximum wood volume in Site I was 308 m 3 /ha, while Sites II and III showed lower results, 286 m 3 /ha and 263 m 3 /ha, respectively. In addition, it was verified that the time for Site III to reach the point of maximum volume growth is between the fourth and the sixth year, while the other Sites are more precocious, reaching this point between the second and third year.