Introduction

The different growth rates of trees in the forest are the physiological, demographic and phylogenetic reactions to the limitations of the biotic and abiotic factors of their surroundings in which each one of them develops (^{Chave, 1999}). It is essential to know the structural relationships between their components to understand the adaptations to the environment of organisms, a forest or a particular ecosystem (^{King, 1996}).

Allometric functions as a means to represent the functional relationship between two or more variables of an individual are useful in understanding the dynamics of the overall growth of a tree, in regard to some of its parts or components (^{Núñez et al., 2010}; ^{Gayon, 2000}). The use of this tool in plant biology has been used to develop the relationships between variables in size, for subsequent application to forest management or understanding the ecosystem ^{(Niklas and Enquist, 2002}).

Examples of this kind of work are the relationships of the normal diameter (dn) depending on the volume (V), biomass (B), leaf area index (LAI) or carbon (C) (^{Martín et al., 1998}); dn relations with the total height (H) (^{Zeide and Vanderschaaf, 2002}); dn-H functions for growth studies (^{Delgado et al., 2005}); or the functional relationship between stump diameter (dt) with dn (^{Pompa et al., 2011}); the H and V (^{Quiñonez-Barraza et al., 2012}; ^{Martínez and Acosta, 2014}); B (^{Návar et al., 2013}) or carbon.

To determine the relationship between dt and dn can be a reference to predictions in variables such as H, which then can be used as a basis for estimating volume (^{Aigbe et al., 2012}).

Since there are not always dimensions of dn for volume calculations; for example, when illegal exploitation occurs; in quantifying the uses on large areas; in the evaluation of silvicultural practices in the forest after harvesting; or simply when inventory information is misplaced, it becomes important to know the allometric relationship and proportional function between dn and dt, as it is used to quantify the removed wood volume (^{Benítez et al., 2004}).

The allometric and proportional relationship between the diameter of the stump and dbh are a useful quantitative tool for managers of forest resources and evaluators of the harvesting practices applied in the forests; therefore, interest arose to generate ratio equations and factors to predict the dn, from dt in Pinus greggii Engelm forest plantations, for restoration purposes, established in Metztitlán, Hidalgo state.

Materials and Methods

The plantations considered in the analysis are located in the Fontezuelas ejido at Metztitlán municipality, Hidalgo. The prevailing climate is temperate to semi-dry (^{García, 1988}; ^{Inegi, 1992}; ^{Hernández et al., 2014}), at an altitude ranging from 2 000 to 2 500 m; they are located between 98°54’ W and 20°29’ N and are established on Haplic feozem and Eutric regosol soil types (^{Inegi, 1992}).

Information was processed from 621 trees with diameter categories in dt of 5-45 cm and the dn of 5-35 cm (Table 1). dn was measured with a diameter tape (Forestry Suppliers P. O. BOX 8397) five meters, 1.30 m, and dt at a variable height, depending on the cutting technique used (tall stump, ht) (^{Bava and López, 2006}).

Table 1 Basic statistics of the sample in Pinus greggii Engelm. plantations at Metztitlán, Hidalgo. 

The assessed models (Table 2) were those proposed by ^{Quiñonez-Barraza et al. (2012)} for Pinus arizonica Engelm., P. ayacahuite Ehrenb. ex Schltdl., P. durangensis Martínez, P. leiophylla Schiede ex Schltdl. & Cham., P. teocote Schiede ex Schltdl. & Cham and Quercus sideroxila Bonpl.; and those of ^{Pompa et al. (2011)} for P. durangensis to estimate dn from dt, as this section has a trunkated neiloide shape type.

Table 2 Models for the estimation of normal diameter from the stump diameter in Pinus greggii Engelm. plantations in Metztitlán, Hidalgo. 

Dn = Normal diameter; dt = Stump diameter; ht = Stump height; b₀ and b₁ = Parameters for estimation.

The fitting of the models was made with the statistical SAS package through the MODEL procedure and the gmm technique; this fit improves the efficiency in the estimation of parameters in the presence of heterocedasticity, since it uses the Generalized Moments Method through the matrix of covariance (^{SAS, 2015}). Ordinary Squared Minimums method (MCO) [for its acronym in Spanish] was used as ^{Diéguez et al. (2003)} and ^{Pompa et al. (2011)} did.

The quality of fitness and the accuracy in the prediction of the equations was achieved with the regular statistics used in forest modeling and biometry (^{Castedo and Álvarez, 2000}; ^{Corral et al., 2007}): the fitted Coefficient of Determination (Raj2), the Root aj. Mean Square Error (REMC), the Bias (), Aggregate Deviation (DA) and DA percentage (DA %) (^{Prodan et al., 1997}). Also the Akaike (AIC) as well as the Bayesian (BIC) information criteria were used, which take into account the greatest verisimilitude and penalize according to the parsimony principle, according to the number of parameters of the model (^{Posada and Rosero, 2007}; ^{Gómez et al., 2013}) with the following equations:

Where:

log Lik = Logarithm of greatest verisimilitude

K = Proportional term to the number of parameters of the model

N = Total data number

A graphic analysis was made with the estimated values by each model in order to confirm if the tendency towards a straight line in regard to the observed values is accomplished (^{Posada and Rosero, 2007}; ^{Augusto et al., 2009}), as well as their own graphic analysis to identify and correct heterocedasticity problems (^{Huang, 2002}).

To statistically assess the normality of the processed data, a Shapiro-Wilk test was applied, since it verifies the unbiased linear estimation of the standard deviation divided into the variance of the errors in regard to the sample (^{Molinero, 2003}).

A calculation of the confidence intervals of the parameters was made in order to know the variation of the expected values if each one of them (Cepeda et al., 2008); thus, the estimated value of the parameter (b_j ), + or - the t value considering the degrees of freedom of the model at 95 % of confidence multiplied by the Approach Standard Error (Eea) as follows:

The tendencies of the predicted values were put in a graphic, in regard to those observed to show the fidelity or the deviation and the fit of them to the used information. Also, the average ratio factor (fp) between dt and dn of the population was determined, as well as the estimations made with the model of less fit.

Results and Discussion

In Table 3 are gathered the parameters and the statistics of the goodness of fit of the assessed models for the estimation of dn through dt in P. greggii trees.

Table 3 Values of the statistics for the goodness of fit and of the parameters obtained in the assessed models in Pinus greggii Engelm. plantations in Metztitlán, Hidalgo.  

Eea = Approximate standard error.

It can be observed that in models (1) and (3) the b₀ parameter and its estimators of fit are identical, from the mathematical structure of the equations, b₀ points out to the interception between dt and dn. Also, it is evident that the values of the approximate standard errors of the b₀ parameter in these two models are higher compared to the value of the same parameter of model (2), which has a different interpretation, as it considers the height at which the measurement of dt was taken.

When comparing the value of the adjusted coefficient of determination, model (2) has the highest setting, as it reaches a value of 0.9486; in this regard, ^{Alder (1980)} and ^{Gujarati (2010)} indicate that it is acceptable when exceeding 0.7 and 0.8, respectively. In this context, it is assumed that all models are satisfactory in their estimates. For this same model, the Raj2 condition coincides with the lowest value of REMC which evaluates the accuracy of the estimates as described by ^{Pompa et al. (2011)} by using a dendrometric model for the estimation of the normal diameter, from the size of the stump, which the best models are those with lower values.

In addition, based on the values of the maximum verisimilitude information and weight by the number of parameters used (AIC and BIC) it can be deducted and confirmed that the model with the best quality of statistical fit is (2) for having the lower values on these criteria, in regard to the four models used; this criterion was also applied by ^{Tamarit et al. (2013)} and ^{Quiñonez-Barraza et al. (2014)} for the election of the best equations in estimating volume and taper.

The use of these criteria tends to select the simplest models, as they penalize the value according to the principle of parsimony (^{Gómez et al., 2013}). However, if the numerical difference between the models in the values of AIC and BIC is very narrow, the complement to the values of RMSE and Raj2 will be the decision criteria for choosing any of them (^{Wintle et al. 2003}; ^{Gómez et al., 2013}).

Table 4 Statistics for the evaluation of the predictive ability of model (2) in Pinus greggii Engelm. plantations in Metztitlán, Hidalgo. 

In Table 4 are summarized the values that assess the predictive ability of model (2), the deviation of their estimations and the results of the Shapiro-Wilk tests of normality of errors.

The deviation of model (2), in regard to the observed values against the predicted ones (E) is low, as well as the numbers of the mean of the residuals (DA). This situation is similar to that described by ^{Benítez et al. (2004)} in a study of site index for P. caribaea Morelet, and to the experience of ^{Barreto et al. (2011)} who, when estimating the normal diameter from the dimensions of the stump in Casuarina equisetifolia L. plantations, used the DA as an indicator of deviations from the models used. The negative value of DA and DA %, is because it has a slight overestimation in the predicted values of dn depending on dt for the P. greggii plantations.

The Shapiro-Wilk test showed normal errors, as it revealed a value of W = 0.9977 (p = 0. 0001) close to 1, while analyzing the dispersion of the residuals showed no heteroscedasticity problems, as when the model was fit, it was corrected with the gmm of SAS procedure (^{SAS, 2015}).

Table 5 shows the confidence intervals (CI) of the parameters to be used to estimate the width of applicability of the model estimates within the range of the information used and verify the values that can be expected in the parameters, as stated by ^{Cepeda et al. (2008)}.

Table 5 Confidence Intervals of the parameters in model (2) in Pinus greggii Engelm. plantations in Metztitlán, Hidalgo. 

In Figure 1, the line of the tendency of the estimated values with model (2) and the observed data can be appreciated. The trend estimates to a straight line (^{Diéguez et al., 2003}; ^{Benítez et al., 2004}) and the values of statistical goodness of fit of the model, in addition to the results of the accuracy test of model (2) make reliable its application within the range of the data used to predict dn, through the dt of P. greggii trees from plantations for restoration in Metztitlán, Hidalgo. However, as recorded by ^{Benítez et al. (2004)}, the results obtained, and according to the function trend as well as of the geometry of the model, this equation may be applicable as a reference outside the range of the data used in the fit, without being specific to the sample used.

Figure 1 Comparative graphic of the estimations of dn depending on dt with model (2) against those observed for trees from Pinus greggii Engelm. plantations in Metztitlán, Hidalgo. 

When analyzing the results by diametric class and determine the ratio factor (fp) between dt and dn, the average fp with the model estimates is 0.6822, that is a conservative value compared to that obtained for the observed data, which was 0.7879. The differences between the use of the model (2) and the proportionality factors when estimating the dn by the dimensions of the stump were less than 1.4 cm of the total sample, a situation that makes reliable estimates (Table 6).

Table 6 Estimation of dn, ratio factors (dt/dn) and differences between the methods to estimate dn by means of the dimensions of dt for trees from Pinus greggii Engelm. plantations in Metztitlán, Hidalgo. 

The use of model (2) for the prediction of dn from dt includes stump height as the independent variable, a situation recommend by ^{Diéguez et al. (2003)} to estimate the normal diameter and volume of the trunk based on the size of the stump for six commercial forest species in Galicia, Spain, because the bole distorts the circular shape as height increases. ^{Corral et al. (2007)} suggest to consider this variable in the development of models when the stump height is known, when used to estimate the diameter and volume in the main species of pine in Durango, Mexico.

To illustrate the application of this model, in a practical manner, the assumption of illegal logging on a plantation with homogeneous characteristics and even-aged is proposed in an area of 0.5 hectares and 300 felled trees, in which, according to the height of the surrounding trees, is classified in a site index (SI) of 16 m (Hernández et al., 2014) and the average of the diameters of the residual stumps is 25 cm, with a stump height around 0.31 m.

Therefore, by applying the proposed equation (2), there is:

The estimated average normal diameter is 18.89 cm. With this value of estimated dn and when applying the total volume equation for P. greggii proposed by Muñoz et al. (2012):

The average total volume would be 0.2356 m³ per tree, which would mean an approximate illegally extracted volume of 70.6776 m³. With this result, combined with a wood products distribution, economic assessments for the extracted amount of timber can be accomplished.

It is worth mentioning that the ideal procedure would be to apply the model to each individual, to estimate dn, to consider the average height and to apply the equations of the total volume, all of which is done so that finally all the results are summed and a more accurate estimation is achieved.

Conclusions

The fit of the model (2) shows a linear trend between stump diameter and normal diameter, a good statistical fit, a reliable prediction of the dependent variable and low bias in the estimates. The inclusion of stump height (ht) as an independent variable in the estimation models of dn through dt, significantly improves the predictions achieved.

Due to the different felling techniques applied in Mexico and the rugged terrain in the harvesting areas of development and cutting fronts, the measurement and inclusion of stump height in works of timber forest inventory is justified.

With the reliable estimate of the normal diameter from the diameter of the stump, it is possible to quantify and a good assessment for cuttings or illegal logging, natural disasters, reconstruction of forest structure before an intervention or evaluation of harvesting practices implemented when there is a loss of information. In addition, it can help to make estimates of the total height, volume, biomass and carbon in a more accurate way.

Conflict of interests

The authors declare no conflict of interest.

Contribution by author

Jonathan Hernandez Ramos: support research, analysis of field data, writing, structure and presentation of the results within the document; Xavier García Cuevas: analysis of field data, statistical analysis and evaluation of fit of the tested models, and document review; José Jesús García Magaña: planning, design and field work supervisor, analysis of the application of results and review of the document; Hipólito Jesús Flores Muñoz: planning, design and supervision of fieldwork, analysis of the application of the results and document review; Juan Cosme Velarde Ramírez: assessment of the applicability of the information obtained in the field, evaluation and support in the fit of equations; Edgar Hugo Olvera Delgadillo: research design, field information survey, analysis of the application of the results and document review.