Transcript Slide 1

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
i
20 40 60 80 100 120
sample size
Population
Real
Ratio of means
2
/N
Real standard error
of mean for a given
sample size across
all 1000 sectors
g
basal area/ha: S or S
2
 ( 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