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Modeling regional variation in the self-thinning boundary line Aaron WeiskittelSean GarberHailemariam Temesgen
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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
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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
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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
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Introduction• Bi (2001; For. Sci. 47, 361) used SFF to examine
the self-thinning surface in Pinus radiata▫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’s are
parameters
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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
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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
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DataSpecies Data
SourceTotal Age Density (#
acre)Site index (ft)
Douglas-fir SMC, SNCC 5-65 92-1208 85.8-164(base age 50)
Red alder HSC 1-17 56-1524 75.4-114.8(base age 30)
Lodgepole pine
BC Ministry of Forests
16-146 136-3638 47.9 – 86.3(base age 50)
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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 componentRandom noise Inefficiency
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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
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Results: Maximum stand densitySpecies Mean Std. Dev. Min Max
Douglas-fir 511 215 213 989
Red alder 484 226 122 1005
Lodgepole pine
725 406 136 1997
• Plot-specific SDImax showed no relationship with any other covariates
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Results: Self-thinning boundary line
Species SFA Quantile regression
Intercept Slope Intercept Slope
Douglas-fir 9.9571(0.2246)
-0.9467(0.0708)
11.2289(0.3604)
-1.3309(0.1256)
Red alder 10.3891(0.3017)
-1.0359(0.1171)
10.6492(0.1849)
-1.1379(0.0666)
Lodgepole pine
10.0975(1.6751)
-0.8564(0.1591)
7.5188(1.5949)
-0.4664(0.5729)
•Stochastic frontier analysis and quantile regression produce significantly different results
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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
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Conclusion• Stochastic frontier functions proved very useful
for this type of analysis and provided insights that other statistical techniques obscure
• SDImax values 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
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Next Steps
•Compare plantation to natural stands
•Use a more extensive red alder database
•Western Hemlock
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Acknowledgements
•Thanks to SMC, SNCC, HSC, BC Ministry of Forests and their supporting members for access to the data