Transcript Defining and fitting matrix population models

Embedding population
dynamics models in
inference
S.T. Buckland, K.B. Newman, L. Thomas
and J Harwood (University of St Andrews)
Carmen Fernández
(Oceanographic Institute, Vigo, Spain)
AIM
A generalized methodology for
defining and fitting matrix
population models that
accommodates process variation
(demographic and environmental
stochasticity), observation error
and model uncertainty
Hidden process models
Special case:
state-space models
(first-order Markov)
States
We categorize animals by their
state, and represent the population
as numbers of animals by state.
Examples of factors that determine state:
age; sex; size class; genotype;
sub-population (metapopulations);
species (e.g. predator-prey models,
community models).
States
Suppose we have m states
at the start of year t. Then
numbers of animals by state are:
 n1,t 
n 
 2 ,t 
n t   n3,t 
 
  
 nm ,t 
 
NB: These numbers
are unknown!
Intermediate states
The process that updates nt to nt+1
can be split into ordered sub-processes.
e.g. survival
ageing
births:
nt  u s,t  ua,t  ub,t  nt 1
This makes model definition much simpler
Survival sub-process
Given nt:
 E(u s ,1,t )  1 0
 E(u )  
0 2
s , 2 ,t 



    

 
E(u s ,m ,t )  0 0
us, j ,t ~ binomial(n j ,t , j )
0   n1,t 
 
 0   n2,t 
    
 
 m  nm ,t 

j  1,, m
NB a model (involving hyperparameters) can be specified for 
or  can be modelled as a random effect
Survival sub-process
Survival
n1,t
n2,t
nm , t
1
2
m
us,1,t
us , 2,t
u s , m ,t
Ageing sub-process
Given
us,t:
No first-year animals left!
 u s ,1,t 
 ua , 2,t  1 0  0 0 

u
s , 2 ,t 
u  


 a , 3, t    0 1  0 0    
        


 
 u s ,m 1,t 
ua ,m ,t  0 0  1 1  u

s
,
m
,
t


NB process is deterministic
Ageing sub-process
Age incrementation
us,1,t
us, 2,t
us,m1,t
u s , m ,t
ua, 2,t
ua,3,t
u a , m ,t
Birth sub-process
Given ua,t:
 E (n1,t 1 ) 2
 n
 
 2,t 1   1
 n3,t 1    0

 
   
 nm ,t 1   0


New first-year
animals
3  m 
0

0
1



0

0

1
  u a , 2 ,t 
  u a , 3, t 


  
 u 
  a ,m,t 
m
e.g. n1,t 1   multinomial(ua , j ,t , p0 j , p1 j ,)
j 2
with
p
ij
i
 1,
 ip
ij
i
NB a model may be specified for 
 j
Birth sub-process
Births
ua, 2,t
ua,3,t
u a , m ,t
2
3
m
n1,t
n2,t
n3,t
nm , t
The BAS model
E(nt 1 | nt , θ)  BASnt
where
 2
1

B0


 0
3  m 
0
1



0


1 0
0 
2
S
 

0 0
0 
0

 
1 
0
 0 
 

 m 

1
0
A


0
0
1

0




λ
θ   
φ
0
0

1
0
0


1
The BAS
model
Leslie matrix
The product BAS is a Leslie projection matrix:
12  2 3

0
1

BAS   0
2


 
 0
0
  m1m  m m 

0
0 

0
0 



 
  m1
 m 
Other processes
Growth:
0
1   1
 
 1 1 2
 0
2
G

 
 0
0

0
 0

0
0

0
0


0

0

  m  2 1   m 1

0
 m 1
0
0
0


0

1
The BGS model with m=2
Lefkovitch matrix
The product BGS is a Lefkovitch projection matrix:
1   1  
BGS  


0 1   
0 1 0 



1  0  2 
(1     )1  2 


1
2 

Sex assignment
 E (u x ,1,t )   
 E (u ) 
1
x , 2 ,t 



 u x , 3, t   0

 
 u x , 4 ,t   0
0 0
 ub ,1,t 

0 0 

u b , 2 ,t 

1 0
 ub ,3,t 
0 1
 u x ,1,t ~ binomial(ub ,1,t ,  ) 


 u x , 2,t  ub ,1,t  u x ,1,t

u


u
b , 2 ,t
 x ,3,t

u


u
x
,
4
,
t
b
,
3
,
t


New-born
Adult female
Adult male
Genotype assignment
Movement
e.g. two age groups in each of two locations
0
 21
0 
1  12
 0

1


0

1 2
21 

V
 12
0
1   21
0 


12
0
1   21 
 0
Movement: BAVS model
Observation equation
E(y t | nt , θ)  Ot nt
e.g. metapopulation with two sub-populations,
each split into adults and young,
unbiased estimates of total abundance
of each sub-population available:
 n 01, t 


 E ( y1, t )  1 1 0 0  n11, t 
E (y t )  


 n 
E
(
y
)
0
0
1
1
2, t 

 02 , t



 n12 , t 
Fitting models to time series of data
• Kalman filter
Normal errors, linear models
or linearizations of non-linear models
• Markov chain Monte Carlo
• Sequential Monte Carlo methods
Elements required for Bayesian
inference
g (θ)
g 0 (n0 | θ)
gt (nt | nt 1 ,...,n 0 , θ)
f t (y t | nt , θ)
Prior for parameters
pdf (prior) for initial state
pdf for state at time t
given earlier states
Observation pdf
Bayesian inference
Joint prior for θ and the nt :
T
g (θ)  g 0 (n 0 | θ)   g t (n t | n t 1 ,, n 0 , θ)
t 1
Likelihood:
T
 f (y
t
t
| n t , θ)
t 1
Posterior:
T
g (n 0 ,, n T , θ | y1 ,, y T ) 
g (θ)  g 0 (n 0 | θ)   g t (n t | n t 1 ,, n 0 , θ)  f t (y t | n t , θ)
t 1
f (y1 ,, y T )
Types of inference
Filtering:
g (nt , θ | y1 ,, y t )
Smoothing:
g (nt , θ | y1 ,, yT )
One step ahead prediction:
g (nt , θ | y1 , , y t 1 )
Generalizing the framework
g (M)
g (θ | M)
g0 (n0 | θ, M)
Model prior
Prior for parameters
pdf (prior) for initial state
gt (nt | nt 1 ,...,n0 , θ, M) pdf for state at time t
given earlier states
ft (y t | nt , θ, M)
Observation pdf
Generalizing the framework
Replace
by
E(nt 1 | nt , θ)  Pt nt
nt 1  Pt (nt )
where Pt (nt )  PK ,t (PK 1,t (P1,t (nt )))
and Pk ,t () is a possibly random operator
Example: British grey seals
British grey seal
breeding colonies
British grey seals
• Hard to survey outside of
breeding season: 80% of
time at sea, 90% of this
time underwater
• Aerial surveys of
breeding colonies since
1960s used to estimate
pup production
• (Other data: intensive
studies, radio tracking,
genetic, counts at haulouts)
• ~6% per year overall
increase in pup
production
Estimated pup production
1980
1990
10000 15000
2000
1960
1970
1980
Y ear
inner hebrides
north sea
Pup count
1990
2000
0
5000
0
2000
10000 15000
Y ear
1990
5000
1970
10000 15000
1960
Pup count
5000
0
5000
Pup count
10000 15000
outer hebrides
0
Pup count
orkney
1960
1970
1980
Y ear
1990
2000
1960
1970
1980
Y ear
Questions
• What is the future population trajectory?
• What types of data will help address this
question?
• Biological interest in birth, survival and
movement rates
Empirical predictions
outer hebrides
10000
2000
1970
1980
1990
2000
2010
1960
1970
1980
1990
Year
Year
inner hebrides
north sea
2000
2010
2000
2010
3000
1000
1500
2500
Pup count
3500
5000
1960
Pup count
6000
Pup count
15000
5000
Pup count
14000
orkney
1960
1970
1980
1990
Year
2000
2010
1960
1970
1980
1990
Year
Population dynamics model
• Predictions constrained to be biologically
realistic
• Fitting to data allows inferences about
population parameters
• Can be used for decision support
• Framework for hypothesis testing (e.g. density
dependence operating on different processes)
Grey seal state model:
states
• 7 age classes
– pups (n0)
– age 1 – age 5 females (n1-n5)
– age 6+ females (n6+) = breeders
• 48 colonies – aggregated into 4 regions
Grey seal state model:
processes
• a “year” starts just after the breeding season
• 4 sub-processes
–
–
–
–
survival
age incrementation
movement of recruiting females
breeding
survival
na,c,t-1
us,a,c,t
age
ui,a,c,t
movement
breeding
um,a,c,t
na,c,t
Grey seal state model:
survival
• density-independent adult survival
us,a,c,t ~ Binomial(na,c,t-1,φadult) a=1-6
• density-dependent pup survival
us,0,c,t ~ Binomial(n0,c,t-1, φ juv,c,t)
where
φ juv,c,t= φ juv.max/(1+βcn0,c,t-1)
Grey seal state model:
age incrementation and sexing
• ui,1,c,t ~Binomial (us,0,c,t , 0.5)
• ui,a+1,c,t = us,a,c,t a=1-4
• ui,6+,c,t = us,5,c,t + us,6+,c,t
Grey seal state model:
movement of recruiting females
• females only move just before breeding for the
first time
• movement is fitness dependent
– females move if expected survival of offspring is
higher elsewhere
• expected proportion moving proportional to
– difference in juvenile survival rates
– inverse of distance between colonies
– inverse of site faithfulness
Grey seal state model:
movement
• (um,5,c→1,t, ... , um,5,c→4,t) ~
Multinomial(ui,5,c,t, ρc→1,t, ... , ρc→4,t)
• ρc→i,t =θc→i,t / Σj θc→j,t
• θc→i,t =
– γsf when c=i
– γdd max([φjuv,i,t-φjuv,c,t],0)/exp(γdistdc,i) when c≠i
Grey seal state model:
breeding
• density-independent
• ub,0,c,t ~ Binomial(um,6+,c,t , α)
Grey seal state model:
matrix formulation
• E(nt|nt-1, Θ) ≈ B Mt A St nt-1
Grey seal state model:
matrix formulation
• E(nt|nt-1, Θ) ≈ Pt nt-1
Grey seal observation model
• pup production estimates normally distributed,
with variance proportional to expectation:
y0,c,t ~ Normal(n0,c,t , ψ2n0,c,t)
Grey seal model:
parameters
•
•
•
•
survival parameters: φa, φjuv.max, β1 ,..., βc
breeding parameter: α
movement parameters: γdd, γdist, γsf
observation variance parameter: ψ
• total 7 + c (c is number of regions, 4 here)
Grey seal model:
prior distributions
Posterior parameter estimates
phi.juv.max 0.734
alpha 0.973
psi 0.07
20
0.93
0.95
10
0.97
0.6
0.8
0.9
0.92
0.96
gamma.dist 0.792
0.06
gamma.sf 0.355
0.07
0.08
0.09
beta.ns 0.000906
2.0
1.0
0.8
0.4
0.20
2
4
6
8
10
14
0.5
2.5
0.2
beta.oh 0.000304
0.6
1.0
1.4
beta.ork 0.000183
0.0008
0.0014
0.0020
0
0
0
500
2000
4000
4000
1000
8000
6000
1500
beta.ih 0.00127
1.5
0
0.0
0.0
0.0
500
0.10
1500
0.30
gamma.dd 3.32
0.7
0
0
0
0
1
10
10
2
20
30
3
30
4
50
30
5
40
phi.adult 0.966
0.0002
0.0004
0.00010
0.00020
0.00030
0.0006
0.0010
0.0014
Smoothed pup estimates
1500
Pups
3500
3000
Inner Hebrides
1500
1990
1995
2000
1985
1990
1995
Year
Outer Hebrides
Orkneys
Pups
12000
Year
2000
6000
8000
Pups
1985
16000
Pups
North Sea
1985
1990
1995
Year
2000
1985
1990
1995
Year
2000
Predicted adults
Inner Hebrides
10000
7000
Adults
13000
9000
2008
2004
2012
2008
Year
Year
Outer Hebrides
Orkneys
2012
25000
40000
Adults
Adults
40000
2004
60000
Adults
North Sea
2004
2008
Year
2012
2004
2008
Year
2012
Seal model
• Other state process models
– More realistic movement models
– Density-dependent fecundity
– Other forms for density dependence
• Fit model at the colony level
• Include observation model for pup counts
• Investigate effect of including additional data
–
–
–
–
data on vital rates (survival, fecundity)
data on movement (genetic, radio tagging)
less frequent pup counts?
index of condition
• Simpler state models
References
Buckland, S.T., Newman, K.B., Thomas, L. and Koesters, N.B.
2004. State-space models for the dynamics of wild animal
populations. Ecological Modelling 171, 157-175.
Thomas, L., Buckland, S.T., Newman, K.B. and Harwood, J.
2005. A unified framework for modelling wildlife population
dynamics. Australian and New Zealand Journal of Statistics
47, 19-34.
Newman, K.B., Buckland, S.T., Lindley, S.T., Thomas, L. and
Fernández, C. 2006. Hidden process models for animal
population dynamics. Ecological Applications 16, 74-86.
Buckland, S.T., Newman, K.B., Fernández, C., Thomas, L.
and Harwood, J. Embedding population dynamics models in
inference. Submitted to Statistical Science.