The Walk Through and Fro Solution The Walk Through and Fro Solution

to the Boundary Overlap Problemto the Boundary Overlap Problem

Jim FlewellingJim Flewelling

flewelling@seattlebiometrics.comflewelling@seattlebiometrics.com

Jacob L. StrunkJacob L. Strunk

jacob.strunk@oregonstate.edujacob.strunk@oregonstate.edu

Presented at:

Western Mensurationists

Missoula, Montana

June 21, 2010

The Bow-Tie Theory of Sampling(alternate title)

OUTLINEOUTLINE

• Some Current Methods

• New Methods

• Any good?

• Theory

• Sources of Variance

• Remove the Center?

• Have Fun !

PROGRESSPROGRESS

• 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 EstimatorHorovitz-Thompson Estimator(before edge effects)(before edge effects)

sk k

ky

Iag

G i

P

i i

i

u A

1 )/(ˆ

ABAFcG ˆ

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 AreaTree 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 estimatorMirage 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 EstimatorWalkthrough EstimatorDucey, Gove and Valentine, 2004Ducey, 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 EstimatorWalkthrough EstimatorDucey, Gove and Valentine, 2004Ducey, Gove and Valentine, 2004

OUT

w = 2

OUTLINEOUTLINE

• 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 FroWalk Through and FroOUTSIDE OF STAND

INSIDE

Tbf bt

S

THROUGHFRO

r

bbr

t f

w

2 2

22

T = Tree Location

S = Sample point

bf, bt = distances to boundaries

R = radius of tree’s plot.

Walk Through and FroWalk Through and Fro

Bounds on two sides

Use same formula

Walk Through and FroWalk Through and FroComplex intersectionComplex intersection

Tr = 10 r = 10

-10 -7 0 3 6 10

IN IN IN

WEIGHT COMPUTATION USES SQUARES OF INTERSECTIONS

64.1]610[]03[]0)7[(

102222222

2

w

throughfro

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

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

OUTSIDE OF STANDINSIDE

Tbf bt

S

THROUGHFRO

r

brf

w 2

2

Simpler formula

Walk To and FroWalk To and Fro

Bounds on two sides

Just measure bf

Walk To and FroWalk To and FroComplex intersectionComplex intersection

Tr = 10 r = 10

-10 -7 0 3 6 10

IN IN IN

WEIGHT COMPUTATION USES SQUARES OF INTERSECTIONS

04.2]0)7[(

1022

2

w

throughfro

OUTLINEOUTLINE

• 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?Any Good?

EVALUATION METHODEVALUATION 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 ConfigurationsExample Configurations

2

1

0 1

× × ×

× Stand is 2 by 2.

Tree Plot has r = 2.

Plot area /stand area = π/4

EVALUATIONEVALUATION

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

Estimator Mean err MSE

Uncorrected -0.154 .256

Tree Concentric .000 .359

Walkthrough .000 .476

Walk Through and Fro .000 .397

Walk To and Fro .000 .647

Corner ConfigurationCorner Configuration

Stand is 2 by 2.

Tree Plot has r = 2.

Plot area /stand area = π/4

2

1

0 1

× × ×

×

EVALUATION - CornerEVALUATION - Corner

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

Estimator Mean err MSE

Uncorrected -.395 .394

Treee Concentric .000 .962

Walkthrough -.104 .809

Walk Through and Fro .000 1.099

Walk To and Fro .000 5.066

Evaluation - ConclusionsEvaluation - 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.

OUTLINEOUTLINE

• 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-tie?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.

TTS

Theory - MathTheory - Math

P

i

m

jij

iji

i Iag

G fA

1 1

ˆ

k

kS

k k a

g

fAG

1

1ˆ

rbbt ff 2

2 2

2

w = 1 / f

Bow-ties

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

Fraction of area in standfor sampled bow tie.

Weights

OUTLINEOUTLINE

• Some current methods• New Methods• Any good?• Theory

• Sources of Variance• Remove the Center? • Have Fun !

Sources of VarianceSources of Variance

Tree Concentric WalkThrough

INSIDE OUTSIDE INSIDE OUTSIDE

W = 1.25 W = 1.0W = 2

MORE VARIANCE

Sources of VarianceSources of Variance

Walk Through and Fro WalkThrough

INSIDE OUTSIDE INSIDE OUTSIDE

W = 1.0W = 2

S

W = constant

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

OUTLINEOUTLINE

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

Remove the CenterRemove 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.

OUTLINEOUTLINE

• 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,