- Open Access
Local and generalized height-diameter models with random parameters for mixed, uneven-aged forests in Northwestern Durango, Mexico
© Corral-Rivas et al.; licensee Springer. 2014
- Received: 15 May 2013
- Accepted: 19 September 2013
- Published: 26 February 2014
We used mixed models with random components to develop height-diameter (h-d) functions for mixed, uneven-aged stands in northwestern Durango (Mexico), considering the breast height diameter (d) and stand variables as predictors.
The data were obtained from 44 permanent plots used to monitor stand growth under forest management in the study area.
The generalized Bertalanffy-Richards model performed better than the other generalized models in predicting the total height of the species under study. For the genera Pinus and Quercus, the models were successfully calibrated by measuring the height of a subsample of three randomly selected trees close to the mean d, whereas for species of the genera Cupressus, Arbutus and Alnus, three trees were also selected, but they are specifically the maximum, minimum and mean d trees.
The presented equations represent a new tool for the evaluation and management of natural forest in the region.
- Conifer and broadleaves forests
- h-d relationship
- Mixed models
Most forests in Durango State (Mexico) are comprised of a mixture of species of the genera Pinus and Quercus with an irregular distribution of trees of all size classes. However, species of the genera Arbutus and Juniperus are also found in most of these forests (Wehenkel et al. 2011). These forests, which cover an area of 5.4 million ha, are considered as the primary forest reserve at a national level, and they provide almost a quarter of the national forest production in Mexico (SRNyMA 2006). The forests also play an important role in providing environmental services, such as protection against soil erosion, biodiversity conservation, carbon capture and protection of water reserves; they also provide recreational areas and represent an important source of income for their owners and local inhabitants.
Forest management requires prediction tools that provide detailed information about the development of mixed, uneven-aged stands. Growth and production models are the most commonly used tools for this purpose. When the breast height diameter (d) and total height (h) are known, application of these models is relatively easy (Sharma and Parton 2007). Measuring diameter is simple, accurate and inexpensive, whereas measuring height is relatively more complex, time-consuming and expensive. Therefore, height-diameter (h-d) functions are often utilized, so that the height of an individual tree can be predicted only from the diameter. These relationships are also very useful for estimating individual volume, site index and for describing growth and production in forest stands over time when the height is not measured (Curtis 1967).
Most h-d functions have been developed for forest plantations (e.g. Soares and Tomé 2002; López Sanchez et al. 2003). However, the relationship between the diameter and height of a tree varies between stands (Calama and Montero 2004) because it depends on stand characteristics such as density and site index (Sharma and Zhang 2004). Moreover, the h-d relationship also varies over time within the same stand (Curtis 1967). Such considerations indicate that stand variables should be used to construct generalized functions that represent all possible conditions in forest stands (Temesgen and Gadow 2004). This is particularly important in mixed, uneven-aged stands in which different species, ages, structures and levels of competition coexist (Vargas-Larreta et al. 2009).
The hierarchical structure of h-d data (i.e. trees grouped in plots and plots grouped in stands) results in a lack of independence between measurements because the observations in each sampling unit will be correlated (Gregoire 1987). Mixed models have been successfully used to address this type of problem (e.g. Lappi 1997; Calama and Montero 2004; Castedo Dorado et al. 2006). This approach simultaneously estimates fixed parameters (parameters that are common to the entire population) and random parameters (parameters that are specific to each plot) within the same model and enables the variability between plots of the same population to be modelled.
The objectives of this study were as follows: i) to compare different local h-d equations for the mixed, uneven-aged forests in north-western Durango; ii) to develop new generalized h-d equations for different groups of species based on the best local model previously fitted iii) to use the local and the generalized equations to study the capacity of mixed models to explain the variability in the h-d relationship; and iv) to determine the most suitable size and type of sample for calibrating the functions fitted with mixed models.
The study was carried out in the Ejido San Diego de Tezains, Municipality of Santiago Papasquiaro, Durango State, Mexico (between 105° 53′ 36″ and 106° 12′ 40″ W and 24° 48′ 16″ and 25° 13′ 32″ N). The predominant vegetation in the area is mixed, uneven-aged forests of Pinus and Quercus. The altitude above sea level of the study area varies between 1,400 and 3,000 m. The prevailing climate is temperate: the annual precipitation ranges between 800 and 1,100 mm and the mean annual temperature varies between 8°C in the highest elevations and 24°C in the lowest elevations (García 1981).
The data were obtained from 44 permanent plots used to monitor the growth and production of the forests in the Ejido San Diego de Tezains. These plots, which were established in 2008, were selected with the aim of representing all types of vegetation, site qualities and diameter distributions in managed stands. The plots, of size 50x50 m, are distributed under a systematic grid sampling approach that varies between 3 and 5 km, and will be remeasured at 5 year intervals. We recorded the following main variables: number of trees, species code, breast height diameter at 1.3 m (d, cm), total tree height (h, m), azimuth (°) and radius (m) from the centre of the plot (point where the diagonals cross) towards all trees of breast height diameter ≥5 cm.
The database included 25 species, which were classified on the basis of their growth patterns into the following 13 groups for posterior analysis: 1 (Pinus arizonica), 2 (P. ayacahuite), 3 (P. durangensis), 4 (P. herrerae), 5 (P. lumholtzii), 6 (P. teocote), 7 (P. douglasina), 8 (Quercus sideroxyla), 9 (other species of Quercus: Q. arizonica, Q. mcvaughii, Q. durifolia, Q. crassifolia, Q. jonesii, Q. rugosa and Q. laeta), 10 Pinus species (all species of the genus Pinus [codes 1 to 7]), 11 Quercus species (all species of the genus Quercus [codes 8 and 9]), 12 other conifer species (Juniperus deppeanna, J. durangensis and Cupresus lusitanica) and 13 other broadleaf species (Arbutus arizonica, A. bicolor, A. madrensis, A. tesselata, A. xalapensis and Alnus firmifolia).
Summary statistics of the database used in fitting the h-d equations
Number of observations
Other conifer spp.
Other broadleaf spp.
The following stand variables were calculated from the trees registered in each plot: number of trees per hectare (N, trees ha-1), stand basal area (G, m2 ha-1), mean square diameter (d g , cm), dominant height (estimated as the mean height of the 100 largest diameter trees per hectare, independently of the species [H0, m]), dominant diameter (estimated as the mean diameter of the 100 largest diameter trees per hectare, independently of the species [D0, cm]) and Hart’s index (%) estimated as follows: .
Comparison of equations
We selected a total of 27 local equations (Huang et al. 2000) for data fitting. We also studied the relationship between the stand variables and the parameters of the local equations that best described the h-d relationship, with the aim of improving the accuracy of the equation and developing new generalized functions.
where h i , and are the observed and estimated heights and the mean of the observed heights, respectively; n is the number of observations used in the fitting; k is the number of parameters in the equation, and ln is the natural logarithm.
As each local equation has different strengths and weaknesses, which may lead to different goodness-of-fit results for each group of species, we used a Qualification Index (QI t ) to evaluate the goodness of fit by considering the values of R2 (with high values representing good fits), Bias (with low absolute values representing good fits) and RMSE and BIC (with low values representing good fits). For this index, a value of 1 is assigned to the equation that was best for each group of species and a value of 0 to the others. The qualifications obtained for each equation and statistics were then summed as follows: ; where QIij is the qualification for the j-th goodness of fit criterion in the i-th group of species.
For the local equation for which the QI total was highest for the defined groups, we used graphical analysis and the CORR procedure in SAS (SAS Institute Inc 2008) to analyse the relation between each of the parameters and the main stand variables, with the aim of testing different forms of generalized equations.
Effect of mixed models
The h-d observations made in plots and stands may be highly correlated, thus violating the principle of independence of error terms (Calama and Montero 2004). One procedure used to deal with correlated observations is to fit mixed models, in which the variability between the sampling units can be explained by including random parameters, which are estimated at the same time as the fixed parameters (Lappi 1997; Calama and Montero 2004).
where Φ j is the parameter vector r × 1 (where r is the total number of parameters in the model) specified for the j-th plot, λ is the vector p × 1 of the common fixed parameters for the whole population (p is the number of fixed parameters in the model), b j is the vector q × 1 of the random parameters associated with the j- th plot (q is the number of random parameters in the model), A j and B j are matrices of size r × p and r × q for specific and random effects for the j-th plot, respectively.
The basic theory of non-linear mixed models says that the residual vector () and the random effects vector (b j ) are often assumed to be uncorrelated and normally distributed with mean zero and variance-covariance matrices R j and D, respectively. The residual vector represents within subject (e.g., plot) variability and the random effects vector represents between subject variability (Littell et al. 1996).
We constructed the non-linear mixed effects model by selecting the local and generalized equations that yielded the best fits for the species groups defined using the NLMIXED procedure in SAS/ETS (SAS Institute Inc 2008). We tested different combinations of fixed and random parameters and compared the fitting statistics (RMSE, R2, Bias and BIC), to determine which parameter(s) should be considered mixed.
where is the matrix q × q of variances-covariances associated with the random parameters (q = number of random parameters included in the model), which is common to all plots and is estimated in the general model fitting procedure; is the m j × m j estimated matrix of variances-covariances of the error term; is the residuals vector m × 1, the components of which are obtained as the difference between the observed height of each tree and the value predicted using the model with fixed parameters only; and is the matrix m × q of the partial derivatives of the random parameters evaluated in .
CR1: Measuring the total height of between 1 and 5 randomly selected trees within each plot that are close (± 10% ) to the mean breast height diameter.
CR2: Measuring the total height of the tree of mean breast height diameter, or measuring the height of two trees – the mean and minimum breast height diameters, or measuring the height of three trees – the mean, minimum and maximum breast height diameters within each plot.
We evaluated these two alternatives in terms of the previously defined goodness-of-fit statistics (RMSE, R 2 and Bias), which we compared with the statistics obtained for the equations fitted by the ONLS and NLMIXED procedures.
Qualification index for the 8 best local equations for the 13 groups of species
The Bertalanffy-Richards equation yielded the highest R2 values for 6 of the 13 species groupings and the lowest RMSE values for 4 of the groups. Finally, comparison of the BIC values indicated that this was the preferred equation only for the Quercus spp. grouping.
where b 0 -b 2 are equation parameters and the rest of variables as defined in the data section.
Considering that the fitting statistics for the different broad groupings (Pinus species, Quercus species, other conifer species and other broadleaf species) are similar to those obtained for each individual species and that some parameters were not significant in the individual fits for some of the species, we decided to use 4 different local equations, one for each broad grouping studied.
Estimated parameters and fitting statistics obtained for the local model with and without mixed effects for the groups of species considered
σ 2 u
σ 2 v
Other conifer species
Other broadleaf species
On relating the parameters in Eq. (7) to the stand variables, we found that in the 4 groups that included all species, parameter b 0 , representative of the asymptote of Eq. (7), was positively correlated (almost 52%) with H0and D0, whereas parameter b 1 , representative of “scale” was only positively correlated by more than 50% with Hart’s index (HI) in the other conifer species and other broadleaf species and, finally, parameter b 2 , representative of “shape” was also positively correlated (~ 51%) with d g and N for the four groups that include all species.
where b 0 -b 2 are equation parameters and the rest of variables as defined in the data section.
Estimated parameters and fitting statistics obtained for the generalized model with and without mixed effects for the groups of species considered
σ 2 u
Other conifer species
Other broadleaf species
The generalized h–d equations selected in this study included dominant stand height. This represents an advantage over equations that include the mean height because less effort is required in conventional inventories to estimate the dominant height than the mean height of the stand (López Sanchez et al. 2003). These functions also include the density of the stand in terms of number of trees per unit of area and mean square diameter. Stand density is the most obvious factor affecting the h-d relationship in a stand (Zeide and VanderSchaaf 2002); in other words, trees of the same diameter are generally taller in denser stands.
Various stand variables have been proposed as predictors of the h-d relationship: stand age (Curtis 1967; Soares and Tomé 2002; López Sanchez et al. 2003); crown competition index (Temesgen et al. 2007); geographic variables (Schmidt et al. 2011); and wind speed (Meng et al. 2008). Although the inclusion of other variables may improve the predictive capacity of the selected functions, this requires great sampling effort and limits the practical application of the functions and therefore we did not take such variables into account.
Effect of mixed models
where b 0 – b 4 are the fixed parameters of the model (common to all plots); (u j , v j ) ~ N(0, τ) are the random parameters (specific to each plot); and and e ij are respectively the height and error estimated by the model for the i-th observation (tree) in the j-th plot.
The values of the parameters and goodness-of-fit statistics for the local mixed models (Eqs 10 and 11) and for the generalized mixed models (Eqs 12 to 15) are shown in Tables 3 and 4, respectively. We compared the RMSE values obtained with the mixed effects equations with those obtained with fixed effects equations (fitted by ONLS); the values obtained with the local mixed model (Eq. 10) and the generalized mixed model (Eq. 12) for the Pinus grouping were 25.0% and 5.2% lower than those obtained with the local model (Eq. 7) and the generalized model (Eq. 8) without random parameters, respectively. For the group of Quercus species, the RMSE values obtained with the local mixed model (Eq. 11) and the generalized mixed model (Eq. 13) were 26.0% and 9.0% lower than those obtained with the local (Eq. 7) and the generalized models without random parameters (Eq. 8), respectively. For the other conifers, the RMSE values obtained with the local mixed model (Eq. 11) and the generalized mixed model (Eq. 14), were 9.8% and 14.3% lower than those obtained with Eqs (7) and (9), respectively. For the group comprising other broadleaf species, the RMSE values were 20.9% and 25.0% lower with the local mixed model (Eq. 11) and the generalized mixed model (Eq. 15) than with Eqs (7) and (9). The results obtained for BIC and R2 were similar to those obtained for RMSE.
On inspecting the graphs of the residuals for the heights estimated by the models for each species grouping, we did not find any anomalies that would suggest non compliance of underlying hypothesis of independence of errors or homogeneity of variance. The magnitude of the bias in the residual values estimated by the two fitting methods (ONLS and NLMIXED) was consistent for all ranges and classes of heights observed by the defined species groupings.
The variation in the value of RMSE with respect to the number of trees used with the two calibration options for the four main groups of species studied is shown in Figure 2. This statistic was also compared with those values obtained when fitting the equations by the NLMIXED (minimum value of RMSE reached only using all trees as a calibration subsample) and ONLS (maximum value of RMSE using only fixed parameters) methods.
In the calibration process, the reduction in the RMSE value was particularly evident with the generalized mixed models for the other broadleaf species and other conifer species (21.0% and 13.0% respectively) compared with the generalized model fitted without random parameters; however, for the Pinus and Quercus groupings, the decrease in the value of this statistic was lower. Both calibration options resulted in an important reduction of RMSE for the local mixed model compared to the same model fitted without random parameters for all the groups analized. In accordance with Trincado et al. (2007), the use of a local mixed model in forest inventories with a subsample of trees to calibrate and then predict the total height of all trees not used in calibration allows retention of a simple model structure (i.e. without the need to include stand predictor variables) and may be an useful alternative to generalized mixed models when there is a lack of data to calculate stand variables.
Two generalized equations (Eqs 8 and 9) were derived from a local equation (Eq. 7) and used to estimate total tree height from breast height diameter and stand variables for the 25 species identified in the sample by using mixed models. The variability between plots is explained in terms of the random effect of each plot and from the stand variables included in the generalized models.
For species in the Pinus and Quercus groups, inclusion of the height measurements of 3 trees close (± 10%) of the mean breast height diameter from each plot improved the predictive capacity of the calibrated model. For the species included in other broadleaf species and other conifer species, the predictive capacity of the model was improved by including the total height measured in a subsample of 3 trees of minimum, mean and maximum breast height diameter. The possibility of using complementary data from the stands to calibrate the mixed models provides a clear advantage over models developed by other procedures, which require large amounts of data or are less accurate.
The present investigation was financially supported by the “Programa de Mejoramiento del Profesorado” (project: Seguimiento y Evaluación de Sitios Permanentes de Investigación Forestal y el Impacto Socioeconómico del Manejo Forestal en Norte de México). The study was conducted during the doctoral studies of the first author at the Universidad de Santiago de Compostela USC (Spain), supported by “Programa Banco Santander – USC” (becas para estancias predoctorales destinadas a docentes e investigadores de América Latina).
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