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The Walk Through and Fro Solution to the Boundary Overlap Problem. Jim Flewelling flewelling@seattlebiometrics.com Jacob L. Strunk jacob.strunk@oregonstate.edu. Presented at: Western Mensurationists Missoula, Montana June 21, 2010. The Bow-Tie Theory of Sampling (alternate title).

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the walk through and fro solution to the boundary overlap problem

The Walk Through and Fro Solution to the Boundary Overlap Problem

Jim Flewelling

flewelling@seattlebiometrics.com

Jacob L. Strunk

jacob.strunk@oregonstate.edu

Presented at:

Western Mensurationists

Missoula, Montana

June 21, 2010

outline
OUTLINE
  • Some Current Methods
  • New Methods
  • Any good?
  • Theory
  • Sources of Variance
  • Remove the Center?
  • Have Fun !
progress
PROGRESS
  • Some current methods
    • Horovitz-Thompson Estimator
    • Tree Concentric
    • Mirage
    • WalkThrough
  • New Methods
  • Any good?
  • Theory
  • Sources of Variance
  • Remove the Center?
  • Have Fun !
horovitz thompson estimator before edge effects
Horovitz-Thompson Estimator(before edge effects)

General

For tree centered plots -

If a sample point falls within this area (ai), the tree is sampled (Ii =1).

Easy and Unbiased

- If ai is entirely in A.

tree concentric direct area
Tree concentric – Direct Area
  • What if πi  (ai/A) ?

IN

OUT

Fix ai OR use weight w = 1 / ( fraction of ai within the stand)

Unbiased, Lowest Variance

mirage estimator
Mirage estimator
  • Each sample location near a boundary is reflected across the boundary. (fence climbing may be required).
  • Effectively double weight part of each tree centered plot.
walkthrough estimator ducey gove and valentine 2004
Walkthrough EstimatorDucey, Gove and Valentine, 2004
  • Walk from plot center to sample tree.
  • Continue walking an equal distance.
  • Double count the tree if now outside of the stand. (w = 2)
  • Method is easy to apply.
  • Unbiased for some boundaries; at least half of each plot must be within the stand.
outline10
OUTLINE
  • Some current methods
  • New Methods
    • Walk Through and Fro
    • Walk To and Fro
  • Any good?
  • Theory
  • Sources of Variance
  • Remove the Center?
  • Have Fun !
walk through and fro
Walk Through and Fro

INSIDE

OUTSIDE OF STAND

r

FRO

THROUGH

T

S

bf

bt

T = Tree Location

S = Sample point

bf, bt = distances to boundaries

R = radius of tree’s plot.

walk through and fro12
Walk Through and Fro

Bounds on two sides

Use same formula

walk through and fro complex intersection
Walk Through and FroComplex intersection

r = 10

r = 10

T

fro

through

IN

IN

IN

6

0

3

10

-10

-7

WEIGHT COMPUTATION USES SQUARES OF INTERSECTIONS

Green = inside of stand. Red = Outside of stand.

walk to and fro less walking s to t then fro to plot or stand bound
Walk To and Fro( less walking. S to T, then fro to plot or stand bound )

INSIDE

Simpler formula

OUTSIDE OF STAND

r

FRO

THROUGH

T

S

bf

bt

walk to and fro
Walk To and Fro

Bounds on two sides

Just measure bf

walk to and fro complex intersection
Walk To and FroComplex intersection

r = 10

r = 10

T

fro

through

IN

IN

IN

6

0

3

10

-10

-7

WEIGHT COMPUTATION USES SQUARES OF INTERSECTIONS

outline17
OUTLINE
  • Some current methods
  • New Methods
  • Any good?
    • Evaluation Method
    • Sample Configurations
    • Overlap a straight boundary
    • Overlap a corner boundary
    • Conclusions
  • Theory
  • Sources of Variance
  • Remove the Center?
  • Have Fun !
any good evaluation method
Any Good?EVALUATION METHOD
  • Very small stand
  • One sample point per stand
  • One tree per stand – circular tree plot
  • Estimate ∑ (tree plot areas) / [Stand Area]
  • Calculate error.
  • Integrate over all possible sample points.
  • Report mean error and MSE.
example configurations

×

1

×

×

×

0

1

2

Example Configurations

Stand is 2 by 2.

Tree Plot has r = 2.

Plot area /stand area = π/4

evaluation
EVALUATION
  • For Tree at (1.5, 1.0) - red circle last slide
corner configuration

×

1

×

×

×

0

1

2

Corner Configuration

Stand is 2 by 2.

Tree Plot has r = 2.

Plot area /stand area = π/4

evaluation corner
EVALUATION - Corner
  • For Tree at (1.8, 1.5) - red circle last slide
evaluation conclusions
Evaluation - Conclusions
  • Tree Concentric always the best!
  • WalkThrough can be biased.
  • Walk Through and Fro is unbiased. Has MSE slightly better than Walkthrough.
  • Walk To and Fro is unbiased. - can have high MSE.
outline24
OUTLINE
  • Some current methods
  • New Methods
  • Any good?
  • Theory
  • Sources of Variance
  • Remove the Center?
  • Have Fun !
what s a bow tie
What’s a bow-tie?

Divide the tree centered plot into m bow-ties, each with angle π/m.

Let m approach infinity.

what s a bow tie26
What’s a bow-tie?

Remember the complex boundary example?

what s half a bow tie fro
What’s half a bow-tie? (Fro)

Keep triangle-shaped portion from the tree towards the sample point.

S

T

T

theory math
Theory - Math

Bow-ties

Sample Trees, with fk referring to sampled bow-tie.

Fraction of area in stand

for sampled bow tie.

w = 1 / f

Weights

outline29
OUTLINE
  • Some current methods
  • New Methods
  • Any good?
  • Theory
  • Sources of Variance
  • Remove the Center?
  • Have Fun !
sources of variance
Sources of Variance

WalkThrough

Tree Concentric

INSIDE

INSIDE

OUTSIDE

OUTSIDE

W = 1.25

W = 1.0

W = 2

MORE VARIANCE

sources of variance31
Sources of Variance

WalkThrough

Walk Through and Fro

INSIDE

INSIDE

OUTSIDE

OUTSIDE

W = constant

W = 1.0

W = 2

S

For any particular through and fro line, w is invariant to sample location.

outline32
OUTLINE
  • Some current methods
  • New Methods
  • Any good?
  • Theory
  • Sources of Variance
  • Remove the Center?
  • Have Fun !
remove the center
Remove the Center
  • Through and fro and to-and-fro may be difficult if the sample point is near coincident with the tree.
  • Directional line can not be established.
  • Either method can be modified by excluding a small central area from the bow-tie.

w formulas modied slightly.

Tree concentric formula used for small center circle.

Will improve MSE of To and Fro extimator.

outline34
OUTLINE
  • Some current methods
  • New Methods
  • Any good?
  • Theory
  • Sources of Variance
  • Remove the Center?
  • Have Fun !
have fun
Have Fun!

Try it. No barbed wired fences to climb !

Thank you,