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Sector sampling – some statistical properties 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 What do sector samples look like? Reduction to partial sectorreduced effort 10% sample Constant angle which has variable area Designed to sample objects inside small, irregular polygons Harvest area edge Pivot point Remaining group Named after Galileo’s Sector 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. Stand boundary For a complete revolution of the sectors, 10% of the total arc length that passes through each tree is swept within the sectors tree a tree b 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 53m2/ha, site index 25m • video_mhatpt3.avi • PSP 81 years, site index 25, plot 10m x 45m, 43 trees and 21m2/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, 36o is 36/360=10 – Don’t need areas • Use ordinary statistics (nothing special): means and variance Expansion Factor Off-centre 15 14 Totals 13 12 11 0 20 40 60 80 100 120 sample size 0.6 Standard error 0.5 2 total basal area, m /patch 2 total basal area, m /patch No area, e.g. total patch basal area 0.4 0.3 Estimates are unbiased [s/C*10=1] off-centre centre systematic A systematic arrangement reduces variance 0.2 0.1 0.0 0 20 40 60 80 100 120 sample size off-centre centre systematic Systematic sample as good as putting in the centre Systematic Centre 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 angle Ratio of means Use usual ratio of means formulas • Random point Mean of ratios Selection with probability proportional to sector size Use standard formulas (standard error)/mean, % Random point selection is more efficient 60 50 40 30 20 Sector selection 10 0 0 20 40 60 80 100 120 groupsize size sample Random angle (real) Random points coefficient of variation Ratio estimator (area known) no advantage to using systematic* 100 80 60 40 20 0 0 20 40 60 80 100 120 sample size Ratio estimator-random Ratio estimator-systematic Random sector (angles) Considering measured area Systematic sample usually balances areas* *antithetic variates 5 4 3 2 1 0 0.000 0.025 0.050 0.075 0.100 area,ha per sector 250 2 basal area per ha, m /ha 2 basal area, m /sector Ratio estimation properties 200 150 100 50 0 0.000 0.025 0.050 0.075 0.100 area,ha per sector 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) 400 Pivot point count 300 200 100 0 0 100200300400500 basal area, m 2ha 2 basal area/ha: S or S 2 basal area/ha: S or S Ratio standard deviation is biased 50 Population 40 30 SD For all 1000 sectors around population mean (no resampling) 20 SE 10 0 0 20 40 60 80 100 120 sample size Population Real Ratio of means Real Calculate ba/ha standard error around population mean from a resampling approach (1000 times) for each sample size 50 40 30 SD Ratio of means variance 20 10 0 0 SE 20 40 60 80 100 120 sample size Population Real Ratio of means ROM estimator for a given sample size around the sample mean averaged over the 1000 resamples. standard error bias percent Bias in the standard error by sample size 40 30 20 10 0 -10 0 20 40 60 80 100 120 sample size Data set 2 Data set 1 For small sample sizes actual se up to 40% larger Each runs 9 times (replicate) Raynor’s method 100 2 Standard deviation, m /ha So let’s correct the bias! Real (‘Actual’) (green) 90 80 70 Fitted line (black) 60 50 40 0 20 40 60 80 100 120 sample size Ordinary Real Ratio Sˆ 2 xa S 2 xr = ( S 2 xo S 2 xr ) n 0.84 Note- there were 6 groups and 9 ‘replicates’ Ordinary: use standard formulae as in simple random sampling Applications layout of sectors in an experimental block 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) GG and WGC spotted in line-up to buy latest version of Sector Sampling Simulator! Fixed area plots The same logic can be applied to small circular fixed plots along a ray extending from the tree cluster center. Equal selection of plot centerline along random ray. Equal area plots. Selection probability is plot area divided by ring area. Relative Weight=distance from pivot point 2 basal area/ha: S or S Ratio standard deviation is biased 50 Population variance (N= 1000 sectors) 40 30 N ( BAha 20 i 1 10 0 0 20 40 60 80 100 120 sample size Population Real Ratio of means i )2 / N Real standard error of mean for a given sample size across all 1000 sectors 2 basal area/ha: S or S g ( BAha 50 j 1 40 j )2 (g) 30 Ratio of means variance (for each sample size, n) 20 10 0 0 20 40 60 80 100 120 sample size Population Real Ratio of means n (ba j 1 j 2 j R j ha j ) /(( n 1)ha ) 2 2 basal area/ha: S or S 2 basal area/ha: S or S Ratio standard deviation is biased 50 Population variance (N= 1000 sectors) 40 30 20 10 0 0 20 40 60 80 100 120 sample size Population Real Ratio of means Real standard error of mean 50 40 30 Ratio of means variance (for each sample size, n) 20 10 0 0 20 40 60 80 100 120 sample size Population Real Ratio of means