motivation and introduction availability models mmpp ... · discussion/questions/future work 1...

Motivation and introductionAvailability models

MMPP application to minke whale survey dataDiscussion/questions/future work

Availability modelling in distance sampling

...Roland Langrock Availability modelling



1 Motivation and introduction

2 Availability modelsHidden Markov modelsPoisson processesMarkov-modulated Poisson processes

3 MMPP application to minke whale survey dataReal dataA simulation study

4 Discussion/questions/future work

Roland Langrock Availability modelling



Why availability modelling?

conventional distance sampling: g(0) = 1

g(0) 6= 1 ⇒ abundance estimates (negatively) biased

e.g. for marine mammals clearly g(0) 6= 1 in many scenarios

→ Laake et al. (1997): g(0) = 0.29 (harbor porpoise)→ Marsh and Sinclair (1989): 83% of dugongs unavailable

aim: estimate g(0) via availability modelling (i.e. replaceg(0) = 1 assumption by knowledge of the availability process)

not discussed here:

perception bias – which may also cause g(0) < 1




Why bother, i.e. why not MRDS?

Mark-recapture distance sampling: combine

DS information – for estimating the detection functionand

MR information – for estimating g(0)

may suffer from bias due to unmodelled heterogeneity, inparticular caused by the availability process

→ suppose whales’ surfacing rates vary substantially→ frequent surfacing whales have stronger effect on g(0)→ g(0) positively biased

logistically more challenging




Availability modelling – basic idea

explicitly model availability process

use separate data – e.g. from radio-tagging – to increaseamount of information on availability process

use both perpendicular and along-track line distances

estimate detection function by integrating over availabilityprocess along track line

in a nutshell: replace MR component by availability modeland separately collected information on availability




Different probabilistic availability models – an overview

static availability

→ animals either available or unavailable for the entire time theyare in detectable range

→ availability model: Bernoulli distribution (correction factorsufficient)

discrete/instantaneous availability

→ animals available for instants between periods of unavailability→ availability model: Poisson process→ often inadequate (clustered availability events)

intermittent availability

→ periods of availability and unavailability alternate→ MMPP (continuous time) or HMM (discrete time)→ note: use of correction factor based only on proportion of time

available inappropriate (see next slide)




0 200 400 600 800 1000

0.0

0.2

0.4

0.6

0.8

1.0

surfacing process 1

distance along track line

avai

labi

lity

0 200 400 600 800 1000

0.0

0.2

0.4

0.6

0.8

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surfacing process 2

distance along track line

avai

labi

lity

0 200 400 600 800 1000

0.0

0.2

0.4

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perp. distance detection prob.

perp. distance

prob

. det

ectio

n

0 200 400 600 800 1000

0.0

0.2

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perp. distance detection prob.

perp. distance

prob

. det

ectio

n

Figure: Two availability processes and resulting detection probabilitiesplotted against perp. distance – same hazard detection function

h(x , y) = 0.1 exp(− x1.25+y1.5

2000

)was used in both cases.




Hidden Markov modelsPoisson processesMarkov-modulated Poisson processes









HMMs for intermittent availability

this is ongoing research (Borchers, Zucchini and others)

aim: explicitly model periods of availability/unavailability

observations from DS survey:

perp. distance x & forward distance y at first detection

model assumptions:

at any time probability of detection depends on x and y and(unobserved) availability statusat any time the availability status depends on (unobserved)behavioural statesat any time the behavioural state depends on the previousbehavioural state (Markov chain)

crucial: time is discretized (finely)





0

Bt−1 0Bt 0 Bt+1

At−1 0At 0 At+1

Dt−1 0Dt 0 Dt+1

. . . (behavioural state)

(availability status)

(detection yes/no)

St+1 St+1

unconditional probability that an animal at perp. distance x isfirst detected at time t:

f (yt , x) = δ(t−1∏k=1

Pc(yk , x)Γ)P(yt , x)1t

(P and Pc determined by the relation states → observations;this part involves a hazard detection function)0





0

Bt−1 0Bt 0 Bt+1

At−1 0At 0 At+1

Dt−1 0Dt 0 Dt+1

. . . (behavioural state)

(availability status)

(detection yes/no)

St+1 St+1

likelihood to be maximized:

L(θ) =n∏

i=1

f (yiti , xi ; θ)∫ W0

∑tmaxk=1 f (yk , x ; θ)dx

some information on behavioural state process and therelation Bt → At required, otherwise identifiability issues

choice of time scale necessary – and a bit arbitrary?





Poisson processes for discrete availability

model in continuous time

scenario: animals available in short, distinct time periods

Skaug & Schweder (1999) consider two model components:

(i) a signal process (signal ' event of availability)→ Poisson process with intensity α

(i.e. time between consecutive signals ∼ Exp(α−1))(ii) hazard probability of detection h(x , y), given a signal

→ e.g. exponential power hazard model

Alternatively, one may merge (i) and (ii) by letting h(x , y)denote the rate of a non-homogeneous Poisson process

likelihood involves integration over h(x , y)

often unrealistic (availability events tend to be clustered)





MMPPs for intermittent availability

Relation to HMMs:

same principle, but model in continuous-timecan in fact be regarded as HMM (not relevant here)

we consider two states only (more states straightforward – intheory!!)

two-state continuous-time Markov chain governs availability:

Q =

(−µ1 µ1

µ2 −µ2

)→ time spent in state i ∼ Exp(µi )

rate of signals (detections) in state i : λi





0

● ●● ●●●●● ●●●●●●●●●●● ●●●●● ●●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●● ●●●●● ●●●●● ●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●

simulated surfacing events

time

MM

PP

PP ●●●●●●●● ●●●● ● ●● ●●● ●●● ●● ● ●● ● ●●● ●●● ● ● ● ●● ● ● ●● ●●● ● ● ● ●●●● ●● ● ●● ● ● ● ● ● ●●● ●● ●●●● ●● ●●● ● ●● ● ●●●●●

more adequate model for availability events (than PP)

unconditional density that an animal at perp. distance x isfirst detected at forward distance y (homogeneous case):

f (l , x) = δ exp((Q− Λ)l

)Λ1t (l = ymax − y)

but: we are interested in the non-homogeneous case – withrates depending on distances x and y ...





MMPPs – the non-homogeneous case

rates of sightings depending on x and y , e.g.

λi (x , y) = µ exp

(−xγ + yγ

σγ

)in general likelihood intractable...

λi (x , y) piecewise constant w.r.t. y ⇒ closed form density:

f (l , x) = δ

K∏k=1

exp((Q− Λ(i∗k , x))(ik − ik−1)

)· exp

((Q− Λ(l , x))(l − iK )

)Λ(l , x)1t





MMPP likelihood (non-homogeneous case)

strategy: assume smooth hazard function, but approximate itby piecewise constant function in the estimation – using veryfine partition of interval [0, ymax ]

likelihood to be maximized:

L(θ) =n∏

i=1

f (yiti , xi ; θ)∫ W0 F (ymax , x ; θ)dx

computationally very demanding...





Knowledge of availability process

using only the first sighting of each animal, we don’t haveenough information to estimate both the hazard function andthe availability process parameters

instead assume that availability process is (partly) known

assume that auxiliary data (e.g. from GPS tagging) gives us

1. full knowledge of the availability process or2. partial knowledge: expected dive cycle duration

we also may want to allow for heterogeneity in the surfacingprocess (across animals)




Real dataA simulation study









Data

35 dive sequences from radiotagged minke whales (about 50surfacing events per sequence)

survey data: 870 shipboard sightings

Analysis

1.) fit MMPP to each dive sequence to obtain information onavailability process (and heterogeneity)

2.) fix estimated MMPP parameters and estimate hazarddetection function parameters

3.) derive abundance estimates (not yet done, butstraightforward)





0

expected dive cycle lengths (in mtrs)

400 600 800 1000 12000.

0000

0.00

100.

0020

when estimating detection function parameters (θd) we usedempirical distribution of individual-specific availability processparameters (θa) to account for heterogeneity:

L(θd) =1

35

35∑i=1

L(θd , θa,i )

⇒ hierarchical model, random effects’ distribution: sample of θa’s





hazard detection rate:

λ1(x , y) =µσγ

(σ2 + x2 + y2)γ/2, λ2(x , y) = 0

(from the MMPP fit: surfacing rate = 0 in state 2)

Table: Hazard detection rate parameter estimates and 95% CIs.

lower estimate upper

µ 0.0020 0.0043 0.0092σ 719 1018 1441γ 5.86 8.25 11.62

g(0) = 0.61 (varies from 0.38 to 0.75 across whales)

computation of ESHW straightforward...





Simulation study

model specification:

hazard rate parameters close to estimates from minke whalesavailability parameters: averages of estimates obtained forminke whale data (i.e. no heterogeneity)

0





Simulation study



0mu, n=870, nsim=300

Fre

quen

cy

0.000 0.002 0.004 0.006 0.008 0.010 0.012

010

2030





Simulation study



0sigma, n=870, nsim=300

Fre

quen

cy

500 1000 1500 2000 2500

010

2030

4050





Simulation study



0gamma, n=870, nsim=300

Fre

quen

cy

5 10 15 20

010

2030

4050

60





Simulation study



0ESHW, n=870, nsim=300

Fre

quen

cy

100 200 300 400 500 600

010

2030

40




the HMM approach seems to produce much more stableestimates than the MMPP approach – why is that??

how much information on availability process is required?(expected dive cycle length seems sufficient if sample size isvery large)

Markov assumption probably inadequate – try hiddensemi-Markov models? can we do anything in the continuouscase?

for MMPP the algebra is more challenging than in the HMMcase – focus on the latter in future?

(The slides of this talk can be found on my web page)


motivation and introduction availability models mmpp ... · discussion/questions/future work 1...

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