unmarked individuals: Guelph

Background Model deﬁnitions Tools Tests

Estimating demographic parameters from
samples of unmarked individuals

Ben Bolker

Departments of Mathematics & Statistics and Biology, McMaster University

8 March 2011

Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals


Outline
1 Background
Ecological/statistical motivations
Study system
2 Model deﬁnitions
3 Tools
State-space models
Markov chain Monte Carlo
Gibbs/block sampling
4 Tests
Perfect observation, trivial dynamics
Perfect observation, non-trivial dynamics
Imperfect observation



Cryptic density-dependence

Environmental variation
+ correlated variation in
density . . .
= cryptic
density-dependence
best possible estimates of
demographic parameters?
(Wilson & Osenberg 2002, Shima &
Osenberg 2003)



Data

Repeated samples of individual animals
. . . Multiple times, observers, locations
Presence and size (imperfect)
Estimate demographic parameters:
birth/immigration
death/emigration
change (growth)
. . . as a function of size, (population density)



Study system

Natural history

Thalassoma hardwicke
(sixbar wrasse)
Settlement: ≈ 5–10 mm
Growth: a few mm per month
Death: by predator
(competition for refuges)
Immigration/emigration:
photo: Leonard Low, Flickr via species.wikimedia.org rare below 40–50 mm



Study system

Where they live

Patch reefs, French
Polynesia (and
throughout the south
Paciﬁc)



Study system

What they look like to me

80
Estimated fish size (mm)

60

40

20

Feb Mar Apr May Jun Jul

Date



Study system

40
What they look like to me

20

Feb Mar Apr


Study system

Outline
1 Background
Study system
3 Tools
State-space models
4 Tests


Demographic parameters as f (size)
Growth Appearance Persistence
1.0

0.8

0.6
Probability
(per day)
0.4

a.settle
l.grow b.settle a.mort
0.2 tot.settle c.mort

0.0
1020304050607080 1020304050607080 1020304050607080
Fish size (mm)



Observation model

Detection Measurement
(probability) (standard deviation)

0.8
3.5

0.6 3.0

2.5 Measurement error
Double−count
0.4 Zero−count
2.0

1.5
0.2
1.0

10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80
Fish size (mm)


State-space models

State-space models

Problem: estimating parameters of systems with un- or
imperfectly-observed states?
Easy if no feedback (measurement error models)
With feedback (dynamic models), brute force approaches
become infeasible . . .



State-space models

Directed acyclic graph (DAG)

parameters

N1 N2 N3 N4 N5

obs1 obs2 obs3 obs4 obs5




Bayesian statistics, in 30 seconds

computational (convenience) Bayesian statistics
(forget all that stuﬀ about philosophy of statistics, subjectivity, incorporating prior information . . . )

have a likelihood, L(θ) = Prob(data|θ)
want a posterior probability, Ppost (θ) = Prob(θ|data)
Bayes’ rule:

L(θ) · prior(θ)
Ppost (θ) =
L(θ ) · prior(θ ) dθ

may want mean, mode, conﬁdence intervals, . . . denominator
very high-dimensional





θA
Posterior probability

MCMC rule:
P(θ A ) J(θ B → θ A )
if =
P(θ B ) J(θ A → θ B )

then stationary distribution =
posterior probability
(provided chain is irreducible etc.)
Parameter value





θA

MCMC rule:
P(θ A ) J(θ B → θ A )
if =
P(θ B ) J(θ A → θ B )

q q
Parameter value





θA

MCMC rule:
P(θ A ) J(θ B → θ A )
if =
P(θ B ) J(θ A → θ B )

q qq
Parameter value





θA

MCMC rule:
P(θ A ) J(θ B → θ A )
if =
P(θ B ) J(θ A → θ B )

q qqq

Parameter value





θA

MCMC rule:
P(θ A ) J(θ B → θ A )
if =
P(θ B ) J(θ A → θ B )

q q q qq
q q qqq

Parameter value





θA

MCMC rule:
P(θ A ) J(θ B → θ A )
if =
P(θ B ) J(θ A → θ B )


Parameter value





θA

MCMC rule:
P(θ A ) J(θ B → θ A )
if =
P(θ B ) J(θ A → θ B )
θB

Parameter value




Metropolis-(Hastings) updating

θA

Metropolis rule:

accept B with probability
P(θ B )
min 1,
P(θ A )

satisﬁes MCMC rule . . . Don’t
need to know denominator!
Candidate distribution
Parameter value




Gibbs sampling

Joint distribution of
parameters often diﬃcult
Condition each element
on “known” (i.e. imputed)
values of all other
elements: P(a, b, c) =
P(a|b, c)P(b, c)
0.5

1
Gibbs sampling: do this
5
2.

1.5
2

repeatedly for each
5
1.

2.5
3

element, or block of
5
0.

3.5
4.5

4 3
4 3.5 2
elements
1




Gibbs sampling

Joint distribution of
parameters often diﬃcult
q
Condition each element
q q

q
q
on “known” (i.e. imputed)
q
q
q q
q

q
values of all other
q q q
q q
q
qq elements: P(a, b, c) =
q q
q
P(a|b, c)P(b, c)
0.5 q

1
Gibbs sampling: do this
5
2.

q
1.5
2

repeatedly for each
5
1.

2.5
3

element, or block of
5
0.

3.5
4.5

4 3
4 3.5 2
elements
1




Back to the DAG

parameters

N1 N2 N3 N4 N5

obs1 obs2 obs3 obs4 obs5




Outline
1 Background
Study system
3 Tools
State-space models
4 Tests



DAG

parameters fates

Demographic parameters|fates:
standard M-H
Fates|parameters
N1 N2




Scenario

One ﬁsh of the same size observed
Lsurv = (1 − m)(1 − p) S on two consecutive days . . .
was it the same individual?
(1 − m)(1 − p)
P(surv|m, p) =
1 − m − p + 2mp
X
Lmort = mp(1 − p)S−1




Updating probabilities for parameters

mortality (m)

1.5 If the ﬁsh survived, then our
1.0 posterior probabilities for the
0.5 parameters are:
type
probability
immigration (p) prior
m ∼ Beta(1, 2)
density
6
posterior (0 mortalities, 1 survival)
5
4 p ∼ Beta(1, S + 1)
3
2
(0 immigrations, S
1 non-immigrations)
0
0.2 0.4 0.6 0.8
probability




Trivial example: results

mortality (m) immigration (p)

4

2(m + S(1 − m))
3 Beta(p,1,S)
S+1
density
2

1

0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
value



P(mortality & no immigration) ≈ 0.16 = 1/(S + 1);
P(survival & immigration) ≈ 0.83 = S/(S + 1);
results are simple and make sense; closed form solution is
possible but ugly . . .
combination of observations and non-observation of other
individuals provides information (could fail with non-detection)




Outline
1 Background
Study system
3 Tools
State-space models
4 Tests



Non-trivial dynamics

Multiple individuals, measured over multiple days
Still simple: no density-dependent rates,
immigration/emigration of large individuals
observations still error-free




Parameter/fate sampling

parameters Parameter sampling: M-H
updating with simple
fates(1,2) fates(2,3)
candidate distributions
Fate sampling: ???
Enumerating
N1 N2 N3 possibilities is tedious
...




M-H fate sampling

6 mm 7 mm dead
5 mm G (5) · (1 − M(5)) X M(5)
6 mm (1 − G (6)) · (1 − M(6)) G (6) · (1 − M(6)) M(6)
absent I (6) I (7) X




Sampling test

6 survives

both grow

w
5 grows Sampling
Fate True
Gibbs
6 grows

both die

10−3.5 10−3 10−2.5 10−2 10−1.5 10−1 10−0.5
Probability




Testbed

Simulate for 80 days, with a settlement rate (tot.settle) of
5% day:
29 distinct individuals, size range 5–30, total of 618 ﬁsh-days
(observations).
Sample fates only (1000 MCMC steps), ﬁxed demographic
parameters
Sampled 1000 unique fates (but only 895 unique likelihoods)




True demography30
25
20
size

15
10
5

0 20 40 60 80

day




Fate-only results

0.05
0.04

Density 0.03
0.02
0.01
0.00 q

−520 −500 −480 −460
Log−likelihood



Sampling fates, parameters, both . . .

0.25

0.20

0.15 type
fate only
Density parameters only
0.10 both

0.05

0.00
−500 −490 −480 −470 −460 −450
Log−likelihood




Parameter estimates 1
lgrow a.mort c.mort
140
3.0 0.8 120
2.5 100
2.0 0.6
80
1.5 0.4 60
1.0 40
0.2
0.5 20

−1.9 −1.7 −1.5 −1.3
−1.8 −1.6 −1.4 −1.2 −4.0 −3.5 −3.0 −2.5 0.010.015.020.025.030
0 0 0 0
density a.settle b.settle tot.settle
0.7 50
0.6 0.5
40
0.5 0.4
0.4 0.3 30
0.3 20
0.2
0.2
0.1 0.1 10

11 12 13 14 1.01.52.02.53.03.54.0 0.020.030.040.050.06




Growth Appearance Persistence
1.0 q q q q 1.00 q qqqqqqqqqqqqqqqq

0.08 a.settle

0.8 b.settle 0.98
q
q q
q
0.06 tot.settle
q
0.6 q q
q
0.96
Probability q

(per day) 0.04 q a.mort
q
0.4 l.grow
q
q q q 0.94 c.mort
q q
q q
q q q q
0.02
0.2 q
q
qqq
q

q q q q
q
q q 0.92

0.0 q 0.00 q qqqqqqqqqqqqqqqqqq q

0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30
Fish size (mm)




a.mort a.settle b.settle
−2.0 16 4.0 q

15 3.5
q
−2.5 q
q
q 14 3.0 q
q
q
13
q
q 2.5
−3.0 q q
q
q
q q q q 2.0 q
q q 12 q q q q

−3.5 q q q q 1.5
11
1.0
−4.0 10 q

c.mort lgrow tot.settle
−1.2 0.045
0.030 q 0.040
0.025 q
0.035
−1.4 q q
0.020 q
q
q
q q 0.030 q
q
q
0.015 q −1.6 q
q
q 0.025 q q
q

q
0.010 q
0.020 q q
q q
−1.8 0.015 q
0.005 q
q
q
q
0.010




Outline
1 Background
Study system
3 Tools
State-space models
4 Tests



Adding errors

parameters

fates(1,2) fates(2,3)
Back to the more typical
N1 N2 N3 graph: incorporate information
from N(t − 1), N(t + 1), and
current observation
obs1 obs2 obs3

obs parameters




New algorithm

For each individual:
Pick true state at N(t), from all possible states, conditioning
on N(t − 1)
Pick identity at time N(t + 1) from feasible set:
update probabilities
Pick corresponding observation from observed(t):
update probabilities
Calculate likelihood, M-H update




Open questions, caveats

Will it work ??
Add complexities:
Multiple populations
Spatial variation, correlation among parameters
Simplify/generalize code
Speed?




Conclusions & musings

MCMC as powerful technology
. . . can it be democratized?
Data limitations shift over time:
hierarchical models are great for “modern” (high-volume,
low-quality) data


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