some thoughts on density surface updating
DESCRIPTION
Some thoughts on density surface updating. Major Updates every X years: refitting models (perhaps new kinds of models) to accumulated data over large areas. - PowerPoint PPT PresentationTRANSCRIPT
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.)
0 2 4 6 8
0.0
0.5
1.0
1.5
Density, abundance etc
Pro
babi
lity
dens
ity
Mean=1, Variance=1
Mean=5, Variance=25
Mean=1.88, Variance=1.46
3
Updating a cell
0 2 4 6 8
0.0
0.5
1.0
1.5
Density, abundance etc
Pro
babi
lity
dens
ity
Mean=1, Variance=1
0 2 4 6 8
0.0
0.5
1.0
1.5
Density, abundance etc
Pro
babi
lity
dens
ity
Mean=1, Variance=1
Mean=5, Variance=25
4
Multi-cell Example
-164 -162 -160 -158 -156 -154 -152
1618
2022
2426
Blainvilles whale: prior
Longitude (dec. deg.)
Latit
ude
(dec
. deg
.)
20
25
30
35
40
45
50
-162 -160 -158 -156 -154
1618
2022
2426
Blainvilles whale: new data
Longitude (dec. deg.)
Latit
ude
(dec
. deg
.)
4
6
8
10
12
5
Multi-cell Example
-164 -162 -160 -158 -156 -154 -152
1618
2022
2426
Blainvilles whale: prior
Longitude (dec. deg.)
Latit
ude
(dec
. deg
.)
20
25
30
35
40
45
50
-162 -160 -158 -156 -154
1618
2022
2426
Blainvilles whale: new data
Longitude (dec. deg.)
Latit
ude
(dec
. deg
.)
4
6
8
10
12
-162 -160 -158 -156 -154
1618
2022
2426
Blainvilles whale: posterior prediction
Longitude (dec. deg.)
Latit
ude
(dec
. deg
.)
10
15
20
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.
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
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.
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
Survey Estimate Current Density
Updated, Unsmoothed
Updated, Smoothed
1. Smoothing Current Density (Example at single location)
0 2 4 6 8 10 12
01
23
45
6
Month
Den
sity
0 2 4 6 8 10 12
01
23
45
6
Month
Den
sity
Current Density Smoothed
0 2 4 6 8 10 12
01
23
45
6
Month
Den
sity
0 2 4 6 8 10 12
01
23
45
6
Month
Den
sity
Smoothing: • Periodic smooth (interpolate current estimates)• Take account of uncertainty in fitting and reflect in fit. (not shown above)
0 2 4 6 8 10 12
01
23
45
6
Month
Den
sity
2. Updating with New Survey Estimate (Example: single location, Month 2)
Current Smoothed Estimate at Month=2
2 4 6 8 10 1 2
24
68
10
Survey
0.20
0.25
0.30
0.35
2 4 6 8 1 0 12
24
68
10
Pr ior
0.2 0
0.2 5
0.3 0
0.3 5
2 4 6 8 10 12
24
68
10
Upda ted 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
Sur vey
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
Up dated 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 1 2
24
68
10
Survey
0.20
0.25
0.30
0.35
2 4 6 8 1 0 12
24
68
10
Pr ior
0.2 0
0.2 5
0.3 0
0.3 5
2 4 6 8 10 12
24
68
10
Upda ted 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
Sur vey
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
Up dated 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
0 2 4 6 8 10 12
01
23
45
6
Month
Den
sity
0 2 4 6 8 10 12
01
23
45
6
Month
Den
sity
Updated estimate
2. Updating with New Survey Estimate (Example: single location, Month 2)
2 4 6 8 10 1 2
24
68
10
Survey
0.20
0.25
0.30
0.35
2 4 6 8 1 0 12
24
68
10
Pr ior
0.2 0
0.2 5
0.3 0
0.3 5
2 4 6 8 10 12
24
68
10
Upda ted 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
Sur vey
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
Up dated 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 1 2
24
68
10
Survey
0.20
0.25
0.30
0.35
2 4 6 8 1 0 12
24
68
10
Pr ior
0.2 0
0.2 5
0.3 0
0.3 5
2 4 6 8 10 12
24
68
10
Upda ted 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
Sur vey
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
Up dated 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
0 2 4 6 8 10 12
01
23
45
6
Month
Den
sity
0 2 4 6 8 10 12
01
23
45
6
Month
Den
sity
2. Updating with New Survey Estimate (Example: single location, Month 2)
0 2 4 6 8 10 12
01
23
45
6
Month
Den
sity
Updated Smooth
Another update and Smooth
0 2 4 6 8 10 12
01
23
45
6
Month
Den
sity
And One More...
0 2 4 6 8 10 12
01
23
45
6
Month
Den
sity
All Together
0 2 4 6 8 10 12
01
23
45
6
Month
Den
sity
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.
0 2 4 6 8 10 12
01
23
45
6
Month
Den
sity
0 5 10 15 20
05
1015
20
lat
lon
Multiple Locations
0 2 4 6 8 10 12
01
23
45
6
Month
Den
sity
0 2 4 6 8 10 120
12
34
56
Month
Den
sity
0 2 4 6 8 10 12
01
23
45
6
Month
Den
sity
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.)
0 2 4 6 8 10 12
01
23
45
6
Month
Den
sity
0 2 4 6 8 10 12
01
23
45
6
Month
Den
sity
0 2 4 6 8 10 12
01
23
45
6
Month
Den
sity