Post on 30-Mar-2015
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
ba
bilit
y de
nsi
ty
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
ba
bilit
y de
nsi
ty
Mean=1, Variance=1
0 2 4 6 8
0.0
0.5
1.0
1.5
Density, abundance etc
Pro
ba
bilit
y de
nsi
ty
Mean=1, Variance=1
Mean=5, Variance=25
4
Multi-cell Example
-164 -162 -160 -158 -156 -154 -152
16
18
20
22
24
26
Blainvilles whale: prior
Longitude (dec. deg.)
La
titu
de
(d
ec.
de
g.)
20
25
30
35
40
45
50
-162 -160 -158 -156 -154
16
18
20
22
24
26
Blainvilles whale: new data
Longitude (dec. deg.)
La
titu
de
(d
ec.
de
g.)
4
6
8
10
12
5
Multi-cell Example
-164 -162 -160 -158 -156 -154 -152
16
18
20
22
24
26
Blainvilles whale: prior
Longitude (dec. deg.)
La
titu
de
(d
ec.
de
g.)
20
25
30
35
40
45
50
-162 -160 -158 -156 -154
16
18
20
22
24
26
Blainvilles whale: new data
Longitude (dec. deg.)
La
titu
de
(d
ec.
de
g.)
4
6
8
10
12
-162 -160 -158 -156 -154
16
18
20
22
24
26
Blainvilles whale: posterior prediction
Longitude (dec. deg.)
La
titu
de
(d
ec.
de
g.)
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
De
nsi
ty
0 2 4 6 8 10 12
01
23
45
6
Month
De
nsi
ty
Current Density Smoothed
0 2 4 6 8 10 12
01
23
45
6
Month
De
nsi
ty
0 2 4 6 8 10 12
01
23
45
6
Month
De
nsi
ty
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
De
nsi
ty2. Updating with New Survey Estimate
(Example: single location, Month 2)
Current Smoothed Estimate at Month=2
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
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
0 2 4 6 8 10 12
01
23
45
6
Month
De
nsi
ty
0 2 4 6 8 10 12
01
23
45
6
Month
De
nsi
ty
Updated estimate
2. Updating with New Survey Estimate (Example: single location, Month 2)
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
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
0 2 4 6 8 10 12
01
23
45
6
Month
De
nsi
ty
0 2 4 6 8 10 12
01
23
45
6
Month
De
nsi
ty2. Updating with New Survey Estimate
(Example: single location, Month 2)
0 2 4 6 8 10 12
01
23
45
6
Month
De
nsi
ty
Updated Smooth
Another update and Smooth
0 2 4 6 8 10 12
01
23
45
6
Month
De
nsi
ty
And One More...
0 2 4 6 8 10 12
01
23
45
6
Month
De
nsi
ty
All Together
0 2 4 6 8 10 12
01
23
45
6
Month
De
nsi
ty
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
De
nsity
0 5 10 15 20
05
10
15
20
lat
lon
Multiple Locations
0 2 4 6 8 10 12
01
23
45
6
Month
De
nsi
ty
0 2 4 6 8 10 120
12
34
56
Month
De
nsi
ty
0 2 4 6 8 10 12
01
23
45
6
Month
De
nsi
ty
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
De
nsi
ty