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Three-Dimensional Crown Mass Distribution via Copulas

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

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Three-Dimensional Crown Mass Distribution via Copulas

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1. Three-Dimensional Crown MassDistribution 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) = Cx,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

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

8. …and my “Fall Guy”

9. 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

10. Crown Reconstruction

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

12. 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

13. Vertical Distribution

14. Horizontal Distribution

16. 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)

17. 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

18. Predicted “Composite” Vertical Distribution

20. Predicted Copula

21. 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)

22. 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

23. Foliage Distributions

24. Plot Reconstruction

25. 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

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