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

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

Structure Generator

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

Spatial Process

Species Distributions

Species Distributions

• Composition• Mixed Weibull distributions– 2-Parameter, left truncated for DBH– 3-Parameter, reversed for Height

Correlations

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

Correlated Pr[A(0,1) D(0,1) H(0,1)]

Correlated Spatial Data

Visualization

• Runs SVS from R to visualize the stand structure

Mark Correlations(Observed vs Simulations)

Distance (r)

Rho_

f(r)