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The Stand Structure Generator - further explorations into the joy of copula Dr. John A. Kershaw, Jr., CF, RPF Professor 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) = C x,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 ProcessThomas 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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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)

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