1 Focus Last Change : June, 2003 C/Clients/Publications/p-plant

2 Point-to-Plant Distances An unbiased and general solution to estimating plant density

3 A historical problem, … which Becomes a polygon problem … then Becomes a circle problem … then Becomes a familiar problem … and then Becomes no problem at all

4 Tract Boundary Plants (trees)

5 Nearest Tree Random x:y point within the tract

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7 More trees  smaller polygons  smaller distances

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10 More trees  smaller polygons  smaller distances

11 What function of distance = n ??

12 Toy SituationReal Situation Toy Solutions Real Solutions N = ?? The real solution to an actual distribution

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14 Toy SituationReal Situation Toy Solutions Real Solutions N=?? Recognize it’s not random, pretend solution works anyway

15 Toy SituationReal Situation Toy Solutions Real Solutions N=?? Pretend it is random, apply known solution

16 Blank blue

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18 Can’t we correct for a biased choice ?

19 We are dealing with Voroni Polygons, whether we see them or not. Insight #1

20 Finding the nearest tree identifies the voroni polygon, but not its size

21 Average size Polygon Tract size. average polygon size = N (number of trees)

22 Sample for the polygon size Method #1

23 Find a random point Method #1

24 Go to the nearest tree

25 Choose a random direction from that tree ?

26 Find the edge of the polygon in that direction Equidistant tree establishes edge

27 RiRi and measure distance Ri

28 (Sometimes, calculus is useful)

29 {Polygon average R i 2 } *  equals polygon area. Equals circle with radius R R is quadratic average of polygon R i

30 Sector plots (last years topic) can start from any interior point.

31 a simplified example Polygon of interest 3 partial circles make up this polygon

32 1/3 red

33 1/3 red + 1/3 green

34 1/3 red + 1/3 green + 1/3 blue = polygon area

35 1/3 red + 1/3 green + 1/3 blue = polygon area (red + green + blue) ÷ 3 = polygon area (and we are sampling for that) ++

36 Sampling the radii as circles establishes the polygon area

37 3/4 of the area in outer 1/2 of radius A constant relationship Outer section of non-square area vanishes (in the limit) Vertex Similar triangles Some wedge properties are intransitive, some vanish at the limit

38 Insight #2 Sampling these rays is the same as sampling circles

39 Method #2 Back off from the closest tree to find polygon edge ?

40 Measure the ray distance By selecting it with a random x:y coordinate

41 Forget they are polygons - they are just a bunch or partial circles which tessellate the area. Insight #3 - we are ONLY sampling circles

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43 Measuring the Rays in polygons is equivalent to Variable Plot Sampling

44 Bitterlich knows how to sample circles

45 Lew Grosenbaugh knows how to expand a variable probability sample

46 Your observation Weight by trees represented per area Now we have estimated the average polygon area, and by extension, number of trees Working with circle area (polygon size) ?

47 Working with volume ? Now we have estimated volume

48 Working with anything...

49 Tree Volume / Circle area = Partial Volume / Sector area

50 Does that kind of thing look familiar ?? “Something divided by (circle) area”

51 Don Bruce and John Bell know how to subsample with VBAR

52 Green : spruce tree is Nearest What part of the area is “covered” by Spruce

53 Percent of cases where Spruce is closest = 30/72 = 42% Let’s just count which species is “closest” on 72 grid points

54 42% of the tract (of known area) is covered by Spruce polygons If we found the Volume/area ratio for those polygons (or sectors) we could estimate total volume BA * VBAR = Volume

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56 riri R Insight #4 : r i 2 is a simple proportion to R 2

57 e{r i 2 }*2 = e{R 2 } riri R

58 Those who asked questions … Bless them all Cottom Fall Smith Bitterlich Grosenbaugh Bell & Bruce

59 We have …  An unbiased general solution (actually, 3 of them) with no limitation on tree distribution.  One method can be seen as a special case of VP sampling.  That method can be extended to other parameters of the plants.  It can also be used to subsample parameters.  One method uses only the simplest distance (r i ) --- but it’s a dog, and way too variable ---

60 A historical problem, … which Becomes a polygon problem … then Becomes a circle problem … then Becomes a familiar problem … and then Becomes no problem at all

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