Bayesian Models for Radio
Telemetry Habitat Data
Megan C. Dailey*
Alix I. Gitelman †
Fred L. Ramsey †
Steve Starcevich ‡
* Department of Statistics, Colorado State University
† Department
of Statistics, Oregon State University
‡ Oregon Department of Fish and Wildlife
Affiliations and funding
FUNDING/DISCLAIMER
The 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 Trout
 Year long radio-telemetry study (Steve Starcevich)
• 2 headwater streams of the John Day River in eastern
Oregon
• 26 trout were tracked ~ weekly from stream side
 Roberts Creek
 Rail Creek
F = 17
F=9
• Winter, Spring, Summer (2000-2001)
 S=3
Habitat association

Habitat inventory of entire creek once per season
•
•
•

Channel unit type
Structural association of pools
Total area of each habitat type
For this analysis:
•
H = 3 habitat classes
1. In-stream-large-wood pool (ILW)
2. Other pool (OP)
3. Fast water (FW)
•
Habitat availability = total area of habitat in creek
Goals of habitat analysis
 Incorporate
– multiple seasons
– multiple streams
– Other covariates
 Investigate “Use vs. Availability”
Radio telemetry data
 Sequences of observed habitat use
WINTER
SPRING
SUMMER
FISH 1
FISH 2
Habitat 1
Habitat 2
Habitat 3
missing
X f 1, s  winter  3,3,3,3,3,3,3,3,1,3,3,1,3,3,3,3,2,2
X f  2, s  winter  0,3,3,3,1,1,3,3,1,1,3,3,3,3,3,1,2,2
Independent Multinomial Selections
Model (IMS)
(McCracken, Manly, & Vander Heyden, 1998)

Product multinomial likelihood with multinomial logit
parameterization

h 
P( X | π)   ni !

i 1 
h 1 yih ! 
F
H
yih
yih
h
= number of sightings of animal i in habitat h
ni
= number of times animal i is sighted
= habitat selection probability (HSP) for habitat h
IMS 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 persistence
Examples of individual habitat use data
WINTER
SPRING
SUMMER
Habitat type
FW
OP
ILW
0
10
20
Sighting
30
Persists and moves
200
200
250
Rail Creek
250
Roberts Creek
50
100
150
Persists
Moves
0
0
50
100
150
Persists
Moves
Winter
Spring
Summer
Winter
Spring
Summer
Persistence Model
(Ramsey & Usner, 2003)
 One parameter extension of the IMS model to
relax assumption of independent sightings
 H-state Markov chain (H = # of habitat types)
 Persistence parameter : 
  1 : equivalent to the IMS model
  1 : greater chance of staying (“persisting”)
Persistence likelihood
 One-step transition probabilities:
pr (move to habitat h)   h
pr (stay in habitat h)  1  (1   h )
 Likelihood
F
H
P( X | π, )   
i 1 h 1
f ih
h
( h )
vihh*
(1  ( (1   h )))
vihh
vihh* = number of moves from habitat h* to habitat h ;
vihh = number of stays in habitat h ; f ih= indicator for initial sighting habitat
Bayesian extensions
I.
Reformulation of the original non-seasonal
persistence model into Bayesian framework: ( ,  h )
II.
Non-seasonal persistence / Seasonal HSPs: ( ,  sh )
III.
Seasonal persistence / Non-seasonal HSPs: ( s ,  h )
IV.
Seasonal persistence / Seasonal HSPs: (s ,  sh )
 sh
 sh
II. Non-seasonal persistence/Seasonal HSPs
( ,  sh )
Likelihood
F
S
H
P ( X | π, )    
f ih
i 1
 sh

 sh
s 1 h 1
sh
vih h* ,n s
 sh
vih h*
(1  ( (1   sh ))) vih h
= habitat selection probability for habitat h in season s
= overall persistence parameter
 sh
 sh
 sh
Multinomial logit parameterization
 Habitat Selection Probability (HSP):  sh
 Multinomial logit parameterization:
  sh
logit(  sh )  ln 
  TR
Arat 
Area sh
Area TR

  ln( Arat )   h   sh

 R  Th   sR  0
T = reference season
R = reference habitat
 sh
s = 1, …, S
h = 1, …, H
i = 1, …, F
 sh
h
h
III. Seasonal persistence / Non-seasonal HSPs
(s ,  h )
Likelihood
F
S
H
P( X | π, η)    f ish ( s h ) vishh* (1  ( s (1   h ))) vishh
i 1 s 1 h 1
h
vishh* = number of moves from habitat h* to habitat h in season s
vishh = number of stays in habitat h in season s
f ish
h
= indicator for initial sighting habitat h in season s
h
s
s
 sh
 sh
IV. Seasonal persistence / Seasonal HSPs
(s ,  sh )
Likelihood
F
S
H
P( X | π, η)   
i 1 s 1 h 1
f ish
sh
( s sh )
vishh*
(1  ( s (1   sh )))
vishh
Priors for all models
s
 sh
 h ~ diffuse normal
  I ( 0 , )
 sh~ diffuse normal
 s  I ( 0 , )
s
 sh
s
s
Estimated persistence parameters:
ROBERTS Persistence Parameter (eta): 95% Posterior Intervals
Non-seasonal persistence / Seasonal HSP model
(
( ,  sh )
)
Seasonal persistence / Non-seasonal HSP model
WINTER
(
)
(
SPRING
(
SUMMER
(s ,  h )
)
)
Seasonal persistence / Seasonal HSP model
WINTER
(
)
(
SPRING
(
SUMMER
0.0
0.2
)
(s ,  sh )
)
0.4
0.6
0.8
1.0
persistence parameter
s
s
Estimated habitat selection probabilities:
Roberts Creek
(s ,  h )
(s ,  sh )
Seasonal Persistence
Non-seasonal Persistence
In-Stream-Large-Wood
In-Stream-Large-Wood
(
)
(
WINTER
(
)
)
SPRING
(
)
)
(
)
)
)
(
WINTER
)
(
)
)
WINTER
SPRING
(
SUMMER
)
SUMMER
Fast Water
Fast Water
(
)
(
0.0
0.2
0.8
1.0
0.0
0.2
HSP
)
0.4
Seasonal Persistence / Non-Seasonal HSPs
(
(
)
In-Stream-Large-Wood
Other Pools
Fast Water
0.0
)
0.2
(
)
0.4
0.6
HSP
0.8
SPRING
SUMMER
0.6
HSP
(s ,  h )
WINTER
)
(
SUMMER
0.6
)
(
SPRING
)
0.4
(
WINTER
)
(
SUMMER
Other Pools
SPRING
(
SPRING
(
SUMMER
Other Pools
(
WINTER
(
1.0
0.8
1.0
BIC comparison
( ,  h )
( ,  sh )
(s ,  h )
( s ,  sh )
MODEL
Persistence
HSP
BIC Roberts
BIC Rail
I
NS
NS
742.6
482.2
II
NS
seasonal
751.2
479.4
III
seasonal
NS
711.6 **
467.8 **
IV
seasonal
seasonal
717.0
469.2
BIC = -2*log(likelihood) + p*log(n)
Conclusions
 Relaxes assumption of independent sightings
 Biological meaningfulness of the persistence parameter
 Provides a single model for the estimation of seasonal
persistence parameters and other estimates of interest
(HSP, (SPR/Arat)), along with their respective uncertainty
intervals
 Allows for seasonal comparisons and the incorporation of
multiple study areas (streams)
 Allows for use of other covariates by changing the
parameterization of the multinomial logit
s
s
 sh
 sh
THANK YOU
 sh
s
s
s
s
s
 sh
s
 sh
s
 sh
s
s
s
s
V. Multiple stream persistence
(cs ,  csh )
Likelihood
C
F
S
H
P( X | π, η)    f icsh (cs csh ) vicshh* (1  (cs (1   csh ))) vicshh
c 1 i 1 s 1 h 1
csh
vicshh* = number of moves from habitat h* to habitat h in season s
in stream c
vicshh = number of stays in habitat h in season s in stream c
f icsh = indicator for initial sighting in habitat h in season s in stream c
Evidence of persistence
Roberts Creek
Percent persists of total sightings
0.8
200
1.0
Number of persists and moves per season
Winter
Spring
Summer
0.0
0
0.2
50
0.4
100
0.6
150
Persists
Moves
Winter
Spring
Summer
Markov chain persistence
One-step Transition Probability Matrix:
1   1  1 
2

1   1  2 
 1
   =  1
2


 
2
1

where
K 1
K 1
1   1  K 1 
K 1
 1
1 

0    min  ,
  h (1   h ) 


K



K

1   1  K  
K
Persistence example
  = 1 -> IMS
  < 1 -> greater chance of remaining in previous habitat
=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
1
2
3
1
0.60
0.15
.25
2
0.10
0.65
.25
3
0.10
0.15
0.75
 = 0.5
SPR
Arat
SPR
Arat
Estimate of Use vs. availability
 Selection Probability Ratio (SPR)
  sh 
  ln( SPR)  ln( Arat)   h   sh
ln 
  TR 
  sh
SPR  
  TR

  Arat  exp( h   sh )

 SPR/(Area Ratio) for Use vs. Availability
SPR
 exp( h   sh )
Arat
SPR
Arat
SPR
Arat
Persistence vs. IMS
Persistence vs. IMS: SPR/AreaRatio
Wald's 95% CIs
(
) Winter RIFFLE-PERS
) Winter RIFFLE-IMS
(
(
) Winter GLIDE-PERS
(
(
)
)
Winter GLIDE-IMS
Winter SCOUR-PERS
(
()
()
)
Winter SCOUR-IMS
Spring RIFFLE-PERS
Spring RIFFLE-IMS
( )
( )
(
(
0
Spring GLIDE-PERS
Spring GLIDE-IMS
)
)
Spring SCOUR-PERS
Spring GLIDE-IMS
5
10
SPR/AR
15
20
Estimated persistence parameters
ROBERTS Persistence Parameter (eta): 95% Posterior Intervals
Hierarchical seasonal model
WINTER
)
(
)
(
SPRING
)
(
SUMMER
)
(
OVERALL Persistence
Non-seasonal persistence, seasonal HSPs model
)
(
0.0
0.2
persistence parameter
0.6
0.4
Eta
0.8
1.0
stuff
BIC = -2*mean(llik[1001:10000]) - p*log(17)
model IV. p = 7 in basemodelROB and
model III. p = 5 in seaspersonlyROB
Priors
 Multinomial logit parameters:
 h ~ diffuse normal
 sh~ diffuse normal
 Non-seasonal persistence:
 Seasonal persistence:
 ~ Unif (0,1)
s ~ Unif (0,1)
 Hierarchical seasonal persistence:
s
~ Beta(a,b)
a,b ~ Unif (0, )