Sector sampling Nick Smith, Kim Iles and Kurt Raynor Partly funded by British Columbia Forest Science Program, Canada; Western Forest Products, Canada with support from ESRI Canada
What do sector samples look like? Reduction to partial sectorreduced effort Harvest area edge 10% sample Pivot point Constant angle which has variable area Designed to sample objects inside small, irregular polygons Remaining group Named after Galileo s Sector
Probability of Selecting Each Tree from a Random Spin = (cumulative angular degrees in sectors)/360 o* Stand boundary Example: total degrees in sectors 36 o or 10% of a circle. tree b tree a 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
Background Designed to sample variable-retention (VR), a new forest harvesting approach in BC, Canada VR definition: half the harvested area within one tree length of a retained tree or group of trees Designed set of large scale experiments (100ha+) to look at impacts of VR
Experimental design:100ha Group Retention Dispersed Retention Group Size Group Removal Supplemental : 30ha Dispersed Retention Mixed Retention
Example of sector plot layout
Simulation Program
Two data sets Data used Variable retention patch PSP
Scenarios: Center of regular polygon Off-centre in regular polygon Irregular polygon Off-centre systematic
Simulation details Select pivot point Split into a large number of sequential sectors Resample for different sample sizes 0 0.000 0.025 0.050 0.075 0.100 area,ha per sector basal area, m 2 /sector 5 4 3 2 1
Two Sorts of Data Tract totals and means (don t know area)-use Expansion Factor Unit area estimates e.g. ba/ha
Expansion factor For example total and mean patch basal area Expansion factor for the sample For each tree, e.g. 36 o is 36/360=10 Don t need areas Use ordinary statistics: means and variance
total basal area, m 2 /patch total basal area, m 2 /patch Expansion Factor 15 14 13 12 11 0 20 40 60 80 100 120 sample size 0.6 0.5 0.4 0.3 0.2 0.1 Totals Standard error 0.0 0 20 40 60 80 100 120 sample size off-centre centre systematic off-centre centre systematic Estimates are unbiased A systematic arrangement reduces variance Systematic sample as good as putting in the centre
Unit area estimates Two approaches 1. Random angles (ROM) E.g. (Basal area)/(hectares) weights sectors proportional to sector area 2. Random points (MOR) probability proportional to sector size (importance sampling)
Per unit area estimates Random angle Random point Ratio of means Use usual ratio of means formulas Mean of ratios Use standard formulas
Random point selection is more efficient (standard error)/mean, % 60 50 40 30 20 10 0 0 20 40 60 80 100 120 group size sample size Sector selection Random angle (real) Random points
Ratio estimator: no advantage to using systematic 100 coefficient of variation 80 60 40 20 0 0 20 40 60 80 100 120 sample size Expansion factor-random Expansion factor-systematic Ratio estimator-random Ratio estimator-systematic Systematic approach:antithetic variates
Ratio estimation bias basal area, m 2 /ha 54 53 52 51 50 49 48 47 0 20 40 60 80 100 120 sample size Corrections: e.g. Hartley Rao and Mickey Means can be highly variable and biased for small sample size off center centre off centre-systematic
Ratio Data Properties Often positively skewed (small areas) basal area, m 2 patch 3 2 1 0 0.00 0.010.02 0.03 0.040.05 0.06 count 400 300 200 100 0 0 100200300400500 basal area, m 2 ha sector size, m 2 ha
basal area/ha: S 2 or S Ratio variance is biased for small 50 40 30 20 10 bias SD SE 0 0 20 40 60 80 100 120 sample size sample sizes Population from population variance (N= 1000 sectors) Population Actual Ratio of means Actual from resampled standard error for a given sample size, n 50 basal area/ha: S 2 or S 40 30 20 10 SD SE 0 0 20 40 60 80 100 120 sample size Population Actual Ratio of means Ratio of means resampled variance for each sample size, n
Bias in the standard error by sample size standard error bias percent 40 30 20 10 0-10 0 20 40 60 80 100 120 sample size For small sample sizes actual se up to 40% larger Data set 2 Data set 1
Standard deviation, m 2 /ha Correct the SD bias: Raynor s method : Ŝ 100 90 80 70 60 50 2 S 40 0 20 40 60 80 100 120 sample size 2 Ordinary 0.84 Actual :green Ordinary Actual Ratio 2 ROM ( S S ) 2 ROM n Fitted line: black
CONCLUSIONS EXPANSION FACTOR AREA put in centre, and/or systematic sample (balanced) Small sample size ROM variance estimator biased: 1) Raynorize 2) Or avoid it (make bias very small) Can use systematic arrangement 3) Or use random points approach (mean of ratios variance estimator is unbiased)
Sub-sampling inside the Sector Sample Any plot shape can be used along a sector sample mid-line( ray ) Reasons For example: natural regeneration may be very dense You may simply prefer fixed area or perhaps prism plots
Example: fixed area plots Circular plots along a ray from the pivot-point plots Equal selection of plot center-line along random ray. Equal area plots: Selection probability is plot area divided by ring area. Relative Weight=distance from pivot-point
Computational cost: weights Each Ray Plots are weighted by distance from pivotpoint (plots at edge have greatest weighting) If plot overlaps boundary then trees in plots must have their selection probabilities corrected Combining Rays Random angle: ray averages are weighted by plot number Random point: no weighting is needed
Variance of weighted means When combining several rays using random angle selection Variance of weighted means: no exact solution Used (ba/ha*no plots)= total ba/ha per ray vs. no. plots per ray in a ROM approach: Mean =weighted ba/ha total ba/ha per ray 3000 2000 1000 0 0 2 4 6 8 10 12 number of plots per ray
Off-centre ray 60 50 mean basal area m 2 /ha 50 40 30 20 10 0 0 20 40 60 80 100 120 sample size standard error 40 30 20 10 0 0 20 40 60 80 100 120 sample size Actual Population ROM
Random Point 60 40 mean basal area m 2 /ha 50 40 30 20 10 0 0 20 40 60 80 100 120 sample size standard error 30 20 10 0 0 20 40 60 80 100 120 sample size Actual Population ROM Random Point v. Random Angle 50 Standard error m 2 /ha 40 30 20 10 0 0 20 40 60 80 100 120 sample size Random Point Random Angle
Conclusions Fixed area plots can be used Easy to establish and measure Weighting/edge effect considerations by distance from pivot point edge trees, could use walk-through method combining rays: weighting by plot number per ray (not needed if random point selection) Other extensions: prism plots, line transects, etc.