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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

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

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

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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

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HT-DBH Simulation Example

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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

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

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…and my “Fall Guy”

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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

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Crown Reconstruction

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

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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

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Vertical Distribution

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Horizontal Distribution

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Radial Distribution

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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)

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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

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Predicted “Composite” Vertical Distribution

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Voronoi Derived Radial Distribution

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Predicted Copula

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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)

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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

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Foliage Distributions

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Plot Reconstruction

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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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