Generalized Algebraic Difference Approach Research Articles

New dynamic site equation that fits best the Schwappach data for Scots pine ( Pinus sylvestris L.) in Central Europe

Using historical growth series data of Scots pine ( Pinus sylvestris L.) in Central Europe we examine all the dynamic site equations previously used for modeling the height growth of this species as well as a new dynamic site equation that has not been used previously in the context of this forestry data. The tested models included two groups of anamorphic and polymorphic dynamic site equations (three-dimensional site–height–age models, such as Y = f( t, t 0, y 0)). One group of the models is based on the algebraic difference approach (ADA) implementation of different, preexisting base equations (two-dimensional equations, such as Y = f( t)). The other group of models is based on newer generalized algebraic difference approach (GADA) formulations of new site–height–age relationships that may use older models only as a part of their structure. The models were selected because they were relevant to Scots pine height growth modeling in other studies. We compared all the models with each other in terms of the sum of square deviations associated with fitting them simultaneously to all sites represented by the Scots pine data. All the fits were based on base-age invariant stochastic regressions, in which the global model parameters that are common to all growth series are estimated simultaneously with the site-specific effects that are different for each of the site productivity series. Cieszewski's model [Cieszewski, C.J., 2005. A new flexible GADA based dynamic site equation with polymorphism and variable asymptotes. PMRC Technical Report 2005-2] best described the data.

Modelling stand basal area growth for radiata pine plantations in Northwestern Spain using the GADA

A stand basal area growth system for radiata pine (Pinus radiata D. Don) plantations in Galicia (Northwestern Spain) was developed from data corresponding to 247 plots measured between one and five times. Six dynamic equations were considered for analysis and both numerical and graphical methods were used to compare alternative models. The equation that best described the data was a dynamic equation derived from the Korf growth function by the generalized algebraic difference approach (GADA) and by considering two parameters as site-specific. This equation was fitted in one stage by the base-age-invariant dummy variables method. The system also incorporated an equation for predicting initial stand basal area, expressed as a function of stand age, site index, and the number of trees per hectare. This information can be used to establish the starting point for the projection equation when no inventory data are available. The effect of thinning on stand basal area growth was also analyzed and the results showed that the same projection equation can be used to obtain reliable predictions of unit-area basal area development in thinned and unthinned stands.

A growth model for Pinus radiata D. Don stands in north-western Spain

A dynamic whole-stand growth model for radiata pine (Pinus radiata D. Don) stands in north-western Spain is presented. In this model, the initial stand conditions at any point in time are defined by three state variables (number of trees per hectare, stand basal area and dominant height), and are used to estimate total or merchantable stand volume for a given projection age. The model uses three transition functions derived with the generalized algebraic difference approach (GADA) to project the corresponding stand state variables at any particular time. These equations were fitted using the base-age-invariant dummy variables method. In addition, the model incorporates a function for predicting initial stand basal area, which can be used to establish the starting point for the simulation. Once the state variables are known for a specific moment, a distribution function is used to estimate the number of trees in each diameter class by recovering the parameters of the Weibull function, using the moments of first and second order of the distribution. By using a generalized height-diameter function to estimate the height of the average tree in each diameter class, combined with a taper function that uses the above predicted diameter and height, it is then possible to estimate total or merchantable stand volume.

Development of a basal area growth system for maritime pine in northwestern Spain using the generalized algebraic difference approach

A basal area growth system for single-species, even-aged maritime pine (Pinus pinaster Ait.) stands in Galicia (northwestern Spain) was developed from data of 212 plots measured between one and four times. Six dynamic equations were considered for analysis, and both numerical and graphical methods were used to compare alternative models. The double cross-validation approach was used to assess the predictive ability of the models. The data were best described by a dynamic equation derived from the Korf growth function using the generalized algebraic difference approach (GADA) by considering two parameters to be site specific. The equation was fitted in one stage using the base-age-invariant dummy variables method. In addition, the system incorporates a function for predicting initial stand basal area, in which the site-related variable was expressed as a power function of site index. This function can be used to establish the starting point for the projection equation when no inventory data are available. The two equations are compatible. The effect of thinning on basal area growth was examined; the results showed that there was no need to use a different equation to reliably predict postthinning basal area development. The nonlinear extra sum of squares method indicated differences in the model parameters for the two ecoregions (coastal and interior) defined for this species in the area of study.

Dynamic Site Model for Loblolly Pine (Pinus taeda L.) Plantations in the United States

Abstract Data from a large, long-term loblolly pine (Pinus taeda L.) plantation study including 186 permanent sample plots were used to evaluate four dynamic site equations derived with the generalized algebraic difference approach (GADA). All of the equations are base-age invariant and directly estimate height and site index from any height and age. The fittings were done in one stage using the base–age invariant dummy variables method. A first-order autoregressive error structure was used to correct the serial correlation of the longitudinal data used in this study. Cieszewski’s model (Cieszewski, C.J. 2002. Comparing fixed- and variable-base–age site equations having single versus multiple asymptotes. For. Sci. 48:7–23) best described the data.

Site quality curves for birch stands in north-western Spain

A model for predicting the height growth of even-aged, birch (Betula pubescens Ehrh.) dominated stands in Galicia (north-western Spain) was developed. Data from stem analysis of 214 trees were used for model construction. Two dynamic site equations derived with the generalized algebraic difference approach (GADA) were tested, which combine compatible site index and height models in one common equation. Both equations are base-age invariant and directly estimate height and site index from any height and age. The fittings were done in one stage using the base-age-invariant dummy variables method. A second-order continuous-time autoregressive error structure was used to correct the inherent autocorrelation of the longitudinal data used in this study. Cieszewski’s model best described the data. This model is therefore recommended for height growth prediction and site classification of birch stands in Galicia.

Modeling dominant height growth of radiata pine ( Pinus radiata D. Don) plantations in north-western Spain

A model for predicting the height growth of radiata pine plantations in Galicia (north-western Spain) was developed. Data from stem analysis of 161 trees were used for model construction. Seven dynamic site equations derived with the generalized algebraic difference approach (GADA) were tested, which constitute compatible site index and height models in one common equation. All of the equations directly estimate height and site index from any other height and age, and are base-age invariant. To obtain unbiased parameter estimates the base-age invariant dummy variables method was used, which recognizes that each measurement is made with error and, thus, fits a curve to the observed individual trends in the data by considering one parameter to be site-specific and the remaining parameters common for all individuals. A second-order continuous-time autoregressive error structure was used to correct the inherent autocorrelation of the longitudinal data used in this study, which allows the model to be applied to irregularly spaced, unbalanced data. The GADA formulation derived on the basis of the Bertalanffy–Richards model by considering the asymptote and the initial pattern parameters as related to site productivity resulted in the best compromise between biological and statistical considerations, producing the most adequate site curves. This model showed superior behavior when compared to the model currently in use. It is, therefore, recommended for height growth prediction and site classification of radiata pine plantations in Galicia.