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
Genest, C. and MacKay, J. (1987). The Joy of Copulas: The Bivariate Distributions with Uniform Marginals. American Statistician, 40, 280-283.
Copula • [kop-yuh-luh] • something that connects or links together
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.
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
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
The Stand Structure Generator • Built in R using tcl/tk interface • Spatial Model • Species Model • Correlation Model • Copula Generator • Visualization
Spatial Model Lattice Process Thomas Process Easy to add additional point process models
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
Species Distributions • Composition • Mixed Weibull distributions • 2-Parameter, left truncated for DBH • 3-Parameter, reversed for Height
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
Visualization • Runs SVS from R to visualize the stand structure
Mark Correlations(Observed vs Simulations) Rho_f(r) Distance (r)