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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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**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**Copula**• [kop-yuh-luh] • something that connects or links together Cupola**Genest, C. and MacKay, J. (1987). The Joy of Copulas: The**Bivariate Distributions with Uniform Marginals. American Statistician, 40, 280-283.**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**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**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**Two Copula Approaches**• “Fitted” based on reconstructed branches • “Predicted” based on tree-level moment-based parameter prediction**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**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)**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**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)**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**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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