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TOPIC: More from the Formula Factory
http://groups.google.com/group/entstrees/browse_thread/thread/0167e5cf3cd8b354?hl=en
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== 1 of 1 ==
Date: Wed, Nov 14 2007 6:50 pm
From: dbhguru
Ed,
Right. Will started the whole thing by sending me a spreadsheet that
consisted of columns A - F, showing the taper of the listed hemlocks
from their breast-high diameters to their diameters at 95.5 feet.
Will was getting a feel for how quickly or gradually the giant
hemlocks tapered in terms of how much diameter they had lost going
from 4.5 to 95.5 feet. I added the remaining columns in order to
predict what the taper would be were the whole form of the tree a
paraboloid, a cone, half way between a paraboloid and a cone (or
conoid as it is sometimes called), and 2/3rd paraboloid and 1/3
cone. The percentage columns show how much the the predicted
deviates from the actual. The way I developed the predictive
equations, I employed radius, thus the "r" columns. I then
convert to circumference, thus the "c" columns. I
frequently bounce between circumference, diameter, and radius
without much thought given. I probably eed to be more consistent for
the sake of the rest of you.
The R, H, h1, and h2 columns figure into the frustum formulas for
100% paraboloid and cone and the hybrids. The direction of the
analysis suggests two things: (1) the huge hemlocks tend toward the
paraboloid form, and (2) there are mavericks among them that don't
behave themselves.
Bob
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TOPIC: OG Hemlock modeling
http://groups.google.com/group/entstrees/browse_thread/thread/ae5d2d51eb90c0d6?hl=en
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Will,
The following table shows the results of testing 38
modeled OG hemlocks against standard shapes. All trees are
ones you've climbed and modeled. The notes in the table explain
how well modeled hemlocks fit the paraboloid, cone, 50% paraboloid
and 50% cone, 67% paraboloid and 33% cone, and 33% paraboloid and
67% cone. More to come.
Bob
| Results
of OG hemlock modeling |
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| Sample
Size |
%
Fit Parabola |
Abs(%)
Parabola |
%
Fit Cone |
Abs(%)
Cone |
Para
-Cone %
50-50 |
Abs(%)
50-50 |
Para
- Cone %
67-33 |
Abs(%)
67-33 |
Para
- Cone %
33-67 |
Abs(%)
33-67 |
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| 38 |
13.4 |
18.5 |
-32.5 |
32.5 |
9.6 |
16.1 |
-1.9 |
14.4 |
-17.2 |
20.4 |
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| Notes: |
Objective
is to predict circumference at a designated height using a
variety of predictive models. |
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utilizing the following formula. The designated
height is 100 feet. |
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c
= C*[f1*SQRT((H-h)/H) + (1-f1)*(H-h)/H] |
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c
= the predicted circumference for the mix of paraboloid
and cone |
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based
on proportions f1 and (1-f1) |
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C
= circumference at 4.5 feet |
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H
= total height |
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h
= height at designated point where c is calculated. |
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%
fit is calculated by:
[(Predicted - Actual)/Actual]*100 |
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The
best fit is 67% paraboloid and 33% cone |
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The
2nd best fit is 50% paraboliid abd 50% cone |
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The
worst fit is the cone. It under-predicts in 100% of the
cases. |
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The
paraboloid over-predicts in 27 of 38 cases. |
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The
67-33 over-predicts in 15 of 38 cases. |
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The
50-50 over-predicts in 6 cases, meaning that it
under-predicts |
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in 32 cases. This means that the correct mix needs
more paraboloid |
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