Nick Smith, Kim Iles and Kurt Raynor

1. 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. 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. Random angle Random point Per unit area estimates e.g. basal area per hectare 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 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)