1. Three-Dimensional Crown Mass Distribution via Copulas Dr. John A. Kershaw, Jr. Professor of Forest Mensuration/Biometrics Faculty of Forestry and Env. Mgmt University of New Brunswick

2. Copula [kop-yuh-luh] something that connects or links together Cupola

3. Genest, C. and MacKay, J. (1987). The Joy of Copulas: The Bivariate Distributions with Uniform Marginals. American Statistician, 40, 280-283.

4. 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 (Inverse) 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 copulas, F() and G() can be any marginal distribution

5. HT-DBH Simulation Example

6. Western Hemlock Crown Data 42 western hemlock trees dissected standing EVERY branch measured for height on stem, azimuth, total length, green length, maximum branch width, and branch basal diameter 10% sample, stratified by height, dissected in 15 cm concentric bands and mass determined for current foliage, older foliage, current wood, and older wood

8. Of course I had a little bit of help from a Sidekick…

9. …and my “Fall Guy”

10. Crown Reconstruction Dissected branches used to build prediction system for all branches Total branch mass by component (current and older foliage, current and older wood – Kershaw and Maguire 1995 CJFR) Horizontal distribution by component (Kershaw and Maguire 1996 CJFR) Refitted to take advantage of nonlinear mixed effects models and SUR

11. Crown Reconstruction

12. Two Copula Approaches “Fitted” based on reconstructed branches “Predicted” based on tree-level moment- based parameter prediction

13. Crown Copula Requirements Vertical Marginal Distribution Horizontal Marginal Distribution Radial Marginal Distribution Correlation Matrix Separate Copula for each Component – Current and Older Foliage Mass – Current and Older Wood Mass

14. Vertical Distribution

15. Horizontal Distribution

16. Radial Distribution

17. Simulation via Normal Copula Generate m standard normal random variates of length n – rnorm() Correlate using partial correlation matrix and Choleski’s decomposition – Chol(X) :: X = A ’ A Strip off Normal marginals using Inverse Normal distribution – pnorm() Apply desired margin using the quantile for the distribution qDIST() The “rdpq”s in R makes this trivial (given a few custom tools)

19. Predicted Copula Estimated Kernel Density Distribution – Overall vertical distribution estimated using Reverse Weibull – Density “peaks” estimated using Wiley’s (1977) Site Index and Height Growth models – Weibull Density distributed via Normal Distribution between Density “peaks” Horizontal Distribution recovered from tree-level mean and CV predictions Radial Distribution estimated using Voronoi polygon Correlations sampled from copula distribution of observed correlations

20. Predicted “Composite” Vertical Distribution

21. Voronoi Derived Radial Distribution

22. Predicted Copula

23. Goodness-of-fit Criterion Needed a Criterion that: – Could be expanded to 3 or more dimensions – Didn’t require binning – Applied to multivariate distributions with mixed margins Two-Sample n-Nearest Neighbor Approach (Narsky 2008)

24. Two Sample n-Nearest Neighbors Two Distributions – Observed – Predicted Interested in how the two distributions conform to one another Randomly select a point from the observed distribution Determine distances to all other Observed and all Predicted points Select the n nearest neighbors Classify n neighbors as belonging to the Observed (i=1) or Predicted (i=0) Distribution I = Sum(i)/n If the two distributions are the same I ≈ 0.50 I = 1 shows no conformity

25. Foliage Distributions

26. Plot Reconstruction

28. Framework for Analyzing LiDAR Copula decomposition of LiDAR – Extract tree locations – Develop a classification of LiDAR points into foliage and wood – Extract the relative 3D distribution via a copula Use allometric equations to predict totals Put them together to get mass distributions