model complexity and model choice for animal movement models
-
Upload
ben-bolker -
Category
Science
-
view
526 -
download
0
Transcript of model complexity and model choice for animal movement models
![Page 1: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/1.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Model complexity and model choice for animalmovement models
Ben Bolker, McMaster UniversityDepartments of Mathematics & Statistics and Biology
Guelph Biomathematics & Biostatistics Symposium
9 June 2016
![Page 2: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/2.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Outline
1 Animal movement
2 Florida panthers
3 Hidden Markov models
4 Basic analysis (van de Kerk et al., 2015)
5 Incorporating diurnal variation (Li, 2015)
6 Broader issues/outlook
![Page 3: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/3.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Acknowledgements
People Michael Li, Madelon van de Kerk,Dave Onorato, Madan Oli; many unnamed fieldbiologists
Agencies US Fish and Wildlife Service, US Geological Survey,US National Park Service
Funding NSERC Discovery grant, NSF IGERT program
![Page 4: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/4.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Outline
1 Animal movement
2 Florida panthers
3 Hidden Markov models
4 Basic analysis (van de Kerk et al., 2015)
5 Incorporating diurnal variation (Li, 2015)
6 Broader issues/outlook
![Page 5: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/5.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Animal movement: data
observations:e.g. mass mark-recapture,longitudinal density, directobservation, telemetry(VHF, GPS)most methods provide asequence of times andlocations for each individual
●
●
●
●
θ1
θ2s1
s2
t1
t2
t3
t4
![Page 6: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/6.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
summaries:home range(convex hull, kerneldensity estimate, etc.)root-mean-squareddisplacementstep length and turningangle
covariates:e.g. habitat map,individual characteristics(sex, age, weight . . . )
0
100
200
300
0 1 2 3
step length (log10m)
coun
t
180
270
0
900
50
100
150
200
250
![Page 7: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/7.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Animal movement: questions
simple descriptionhow do animals’ movements change as a function of their(internal or external) environment?what does that tell us about their biology?how might animals’ distributions, etc. change when conditions(density, habitat, ...) change?
![Page 8: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/8.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Outline
1 Animal movement
2 Florida panthers
3 Hidden Markov models
4 Basic analysis (van de Kerk et al., 2015)
5 Incorporating diurnal variation (Li, 2015)
6 Broader issues/outlook
![Page 9: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/9.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Biological/conservation issues
Florida panther: Pumaconcolor coryiendangered subspeciesseverely reduced habitatsmall, isolated populationcurrently recovering
www.peer.org
![Page 10: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/10.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Panther movement questions
movement variation by sexand life history stage(juvenile, adult, mom withkittens . . . )effects of movement onthreats(intraspecific aggression,roadkill) ?predicting the effects offuture changes in populationdensity / populationstructure / habitat
![Page 11: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/11.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Panther movement data
panthers tracked, capturedGPS collars18 males (13 male, 5 female,1-15 years old)3200 panther days,hourly/bihourly; 49000locations?? per panther
![Page 12: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/12.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
example movement tracks
CatID: 48 CatID: 131 CatID: 189
0
10
20
30
0 5 10 15 20 0 5 10 15 20 0 5 10 15 20x (km)
y (k
m)
![Page 13: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/13.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Outline
1 Animal movement
2 Florida panthers
3 Hidden Markov models
4 Basic analysis (van de Kerk et al., 2015)
5 Incorporating diurnal variation (Li, 2015)
6 Broader issues/outlook
![Page 14: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/14.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Hidden Markov models
finite mixture model with temporal dependencediscrete time stepsdiscrete latent state; transition matrixobservations from emission distributions(continuous or discrete, univariate or multivariate)multiphasic movement (Fryxell et al., 2008; Langrock et al.,2012)
![Page 15: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/15.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Hidden Markov models (cont.)
state:
St ∼ Multinomial(St−1, µS ,t)
µS ,t = multi-logistic(XS ,tβS)
emission:
Zt ∼ {Dist1(µZ1,St ), . . . ,Distn(µZn,St )}µZi ,St = g−1 (XZi ,tβZi ,St )
![Page 16: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/16.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Hidden Markov models (part 3)
forward-backward algorithm for estimating parametersViterbi algorithm for estimating most probable state sequencesdepmixS4 package (Visser and Speekenbrink, 2010) (alsomoveHMM (Michelot et al., 2016))hidden semi-Markov models: allow for non-geometric dwelldistributions (Langrock, 2011; Augustine, 2016): move.HMM
![Page 17: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/17.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Outline
1 Animal movement
2 Florida panthers
3 Hidden Markov models
4 Basic analysis (van de Kerk et al., 2015)
5 Incorporating diurnal variation (Li, 2015)
6 Broader issues/outlook
![Page 18: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/18.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
State distributions
0.00
0.25
0.50
0.75
0.01 0.10 1.00 10.00Step length (km)
0.2
0.4
0.6
−2 0 2Turning angle (radians)
0.0
0.2
0.4
0.6
4 8 12Dwell time (hrs)
State 1 2 3
![Page 19: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/19.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Parameter estimates
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
● ●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●●
●●
●
●
●
●
●
●●
● ●
●
● ●
●
● ●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
● ●
●
●
●
●
dwell time(scale)
dwell time(shape)
step length(scale)
step length(shape)
turn angle(direction)
turn angle(dispersion)
0.25
0.50
0.75
1.00
0
1
2
3
4
5
0.0
0.5
1.0
1.5
1.0
1.5
2.0
2.5
0
2
4
6
0.0
0.2
0.4
0.6
1 2 3 1 2 3 1 2 3State
Val
ue
Sex●
●
F
M
![Page 20: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/20.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Tracks with Viterbi estimates
CatID: 48 CatID: 131 CatID: 189
0
10
20
30
0 5 10 15 20 0 5 10 15 20 0 5 10 15 20x (km)
y (k
m)
factor(vitstates)1
2
3
![Page 21: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/21.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Diurnal variation
![Page 22: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/22.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
what can we conclude so far?
good news
basic biology: males move faster, fartherthree states are identifiable, sensibledwell distributions approximately geometric(HSMM → HMM)
bad newsdiurnal variation in Viterbi results - but it’s not in the model!estimates of model complexity are too high
![Page 23: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/23.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
what can we conclude so far?
good news
basic biology: males move faster, fartherthree states are identifiable, sensibledwell distributions approximately geometric(HSMM → HMM)
bad newsdiurnal variation in Viterbi results - but it’s not in the model!estimates of model complexity are too high
![Page 24: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/24.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Model complexity (bad news)
●
●●
●
● ●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●● ●
●
●
●
●
●
●
●
●
●
●●●
●
●
● ●
●
●
●● ●
CatID: 139 CatID: 157 CatID: 185 CatID: 165
CatID: 137 CatID: 167 CatID: 155 CatID: 189
CatID: 48 CatID: 131 CatID: 94 CatID: 130
0
100
0
100
0
100
200
0
100
200
0
100
200
0
500
0
500
0
500
0
500
0
500
0
500
1000
0
500
1000
2 3 4 5 6 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6Number of latent states
∆BIC
Sex●
●
F
M
![Page 25: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/25.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Model complexity (Manx shearwaters, Dean et al. (2013))
●
●●
●● ● ● ● ●
0
1000
2000
3000
2 4 6 8 10number of latent states
∆ ne
gativ
e lo
g−lik
elih
ood
●●●
min AIC
min BIC
![Page 26: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/26.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Outline
1 Animal movement
2 Florida panthers
3 Hidden Markov models
4 Basic analysis (van de Kerk et al., 2015)
5 Incorporating diurnal variation (Li, 2015)
6 Broader issues/outlook
![Page 27: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/27.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Expanding the model
Attempting to fix these problems:extend the model to allow covariatesspecifically, allow for diurnal variation
simplify model (log-Normal step length only)fixed state-specific emissions parameters(step length mean and std dev)time-varying transition parametersalso try finite mixture models(independent occupancy)
how much does this help?
![Page 28: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/28.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Expanding the model
Attempting to fix these problems:extend the model to allow covariatesspecifically, allow for diurnal variation
simplify model (log-Normal step length only)fixed state-specific emissions parameters(step length mean and std dev)time-varying transition parametersalso try finite mixture models(independent occupancy)
how much does this help?
![Page 29: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/29.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Temporal models
0 6 12 18 24time
modelblock
hourly
quad
sin
![Page 30: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/30.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Temporal patterns (step length)
1.6
2.0
2.4
0 5 10 15 20Time of day
Mea
n st
ep le
ngth
(log
10m
)
modelhomogeneous (n=6)
block (n=4)
quadratic (n=5)
sin (n=5)
hourly (n=3)
observed
![Page 31: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/31.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Temporal patterns (autocorrelation)
0.00
0.25
0.50
0.75
1.00
0 10 20 30 40Time of day
Aut
ocor
rela
tion
modelhomogeneous (n=6)
block (n=4)
quadratic (n=5)
sin (n=5)
hourly (n=3)
observed
![Page 32: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/32.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Goodness of fit/model complexity
●
●
● ● ●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●●
0
500
1000
1500
3 4 5 6 7number of latent states
∆BIC
model●
●
●
●
●
homogeneous
quadratic
sinusoid
block
hourly
# of parameters●
●
100
200
![Page 33: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/33.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Model complexity: simulation
● ●
●
●●
●
●
●
●
●
●
●
●
●
0
200
400
600
800
2 3 4 5number of latent states
∆BIC
type●
●
●
●
homogeneous
quadratic
sin
hourly
![Page 34: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/34.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Diurnal model: conclusions
diurnal structure greatly improves fit (∆BIC ≈ 500)slightly improves latent-state issue (n = 6→ 5)lots left to do!
seasonal variationincorporate habitat, home range behaviouretc. etc. etc.
![Page 35: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/35.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Outline
1 Animal movement
2 Florida panthers
3 Hidden Markov models
4 Basic analysis (van de Kerk et al., 2015)
5 Incorporating diurnal variation (Li, 2015)
6 Broader issues/outlook
![Page 36: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/36.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Big data and small models
simple model families +model misspecification →overparameterizationGelman: “Sample sizes are never large”: (blog post)
N is never enough because if it were “enough” you’dalready be on to the next problem for which youneed more data.
![Page 37: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/37.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
An aside on AIC vs BIC
“should I use AIC or BIC? Iheard that AIC isinconsistent ...”complexity penalty = 2(AIC) vs log(n) (BIC)best prediction vs. modelidentification (Yang, 2005)
effect size spectrum:tapering or discrete?
effect
effe
ct s
ize type
discrete
tapering
![Page 38: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/38.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Animal movement: open challenges
Cognition/memory (Braciset al., 2015)
Intraspecificinteraction/collectivemovement (Delgado et al.,2014)
Continuous-time movementmodels (Calabrese et al., 2016)Edges, barriers, and corridors(Beyer et al., 2016)
Efficient (big-data)approaches (Brillinger et al.,2008)
Putting it all together ...
![Page 39: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/39.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
Tools needed
cross-validation (Wenger and Olden, 2012)
protocols and tools for model checking (Potts et al., 2014);score tests?flexible computational frameworks(ecologists can’t afford consultants/there are too many species out there)
![Page 40: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/40.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
http://tinyurl.com/panthermoves;http://www.slideshare.net/bbolker
![Page 41: model complexity and model choice for animal movement models](https://reader031.fdocuments.us/reader031/viewer/2022030313/58a6a2c91a28ab0a7a8b49cb/html5/thumbnails/41.jpg)
Animal movement Panthers HMM Basic analysis Diurnal model Broader issues/outlook References
References
Augustine, B., 2016. Flexible, user-friendly hidden(semi) Markov models for animal movement data.
Beyer, H.L., Gurarie, E., et al., 2016. Journal ofAnimal Ecology, 85(1):43–53. ISSN 00218790.doi:10.1111/1365-2656.12275.
Bracis, C., Gurarie, E., et al., 2015. PLOS ONE,10(8):e0136057. ISSN 1932-6203.doi:10.1371/journal.pone.0136057.
Brillinger, D.R., Stewart, B.S., et al., 2008. InProbability and statistics: Essays in honor of DavidA. Freedman, pages 246–264. Institute ofMathematical Statistics.
Calabrese, J.M., Fleming, C.H., and Gurarie, E., 2016.Methods in Ecology and Evolution, pages n/a–n/a.ISSN 2041-210X. doi:10.1111/2041-210X.12559.
Dean, B., Freeman, R., et al., 2013. Journal of TheRoyal Society Interface, 10(78):20120570. ISSN1742-5689, 1742-5662.doi:10.1098/rsif.2012.0570.
Delgado, M.d.M., Penteriani, V., et al., 2014. Methodsin Ecology and Evolution, 5(2):183–189.
Fryxell, J.M., Hazell, M., et al., 2008. Proceedings ofthe National Academy of Sciences,105(49):19114–19119. ISSN 0027-8424,1091-6490. doi:10.1073/pnas.0801737105.
Langrock, R., 2011. Computational Statistics and DataAnalysis, 55(1):715–724. ISSN 01679473.
Langrock, R., King, R., et al., 2012. Ecology,93(11):2336–2342. ISSN 0012-9658.doi:10.1890/11-2241.1.
Li, M., 2015. Incorporating Temporal Heterogeneity inHidden Markov Models For Animal Movement.Master’s thesis.
Michelot, T., Langrock, R., and Patterson, T.A., 2016.Methods in Ecology and Evolution, in press.doi:10.1111/2041-210X.12578.
Potts, J.R., Auger-Méthé, M., et al., 2014. Methods inEcology and Evolution, 5(10):1012–1022.
van de Kerk, M., Onorato, D.P., et al., 2015. Journalof Animal Ecology, 84(2):576–585.
Visser, I. and Speekenbrink, M., 2010. Journal ofStatistical Software, 36(7):1–21.
Wenger, S.J. and Olden, J.D., 2012. Methods inEcology and Evolution, 3(2):260–267. ISSN2041210X.doi:10.1111/j.2041-210X.2011.00170.x.
Yang, Y., 2005. Biometrika, 92(4):937–950.doi:10.1093/biomet/92.4.937.