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Height-diameter-age equation systems for Pinus arizonica Engelmann and Pinus durangensis Martinez in mixed-species stands in Durango, Mexico

Abstract Introduction: Total height (H) and diameter at breast height (DBH) are important variables in forest inventory and they are the basis for growth and yield systems. Objective: To generate three prediction and projection equation systems for Pinus arizonica Engelmann (Pa) and Pinus durangensis Martinez (Pd) in mixed-species stands in Durango, Mexico. Materials and methods: The outside-bark DBH equations as functions of the inside-bark DBH were developed and the H-DBH relationship was extended to three relationships with the use of age (A): H-DBH, H-A and DBH-A. The equation systems of H-DBH-A were developed from a database of 46 and 66 stem analysis trees with 601 and 760 longitudinal measurements of Pa and Pd, respectively. The equations were fitted with seemingly unrelated regression and Dummy variables approach with common and specific parameters. Results and discussion: The relationships showed significant accuracy in the assessed fitting statistics (adjusted coefficient of determination, root mean square error, Akaike’s information criterion, standard error of the estimate and bias). The inverse equations of the three relationships formed a global system of prediction and projection equations. Conclusions: The equations are useful for predicting and projecting H and DBH and they can be used as input variables in growth and yield models.

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Compatible Taper and Stem Volume Equations for Pine Species in Mixed-Species Forests

Ten systems of compatible taper and outside-bark volume equations derived from upper-height-based volume ratio equations were compared with a used segmented-stem system of compatible taper and volume equations. All the systems were simultaneously fitted to cumulative volume data and taper data for Arizona pine, Aztec pine, Durango pine, Mexican white pine, and Smooth-leaved pine species in naturally regenerated mixed-species forests in Mexico. The systems derived from volume ratio equations performed better than the segmented-stem system for most species. The resultant top systems might be easier to implement than the segmented-stem system for predicting upper-stem height, upper-stem diameter, merchantable volume, and total stem volume for considered species.

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Compatible taper, volume, green weight, biomass and carbon concentration system for Quercus sideroxyla Bonpl./Sistema compatible de ahusamiento, volumen, peso verde, biomasa y concentración de carbono para Quercus sideroxyla Bonpl.

Abstract
Introduction: Estimation of total and merchantable tree volume, as well as of biomass and carbon, implies the generation of biometric tools essential in forest management and planning.
Objectives: To fit a compatible taper, volume, green weight, dry biomass and carbon concentration system for Quercus sideroxyla Bonpl. species using wood density.
Materials and methods: A database of 522 diameter-height measurements, obtained from 37 trees, was used in the fitting equations. The compatible system (CS) was integrated by 34 equations, which were simultaneously fitted by generalized nonlinear least squares. Taper and volume were the base variables for estimating green weight, dry biomass and carbon concentration.
Results and discussion: All equations were compatible with the stem volume equation, and the merchantable equations with the taper and merchantable volume equations. The fit statistics showed the efficiency of the equations in global terms and by relative height classes.
Conclusions: The CS has the property of estimating taper, merchantable volume, green weight, dry biomass and carbon concentration at upper-height and by components (stem, total tree and branches).
Resumen
Introducción: La estimación de volumen total y comercial de árboles, así como la de biomasa y carbono, implica la generación de herramientas biométricas esenciales en el manejo y planeación forestal.
Objetivos: Ajustar un sistema compatible (SC) de ahusamiento, volumen, peso verde, biomasa seca y concentración de carbono para la especie Quercus sideroxyla Bonpl., con el uso de la densidad de la madera.
Materiales y métodos: Una base de datos de 522 pares de diámetro-altura, obtenida de 37 árboles, se utilizó en el ajuste. El SC se conformó de 34 ecuaciones ajustadas simultáneamente por mínimos cuadrados generalizados no lineales. El ahusamiento y volumen fueron las variables base para la estimación del peso verde, biomasa seca y concentración de carbono.
Resultados y discusión: Todas las ecuaciones fueron compatibles con la ecuación de volumen de fuste, y las ecuaciones comerciales, con los parámetros del ahusamiento y volumen comercial. Los estadísticos de ajuste mostraron la eficiencia de las ecuaciones en términos globales y por clases de altura relativa.
Conclusiones: El SC posee la cualidad de estimar el ahusamiento, volumen comercial, peso verde, biomasa seca y concentración de carbono a una altura comercial y por componentes (fuste, total árbol y ramas).

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How to correct the heteroscedasticity and autocorrelation of residuals in taper and height growth models?

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Abstract

In modeling of taper functions and dominant height growth with time series data, the presence of heteroscedasticity and autocorrelation in residuals is common. Variance Functions (varFunc) and correlation structures (corStruct) were used to correct heteroscedasticity and autocorrelation; both were combined and evaluated through taper and height growth equations for Pinus teocote in Durango, Mexico. A dataset of 51 stems analysis with 768 taper observations and 634 height growth observations was used. The varFuncs applied were: 1) power function (varPower); 2) exponential function (varExp); 3) constant plus power function (varConstPower); and 4) a combination of power and exponential functions (varComb). The corStructs were: compound symmetry (corCompSymm), autoregressive of order 1 (corAR1), continuous-time autoregressive of order 1 (corCAR1), autoregressive-moving average (corARMA2-0), corARMA1-1, corARMA2-1, corARMA2-2, corARMA3-1 and corARMA3-2. To fit the equations, the generalized nonlinear least squares method was used and evaluated with a rating system through: RMSE, R^2, AIC, BIC, LogLik, VC and average bias. According to the rating system, the best combinations for taper and height growth equations were 1-9, 2-5, 3-8 and 4-6 and 1-6, 2-9, 3-7 and 4-4,respectively. In the taper equation, only the combination 2-5 was homoscedastic with independent residuals, and the selected height growth equations were homoscedastic with independent residuals; the varFunc and corStruct had influence on the trajectories of site index curves.

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