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demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
demography of unmarked individuals
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demography of unmarked individuals

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McMaster math colloquium, 19 Nov 2010

McMaster math colloquium, 19 Nov 2010

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  1. Background Model definitions Tools Tests Estimating demographic parameters from samples of unmarked individuals Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University 19 November 2010Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  2. Background Model definitions Tools TestsOutline 1 Background Ecological/statistical motivations Study system 2 Model definitions 3 Tools State-space models Markov chain Monte Carlo Gibbs/block sampling 4 Tests Perfect observation, trivial dynamics Perfect observation, non-trivial dynamics Imperfect observationBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  3. Background Model definitions Tools TestsEcological/statistical motivationsWhy do ecologists care about demography? Determines “distribution and abundance of organisms” Basic: what regulates populations? effects of population density & individual characteristics? Applied: predict population dynamics use this information to manage them sustainably effects of interventions (e.g. attraction-production controversy)Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  4. Background Model definitions Tools TestsEcological/statistical motivationsWhy do ecologists care about demography? Determines “distribution and abundance of organisms” Basic: what regulates populations? effects of population density & individual characteristics? Applied: predict population dynamics use this information to manage them sustainably effects of interventions (e.g. attraction-production controversy)Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  5. Background Model definitions Tools TestsEcological/statistical motivationsData Repeated samples of individual animals . . . Multiple times, observers, locations Presence and size (imperfect) Estimate demographic parameters: birth/immigration death/emigration change (growth) . . . as a function of size, (population density)Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  6. Background Model definitions Tools TestsEcological/statistical motivationsData Repeated samples of individual animals . . . Multiple times, observers, locations Presence and size (imperfect) Estimate demographic parameters: birth/immigration death/emigration change (growth) . . . as a function of size, (population density)Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  7. Background Model definitions Tools TestsEcological/statistical motivationsStatistical motivations Challenging (=fun) General problem (individual identification is often difficult) Impossible with current methods Novel application of standard building blocksBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  8. Background Model definitions Tools TestsStudy systemNatural history Thalassoma hardwicke (sixbar wrasse) Settlement: ≈ 5–10 mm Growth: a few mm per month Death: by predator (competition for refuges) Immigration/emigration: photo: Leonard Low, Flickr via species.wikimedia.org rare below 40–50 mmBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  9. Background Model definitions Tools TestsStudy systemWhere they live Patch reefs, French Polynesia (and throughout the south Pacific)Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  10. Background Model definitions Tools TestsStudy systemWhat they look like to me 80 Estimated fish size (mm) 60 40 20 Feb Mar Apr May Jun Jul DateBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  11. Background Model definitions Tools TestsStudy system40What they look like to me 20 Feb Mar AprBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  12. Background Model definitions Tools TestsStudy systemOutline 1 Background Ecological/statistical motivations Study system 2 Model definitions 3 Tools State-space models Markov chain Monte Carlo Gibbs/block sampling 4 Tests Perfect observation, trivial dynamics Perfect observation, non-trivial dynamics Imperfect observationBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  13. Background Model definitions Tools TestsOutline 1 Background Ecological/statistical motivations Study system 2 Model definitions 3 Tools State-space models Markov chain Monte Carlo Gibbs/block sampling 4 Tests Perfect observation, trivial dynamics Perfect observation, non-trivial dynamics Imperfect observationBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  14. Background Model definitions Tools TestsDemographic parameters as f (size) Growth Appearance Persistence 1.0 0.8 0.6 Probability (per day) 0.4 a.settle l.grow b.settle a.mort 0.2 tot.settle c.mort 0.0 1020304050607080 1020304050607080 1020304050607080 Fish size (mm)Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  15. Background Model definitions Tools TestsObservation model Detection Measurement (probability) (standard deviation) 0.8 3.5 0.6 3.0 2.5 Measurement error Double−count 0.4 Zero−count 2.0 1.5 0.2 1.0 10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80 Fish size (mm)Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  16. Background Model definitions Tools TestsOutline 1 Background Ecological/statistical motivations Study system 2 Model definitions 3 Tools State-space models Markov chain Monte Carlo Gibbs/block sampling 4 Tests Perfect observation, trivial dynamics Perfect observation, non-trivial dynamics Imperfect observationBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  17. Background Model definitions Tools TestsState-space modelsState-space models Problem: estimating parameters of systems with un- or imperfectly-observed states? Easy if no feedback (measurement error models) With feedback (dynamic models), brute force approaches become infeasible . . .Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  18. Background Model definitions Tools TestsState-space modelsDirected acyclic graph (DAG) parameters N1 N2 N3 N4 N5 obs1 obs2 obs3 obs4 obs5Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  19. Background Model definitions Tools TestsMarkov chain Monte CarloBayesian statistics, in 30 seconds computational (convenience) Bayesian statistics (forget all that stuff about philosophy of statistics, subjectivity, incorporating prior information . . . ) have a likelihood, L(θ) = Prob(data|θ) want a posterior probability, Ppost (θ) = Prob(θ|data) Bayes’ rule: L(θ) · prior(θ) Ppost (θ) = L(θ) · prior(θ) dθ may want mean, mode, confidence intervals, . . . denominator very high-dimensionalBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  20. Background Model definitions Tools TestsMarkov chain Monte CarloMarkov chain Monte Carlo θA Posterior probability MCMC rule: P(θ A ) J(θ B → θ A ) if = P(θ B ) J(θ A → θ B ) then stationary distribution = posterior probability (provided chain is irreducible etc.) Parameter valueBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  21. Background Model definitions Tools TestsMarkov chain Monte CarloMarkov chain Monte Carlo θA Posterior probability MCMC rule: P(θ A ) J(θ B → θ A ) if = P(θ B ) J(θ A → θ B ) then stationary distribution = posterior probability (provided chain is irreducible etc.) q q Parameter valueBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  22. Background Model definitions Tools TestsMarkov chain Monte CarloMarkov chain Monte Carlo θA Posterior probability MCMC rule: P(θ A ) J(θ B → θ A ) if = P(θ B ) J(θ A → θ B ) then stationary distribution = posterior probability (provided chain is irreducible etc.) q qq Parameter valueBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  23. Background Model definitions Tools TestsMarkov chain Monte CarloMarkov chain Monte Carlo θA Posterior probability MCMC rule: P(θ A ) J(θ B → θ A ) if = P(θ B ) J(θ A → θ B ) then stationary distribution = posterior probability (provided chain is irreducible etc.) q qqq Parameter valueBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  24. Background Model definitions Tools TestsMarkov chain Monte CarloMarkov chain Monte Carlo θA Posterior probability MCMC rule: P(θ A ) J(θ B → θ A ) if = P(θ B ) J(θ A → θ B ) then stationary distribution = posterior probability (provided chain is irreducible etc.) q q q qq q q qqq Parameter valueBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  25. Background Model definitions Tools TestsMarkov chain Monte CarloMarkov chain Monte Carlo θA Posterior probability MCMC rule: P(θ A ) J(θ B → θ A ) if = P(θ B ) J(θ A → θ B ) then stationary distribution = posterior probability (provided chain is irreducible etc.) Parameter valueBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  26. Background Model definitions Tools TestsMarkov chain Monte CarloMarkov chain Monte Carlo θA Posterior probability MCMC rule: P(θ A ) J(θ B → θ A ) if = P(θ B ) J(θ A → θ B ) θB then stationary distribution = posterior probability (provided chain is irreducible etc.) Parameter valueBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  27. Background Model definitions Tools TestsMarkov chain Monte CarloMetropolis-(Hastings) updating θA Posterior probability Metropolis rule: accept B with probability P(θ B ) min 1, P(θ A ) satisfies MCMC rule . . . Don’t need to know denominator! Candidate distribution Parameter valueBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  28. Background Model definitions Tools TestsMarkov chain Monte CarloMetropolis-(Hastings) updating θA Posterior probability Metropolis-Hastings rule: accept B with probability P(θ B ) C (θ B , θ A ) min 1, · P(θ A ) C (θ A , θ B ) satisfies MCMC rule . . . Don’t need to know denominator! Candidate distribution Parameter valueBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  29. Background Model definitions Tools TestsGibbs/block samplingGibbs sampling Joint distribution of parameters often difficult Condition each element on “known” (i.e. imputed) values of all other elements: P(a, b, c) = P(a|b, c)P(b, c) 0.5 1 Gibbs sampling: do this 5 2. 1.5 2 repeatedly for each 5 1. 2.5 3 element, or block of 5 0. 3.5 4.5 4 3 4 3.5 2 elements 1Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  30. Background Model definitions Tools TestsGibbs/block samplingGibbs sampling Joint distribution of parameters often difficult q Condition each element q q q q on “known” (i.e. imputed) q q q q q q values of all other q q q q q q qq elements: P(a, b, c) = q q q P(a|b, c)P(b, c) 0.5 q 1 Gibbs sampling: do this 5 2. q 1.5 2 repeatedly for each 5 1. 2.5 3 element, or block of 5 0. 3.5 4.5 4 3 4 3.5 2 elements 1Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  31. Background Model definitions Tools TestsGibbs/block samplingGibbs sampling Joint distribution of parameters often difficult q Condition each element q q q q on “known” (i.e. imputed) q q q q q q values of all other q q q q q q qq elements: P(a, b, c) = q q q P(a|b, c)P(b, c) 0.5 q 1 Gibbs sampling: do this 5 2. q 1.5 2 repeatedly for each 5 1. 2.5 3 element, or block of 5 0. 3.5 4.5 4 3 4 3.5 2 elements 1Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  32. Background Model definitions Tools TestsGibbs/block samplingBack to the DAG parameters N1 N2 N3 N4 N5 obs1 obs2 obs3 obs4 obs5Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  33. Background Model definitions Tools TestsOutline 1 Background Ecological/statistical motivations Study system 2 Model definitions 3 Tools State-space models Markov chain Monte Carlo Gibbs/block sampling 4 Tests Perfect observation, trivial dynamics Perfect observation, non-trivial dynamics Imperfect observationBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  34. Background Model definitions Tools TestsPerfect observation, trivial dynamicsOutline 1 Background Ecological/statistical motivations Study system 2 Model definitions 3 Tools State-space models Markov chain Monte Carlo Gibbs/block sampling 4 Tests Perfect observation, trivial dynamics Perfect observation, non-trivial dynamics Imperfect observationBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  35. Background Model definitions Tools TestsPerfect observation, trivial dynamicsDAG parameters fates Demographic parameters|fates: standard M-H Fates|parameters N1 N2Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  36. Background Model definitions Tools TestsPerfect observation, trivial dynamicsScenario One fish of the same size observed Lsurv = (1 − m)(1 − p) S on two consecutive days . . . was it the same individual? (1 − m)(1 − p) P(surv|m, p) = 1 − m − p + 2mp X Lmort = mp(1 − p)S−1Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  37. Background Model definitions Tools TestsPerfect observation, trivial dynamicsUpdating probabilities for parameters mortality (m) 1.5 If the fish survived, then our 1.0 posterior probabilities for the 0.5 parameters are: type probability immigration (p) prior m ∼ Beta(1, 2) density 6 posterior (0 mortalities, 1 survival) 5 4 p ∼ Beta(1, S + 1) 3 2 (0 immigrations, S 1 non-immigrations) 0 0.2 0.4 0.6 0.8 probabilityBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  38. Background Model definitions Tools TestsPerfect observation, trivial dynamicsTrivial example: results mortality (m) immigration (p) 4 2(m + S(1 − m)) 3 Beta(p,1,S) S+1 density 2 1 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 valueBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  39. Background Model definitions Tools TestsPerfect observation, trivial dynamics P(mortality & no immigration) ≈ 0.16 = 1/(S + 1); P(survival & immigration) ≈ 0.83 = S/(S + 1); results are simple and make sense; closed form solution is possible but ugly . . . combination of observations and non-observation of other individuals provides information (could fail with non-detection)Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  40. Background Model definitions Tools TestsPerfect observation, non-trivial dynamicsOutline 1 Background Ecological/statistical motivations Study system 2 Model definitions 3 Tools State-space models Markov chain Monte Carlo Gibbs/block sampling 4 Tests Perfect observation, trivial dynamics Perfect observation, non-trivial dynamics Imperfect observationBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  41. Background Model definitions Tools TestsPerfect observation, non-trivial dynamicsNon-trivial dynamics Multiple individuals, measured over multiple days Still simple: no density-dependent rates, immigration/emigration of large individuals observations still error-freeBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  42. Background Model definitions Tools TestsPerfect observation, non-trivial dynamicsParameter/fate sampling parameters Parameter sampling: M-H updating with simple fates(1,2) fates(2,3) candidate distributions Fate sampling: ??? Enumerating N1 N2 N3 possibilities is tedious ...Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  43. Background Model definitions Tools TestsPerfect observation, non-trivial dynamicsM-H fate sampling 6 mm 7 mm dead 5 mm G (5) · (1 − M(5)) X M(5) 6 mm (1 − G (6)) · (1 − M(6)) G (6) · (1 − M(6)) M(6) absent I (6) I (7) XBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  44. Background Model definitions Tools TestsPerfect observation, non-trivial dynamicsM-H fate sampling 6 mm 7 mm dead 5 mm G (5) · (1 − M(5)) X M(5) 6 mm (1 − G (6)) · (1 − M(6)) G (6) · (1 − M(6)) M(6) absent I (6) I (7) XBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  45. Background Model definitions Tools TestsPerfect observation, non-trivial dynamicsM-H fate sampling 6 mm 7 mm dead 5 mm G (5) · (1 − M(5)) X M(5) 6 mm (1 − G (6)) · (1 − M(6)) G (6) · (1 − M(6)) M(6) absent I (6) I (7) XBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  46. Background Model definitions Tools TestsPerfect observation, non-trivial dynamicsM-H fate sampling 6 mm 7 mm dead 5 mm G (5) · (1 − M(5)) X M(5) 6 mm (1 − G (6)) · (1 − M(6)) G (6) · (1 − M(6)) M(6) absent I (6) I (7) XBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  47. Background Model definitions Tools TestsPerfect observation, non-trivial dynamicsM-H fate sampling 6 mm 7 mm dead 5 mm G (5) · (1 − M(5)) X M(5) 6 mm (1 − G (6)) · (1 − M(6)) G (6) · (1 − M(6)) M(6) absent I (6) I (7) XBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  48. Background Model definitions Tools TestsPerfect observation, non-trivial dynamicsM-H fate sampling 6 mm 7 mm dead 5 mm G (5) · (1 − M(5)) X M(5) 6 mm (1 − G (6)) · (1 − M(6)) G (6) · (1 − M(6)) M(6) absent I (6) I (7) XBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  49. Background Model definitions Tools TestsPerfect observation, non-trivial dynamicsSampling test 6 survives both grow w 5 grows Sampling Fate True Gibbs 6 grows both die 10−4 10−3 10−2 10−1 ProbabilityBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  50. Background Model definitions Tools TestsPerfect observation, non-trivial dynamicsTestbed Simulate for 80 days, with a settlement rate (tot.settle) of 5% day: 29 distinct individuals, size range 5–30, total of 618 fish-days (observations). Sample fates only (1000 MCMC steps), fixed demographic parameters Sampled 1000 unique fates (but only 895 unique likelihoods)Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  51. Background Model definitions Tools TestsPerfect observation, non-trivial dynamicsTestbed Simulate for 80 days, with a settlement rate (tot.settle) of 5% day: 29 distinct individuals, size range 5–30, total of 618 fish-days (observations). Sample fates only (1000 MCMC steps), fixed demographic parameters Sampled 1000 unique fates (but only 895 unique likelihoods)Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  52. Background Model definitions Tools TestsPerfect observation, non-trivial dynamicsTrue demography30 25 20 size 15 10 5 0 20 40 60 80 dayBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  53. Background Model definitions Tools TestsPerfect observation, non-trivial dynamicsFate-only results 0.05 0.04 Density 0.03 0.02 0.01 0.00 q −520 −500 −480 −460 Log−likelihoodBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  54. Background Model definitions Tools TestsPerfect observation, non-trivial dynamicsSampling fates, parameters, both . . . 0.25 0.20 0.15 type fate only Density parameters only 0.10 both 0.05 0.00 −500 −490 −480 −470 −460 −450 Log−likelihoodBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  55. Background Model definitions Tools TestsPerfect observation, non-trivial dynamicsParameter estimates 1 lgrow a.mort c.mort 140 3.0 0.8 120 2.5 100 2.0 0.6 80 1.5 0.4 60 1.0 40 0.2 0.5 20 −1.9 −1.7 −1.5 −1.3 −1.8 −1.6 −1.4 −1.2 −4.0 −3.5 −3.0 −2.5 0.010.015.020.025.030 0 0 0 0 density a.settle b.settle tot.settle 0.7 50 0.6 0.5 40 0.5 0.4 0.4 0.3 30 0.3 20 0.2 0.2 0.1 0.1 10 11 12 13 14 1.01.52.02.53.03.54.0 0.020.030.040.050.06Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  56. Background Model definitions Tools TestsPerfect observation, non-trivial dynamicsParameter estimates 2 Growth Appearance Persistence 1.0 q q q q 1.00 q qqqqqqqqqqqqqqqq 0.08 a.settle 0.8 b.settle 0.98 q q q q 0.06 tot.settle q 0.6 q q q 0.96 Probability q (per day) 0.04 q a.mort q 0.4 l.grow q q q q 0.94 c.mort q q q q q q q q 0.02 0.2 q q qqq q q q q q q q q 0.92 0.0 q 0.00 q qqqqqqqqqqqqqqqqqq q 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Fish size (mm)Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  57. Background Model definitions Tools TestsPerfect observation, non-trivial dynamicsParameter estimates 3 a.mort a.settle b.settle −2.0 16 4.0 q 15 3.5 q −2.5 q q q 14 3.0 q q q 13 q q 2.5 −3.0 q q q q q q q q 2.0 q q q 12 q q q q −3.5 q q q q 1.5 11 1.0 −4.0 10 q c.mort lgrow tot.settle −1.2 0.045 0.030 q 0.040 0.025 q 0.035 −1.4 q q 0.020 q q q q q 0.030 q q q 0.015 q −1.6 q q q 0.025 q q q q 0.010 q 0.020 q q q q −1.8 0.015 q 0.005 q q q q 0.010Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  58. Background Model definitions Tools TestsImperfect observationOutline 1 Background Ecological/statistical motivations Study system 2 Model definitions 3 Tools State-space models Markov chain Monte Carlo Gibbs/block sampling 4 Tests Perfect observation, trivial dynamics Perfect observation, non-trivial dynamics Imperfect observationBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  59. Background Model definitions Tools TestsImperfect observationAdding errors parameters fates(1,2) fates(2,3) Back to the more typical N1 N2 N3 graph: incorporate information from N(t − 1), N(t + 1), and current observation obs1 obs2 obs3 obs parametersBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  60. Background Model definitions Tools TestsImperfect observationNew algorithm For each individual: Pick true state at N(t), from all possible states, conditioning on N(t − 1) Pick identity at time N(t + 1) from feasible set: update probabilities Pick corresponding observation from observed(t): update probabilities Calculate likelihood, M-H updateBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  61. Background Model definitions Tools TestsImperfect observationOpen questions, caveats Will it work ?? Add complexities: Multiple populations Spatial variation, correlation among parameters Simplify/generalize code Speed?Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals
  62. Background Model definitions Tools TestsImperfect observationConclusions & musings MCMC as powerful technology . . . can it be democratized? Data limitations shift over time: hierarchical models are great for “modern” (high-volume, low-quality) dataBen Bolker Departments of Mathematics & Statistics and Biology, McMaster UniversityEstimating unmarked individuals

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