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# The Walk Through and Fro Solution to the Boundary Overlap Problem - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about 'The Walk Through and Fro Solution to the Boundary Overlap Problem' - louie

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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

(alternate title)

OUTLINE Problem

• Some Current Methods

• New Methods

• Any good?

• Theory

• Sources of Variance

• Remove the Center?

• Have Fun !

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 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.

• 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 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 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 ProblemDucey, Gove and Valentine, 2004

OUT

w = 2

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 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 Fro Problem

Bounds on two sides

Use same formula

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 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 Problem

Bounds on two sides

Just measure bf

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

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? 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.

× Problem

1

×

×

×

0

1

2

Example Configurations

Stand is 2 by 2.

Tree Plot has r = 2.

Plot area /stand area = π/4

EVALUATION Problem

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

× 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 Problem

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

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.

OUTLINE Problem

• Some current methods

• New Methods

• Any good?

• Theory

• Sources of Variance

• Remove the Center?

• Have Fun !

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-tie? Problem

Remember the complex boundary example?

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

S

T

T

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

OUTLINE Problem

• Some current methods

• New Methods

• Any good?

• Theory

• Sources of Variance

• Remove the Center?

• Have Fun !

Sources of Variance Problem

WalkThrough

Tree Concentric

INSIDE

INSIDE

OUTSIDE

OUTSIDE

W = 1.25

W = 1.0

W = 2

MORE VARIANCE

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.

OUTLINE Problem

• Some current methods

• New Methods

• Any good?

• Theory

• Sources of Variance

• Remove the Center?

• Have Fun !

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.

OUTLINE Problem

• Some current methods

• New Methods

• Any good?

• Theory

• Sources of Variance

• Remove the Center?

• Have Fun !

Have Fun! Problem

Try it. No barbed wired fences to climb !

Thank you,