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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 l.jpg

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


Slide2 l.jpg

The Bow-Tie Theory of Sampling Problem

(alternate title)


Outline l.jpg
OUTLINE Problem

  • Some Current Methods

  • New Methods

  • Any good?

  • Theory

  • Sources of Variance

  • Remove the Center?

  • Have Fun !


Progress l.jpg
PROGRESS Problem

  • 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 l.jpg
Horovitz-Thompson Estimator Problem(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 l.jpg
Tree concentric – Direct Area Problem

  • 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 l.jpg
Mirage estimator Problem

  • 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 l.jpg
Walkthrough Estimator ProblemDucey, 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.


Walkthrough estimator ducey gove and valentine 20049 l.jpg
Walkthrough Estimator ProblemDucey, Gove and Valentine, 2004

OUT

w = 2


Outline10 l.jpg
OUTLINE Problem

  • 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 l.jpg
Walk Through and Fro Problem

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 l.jpg
Walk Through and Fro Problem

Bounds on two sides

Use same formula


Walk through and fro complex intersection l.jpg
Walk Through and Fro ProblemComplex 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 l.jpg
Walk To and Fro Problem( 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 l.jpg
Walk To and Fro Problem

Bounds on two sides

Just measure bf


Walk to and fro complex intersection l.jpg
Walk To and Fro ProblemComplex intersection

r = 10

r = 10

T

fro

through

IN

IN

IN

6

0

3

10

-10

-7

WEIGHT COMPUTATION USES SQUARES OF INTERSECTIONS


Outline17 l.jpg
OUTLINE Problem

  • 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 l.jpg
Any Good? ProblemEVALUATION 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 l.jpg

× Problem

1

×

×

×

0

1

2

Example Configurations

Stand is 2 by 2.

Tree Plot has r = 2.

Plot area /stand area = π/4


Evaluation l.jpg
EVALUATION Problem

  • For Tree at (1.5, 1.0) - red circle last slide


Corner configuration l.jpg

× Problem

1

×

×

×

0

1

2

Corner Configuration

Stand is 2 by 2.

Tree Plot has r = 2.

Plot area /stand area = π/4


Evaluation corner l.jpg
EVALUATION - Corner Problem

  • For Tree at (1.8, 1.5) - red circle last slide


Evaluation conclusions l.jpg
Evaluation - Conclusions Problem

  • 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 l.jpg
OUTLINE Problem

  • Some current methods

  • New Methods

  • Any good?

  • Theory

  • Sources of Variance

  • Remove the Center?

  • Have Fun !


What s a bow tie l.jpg
What’s a bow-tie? Problem

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

Let m approach infinity.


What s a bow tie26 l.jpg
What’s a bow-tie? Problem

Remember the complex boundary example?


What s half a bow tie fro l.jpg
What’s half a bow-tie? (Fro) Problem

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

S

T

T


Theory math l.jpg
Theory - Math Problem

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 l.jpg
OUTLINE Problem

  • Some current methods

  • New Methods

  • Any good?

  • Theory

  • Sources of Variance

  • Remove the Center?

  • Have Fun !


Sources of variance l.jpg
Sources of Variance Problem

WalkThrough

Tree Concentric

INSIDE

INSIDE

OUTSIDE

OUTSIDE

W = 1.25

W = 1.0

W = 2

MORE VARIANCE


Sources of variance31 l.jpg
Sources of Variance Problem

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 l.jpg
OUTLINE Problem

  • Some current methods

  • New Methods

  • Any good?

  • Theory

  • Sources of Variance

  • Remove the Center?

  • Have Fun !


Remove the center l.jpg
Remove the Center Problem

  • 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 l.jpg
OUTLINE Problem

  • Some current methods

  • New Methods

  • Any good?

  • Theory

  • Sources of Variance

  • Remove the Center?

  • Have Fun !


Have fun l.jpg
Have Fun! Problem

Try it. No barbed wired fences to climb !

Thank you,