Three-Dimensional Crown MassDistribution via Copulas

Dr. John A. Kershaw, Jr.Professor of Forest Mensuration/BiometricsFaculty of Forestry and Env. MgmtUniversity 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

HT-DBH Simulation Example

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

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

…and my “Fall Guy”

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

Crown Reconstruction

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

Vertical Distribution

Horizontal Distribution

Radial Distribution

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

Predicted “Composite” Vertical Distribution

Voronoi Derived Radial Distribution

Predicted Copula

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

Foliage Distributions

Plot Reconstruction

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