Bayesian Models for Radio Telemetry Habitat Data

Bayesian Models for Radio Bayesian Models for Radio Telemetry Habitat DataTelemetry Habitat Data

Megan C. Dailey*Megan C. Dailey*Alix I. GitelmanAlix I. GitelmanFred L. RamseyFred L. RamseySteve StarcevichSteve Starcevich

* * Department of Statistics, Colorado State UniversityDepartment of Statistics, Colorado State UniversityDepartment of Statistics, Oregon State UniversityDepartment of Statistics, Oregon State University

Oregon Department of Fish and WildlifeOregon Department of Fish and Wildlife

†

‡

†

†

‡

Affiliations and fundingAffiliations and funding

FUNDING/DISCLAIMERThe work reported here was developed under the STAR Research Assistance Agreement CR-829095

awarded by the U.S. Environmental Protection Agency (EPA) to Colorado State University. This

presentation has not been formally reviewed by EPA. The views expressed here are solely those of the

authors and STARMAP, the Program they represent. EPA does not endorse any products or

commercial services mentioned in this presentation.

CR-829095

Westslope Cutthroat TroutWestslope Cutthroat Trout Year long radio-telemetry study (Year long radio-telemetry study (Steve Starcevich)Steve Starcevich)

• 2 headwater streams of the John Day River in eastern 2 headwater streams of the John Day River in eastern OregonOregon

• 26 trout were tracked ~ weekly from stream side26 trout were tracked ~ weekly from stream side Roberts CreekRoberts Creek F = 17F = 17 Rail CreekRail Creek F = 9F = 9

• Winter, Spring, Summer (2000-2001)Winter, Spring, Summer (2000-2001) S=3S=3

Habitat associationHabitat association Habitat inventory of entire creek once per seasonHabitat inventory of entire creek once per season

• Channel unit typeChannel unit type• Structural association of poolsStructural association of pools• Total area of each habitat typeTotal area of each habitat type

For this analysis: For this analysis: • H = 3 habitat classesH = 3 habitat classes

1.1. In-stream-large-wood pool (ILW)In-stream-large-wood pool (ILW)

2.2. Other pool (OP)Other pool (OP)

3.3. Fast water (FW)Fast water (FW)

• Habitat availability = total area of habitat in creekHabitat availability = total area of habitat in creek

Goals of habitat analysisGoals of habitat analysis IncorporateIncorporate

– multiple seasonsmultiple seasons– multiple streamsmultiple streams– Other covariatesOther covariates

Investigate “Use vs. Availability”Investigate “Use vs. Availability”

Radio telemetry dataRadio telemetry data Sequences of observed habitat useSequences of observed habitat use

SUMMERWINTER SPRING

FISH 2

FISH 1

Habitat 1 Habitat 3Habitat 2 missing

2,2,1,3,3,3,3,3,1,1,3,3,1,1,3,3,3,0winter,2 sfX

2,2,3,3,3,3,1,3,3,1,3,3,3,3,3,3,3,3winter,1 sfX

Independent Multinomial Selections Independent Multinomial Selections Model (IMS)Model (IMS)

(McCracken, Manly, & Vander Heyden, 1998)(McCracken, Manly, & Vander Heyden, 1998)

Product multinomial likelihood with multinomial logit parameterization

F

i

H

h ih

yh

i ynP

ih

1 1 !!)|(

πX

= number of sightings of animal i in habitat hihy

h = habitat selection probability (HSP) for habitat h

= number of times animal i is sightedin

IMS Model: AssumptionsIMS Model: Assumptions

Repeat sightings of same animal represent independent habitat selections

Habitat selections of different animals are independent

All animals have identical multinomial habitat selection probabilities

Evidence of persistenceEvidence of persistence

Examples of individual habitat use data

Sighting

Ha

bita

t ty

pe

0 10 20 30

WINTER SPRING SUMMER

ILW

OP

FW

Persists and movesPersists and moves0

50

10

01

50

20

02

50

Roberts Creek

Winter Spring Summer

PersistsMoves

05

01

00

15

02

00

25

0

Rail Creek


PersistsMoves

Persistence ModelPersistence Model

(Ramsey & Usner, 2003)(Ramsey & Usner, 2003)

One parameter extension of the IMS model to One parameter extension of the IMS model to relax assumption of independent sightingsrelax assumption of independent sightings

H-state Markov chain H-state Markov chain (H = # of habitat types)(H = # of habitat types)

Persistence parameter :Persistence parameter :

11 : equivalent to the IMS model

: greater chance of staying (“persisting”)

Persistence likelihoodPersistence likelihood

One-step transition probabilities:One-step transition probabilities:

LikelihoodLikelihood

hpr h)habitat tomove(

)1(1h)habitat in stay ( hpr

F

i

H

h

vh

vh

f

h

ihhihhihP1 1

)))1((1()(),|( * πX

= number of moves from habitat h* to habitat h ;

*ihhv

ihf = indicator for initial sighting habitat= number of stays in habitat h ;ihhv

Bayesian extensions

I.I. Reformulation of the original non-seasonal Reformulation of the original non-seasonal persistence model into Bayesian framework:persistence model into Bayesian framework:

II.II. Non-seasonal persistence / Seasonal HSPs:Non-seasonal persistence / Seasonal HSPs:

III.III. Seasonal persistence / Non-seasonal HSPs:Seasonal persistence / Non-seasonal HSPs:

IV.IV. Seasonal persistence / Seasonal HSPs:Seasonal persistence / Seasonal HSPs:

),( sh

),( shs

),( hs

),( h

II. Non-seasonal persistence/Seasonal HSPsII. Non-seasonal persistence/Seasonal HSPs


),( sh

F

i

S

s

H

h

vsh

vsh

vf

sh

ihhihhnsihhihP1 1 1

)))1((1(),|( **, πX

sh

shsh

sh

= habitat selection probability for habitat h in season ssh = overall persistence parameter

Multinomial logit Multinomial logit parameterizationparameterization

Habitat Selection Probability (HSP):Habitat Selection Probability (HSP):

Multinomial logit parameterization:Multinomial logit parameterization:

sh

shsh

shhTR

shsh

)Aratln(ln)logit(

TR

sh

Area

AreaArat

s = 1, …, S h = 1, …, H i = 1, …, F

T = reference seasonR = reference habitat

0 sRThR

sh

sh

IIIIII.. Seasonal persistence / Non-seasonal HSPsSeasonal persistence / Non-seasonal HSPs

F

i

S

s

H

h

vhs

vhs

f

h

ishhishhishP1 1 1

)))1((1()(),|( * ηπX

h h

hh


),( hs

ishf = indicator for initial sighting habitat h in season s

= number of stays in habitat h in season sishhv= number of moves from habitat h* to habitat h in season s*ishhv

IV. Seasonal persistence / Seasonal HSPsIV. Seasonal persistence / Seasonal HSPs),( shs

F

i

S

s

H

h

vshs

vshs

f

sh

ishhishhishP1 1 1

)))1((1()(),|( * ηπX

sh sh

shsh


Priors for all modelsPriors for all models

h ~ diffuse normal

sh ~ diffuse normal

),0( I

),0( Is

ss

ss

Estimated persistence parameters:Estimated persistence parameters:s

ss

s

ROBERTS Persistence Parameter (eta): 95% Posterior Intervals

persistence parameter

0.0 0.2 0.4 0.6 0.8 1.0

Non-seasonal persistence / Seasonal HSP model

( )

Seasonal persistence / Non-seasonal HSP model

( )WINTER

( )SPRING

( )SUMMER

Seasonal persistence / Seasonal HSP model

( )WINTER

( )SPRING

( )SUMMER

),( sh

),( shs

),( hs

Estimated habitat selection probabilities:Estimated habitat selection probabilities:Roberts CreekRoberts Creek

0.0 0.2 0.4 0.6 0.8 1.0

Non-seasonal Persistence

HSP

In-Stream-Large-Wood

( ) WINTER

( ) SPRING

( ) SUMMER

Other Pools

( ) WINTER

( ) SPRING

( ) SUMMER

Fast Water

( ) WINTER

( ) SPRING

( ) SUMMER

0.0 0.2 0.4 0.6 0.8 1.0

Seasonal Persistence

HSP

In-Stream-Large-Wood

( ) WINTER

( ) SPRING

( ) SUMMER

Other Pools

( ) WINTER

( ) SPRING

( ) SUMMER

Fast Water

( ) WINTER

( ) SPRING

( ) SUMMER

),( shs

),( hs

),( hs

BIC comparisonBIC comparison

MODELMODEL PersistencePersistence HSPHSP BIC RobertsBIC Roberts BIC RailBIC Rail

II NS NS 742.6 482.2482.2

IIII NS seasonal 751.2 479.4479.4

IIIIII seasonal NS 711.6 ** 467.8 **467.8 **

IVIV seasonal seasonal 717.0 469.2469.2

),( sh ),( shs ),( hs

BIC = -2*log(likelihood) + p*log(n)

),( h

ConclusionsConclusions Relaxes assumption of independent sightingsRelaxes assumption of independent sightings

Biological meaningfulness of the persistence parameterBiological meaningfulness of the persistence parameter

Provides a single model for the estimation of seasonal Provides a single model for the estimation of seasonal persistence parameters and other estimates of interest persistence parameters and other estimates of interest (HSP, (SPR/Arat)), along with their respective uncertainty (HSP, (SPR/Arat)), along with their respective uncertainty intervalsintervals

Allows for seasonal comparisons and the incorporation of Allows for seasonal comparisons and the incorporation of multiple study areas (streams)multiple study areas (streams)

Allows for use of other covariates by changing the Allows for use of other covariates by changing the parameterization of the multinomial logitparameterization of the multinomial logit

THANK YOUTHANK YOUs

sh

sh

sh

sh

sh

shs

s

s

s

s

s s

s

s

ss s

V.V. Multiple stream persistence Multiple stream persistence

C

c

F

i

S

s

H

h

vcshcs

vcshcs

f

csh

icshhicshhicshP1 1 1 1

)))1((1()(),|( * ηπX


icshf = indicator for initial sighting in habitat h in season s in stream c

= number of stays in habitat h in season s in stream cicshhv

= number of moves from habitat h* to habitat h in season s in stream c

*icshhv

),( cshcs

Evidence of persistenceEvidence of persistenceRoberts CreekRoberts Creek

05

01

00

15

02

00

Number of persists and moves per season


PersistsMoves


0.0

0.2

0.4

0.6

0.8

1.0

Percent persists of total sightings

Markov chain persistenceMarkov chain persistence

One-step Transition Probability Matrix:One-step Transition Probability Matrix:

1 2 K 1 K

1 2 K 1 K

1 2

K 1 K

1 2 K 1 K

1 1

1 1

=

1 1

1 1

)1(

1,

1min0

hh where

Persistence examplePersistence example

= 1 1 2 3

1 0.2 0.3 0.5

2 0.2 0.3 0.5

3 0.2 0.3 0.5

= 0.5 1 2 3

1 0.60 0.15 .25

2 0.10 0.65 .25

3 0.10 0.15 0.75

= 1 -> IMS= 1 -> IMS < 1 -> greater chance of remaining in previous habitat< 1 -> greater chance of remaining in previous habitat

Estimate of Use vs. availabilityEstimate of Use vs. availability Selection Probability Ratio (SPR)Selection Probability Ratio (SPR)

SPR/(Area Ratio) for Use vs. AvailabilitySPR/(Area Ratio) for Use vs. Availability

shhTR

sh SPR

Arat)ln()ln(ln

Arat

SPR

)exp( shhArat

SPR

)exp(Arat shhTR

shSPR

Arat

SPRArat

SPR

Arat

SPR

Persistence vs. IMSPersistence vs. IMS

Persistence vs. IMS: SPR/AreaRatio Wald's 95% CIs

SPR/AR

0 5 10 15 20

( ) Winter RIFFLE-PERS( ) Winter RIFFLE-IMS

( ) Winter GLIDE-PERS( )

Winter GLIDE-IMS( ) Winter SCOUR-PERS

( )Winter SCOUR-IMS

() Spring RIFFLE-PERS() Spring RIFFLE-IMS

( ) Spring GLIDE-PERS( ) Spring GLIDE-IMS

( ) Spring SCOUR-PERS( ) Spring GLIDE-IMS

Estimated persistence parametersEstimated persistence parameters

0.0 0.2 0.4 0.6 0.8 1.0

ROBERTS Persistence Parameter (eta): 95% Posterior Intervals

Eta

Hierarchical seasonal model

( )WINTER

( )SPRING

( )SUMMER

( )OVERALL Persistence

Non-seasonal persistence, seasonal HSPs model

( ) persistence parameter

stuffstuff

BIC = -2*mean(llik[1001:10000]) - p*log(17)

model IV. p = 7 in basemodelROB and

model III. p = 5 in seaspersonlyROB

PriorsPriors

Multinomial logit parameters:Multinomial logit parameters:

Non-seasonal persistence:Non-seasonal persistence:

Seasonal persistence:Seasonal persistence:

Hierarchical seasonal persistence:Hierarchical seasonal persistence:

h ~ diffuse normal

sh ~ diffuse normal

s )1,0(Unif~

)1,0(Unif~

s ~ Beta(a,b) a,b ),0(Unif~

Bayesian Models for Radio Telemetry Habitat Data

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