Nick Smith, Kim Iles and Kurt Raynor Partly funded by BC Forest Science Program and Western Forest Products 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

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

Stand boundary tree a tree b Probability of Selecting Each Tree from a Random Spin = (cumulative angular degrees in sectors)/360 o* Example: total degrees in sectors 36 o 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 Sectors *= s/C (sector arc length/circumference)

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

Simulation Program

Data used Variable retention patch 288 trees in a 0.27 ha patch, basal area 53m 2 /ha, site index 25m video_mhatpt3.avi PSP 81 years, site index 25, plot 10m x 45m, 43 trees and 21m 2 /ha.

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,1,2,3,4,5,10,15,20,25,3 0,50,100 –We know actual patch means and totals

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

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

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)

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

Random point selection is more efficient sample size

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

Ratio estimation properties

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

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

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

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

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

layout of sectors in an experimental block Applications

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)

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

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

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