1. Nick Smith, Kim Iles and Kurt Raynor
Sector sampling – some statistical properties
Nick Smith, Kim Iles and Kurt Raynor
Partly funded by BC Forest Science Program and Western Forest Products

2. Sector sampling – some statistical properties
Overview
What is sector sampling?
Sector sampling description
Some statistical properties
no area involved, e.g. basal area per retention patch
values per unit area, e.g. ba/ha
Random, pps and systematic sampling
Implications and recommendations
Applications

3. What do sector samples look like?
Reduction to partial sector- reduced effort
Harvest area edge
Pivot point
10% sample
Constant angle which has variable area
Remaining group
Designed to sample objects inside small, irregular polygons
Named after Galileo’s Sector

5. *= s/C (sector arc length/circumference)
Probability of Selecting Each Tree from a Random Spin = (cumulative angular degrees in sectors)/360o*
Example: total degrees in sectors 36o or 10% of a circle.
For a complete revolution of the sectors, 10% of the total arc length that passes through each tree is swept within the sectors
Stand boundary
tree a
tree b
Sectors
*= s/C (sector arc length/circumference)

6. The probability of selecting each tree is the same irrespective of where the ‘pivot-point’ is located within the polygon
Stand boundary

7. Simulation Program

8. Data used
Variable retention patch 288 trees in a 0.27 ha patch, basal area 53m2/ha, site index 25m
video_mhatpt3.avi
PSP 81 years, site index 25, plot 10m x 45m, 43 trees and 21m2/ha.

9. Simulation details Random angles Select pivot point and sector size
Split sequentially into a large number of sectors (N=1000)
Combine randomly (1000 resamples, with replacement) into different sample sizes,1,2,3,4,5,10,15,20,25,30,50,100
We know actual patch means and totals

10. Expansion factor-for totals and means
To derive for example total and mean patch basal area
Expansion factor for the sample
For each tree, 36o is 36/360=10
Don’t need areas
Use ordinary statistics (nothing special): means and variance

11. Expansion Factor Off-centre No area, e.g. total patch basal area
Estimates are unbiased [s/C*10=1]
Totals
Systematic
A systematic arrangement reduces variance
Standard error
Centre
Systematic sample as good as putting in the centre

12. Unit area estimates To derive for example basal area per hectare
Two approaches
Random angles (ratio of means estimate)
(Basal area)/(hectares)
ROM weights sectors proportional to sector area
Random points (mean of ratios estimate)
Selection with probability proportional to sector size (importance sampling)

13. Per unit area estimates e.g. basal area per hectare
Random angle
Random point
Ratio of means
Mean of ratios
Selection with probability proportional to sector size
Use usual ratio of means formulas
Use standard formulas

14. Random point selection is more efficient
sample size

15. Ratio estimator (area known) no advantage to using systematic*
Random sector (angles)
Considering measured area
Systematic sample usually balances areas*
*antithetic variates

16. Ratio estimation properties

17. Ratio estimation properties
Means can be biased (well known)
Corrections: e.g. Hartley Ross and Mickey

19. Ratio Data Properties
Often positively skewed- extreme data example (N=1000 sequential sectors)
Pivot point

20. Ratio standard deviation is biased
Population SD
For all 1000 sectors around population mean (no resampling) SE
Real
Calculate ba/ha standard error around population mean from a resampling approach (1000 times) for each sample size SD
Ratio of means variance SE
ROM estimator for a given sample size around the sample mean averaged over the 1000 resamples.

21. Bias in the standard error by sample size
For small sample sizes actual se up to 40% larger
Each runs 9 times (replicate)

22. So let’s correct the bias! Raynor’s method
Real (‘Actual’) (green)
Fitted line (black) =
Note- there were 6 groups and 9 ‘replicates’
Ordinary: use standard formulae as in simple random sampling

23. Applications
layout of
sectors in an
experimental
block

24. CONCLUSIONS Don’t consider area? put in centre, and/or
systematic (balanced)
Do consider area?
Small sample size ratio of means variance estimator needs to dealt with:
1) Raynorize it
2) Avoid it (make bias very small)
Can use systematic arrangement
3) Or, use random points approach (mean of ratios variance estimator is unbiased)

25. GG and WGC spotted in line-up to buy latest version of Sector Sampling Simulator!

26. Fixed area plots Relative Weight=distance from pivot point
Equal selection of plot centerline along random ray.
The same logic can be applied to small circular fixed plots along a ray extending from the tree cluster center.
Equal area plots. Selection probability is plot area divided by ring area.
Relative Weight=distance from pivot point

27. Ratio standard deviation is biased
Population variance (N= 1000 sectors)
Real standard error of mean for a given sample size across all 1000 sectors
Ratio of means variance (for each sample size, n)

28. Ratio standard deviation is biased
Population variance (N= 1000 sectors)
Real standard error of mean
Ratio of means variance (for each sample size, n)