The Walk Through and Fro Solution The Walk Through and Fro Solution
to the Boundary Overlap Problemto the Boundary Overlap Problem
Jim FlewellingJim Flewelling
[email protected]@seattlebiometrics.com
Jacob L. StrunkJacob L. Strunk
[email protected]@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,
