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Modeling regional variation in the self-thinning boundary line . Aaron Weiskittel Sean Garber Hailemariam Temesgen. Introduction. Although self-thinning constraints may not be needed for individual tree growth models (Monserud et al. 2005; For. Sci. 50: 848), they are still important for:

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modeling regional variation in the self thinning boundary line

Modeling regional variation in the self-thinning boundary line

Aaron Weiskittel

Sean Garber

Hailemariam Temesgen

introduction
Introduction
  • Although self-thinning constraints may not be needed for individual tree growth models (Monserud et al. 2005; For. Sci. 50: 848), they are still important for:
    • Stand-level projections
    • Developing stand management diagrams
    • Understanding basic stand dynamics
introduction3
Introduction
  • Size-density relations have been quantified for a variety of species and it has been suggested that:
    • A universal slope exists (-3/2)
    • Intercept varies by species, but is not influenced by other factors
  • Previous analyses have relied on ordinary least squares (OLS) or principal components analysis (PCA) to examine trends
    • Assumptions are violated and tests of parameter significance are invalid
introduction4
Introduction
  • Zhang et al. (2005; CJFR 35: 1507) compared several different methods for estimating the self-thinning boundary line
    • OLS and PCA performed the poorest
      • sensitive to the data subjectively selected for fitting
      • may produce lines with the inappropriate slope
    • Statistical inference is difficult with quantile regression and deterministic frontier functions
    • Stochastic frontier functions (SFF) performed the best
introduction5
Introduction
  • Bi (2001; For. Sci. 47, 361) used SFF to examine the self-thinning surface in Pinusradiata
    • SFF successfully separated the effects of density-dependent and density-independent mortality
    • SFF allows statistical inferences on the model coefficients
    • Generalized model form proposed:
      • B = β0Sβ1Nβ2
      • where B is stand biomass per unit area, N is stand density, S is relative site index, and βi’sareparameters
objectives
Objectives
  • Utilize SFF to examine maximum size-density relations in coastal Douglas-fir, red alder, and lodgepole pine
    • Test the generality of Bi’s (2001) model
    • Examine the influence of other covariates
    • Compare the results to a more traditional approach
analysis
Analysis
  • Used Frontier v4.1 (Coelli 1996) and R library micEcon to fit the SFF
    • ln(TPA) = β10 - β11ln(QMD) + ε11
      • QMD is quadratic mean diameter and TPA is trees per acre
  • Compared to fits obtained using quantile regression
  • Maximum stand density index (SDImax) was estimated for each plot and regressed on other covariates similar to Hann et al. (2003)
  • Significance of covariates evaluated using log-likelihood ratio tests
stochastic frontier analysis
Stochastic frontier analysis
  • Used in econometrics to study firm efficiency and cost & profit frontiers
  • Model error has two components
    • Random symmetrical statistical noise
    • Systematic deviations from the frontier
  • Qit = exp(ß0 + ß1 ln(xit)) * exp(vit) * exp(-uit)

Deterministic component

Inefficiency

Random noise

stochastic frontier analysis10
Stochastic frontier analysis
  • Fit using maximum likelihood
  • u and v are assumed to be distributed independently of each other and the regressors
  • u represents the difference in stand density at any given point and the estimated maximum density
    • Eliminates the subjectively of choosing stands that other techniques rely on
results maximum stand density
Results: Maximum stand density
  • Plot-specific SDImax showed no relationship with any other covariates
results self thinning boundary line
Results: Self-thinning boundary line
  • Stochastic frontier analysis and quantile regression produce significantly different results
results self thinning boundary line14
Results: Self-thinning boundary line
  • Likelihood ratio tests indicated that the inclusion of site index improved the model for Douglas-fir and red alder, but not for lodgepole pine
  • The effect of fertilization in Douglas-fir was insignificant
  • Red alder was also influenced by slope and aspect as well as soil water holding capacity
conclusion
Conclusion
  • Stochastic frontier functions proved very useful for this type of analysis and provided insights that other statistical techniques obscure
  • SDImaxvalues higher in this analysis slightly different than previously published values
    • Lower for Douglas-fir, but higher for red alder and lodgepole pine
  • Douglas-fir and red alder support Bi’s general model, but lodgepole does not
    • Site index only capture some of the variation for red alder
next steps
Next Steps
  • Compare plantation to natural stands
  • Use a more extensive red alder database
  • Western Hemlock
acknowledgements
Acknowledgements
  • Thanks to SMC, SNCC, HSC, BC Ministry of Forests and their supporting members for access to the data