Some thoughts on density surface updating 1.Major Updates every X years: refitting models (perhaps...
18
Some thoughts on density surface updating 1. Major Updates every X years: refitting models (perhaps new kinds of models) to accumulated data over large areas. 2. Minor Updates as and when new (reliable) survey estimates become available: Use new estimates to modify existing surfaces locally (spatially and temporally). a. Should be quick and easy. b. Must accommodate any kind of density surface model (including stratified density estimates).
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Transcript of Some thoughts on density surface updating 1.Major Updates every X years: refitting models (perhaps...
Some thoughts on density surface updating
1. Major Updates every X years: refitting models (perhaps new kinds of models) to accumulated data over large areas.
2. Minor Updates as and when new (reliable) survey estimates become available: Use new estimates to modify existing surfaces locally (spatially and temporally).
a. Should be quick and easy.b. Must accommodate any kind of density surface
model (including stratified density estimates).
Bayesian Update?
• Problem 1: How accommodate any kind of density surface model? (Models could have few/no common parameters.)
• Solution 1: Update densities cell-by-cell rather than parameters.– Assume (say) lognormal distribution of density.– Need variance-covariance matrix of cell densities.– (End product is a weighted average.)
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Density, abundance etc
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Mean=1, Variance=1
Mean=5, Variance=25
Mean=1.88, Variance=1.46
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Updating a cell
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Density, abundance etc
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Mean=1, Variance=1
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0.0
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Density, abundance etc
Pro
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Mean=1, Variance=1
Mean=5, Variance=25
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Multi-cell Example
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Blainvilles whale: prior
Longitude (dec. deg.)
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Blainvilles whale: new data
Longitude (dec. deg.)
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g.)
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Multi-cell Example
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Blainvilles whale: prior
Longitude (dec. deg.)
La
titu
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(d
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g.)
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30
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Blainvilles whale: new data
Longitude (dec. deg.)
La
titu
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(d
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de
g.)
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Blainvilles whale: posterior prediction
Longitude (dec. deg.)
La
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(d
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g.)
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Bayesian Update?
• Problem 2: How get smooth edges?
• Solution 2: Smooth via weights decaying with distance from edge– Easy to smooth; not so easy to
find the best amount of smoothing.
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Survey
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Prior
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Updated Surface on Prior
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Smoothed Surface on Prior
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UPDATE
SMOOTHING
Another Example
• Survey Estimates update current surface at single point in time.
• Two-stage updating:– Bayesian Update within survey
region.– Smoothing across survey region
boundary.
• Current surface and Updated surface apply for whole season (quarter), then jump/drop suddenly to that for next season.
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Survey
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Prior
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Updated Surface on Prior
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Smoothed Surface on Prior
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UPDATE
SMOOTHING
Survey Estimate Current Density
Updated, Unsmoothed
Updated, Smoothed
1. Smoothing Current Density (Example at single location)
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Current Density Smoothed
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Smoothing: • Periodic smooth (interpolate current estimates)• Take account of uncertainty in fitting and reflect in fit. (not shown above)
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Month
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ty2. Updating with New Survey Estimate
(Example: single location, Month 2)
Current Smoothed Estimate at Month=2
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Survey
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Prior
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Updated Surface on Prior
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Smoothed Surface on Prior
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UPDATE
SMOOTHING
2 4 6 8 10 12
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Survey
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Prior
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Updated Surface on Prior
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0.35
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Smoothed Surface on Prior
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0.30
0.35
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UPDATE
SMOOTHING
2 4 6 8 10 12
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68
10
Survey
0.20
0.25
0.30
0.35
2 4 6 8 10 12
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68
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Prior
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0.30
0.35
2 4 6 8 10 12
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Updated Surface on Prior
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0.30
0.35
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Smoothed Surface on Prior
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0.30
0.35
0.20
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0.30
0.35
UPDATE
SMOOTHING
2 4 6 8 10 12
24
68
10
Survey
0.20
0.25
0.30
0.35
2 4 6 8 10 12
24
68
10
Prior
0.20
0.25
0.30
0.35
2 4 6 8 10 12
24
68
10
Updated Surface on Prior
0.20
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0.30
0.35
0.20
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0.30
0.35
2 4 6 8 10 12
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68
10
Smoothed Surface on Prior
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0.35
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0.25
0.30
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UPDATE
SMOOTHING
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0 2 4 6 8 10 12
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Updated estimate
2. Updating with New Survey Estimate (Example: single location, Month 2)
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Survey
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Prior
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0.35
2 4 6 8 10 12
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Updated Surface on Prior
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0.30
0.35
2 4 6 8 10 12
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Smoothed Surface on Prior
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0.30
0.35
0.20
0.25
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0.35
UPDATE
SMOOTHING
2 4 6 8 10 12
24
68
10
Survey
0.20
0.25
0.30
0.35
2 4 6 8 10 12
24
68
10
Prior
0.20
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0.30
0.35
2 4 6 8 10 12
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68
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Updated Surface on Prior
0.20
0.25
0.30
0.35
0.20
0.25
0.30
0.35
2 4 6 8 10 12
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68
10
Smoothed Surface on Prior
0.20
0.25
0.30
0.35
0.20
0.25
0.30
0.35
UPDATE
SMOOTHING
2 4 6 8 10 12
24
68
10
Survey
0.20
0.25
0.30
0.35
2 4 6 8 10 12
24
68
10
Prior
0.20
0.25
0.30
0.35
2 4 6 8 10 12
24
68
10
Updated Surface on Prior
0.20
0.25
0.30
0.35
0.20
0.25
0.30
0.35
2 4 6 8 10 12
24
68
10
Smoothed Surface on Prior
0.20
0.25
0.30
0.35
0.20
0.25
0.30
0.35
UPDATE
SMOOTHING
2 4 6 8 10 12
24
68
10
Survey
0.20
0.25
0.30
0.35
2 4 6 8 10 12
24
68
10
Prior
0.20
0.25
0.30
0.35
2 4 6 8 10 12
24
68
10
Updated Surface on Prior
0.20
0.25
0.30
0.35
0.20
0.25
0.30
0.35
2 4 6 8 10 12
24
68
10
Smoothed Surface on Prior
0.20
0.25
0.30
0.35
0.20
0.25
0.30
0.35
UPDATE
SMOOTHING
New Survey Estimate
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Month
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0 2 4 6 8 10 12
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Month
De
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ty2. Updating with New Survey Estimate
(Example: single location, Month 2)
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Updated Smooth
Another update and Smooth
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And One More...
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All Together
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Cumulative Effect
• Smooth Temporal Updating:– Degree of Smoothness: more flexible with more data.– Effect of update decays with temporal “distance”.– Uncertainty in new smooth (not shown above) updated smoothly
in time too.
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nsity
0 5 10 15 20
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lat
lon
Multiple Locations
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0 2 4 6 8 10 120
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34
56
Month
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etc....
3. Spatio-Temporal Interaction
• How similar in space should temporal effects be? Constraining temporal smooths closer together to be more similar will prevent inconsistent temporal trends at locations near each other. (Unclear in advance how necessary this will prove to be.)
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