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

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68

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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.

2 4 6 8 10 12

24

68

10

Survey

0.20

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0.35

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68

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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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0.35

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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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45

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Den

sity

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Current Density Smoothed

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sity

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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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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

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24

68

10

Upda ted Surface on Prior

0.20

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2 4 6 8 10 12

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68

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Smoothed Surface on Prior

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0.35

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0.30

0.35

UPDATE

SMOOTHING

2 4 6 8 10 12

24

68

10

Sur vey

0.20

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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

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0.30

0.35

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2 4 6 8 10 12

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68

10

Smoothed Surface on Prior

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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

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0.35

2 4 6 8 10 12

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68

10

Up dated Surface on Prior

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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

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0.25

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0.35

2 4 6 8 1 0 12

24

68

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Pr ior

0.2 0

0.2 5

0.3 0

0.3 5

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24

68

10

Upda ted Surface on Prior

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0.35

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0.30

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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

Sur vey

0.20

0.25

0.30

0.35

2 4 6 8 10 12

24

68

10

Prior

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0.35

2 4 6 8 10 12

24

68

10

Up dated Surface on Prior

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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

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24

68

10

Upda ted Surface on Prior

0.20

0.25

0.30

0.35

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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

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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

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

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Month

Den

sity

2. 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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0 5 10 15 20

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1015

20

lat

lon

Multiple Locations

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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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