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![Page 1: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/1.jpg)
Rachel Fewster Department of Statistics, University of Auckland
Variance estimation for systematic designs
in spatial surveys
![Page 2: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/2.jpg)
• Method of estimating density of objects in a survey region.
Line transect sampling
![Page 3: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/3.jpg)
Line transect sampling
D
# detections per unit area
= p
![Page 4: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/4.jpg)
D
# detections per unit area
= p
Line transect sampling
Density, D
![Page 5: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/5.jpg)
Estimate the variance of the ratio by the Delta method: “squared CVs add”
D
# detections per unit area
= p
ENCOUNTER RATE
easy
ENCOUNTER RATE VARIANCE: Largest and most difficult component Usually >70% of total variance
![Page 6: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/6.jpg)
• Encounter rate estimates mean detections per unit line length
Encounter Rate and its variance
![Page 7: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/7.jpg)
Inferential framework: which Var(n/L)?
Animals from spatial p.d.f. Select lines
Detect animals
• Variance is defined over conceptual survey repeats
Find n/L
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Inferential framework: which Var(n/L)?
Animals from spatial p.d.f. Select lines
Detect animals
• Variance is defined over conceptual survey repeats
Find n/L
Gained value of n/L from first survey
![Page 9: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/9.jpg)
Same animals, new positions
Second survey:
Inferential framework: which Var(n/L)?
![Page 10: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/10.jpg)
Select new linesSame animals, new positions
Detect new animalsFind new n/L
Inferential framework: which Var(n/L)?Second survey:
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Select new linesSame animals, new positions
Detect new animalsFind new n/L
Inferential framework: which Var(n/L)?
Gained value of n/L from second survey
Overall, gives var(n/L) across the repeated surveys
This is our ENCOUNTER RATE VARIANCE.
![Page 12: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/12.jpg)
• To estimate a variance, use repeated observations with the same variance
Random-line estimator:• makes no assumptions about the unknown distribution of objects;
How to estimate Var(n/L)?
![Page 13: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/13.jpg)
• To estimate a variance, use repeated observations with the same variance
Random-line estimator:• makes no assumptions about the unknown distribution of objects;• random variables are IID with respect to the design.
How to estimate Var(n/L)?
)(/)( lEnEln ii
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Systematic Survey DesignsSurveys usually use SYSTEMATIC transect lines, instead of random lines.
Grid has random start-point
![Page 15: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/15.jpg)
Systematic lines give LOWER VARIANCE than random lines in trended populations
But the variance is HARD TO ESTIMATE
Systematic Survey DesignsSurveys usually use SYSTEMATIC transect lines, instead of random lines.
Grid has random start-point
A systematic sample has
NO REPETITION: it is a sample
of size 1!
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Variance for systematic designs
• There is no general design-unbiased variance estimator for data from a single systematic sample
• Approaches to systematic variance estimation are:
1. Ignore the problem and use estimators for random lines
2. Use some form of post-stratification
3. Model the autocorrelation in the systematic sample
Approach used to date
![Page 17: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/17.jpg)
Variance for systematic designs
• There is no general design-unbiased variance estimator for data from a single systematic sample
• Approaches to systematic variance estimation are:
1. Ignore the problem and use estimators for random lines
2. Use some form of post-stratification
3. Model the autocorrelation in the systematic sample
Approach in Fewster et al, Biometrics, 2009
![Page 18: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/18.jpg)
![Page 19: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/19.jpg)
But the stratified estimators are still biased sometimes – e.g. high sampling fraction, or population clustering
Stratified variance estimators: results
Can we do better…?
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Variance for systematic designs
• There is no general design-unbiased variance estimator for data from a single systematic sample
• Approaches to systematic variance estimation are:
1. Ignore the problem and use estimators for random lines
2. Use some form of post-stratification
3. Model the autocorrelation in the systematic sample
![Page 21: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/21.jpg)
Historical Note
• Many estimators for systematic designs originated in social statistics
– discrete surveys
Correlation will clearly exist in responses of neighbours, but modelling the correlation is hard!
![Page 22: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/22.jpg)
But space is continuous!
As a strip changes position very slightly...
... it still covers many of the same objects.
![Page 23: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/23.jpg)
But space is continuous!
As a strip changes position very slightly...
... it still covers many of the same objects.
Idea:1. Divide the region into hundreds of tiny
‘striplets’2. Allow the number of objects available in each
striplet to be random variables X1 , X2 , …, XJ
3. The number of objects available in any full strip is the sum of the objects in the constituent striplets
![Page 24: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/24.jpg)
1. Divide the region into hundreds of tiny ‘striplets’2. Number of objects available in striplets 1, 2, …, J
is X1 , X2 , …, XJ
3. Number of objects available in any full strip is the sum of the objects in the constituent striplets.
Expected number of objects per striplet
Random number of objects per striplet, X1 , X2 , …, XJ
~ Multinomial
Str
iple
t #
ob
ject
s availa
ble
striplet position 0
1
2
3
4
![Page 25: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/25.jpg)
1. Divide the region into hundreds of tiny ‘striplets’2. Number of objects available in striplets 1, 2, …, J
is X1 , X2 , …, XJ
3. Number of objects available in any full strip is the sum of the objects in the constituent striplets.
Str
iple
t #
ob
ject
s availa
ble
striplet position 0
1
2
3
4
Full strip at this position: 10 objects
Full strip at next position: 7 objects
Full strip at next position: 8 objects
... etc
![Page 26: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/26.jpg)
Recap:We want the variance in the
encounter rate, n/L, over:1. Moving grid;2. Moving objects;3. Detections
Account for:1. Large-scale trends2. Small-scale noise
![Page 27: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/27.jpg)
1. Trends in object density across the region Observed number of detections per unit search area
#d
ete
ctio
ns
/ u
nit
are
a Points correspond to observed transects
Fit a GAM to give a fitted object density for any search strip in the region
x-coordinate
![Page 28: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/28.jpg)
#d
ete
ctio
ns
/ u
nit
are
a
x-coordinate
1. Trends in object density across the region
Fit a GAM to give a fitted object density for any search strip in the region
For any striplet j, we now have an expected number of objects available, j
![Page 29: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/29.jpg)
Expected number of objects per striplet, j
Str
iple
t #
ob
ject
s availa
ble
striplet position 0
1
2
3
4
Account for:1. Large-scale trends
![Page 30: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/30.jpg)
Str
iple
t #
ob
ject
s availa
ble
striplet position 0
1
2
3
4
Account for:2. Small-scale noise
Random number of objects per striplet, X1 , X2 , …, XJ
~ Multinomial(N, j/N)
Striplet idea means we correctly model the autocorrelation between systematic grids
![Page 31: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/31.jpg)
Str
iple
t #
ob
ject
s availa
ble
striplet position 0
1
2
3
4
Account for:2. Small-scale noise
![Page 32: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/32.jpg)
Recap:We want the variance in the
encounter rate, n/L, over:1. Moving grid;2. Moving objects;3. Detections
Variance in number of objects available is taken care of (1 & 2)
Variance in detections is Binomial given #objects available (1 & 2)
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Law of Total Variance:
b is the grid placement: Mean and variance of
#detections, n, given grid placement, is all that’s needed.
![Page 34: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/34.jpg)
Striplet variance estimator:
![Page 35: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/35.jpg)
Simulation Results:
3 habitat types but no clustering
Clustering included
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Simulation Results:Red lines give correct answers
![Page 37: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/37.jpg)
Simulation Results:Ignoring the systematic design:appalling performance!
![Page 38: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/38.jpg)
Simulation Results:Post-stratification:improvement but still clear bias
![Page 39: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/39.jpg)
Simulation Results:Striplet method: huge improvement!
![Page 40: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/40.jpg)
Striplet method: huge improvement!
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Spotted Hyena in the Serengeti
![Page 43: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/43.jpg)
Spotted Hyena in the Serengeti
Short grass plains: prey herds congregate in wet season
Long grass plains: unattractive in wet season
![Page 44: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/44.jpg)
Spotted Hyena in the Serengeti
Wet season: non-territorial ‘commuters’ (n=186)
Dry season: territorial residents (n=53)
![Page 45: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/45.jpg)
Wet season: highly clustered.cv(n/L) is:
- 17% ignoring systematic design- 14% using poststratification- 7% using striplets!
Overall cv(D) is:- 20% ignoring systematic design- 17% using poststratification- 11% using striplets
The estimator matters!
![Page 46: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/46.jpg)
Dry season: not clustered; small ncv(n/L) is:
- 15% ignoring systematic design- 12% using poststratification- 13% using striplets
Overall cv(D) is:- 23% ignoring systematic design- 20% using poststratification- 21% using striplets
Not much difference
![Page 47: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/47.jpg)
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In Revision, Biometrics
![Page 49: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/49.jpg)
1. For a systematic design, variance estimators based on random lines are not adequate for trended or clustered populations
2. Post-stratification improves estimation for trended pops, but far from perfect
3. New ‘striplet’ method huge improvement in all line/strip situations trialled to date
Variance can be highly overestimated
Conclusions
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Striplet variance estimator:
B is the number of possible grids, in discrete approximation
j is fitted #objects in striplet j
gj(b) is fitted P(detection) in striplet j
![Page 52: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/52.jpg)
Williams & Thomas, JCRM 2008
Application: British Columbia multi-species marine survey
Select species with greatest and least trends in encounter rate for illustration
![Page 53: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/53.jpg)
Greatest trend: Dall’s Porpoise
Highest encounter
rates on short lines
Worst case!
![Page 54: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/54.jpg)
Least trend: floating plastic garbage
No trend in encounter rate with line length
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ResultsDall’s Porpoise: previous reported
CV=31%Stratified methods: reported CV=19%
Estimated CV=31% using Poisson-based estimator with no adjustment for systematic lines
Estimated CV=19% using design-based estimator with post-stratification and overlapping strata
![Page 56: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/56.jpg)
ResultsFloating garbage: previous reported
CV=15%Stratified methods: reported CV=14%
For untrended population, there is little difference in the different estimators
![Page 57: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/57.jpg)
![Page 58: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/58.jpg)
But space is continuous!
As a strip changes position very slightly...
... it still covers many of the same objects.
![Page 59: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/59.jpg)
But space is continuous!
As a strip changes position very slightly...
... it still covers many of the same objects.
Idea:1. Divide the region into hundreds of tiny
‘striplets’2. Allow the number of objects available in each
striplet to be random variables X1 , X2 , …, XJ
3. The number of objects available in any full strip is the sum of the objects in the constituent striplets
![Page 60: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/60.jpg)
1. Divide the region into hundreds of tiny ‘striplets’2. Number of objects available in striplets 1, 2, …, J
is X1 , X2 , …, XJ
3. Number of objects available in any full strip is the sum of the objects in the constituent striplets.
Str
iple
t #
ob
ject
s availa
ble
striplet position 0
1
2
3
4
Expected number of objects per striplet
Random number of objects per striplet, X1 , X2 , …, XJ
~ Multinomial
![Page 61: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/61.jpg)
1. Divide the region into hundreds of tiny ‘striplets’2. Number of objects available in striplets 1, 2, …, J
is X1 , X2 , …, XJ
3. Number of objects available in any full strip is the sum of the objects in the constituent striplets.Full strip at this
position: 10 objects
Full strip at next position: 7 objects
Full strip at next position: 8 objects
... etc
Str
iple
t #
ob
ject
s availa
ble
striplet position 0
1
2
3
4
![Page 62: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/62.jpg)
1. Trends in object density across the region Observed number of detections per unit search area
#d
ete
ctio
ns
/ u
nit
are
a
Points correspond to observed transects
Fit a GAM to give a fitted object density for any search strip in the region
x-coordinate
![Page 63: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/63.jpg)
1. Trends in object density across the region#
dete
ctio
ns
/ u
nit
are
a
Fit a GAM to give a fitted object density for any search strip in the region
x-coordinate
For any new grid placement, we now have an expected number of objects available for that grid
![Page 64: Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649cd65503460f9499d143/html5/thumbnails/64.jpg)
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