Introduction

Precise estimation of volumetric stocks and product distribution are essential for planning the application of silvicultural practices at the right intensities, and in a timely manner, and indispensable to obtain the greatest possible economic return from the investment made in the establishment, maintenance and utilization of commercial forest plantations (PFCs), or in the activities of natural forest use (^{Cancino, 1993}; ^{De la Fuente et al., 1998}, ^{Zepeda and Acosta, 2000}).

The correct use of forestry tools, precise for the management oriented to obtain the maximum yield of the products, allows the evaluation of the silvicultural activities, at any moment of its development (volume and income), and determines the optimum point of harvest (cutting cycle and turn) (^{Cancino, 1993}; ^{De la Fuente et al., 1998}; ^{Zepeda and Acosta, 2000}; ^{Tamarit et al., 2013}).

In calculating the total volume or by type of product in each surface unit, prior to harvesting, it is necessary to have an accurate tree stand system and an equation that determines the cutting pattern, with the purpose of maximizing the volume or value of the products obtained in natural forests or PFCs (^{Cancino, 1993}; ^{De la Fuente et al., 1998}; ^{Tamarit et al., 2013}). This system is a basic tool in forest inventories and in the planning of management activities (^{Corral-Rivas and Návar- Cháidez, 2009}).

From the taper functions it is feasible to describe the shape of the shaft, calculate the commercial height for any desired limit diameter, estimate the total volume, commercial volume and volume per individual log per tree; and generally, for a species or plantations. This allows calculations of volumetric stocks per unit area or stand (^{Clutter et al., 1983}; ^{Cruz-Cobos et al., 2008}).

According to their mathematical complexity, these functions are classified by the number of variables used and the number of coefficients involved (^{Quiñonez-Barraza et al., 2014}), in trigonometric, exponential and segmented polynomial-based models (^{Tamarit et al., 2013}). Simultaneously adjusting the taper with the total volume results in an estimation of the commercial volume in a compatible way, since the values of the obtained parameters integrate the shape of the tree and the volume corresponding to different sections (^{Demaerschalk, 1972}; ^{Clutter, 1980}).

From the importance of product distribution projections in management planning and economic valuation for natural forests or PFCs; in addition to the mathematical robustness of the taper and commercial volume models when adjusted simultaneously, they have been used to model in Mexico species such as P. arizonica Engelm., in Chihuahua (^{Pompa et al., 2009}); P. pseudostrobus Lindl., in Nuevo León (^{Tapia and Návar, 2011}); P. patula Schiede ex Schltdl. & Cham., in Hidalgo (^{Hernández et al., 2013}; ^{Uranga-Valencia et al., 2015}); and P. arizonica, P. ayacahuite Ehrenb. ex Schltdl., P. durangensis Martínez, P. leiophylla Schiede ex Schltdl. & Cham., P. teocote Schiede ex Schltdl. & Cham. (^{Quiñonez-Barraza et al., 2014}), P. oocarpa Schiede ex Schltdl. and P. douglasiana Martínez (^{López et al., 2015}), in Durango.

The objective was to generate a compatible commercial taper-volume (d-Vc) system for individual trees of Pinus greggii Engelm. in forest plantations of Metztitlán, Hidalgo State, Mexico. The compatible system consists of a taper model, one of total volume (Vt) and one of commercial volume (Vc).

Materials and Methods

Pinus greggii forest plantations are located in Fontezuelas ejido, Hidalgo State, between 20°29 ́ N and 98°54 ́ W at 2 000 to 2 500 masl. They are found inside the Transverse Neo-Volcanic Axis and the Carso Huasteco subprovince (^{Inegi, 1985}); with Haplic phaeozem and Eutric regosol units (^{Inegi, 1992}). Climate is semi- dry temperate (^{García, 1988}); the hydrologic region belongs to RH26 Panuco River, Moctezuma river sub-basin and Hermosillo-Fontezuelas microbasin (^{Inegi, 1985}).

Data were obtained from the trunk analysis of 24 P. greggii trees strategically distributed to cover all the variability of shapes and sizes resulting from their growth, as ^{Torres and Magaña (2001)} indicate, in order to improve the application interval of the equations generated.

Data collection was carried out by destructive sampling, similar to that carried out by ^{Tamarit et al. (2013)} and ^{Hernández et al. (2013)}. It consisted in the felling and cutting of trees; the measurements were normal diameter (Dn), diameter with bark (d) and heights (Hm) at distances of 1, 1.5, 2 and 2.55 m according to the tree shape conditions and commercial products that may result; the height of the stump (Ht) was started up to the total height (H). In the case of individual logs, the Newton formula and cone tip were used, while the total volume per individual was estimated using the overlapping log method proposed by ^{Bailey (1995)}.

The database was audited, corrected and refined using graphs of the taper variables and accumulated volume. The behavior of the information and the logical tendency were observed, as was done by ^{Pompa et al. (2009)}, ^{Tamarit et al. (2013)} and ^{Hernández et al. (2013)}. Three compatible commercial taper-volume systems selected from the literature (^{Fang and Bailey, 1999}; ^{Lenhart and Clutter, 1971}) were tested and evaluated to describe the stem profile in P. greggii trees; and are shown below:

1)^{Fang and Bailey (1999)} case 1-a

2)^{Fang and Bailey’s general (1999)}

3)^{Lenhart and Clutter (1971)}

These compatible systems of taper-commercial volume satisfy the condition of h = H when d = 0 and may be used for prediction diameter at any tree height; where h = H-Hm and k = π/40000 when the diameter is in meters.

Fit was made as recommended by ^{Fang and Bailey (1999)} and ^{Fang et al. (2000)}, with Schumacher-Hall’s volume model of total n to improve the estimation of the parameters, to obtain in a faster way the convergence of the system and to the convergence of the system and to make more accurate the significance of the estimators. To avoid the convergence problems in the statistical fit in the estimation of the parameters, a numerical value of delta = 0.001 was used, which, along with an indicator variable in the tip of the tre for h = H when d = 0 avoided that the model fitted to zero and a data was lost; therefore, the ability of fit near the total height or zero diameter.

Compatible systems were fitted using the MODEL procedure and the maximum likelihood method (FIML) in the SAS / ETS^® statistical package (^{SAS, 2008}); A procedure that allows the estimation of the parameters simultaneously, and shows the normality of the frequency of the residues, minimizes the variance and assumes a mean equal to zero (^{Gujarati, 2004}), based on a normal distribution and independence in errors (^{Bruce et al., 2003}).

The evaluation and selection of the best compatible system was carried out with commonly used goodness of fit statisticals, such as the adjusted determination coefficient(R²_aj) and the root mean square error (RCME) (^{Prodan et al., 1997}; ^{Corral-Rivas et al., 2007}).

Once the best model has been selected, the homoscedasticity of the residues was graphically verified. The problems of heteroscedasticity were corrected with a function that weights the variance of the residuals (^{Crecente et al., 2009}), by means of an exponential function based on the combined variable (Residual/((dn²H)^φ)^0.5) (according to the methodology suggested by ^{Harvey (1976)}, in which the value of the parameter φ comes from the regression line of the natural logarithm of the residuals as a function of the combined variable (Dn²H).

Autocorrelation of the errors was measured with the Durbin-Watson (DW) test, as suggested by ^{Augusto et al. (2009)}; ^{Pompa-García et al. (2009)} and ^{Hernández et al. (2013)}; and to correct the problem of autocorrelation in the models that required it, a self-regressive model was included in continuous time CAR (X) (^{Zimmerman and Nuñez-Antón, 2001}), applying delays (pxr) in both taper and volume with the purpose of obtaining values of the Durbin-Watson (DW) statistic near 2 (^{Barrio et al., 2014}), in one of the two variables. In addition, the observed data against the predicted taper and commercial volume were plotted to verify the percentage bias of the sample, with respect to the estimates.

Results and Discussion

The parameters in all compatible systems were highly significant with a reliability level greater than 95 % and approximate standard error (Eea) errors. Situation similar to that obtained by ^{Hernández et al. (2013)} when using the same systems for P. patula in Zacualtipán, Hidalgo (Table 1).

Table 1 Values of the parameters in the commercial taper-volume compatible systems used for Pinus greggii Engelm. in Metztitlán, Hidalgo. 

The explicit model of Schumacher-Hall’s total volume in the three compatible systems had very similar values in the parameters, only with differences attributable to the simultaneous fit of the models. When comparing the total calculated volume for the sample and the estimation of the total volume through the a ₀ , a ₁ and a ₂ parameters of the Schumacher-Hall equation, ^{Fang and Bailey’s (1999)} general system brings more accurate values, since it substituted only 0.02 % of the total volume, in the total of the sample, while ^{Lenhart and Clutter’s system (1971)}did in 0.87 % and that of ^{Fang and Bailey (1999)} case 1-a overestimates it in 0.16 %. In the three compatible system, the average bias in the estimation of the total volume in regard to the real volume is under 0.0022 m³ per individual.

Based upon the criteria of goodness of fit it can be observed that the general model of ^{Fang and Bailey (1999)} is statistically more stable for the used data, in description of the stem as in the total and commercial value (Table 2). This model explains in a better way the total variability of both variables, since the fitted coefficients of determination, in both cases, are higher than the other systems, as well as they show the lowest values of SCE and RCME. Those values are similar to those of ^{Pompa et al. (2009)} when fitting the compatible systems to P. arizonica in southwest Chihuahua.

SCE = Sum of error squares; RCME = Root mean square error; R²_aj = adjusted determination coefficient; DW = Durbin-Watson test value

Table 2 Statistics of the goodness of fit of the three taper-commercial volume systems used for Pinus greggii Engelm. in Metztitlán, Hidalgo State. 

When checking the regression assumptions of the ^{Fang and Bailey’s (1999)} general model which proved to be statistically the best, the value of the Shapiro-Wilk (SW) test indicates normality of the residue frequency (SW> 0.93); however, commercial volume residuals were distributed heteroscedastically and the value of the Durbin-Watson (DW) autocorrelation test was 0.43 for the taper and 0.60 for the commercial volume; in both cases less than 1.5, which is a violation of these regression assumptions.

Once the problems of heteroscedasticity and autocorrelation were corrected, the general model of ^{Fang and Bailey (1999)} was again adjusted, thus a considerable statistical improvement was obtained (Table 3). The value of the Shapiro-Wilk normality test was 0.98 (α = <0.0001) and the residuals behaved homoscedastically, while, the DW test to detect autocorrelation was 1.95 for taper and 1.16 for commercial volume.

SCE = Sum of error squares; RCME = Root mean square error; R²_aj = adjusted determination coefficient; p1r and p2r = Corresponding parameters corresponding to the autoregresive model of continuous time (CAR(2)) applied to correct the autocorrelation of the taper-commercial volume compatible system.

Table 3 Statistics of the goodness of fit and values of the parameters of ^{Fang and Bailey’s (1999)} general compatible system for Pinus greggii Engelm. in Metztitlán, Hidalgo State. 

When plotting the estimates made with the compatible system, an appropriate trend of taper and commercial volume data (Figure 1a and b), similar to that reported by ^{Rodríguez and Broto (2003)}, is presented when analyzing various models of the shaft profile and to do cylindrical studies of the trees of Populus x euroamericana (Dode) Guinier in Navarra, Spain; as well as to that cited by ^{Fassola et al., (2007)} when adjusting variable exponent profile functions and subsequent adjustment of compatible volume models for Eucalyptus grandis W. Hill. ex Maiden in Mesopotamia, Argentina; and by ^{Lara (2011)}, who applied taper functions to later derive the compatible commercial volume model for Tectona grandis L. in the coastal zone of Ecuador.

Figure 1 Estimated diameter and accumulated commercial volume in regard to height over stem (b) for Pinus greggii Engelm. in Metztitlán, Hidalgo State. 

The predicted values with ^{Fang and Bailey’s (1999)} general in regard to the observed date show a taper straight line trend (^{Fassola et al., 2007}; ^{Lara, 2011}) and the commercial value (^{Tamarit et al., 2013}) (Figure 1a and b), which is desirable in this kind of studies (Figure 2a and 2b).

Figure 2 Comparison of the observed diameter and volume against the predicted of diameter (a) and commercial volume (b) of ^{Fang and Bailey’s general (1999)} compatible system for Pinus greggii Engelm. in Metztitlán, Hidalgo State. 

The coefficient of determination between the predicted and observed values indicates that the description of the profile of the trees by the system has an aggregate difference in percent for the whole sample of 3.8 in taper and of 4.6 for commercial volume; with an average in bias by estimation of 0.017 and 0.020, respectively for each variable (Figure 2a and 2b). The biases are lower than those cited for the tapering by ^{Návar and Domínguez (1997)} when using four models of stem profile for Pinus brutia Ten., P. halepensis Mill., P. eldarica Medw. and P. estevezii (Martínez) F. P. Perry; ^{Corral et al. (1999)} when using five profile models for P. cooperi C. E. Blanco, P. durangensis, P. engelmannii Carr., P. leiophylla and P. herrerae Martínez; ^{Tapia and Návar (2011)} in P. pseudostrobus with five taper models.

The bias for commercial volume is similar to that documented by ^{Quiñonez et al. (2014)} with compatible d-Vc systems for P. arizonica, P. ayacahuite, P. durangensis, P. leiophyla and P. teocote.

In the selected compatible system, it was observed that the prediction of the diameter at different heights tends to present problems at heights less than or equal to 0.3 m (stump height), and to slightly underestimate the diameters below the first third of the tree, similar situation to that observed in Ecuador by ^{Lara (2011)} and in Mexico by ^{Tamarit et al. (2013)} in Tectona grandis. Considering that the lower parts of the tree can be measured directly more accurately, it is ideal to have tools that make these precise estimates in the upper parts to know the product distribution, the minimum diameter and the resulting volume (^{Rodríguez and Broto, 2003}; ^{Tamarit et al., 2013}).

With the results of the present study, obtained in the study can be made descriptions of the tree profile, as well as accurate estimates of the total volume and commercial volume in plantations of P. greggii, for the projection of the distribution of timber products destined to a differentiated market.

Conclusions

The ^{Fang and Bailey (1999)} compatible taper and volume system is the best estimate to describe the profile of Pinus greggii trees in the evaluated PFCs and can be used reliably to predict the diameter at any height and its respective volume; it is also compatible with the equation of total volume.

The information obtained is a valuable tool for the analysis of data from forest inventories, particularly in the prediction of the distribution of products per tree or unit area of plantations, and the economic valuation within a differentiated market of timber products of forest use.

Because the silvicultural activities applied in PFCs affect the shape of the tree and, consequently, the distribution of products, the constant updating of this information is fundamental in the planning of sustainable forest management for P. greggii