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The Stand Structure Generator -further explorations into the joy of copula
Dr. John A. Kershaw, Jr., CF, RPFProfessor of Forest Mensuration
UNB, Faculty of Forestry
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Genest, C. and MacKay, J. (1987). The Joy of Copulas: The Bivariate
Distributions with Uniform Marginals. American Statistician, 40, 280-283.
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Copula
• [kop-yuh-luh]• something that connects or links together
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Sklar's theorem
• Given a joint distribution function H for p variables, and respective marginal distribution functions, there exists a copula C such that the copula binds the margins to give the joint distribution.
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Gaussian Copula
• H(x,y) is a joint distribution• F(x) is the marginal distribution of x• G(y) is the marginal distribution of y• H(x,y) = Cx,y,p[Φ-1(x),Φ-1(y)]• Φ is the cumulative Normal distribution• p is the correlation between x and y• So dependence is specified in the same manner as
with a multivariate Normal, but, like all copulae, F() and G() can be any marginal distribution
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Simulating Spatially Correlated Stand Structures
• Start with spatial point process• Mixed Weibull distributions for dbh and height• Correlate dbh, ht, and spatial point process via
the Gaussian copula
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The Stand Structure Generator
• Built in R using tcl/tk interface– Spatial Model– Species Model– Correlation Model– Copula Generator– Visualization
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Structure Generator
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Spatial Model
Lattice Process Thomas Process
Easy to add additional point process models
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Spatial Model
• Specify the region, spatial model, and density• Generates the point process• Calculates Voronoi polygons and associated
polygon areas• Empirical polygon area distribution is
standardized to a mean of 0 and variance of 1• Std(0,1) is used as Normal marginal for Area
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Spatial Process
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Species Distributions
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Species Distributions
• Composition• Mixed Weibull distributions– 2-Parameter, left truncated for DBH– 3-Parameter, reversed for Height
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Correlations
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Generation Process
• Spatial Pattern -> Voronoi Area -> Std(0,1) area A(0,1)• Randomly sample N(0,1) for DBH D(0,1)• Randomly sample N(0,1) for Height H(0,1)• Correlate [A(0,1) D(0,1) H(0,1)] using the correlation
matrix– Choleski’s decomposition
• Strip off Normal marginals by applying Inverse Normal: Pr[A(0,1) D(0,1) H(0,1)]
• Apply DBH and Height marginal distributions
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Correlated Pr[A(0,1) D(0,1) H(0,1)]
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Correlated Spatial Data
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Visualization
• Runs SVS from R to visualize the stand structure
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Mark Correlations(Observed vs Simulations)
Distance (r)
Rho_
f(r)