Occupancy models. Gurutzeta Guillera-Arroita University of Kent, UK National Centre for Statistical Ecology

Transcription

1 Occupancy models Gurutzeta Guillera-Arroita University of Kent, UK National Centre for Statistical Ecology Advances in Species distribution modelling in ecological studies and conservation Pavia and Gran Paradiso, Italy Sept. 2011

2 Outline Session 1: Introduction to occupancy modelling S1.1 Introduction S1.2 Statistical background S1.3 Single-season occupancy model Session 2: Occupancy modelling in practice S2.1 Practical: single-season S2.2 Study design Session 3: Occupancy modelling developments S3.1 Multiple-season occupancy model S3.2 Practical: multi-season S3.3 Further models 2

3 SESSION 3 Occupancy modelling developments S3.1 - Multiple-season occupancy model

4 Multi-season: observing the process Occupancy dynamics are of interest in many areas of ecology e.g. metapopulation studies Single-season occupancy modelling snapshot of the population at a single point in time But, often many processes can result in the same pattern may not possible to infer process from pattern to reliably understand processes need to observe how the system behaves over time MacKenzie et al. (2003) 4

5 Multi-season: observing the process Multiple-season occupancy modelling estimate occupancy over multiple seasons understand underlying population dynamics that cause changes in ψ Need to account for imperfect detection, otherwise negatively biased estimates of occupancy biased estimates of extinction and colonization 5

6 Multi-season sampling Effectively, a sequence of single-season studies conducted at usually the same sites for multiple seasons colonisation extinction colonisation... extinction survey 1,1 1,2 1,3... 1,k ,1 2,2 2,3... 2,k 2 T,1 T,2 T,3... T,k T season 1 (closed) season 2 (closed) season T (closed) 6

7 Multi-season sampling Sampling at two scales: Larger scale: Τ seasons Smaller scale: k t surveys per site per season colonisation extinction colonisation... extinction survey 1,1 1,2 1,3... 1,k ,1 2,2 2,3... 2,k 2 T,1 T,2 T,3... T,k T season 1 (closed) season 2 (closed) season T (closed) 7

8 Multi-season sampling Occupancy status of a site: can change between seasons remains constant within seasons colonisation extinction colonisation... extinction survey 1,1 1,2 1,3... 1,k ,1 2,2 2,3... 2,k 2 T,1 T,2 T,3... T,k T season 1 (closed) season 2 (closed) season T (closed) 8

9 Multi-season detection history e.g. resulting data set h Season 1 Season 2 Season 3... Season T Site Site Site Site Site Site s k replicates 9

10 Multi-season modelling approaches Two general approaches for modelling multi-season data Implicit dynamics model Apply a single-season model to each season Occupancy status in previous season not relevant Only the level of occupancy in each season is modelled Explicit dynamics model (MacKenzie et al. 2003) Model that explicitly incorporates the dynamic process underlying occupancy status Occupancy status at a site depends on previous season s status 10

11 Multi-season implicit model e.g. history for site i Season 1: site occupied, species detected in surveys 1 & 2, missed in 3 Season 2: site occupied and species not detected in any of the surveys OR site not occupied Pr h i = = ψ 1 p 1,1 p 1,2 1 p 1,3 1 ψ 2 + ψ 2 1 p 2,1 1 p 2,2 1 p 2,3 In practice, multiplication of various single-season Can multiply because assume independence 11

12 Multi-season explicit model Changes in the occupancy status at a site across seasons: 1-ε ε 1-γ Site occupied Site not occupied (Markov model) γ ε = probability that an occupied site becomes unoccupied (local extinction) γ = probability that an unoccupied site becomes occupied (colonisation) 12

13 Multi-season explicit model Changes in the occupancy status at a site across seasons: 1-ε ε 1-γ p Site occupied Site not occupied 1 1-p 0 γ 0 We do not observe the latent site occupancy status perfectly Detection probability <1 Hidden Markov model 13

14 Multi-season explicit model Model parameters: Local extinction ε Colonization γ Initial occupancy ψ 1 Detection probability p Alternative parameterizations possible e.g. (ε t, γ t ) (ψ t, γ t ) or (ψ t, ε t ) Extinction and colonization can be season specific (ε t, γ t ) Not assuming that the system is in equilibrium 14

15 Multi-season explicit model 1 1 or 0? e.g. history for site i Site occupied in season 1, and species detected in surveys 1 and 2, and missed in survey 3. Between seasons 1 and 2 the site continued occupied and the species was not detected in any of the three surveys OR the species became locally extinct Pr h i = = ψ 1 p 1,1 p 1,2 1 p 1,3 ε ε 1 1 p 2,1 1 p 2,2 1 p 2,3 15

16 Multi-season explicit model 1 or 0? 1 e.g. history for site i Site occupied in season 1, but species never detected, and species did not go locally extinct between seasons OR site not occupied in season 1 and colonized the site between seasons In season 2 the species was detected in the second survey and missed in the other two Pr h i = = ψ 1 1 p 1,1 1 p 1,2 1 p 1,3 1 ε ψ 1 γ 1 1 p 2,1 p 2,2 1 p 2,3 16

17 Multi-season explicit model 1 or 0? 1 or 0? e.g. history for site i ,1 Site occupied in season 1 and species never detected; no local extinction between seasons and species never detected in season 2 1, 0 Site occupied in season 1, but species never detected and species went locally extinct between seasons 0, 1 Site not occupied in season 1 and colonized between seasons and species never detected in season 2 0, 0 Site not occupied in season 1 and not colonized between seasons Pr h i = = ψ 1 1 p 1,1 1 p 1,2 1 p 1,3 1 ε ψ 1 γ 1 1 p 2,1 1 p 2,2 1 p 2,3 + ψ 1 1 p 1,1 1 p 1,2 1 p 1,3 ε ψ 1 1 γ 1 17

18 Multi-season explicit model History probabilities conveniently expressed with matrices π 1 = initial conditions φ t = matrix of transition probabilities p h,t = vector of probabilities for observed history in season t, conditional upon each occupancy state D(v): diagonal matrix with elements of v as main diagonal T 1 Pr h i = π 1 D p hit,t φ t p hit,t t=1 18

19 Multi-season explicit model History probabilities conveniently expressed with matrices π 1 = initial conditions φ t = matrix of transition probabilities p h,t = vector of probabilities for observed history in season t, conditional upon each occupancy state D(v): diagonal matrix with elements of v as main diagonal T 1 Pr h i = π 1 t=1 D p hit,t φ t p hit,t π 1 = 1 ψ 1 ψ 1 19

20 Multi-season explicit model History probabilities conveniently expressed with matrices π 1 = initial conditions φ t = matrix of transition probabilities p h,t = vector of probabilities for observed history in season t, conditional upon each occupancy state D(v): diagonal matrix with elements of v as main diagonal T 1 Pr h i = π 1 t=1 D p hit,t φ t p hit,t φ t = 1 γ t ε t γ t 1 ε t 20

21 Multi-season explicit model History probabilities conveniently expressed with matrices π 1 = initial conditions φ t = matrix of transition probabilities p h,t = vector of probabilities for observed history in season t, conditional upon each occupancy state D(v): diagonal matrix with elements of v as main diagonal Pr h i = π 1 T 1 D p hit,t φ t p hit,t p 101,t = 0 p t1 1 p t2 p t3 t=1 p 000,t = p tj j =1 21

22 Multi-season explicit model History probabilities conveniently expressed with matrices π 1 = initial conditions φ t = matrix of transition probabilities p h,t = vector of probabilities for observed history in season t, conditional upon each occupancy state D(v): diagonal matrix with elements of v as main diagonal T 1 Pr h i = π 1 D p hit,t φ t p hit,t t=1 Likelihood L ψ 1, γ, ε, p h = Pr h i s i=1 22

23 Multi-season explicit model Derived parameters Occupancy for different seasons π t+1 = π t φ t π t = 1 ψ t ψ t ψ t+1 = ψ t 1 ε t + 1 ψ t γ t Rate of change in occupancy λ t = ψ t+1 ψ t 23

24 Multi-season explicit model Missing data easily accounted for Within a season: e.g. 11- Vector p h,t adjusted by removing the corresponding survey parameters p 11,t = 0 p t1 p t2 Whole season missing for a site: e.g. h i = Omit corresponding p h,t entirely (i.e. p h,2 ) φ 2 still there, processes continue happening even if not surveyed Pr h i = = π 1 D p h101,1 φ 1 φ 2 p h111,3 24

25 Multi-season explicit model: covariates Model assumes that there is no unmodelled heterogeneity in the model parameters Failure to meet this assumption may induce bias in the estimators The relationship between covariates and certain parameters may actually be of primary interest for the study Covariates can be incorporated into the model by use of an appropriate link function (e.g. logit link) 25

26 Multi-season explicit model: covariates Occupancy, extinction and colonization as function of Season-specific covariates, e.g. habitat type, patch size, elevation (note season-specific includes site-specific covariates e.g. elevation) Detection probability as function of Season-specific covariates, e.g. habitat type Survey-specific covariates, e.g. observer, recent rainfall, temperature 26

27 Multi-season explicit model: equilibrium 2 types of equilibrium Constant ψ Constant ε and γ (stationary Markov process) Can test hypothesis regarding these Note they are different concepts Can have constant ψ with variable ε and γ (if they compensate) Can have variable ψ with constant ε and γ (if in transient) 27

28 psi Multi-season explicit model Constant ε and γ do not imply constant ψ Eventually reach an stationary ψ but may be observing the transient ε > γ does not imply the species is going to disappear Only that stationary ψ < 0.5 ψ could be increasing! gamma = 0.11, epsilon = 0.17 ψ eq = γ γ + ε = ψ t+1 = ψ t 1 ε t + 1 ψ t γ t = 0.72ψ t time 28

29 Explicit vs. implicit multi-season models May think that the implicit model does not involve considerations on extinction and colonization but in practice a fairly restrictive assumption is made ε t = 1 γ t ε t 1-ε t 1-γ t Site occupied Site not occupied γ t 29

30 Explicit vs. implicit multi-season models May think that the implicit model does not involve considerations on extinction and colonization but in practice a fairly restrictive assumption is made ε t = 1 γ t ε t 1-ε t 1-γ t = ε t Site occupied Site not occupied γ t = 1-ε t 30

31 Explicit vs. implicit multi-season models e.g. history under explicit model Pr h i = = ψ 1 p 2 1 p ε 1 + (1 ε 1 ) 1 p 3 If impose ε t = 1 γ t Pr h i = = ψ 1 p 2 1 p ε 1 + γ 1 1 p 3 = ψ 1 p 2 1 p 1 ψ 2 + ψ 2 1 p 3 ψ t+1 = ψ t 1 ε t + ψ t γ t = ψ t γ t + 1 ψ t γ t = γ t Pr h i = = ψ 1 p 2 1 p 1 ψ 2 + ψ 2 1 p 3 Likelihood for implicit model! 31

32 Single-season as a multi-season model The multi-season model falls back to the single-season model as we impose the restriction that no colonization or extinction takes place ε t = γ t = 0 (i.e. closure) ε t 1-ε t 1-γ t Site occupied Site not occupied γ t 32

33 Single-season as a multi-season model The multi-season model falls back to the single-season model as we impose the restriction that no colonization or extinction takes place ε t = γ t = 0 (i.e. closure) Site occupied Site not occupied 0 33

34 Single-season as a multi-season model e.g. history under explicit model Pr h i = = ψ 1 p 2 1 p ε 1 + (1 ε 1 ) 1 p 3 If impose ε t = γ t = 0 Pr h i = = ψ 1 p 2 1 p p 3 = ψ 1 p 2 1 p 4 Likelihood for single-season model 34

35 Markovian, random and no changes in occupancy Can test different hypothesis regarding how the occupancy status of sites changes over time comparing the three models: Markovian changes Multi-season explicit model (i.e. ε t, γ t ) Random changes Multi-season implicit model (i.e. ε t = 1 γ t ) No changes Single-season model (i.e. ε t = γ t = 0 ) 35

36 Survey design considerations in multi-season models Time interval between season Time-scale at which extinction and colonization are believed to operate Study objectives Advisable to keep a consistent time between seasons Same sampling sites visited Explicit dynamics model relies on information from same sites surveyed each season (at least a reasonable fraction of them) Implicit dynamics model can use same or different sites Study timeframe vs. resources More sampling effort needed per season if precise information about a occupancy trend needed within a short-time frame 36

37 Exploring occupancy dynamics Case study shown in MacKenzie et al. (2006) Investigates six different hypothesis about occupancy dynamics for the northern spotted owl (Strix occidentalis caurina) Territorial species Protocol: S=55 potential breeding sites in California T=5 breeding seasons ( ) k max = 8 visits/season (k mean =5.3) Consistent survey protocol across seasons Assumed: p(year), constant within seasons J. & K. Hollingsworth 37

38 Exploring occupancy dynamics Hypothesis 1: No change Owl territories well established and occupancy status does not change in this 5 years Constant occupancy of the sites ε =γ=0 Equivalent to analysing the complete data set as a single season ψ(.)p(year) 38

39 Exploring occupancy dynamics Hypothesis 2: Random dynamics, population equilibrium Low site fidelity: owl pairs change breeding sites from year to year at random Not expected to be the case for this species Random changes in occupancy of the sites ε =(1-γ) Population at equilibrium : constant γ and ε ψ(1997)γ(.){ε =(1-γ)}p(year) 39

40 Exploring occupancy dynamics Hypothesis 3: Random dynamics, no population equilibrium Same as hypothesis 2 (random changes in occupancy) but population not in equilibrium Population not at equilibrium: γ and ε may change over time ψ(1997)γ(year){ε =(1-γ)}p(year) 40

41 Exploring occupancy dynamics Hypothesis 4: Markovian dynamics, population equilibrium Previous CMR studies suggest high site fidelity and little migration Markovian changes in occupancy (depends on previous year s state) Now ε and γ estimated independently Population in equilibrium (constant ε and γ) ψ(1997)γ(.)ε(.)p(year) 41

42 Exploring occupancy dynamics Hypothesis 5: Random dynamics, no population equilibrium Same as hypothesis 4 (Markovian dependency) but population NOT in equilibrium: ε and γ depend on the year ψ(1997)γ(year)ε(year)p(year) 42

43 Exploring occupancy dynamics Hypothesis 6: stationary Markov process in equilibrium Probability of occupancy constant over the 5 seasons Colonisation γ also constant extinction ε must also be constant ε defined by ψ and γ: γ(1 ψ) ε = ψ stationary Markov process that has reached stationary distribution ψ(.)γ(.)p(year) 43

44 Exploring occupancy dynamics Results: Markovian dynamics well supported No support for random or no changes in occupancy state (MacKenzie et al. 2006) Good support for some form of equilibrium state (90% combined weight) Model-averaged estimates: ψ(1997)=0.61, γ~0.2, ε~0.14 Equilibrium occupancy would be ~ 0.2/( )=

45 SESSION 3 Occupancy modelling developments S3.2 Practical: multiple-season

46 Sample data set: grand skink Grand skink Oligosoma grande Endemic to the central Otago region of New Zealand Specific locations, typically limited to large rock outcroppings Endangered. Although once widespread, land use change and introduced mammalian predators have led to population decline. Wikimedia commons 46

47 Sample data set: grand skink 5-year data set (Grand_Skinks.xls) Sampling protocol s = 352 sites surveyed, up to k =3 replicates per site and year Each site: rock outcrop (not all surveyed each year) Covariate: surrounding habitat type (farm pasture vs natural grassland) Wikimedia commons Wikimedia commons 47

48 Practical 4 - results Part 4a: Model AIC deltaaic AIC wgt Model Likno.Par. -2loglik psi(h),gamma(t+h),eps(t+h),p(t) psi(h),gamma(t+h),eps(t),p(t) psi(h),gamma(t),eps(t+h),p(t) psi(h),gamma(t),eps(t),p(t) psi(),gamma(t+h),eps(t+h),p(t) psi(),gamma(t+h),eps(t),p(t) psi(),gamma(t),eps(t+h),p(t) psi(),gamma(t),eps(t),p(t) psi(),gamma(),eps(),p() H = pasture vs grassland, T = year 48

49 Practical 4 - results Part 4a: Model AIC deltaaic AIC wgt Model Likno.Par. -2loglik psi(h),gamma(t+h),eps(t+h),p(t) psi(h),gamma(t+h),eps(t),p(t) psi(h),gamma(t),eps(t+h),p(t) psi(h),gamma(t),eps(t),p(t) psi(),gamma(t+h),eps(t+h),p(t) psi(),gamma(t+h),eps(t),p(t) psi(),gamma(t),eps(t+h),p(t) psi(),gamma(t),eps(t),p(t) psi(),gamma(),eps(),p() H = pasture vs grassland, T = year Support for H: γ (0.89), ε (0.73), ψ1 (0.99) 49

50 Extinction Colonization Practical 4 - results Part 4a: pasture grassland pasture grassland year year 50

51 Practical 4 - results Part 4b: Model AIC deltaaic AIC wgt Model Likno.Par. -2loglik psi(h),gamma(h),eps(h),p(t) psi(h),gamma(h),eps(),p(t) psi(h),gamma(t+h),eps(t+h),p(t) psi(h),gamma(),eps(h),p(t) psi(h),gamma(t+h),eps(t),p(t) psi(h),gamma(),eps(),p(t) psi(h),gamma(t),eps(t+h),p(t) psi(h),gamma(t),eps(t),p(t) psi(),gamma(h),eps(h),p(t) psi(),gamma(t+h),eps(t+h),p(t) psi(),gamma(h),eps(),p(t) psi(),gamma(t+h),eps(t),p(t) psi(),gamma(),eps(h),p(t) psi(),gamma(t),eps(t+h),p(t) psi(),gamma(),eps(),p(t)

52 Practical 4 - results Part 4b: Model AIC deltaaic AIC wgt Model Likno.Par. -2loglik psi(h),gamma(h),eps(h),p(t) psi(h),gamma(h),eps(),p(t) psi(h),gamma(t+h),eps(t+h),p(t) psi(h),gamma(),eps(h),p(t) psi(h),gamma(t+h),eps(t),p(t) psi(h),gamma(),eps(),p(t) psi(h),gamma(t),eps(t+h),p(t) psi(h),gamma(t),eps(t),p(t) psi(),gamma(h),eps(h),p(t) psi(),gamma(t+h),eps(t+h),p(t) psi(),gamma(h),eps(),p(t) Support psi(),gamma(t+h),eps(t),p(t) for T: in γ and ε (0.20) psi(),gamma(),eps(h),p(t) Support for H: γ (0.85), ε (0.72), ψ 1 (0.99) psi(),gamma(t),eps(t+h),p(t) psi(),gamma(),eps(),p(t)

53 Extinction Colonization Practical 4 - results Part 4b: 0.5 pasture 0.5 pasture 0.4 grassland 0.4 grassland

54 Psi Practical 4 - results Part 4b: pasture grassland psi year pasture grassland Equilibrium 1: ε and γ are constant in time (stationary Markov process) Equilibrium 2: ψ is constant (stationary distribution) 54

55 SESSION 3 Occupancy modelling developments S3.3 Further models

56 Further models Multiple-detection methods, multi-scale occupancy Species co-occurrence Multi-state occupancy Integrated model of habitat and species occurrence dynamics Replicated count data Continuous sampling protocols 56

57 Multiple detection methods, multi-scale occupancy

58 Multiple detection methods, multi-scale occupancy Multiple independent detection methods at each site The species can be detected by one or more of them Camera trap Hair snare Scat search Nichols et al. (2008) 58

59 Multiple detection methods, multi-scale occupancy Estimation of method-specific detection probability (p s) Allows occupancy estimation at two spatial scales: Larger scale: use of sampling unit (ψ) Smaller scale: presence at sampling unit (θ) Camera trap Hair snare Scat search Nichols et al. (2008) 59

60 (Recall) Assumptions: closure If changes at random: occupancy estimator remains unbiased ψ interpreted as use p is smaller as it involves two components Pr species detected at site i during survey j = Pr(species uses site i) ψ Pr(species present at site i during survey j uses site i) = 1 if closure p Pr(species detected during survey j present at site i during survey j) 60

61 (Recall) Assumptions: closure If changes at random: occupancy estimator remains unbiased ψ interpreted as use p is smaller as it involves two components the multiple-detection model allows to separate these components Pr species detected at site i during survey j = Pr(species uses site i) ψ Pr(species present at site i during survey j uses site i) = 1 if closure θ Pr(species detected during survey j present at site i during survey j) p 61

62 Multiple detection methods, multi-scale occupancy e.g. history Survey 1 Survey 2... Survey Κ Site Site Site Site Site Site s L detection methods 62

63 Multiple detection methods, multi-scale occupancy e.g. 3 detection methods and 2 occasions history for site i = species uses the site 1 st occasion: species present, detected by method 2 and missed by methods 1 and 3 2 nd occasion: species present, missed by method 2 and detected by methods 1 and 3 Pr h i = = ψ θ 1 1 p 1 (1) p 1 (2) 1 p 1 (3) θ 2 p 2 (1) 1 p 2 (2) p 2 (3) 63

64 Multiple detection methods, multi-scale occupancy e.g. 3 detection methods and 2 occasions history for site i = species uses the site 1 st occasion: species present, detected by method 2 and missed by methods 1 and 3 2 nd occasion: species present and missed by the three methods OR species not present Pr h i = = ψ θ 1 1 p 1 (1) p 1 (2) 1 p 1 (3) 1 θ 2 + θ 2 1 p 2 (1) 1 p 2 (2) 1 p 2 (3) 64

65 Multiple detection methods, multi-scale occupancy To distinguish the two occupancy scales, need some time interval between sampling occasions i.e. need to let the species come and go The model can also be applied to a single detection method sampled at two different time-scales ( robust design ) smaller time scale multiple detection methods 65

66 Species co-occurrence

67 Species co-occurrence Assess whether some species tend to occur more or less often together that expected assuming independence Or Need to account for imperfect detection, otherwise If p independent of the other species presence, the direction of the interaction may be correct but its magnitude will be underestimated If p depends on the other species presence, the direction of the interaction may be misleading (e.g. predator-prey) MacKenzie et al. (2004) 67

68 Species co-occurrence: occupancy states ψ A ψ B ψ Ab ψ AB ψ ab ψ ab ψ A = Pr(site occupied by species A) ; ψ B = Pr(site occupied by species B) ψ AB = Pr(site occupied by species A and B) 68

69 Species co-occurrence: occupancy states ψ A ψ B ψ Ab ψ AB ψ ab ψ ab ψ AB = ψ A B ψ B = ψ A ψ B If independence 69

70 Species co-occurrence: occupancy states ψ A = 0.4, ψ B = 0.5 ψ A ψ B ψ A ψ B ψ A ψ B ψ AB ψ AB ψ AB ψ AB = 0.2 = ψ A ψ B independence ψ AB = 0.1 < ψ A ψ B avoidance ψ AB = 0.3 > ψ A ψ B co-occurrence 70

71 Species co-occurrence: detection probabilities p i = Pr(detect species i given it is the only one present) p A, p B Given both species are present: r AB = Pr(detect species A and B) r Ab = Pr(detect species A and not B) r ab = Pr(detect species B and not A) (r ab = Pr(miss species A and B) ) 71

72 Species co-occurrence: likelihood construction e.g. history for site i species A = 11 species B = 10 Both species present Both species detected in survey 1 Only species A detected in survey 2 Pr h i A = 11, h i B = 10 = ψ AB r AB r Ab Parameterization: ψ AB, ψ A, ψ B, p A, p B, r AB, r Ab, r ab 72

73 Species co-occurrence: likelihood construction e.g. history for site i species A = 11 species B = 00 Both species present and A detected in survey 1 and 2 OR only A present and detected in survey 1 and 2 Pr h i A = 11, h i B = 00 = ψ AB r Ab r Ab + ψ Ab p A p A ψ Ab = ψ A ψ AB Parameterization: ψ AB, ψ A, ψ B, p A, p B, r AB, r Ab, r ab 73

74 Species co-occurrence: likelihood construction e.g. history for site i species A = 00 species B = 00 Both species present and missed in both surveys OR only A present and missed in both surveys OR only B present and missed in both surveys OR both species absent Pr h A i = 00, h B i = 00 = ψ AB r 2 ab + ψ Ab 1 p 2 A + ψ ab 1 p 2 B + ψ ab r ab = 1 r Ab r ab r AB ψ Ab = ψ A ψ AB ψ ab = ψ B ψ AB ab = 1 ψ B ψ A + ψ AB Parameterization: ψ AB, ψ A, ψ B, p A, p B, r AB, r Ab, r ab 74

75 Species co-occurrence: other parameterizations Alternative parameterizations are possible E.g. instead of ψ A, ψ B, ψ AB use ψ A, ψ B, φ φ = ψ AB ψ A ψ B level of co-occurrence φ = 1 independent φ < 1 avoidance φ > 1 co-occurrence Same idea for detection probabilities, i.e. use r A, r B, δ where r AB = r A r B δ 75

76 Species co-occurrence: hypotheses The model allows testing three interesting biological hypotheses about the system by constraining various parameter values: level of co-occurrence between species (φ = 1?) independence of detecting the species (δ = 1?) whether detection of each species depends on the presence of the other species (r i = p i?) The model can be extended to multiple-season to test hypotheses about dynamics 76

77 Species co-occurrence: considerations Note the model assesses statistical co-occurrence, not necessarily reflecting true species interactions Could be reflecting an unmodelled habitat relationship! Could be a real ecological co-occurrence (e.g. plant pollinator) Impractical if several species Very large number of parameters required Mathematics get really cumbersome Interpretation of high-order interactions becomes difficult In most cases, insufficient data for estimation 77

78 Multi-state occupancy models

79 Multi-state occupancy Basic idea: generalize the present - absent setup Model more than 2 states, e.g.: Absent / Present & healthy / Present & infected Absent / Present & non-breeding / Present & breeding Absent / Present in low numbers / Abundant... and true state does not change within season (closure) Nichols et al. (2007); MacKenzie et al. (2009) 79

80 state 0 state 1 state 2 Multi-state occupancy Key idea: the top state is identifiable without error: If disease detected (state 2) species present (site occupied) If species detected (state 1 or 2) site occupied If species not detected, is the site: Empty (state 0)? Occupied, with no disease (state 1)? Occupied, with disease (state 2)? Species detected, disease detected Species detected, disease not detected If species detected but disease not, is the site: Occupied, with no disease (state 1)? Occupied, with disease (state 2)? Species not detected 80

81 Multi-state occupancy e.g. data set h Species detected, infection detected State 0: absent State 1: present & no disease State 2: present & at least some animals diseased k Site Site Site Site Site Site s Species detected, infection not detected Species not detected 81

82 True state Parameter definitions Notation as in MacKenzie et al. (2009) State probability: Pr(state 1) = φ [1] : Pr(site i occupied by non-diseased individuals) Pr(state 2) = φ [2] : Pr(site i occupied and disease present) Pr(state 0) = φ [0] = 1 φ [1] φ [2] : Pr(site i empty) Detection: p ij n,m State 0: absent State 1: present & no disease State 2: present & at least some animals diseased : Pr(observing site i at survey j in state n given the true state is m) Observed state p ij 1,1 1 p ij 1,2 p ij 2,2 p ij 1,1 p ij 1,2 0 p ij 2,2 82

83 Probability of detection history State 0: absent State 1: present & no disease State 2: present & at least some animals diseased Pr h i = 120 = φ 2 p 1 1,2 p 2 2,2 (1 p 3 1,2 p 3 2,2 ) we know site is occupied and infected (state 2) Pr h i = 011 = φ 2 1 p 1 1,2 p 1 2,2 p 2 1,2 p 3 1,2 we know the site is occupied but can t be sure if infected (states 1 or 2) +φ 1 1 p 1 1,1 p 2 1,1 p 3 1,1 83

84 Probability of detection history State 0: absent State 1: present & no disease State 2: present & at least some animals diseased occupied and infected Pr h i = 000 = φ 2 1 p 1 1,2 p 1 2,2 1 p 2 1,2 p 2 2,2 1 p 3 1,2 p 3 2,2 +φ 1 1 p 1 1,1 1 p 2 1,1 1 p 3 1,1 occupied but not infected +1 φ 1 φ 2 absent 84

85 Parameter definitions (2) State 0: absent State 1: present & no disease State 2: present & at least some animals diseased Alternative parameterisation in Nichols et al. (2007) State probability: ψ = φ [1] + φ [2] : probability of occupancy R = φ [2] /ψ : Pr(site i in state 2 given it s occupied) Pr(state 0): 1 ψ Pr(state 1): ψ 1 R) Pr(state 2): ψr 85

86 True state Parameter definitions (2) State 0: absent State 1: present & no disease State 2: present & at least some animals diseased Alternative parameterisation in Nichols et al. (2007) Detection: p [m] : Pr(detecting species given true state m) p [1] = p [1,1] p [2] = p [1,2] + p [2,2] δ = p [2,2] / p [2] : Pr(correctly classifying state 2) Observed state p 1 p p 2 p 2 (1 δ) p 2 δ 86

87 Probability of detection history (2) State 0: absent State 1: present & no disease State 2: present & at least some animals diseased Pr h i = 120 = ψrp δ1 p 2 2 δ2 (1 p 3 [2] ) we know site is occupied and infected (state 2) Pr h i = 011 = ψr(1 p 1 2 )p2 2 1 δ2 p 3 2 (1 δ3 ) +ψ(1 R) 1 p 1 [1] p 2 [1] p3 [1] we know the site is occupied but can t be sure if infected (states 1 or 2) 87

88 Probability of detection history (2) State 0: absent State 1: present & no disease State 2: present & at least some animals diseased occupied and infected Pr h i = 000 = ψr(1 p 1 [2] )(1 p2 [2] )(1 p3 [2] ) +ψ(1 R)(1 p 1 [1] )(1 p2 [1] )(1 p3 [1] ) occupied but not infected +(1 ψ) absent 88

89 Multiple seasons and multiple states If more than one season, we can model explicitly the transitions between states (transition probability matrix): φ t = e.g. 2 0 : Pr(state 0 given site was in state 2 in previous season) 1 0 : Pr(state 0 given site was in state 1 in previous season) Note the connection to presence-absence multi-season model (only 2 states) 89

90 Multiple seasons and multiple states Transitions between states: φ [0,0] Not occupied (state 0) φ [0,2] φ [0,1] φ [1,0] φ [2,1] Occupied, no disease (state 1) φ [1,1] φ [2,0] Occupied, disease (state 2) φ [1,2] φ [2,2] 90

91 Multiple seasons and multiple states 2 parameterizations: Multinomial φ t = φ t 0,0 φ t 1,0 φ t 2,0 φ t 0,1 φ t 1,1 φ t 2,1 φ t 0,2 φ t 1,2 φ t 2,2 φ t n,m =Pr(transition from state n at time t to state m at t+1) Conditional binomial φ t = 0 1 ψ t ψ t+1 [2] 1 ψ t+1 0 ψ t+1 1 ψ t+1 2 ψ t R t R t R t ψ t+1rt ψ t+1rt ψ t+1rt+1 m ψ t+1 m R t+1 =occupied at t+1 given state m at t) =Pr(infection at t+1 given state m at t) 91

92 Some considerations Flexible framework for many different models Occupancy models are a particular case of multi-state models Single-season occupancy = single-season multistate (with 2 states) Multi-season occupancy = multi-season multistate (with 2 states) Other models can also be interpreted as multi-state models e.g. species co-occurrence model (presented before) e.g. integrated model of habitat and occupancy dynamics (coming next) 92

93 Integrated model of habitat and species occurrence dynamics

94 Joint habitat & occupancy dynamics Probability of occupancy related to habitat type Basic idea: model simultaneously the dynamics of habitat and of species occupancy, over several seasons Habitat dynamics: Discrete habitat types or states Transition between types depend on previous type (1 st order Markovian) The presence of the species can affect habitat change Occupancy dynamics: Like in multi-season model (1 st order Markovian) Extinction and colonisation can depend on habitat change Probability of occupancy depends on habitat type MacKenzie et al. (2011) 94

95 Joint habitat & occupancy dynamics Example: 2 habitat types, 4 seasons, k=3 surveys Season: t1 t2 t3 t4 Habitat: A A B A Occupancy: Detections: Habitat type: no uncertainty Occupancy: uncertainty (as usual) repeated surveys MacKenzie et al. (2011) 95

96 Joint habitat & occupancy dynamics Model parameters: π [H] : Pr (habitat H in 1 st season) ψ [H] : Pr(occupied at habitat H in 1 st season) X η t,h t,h t+1 t : Pr(habitat change H t H t+1, given species was either present (X=1) or absent (X=0) at the site in season t) Allows the presence of the species to affect habitat change! MacKenzie et al. (2011) 96

97 Joint habitat & occupancy dynamics Model parameters: γ t H t,h t+1 : Pr(colonisation between seasons t and t+1 given habitat transition H t H t+1 ) ε t H t,h t+1 : Pr(local extinction between seasons t and t+1 given habitat transition H t H t+1 ) H p t jt : Pr(detect the species in survey j of season t, given habitat H in season t) MacKenzie et al. (2011) 97

98 Example of Pr(history) Season: Habitat H: A A B Detections h: Absent? extinction between 1 and 2, colonization between 2 and 3 Occupied (undetected)? no extinction between 1 and 2, no colonization between 2 and 3 Pr H, h = π A ψ A A p 1,1 1 p 1,1 A 1 η 1 1,A,B 1 ε 1 A,A A 1 p 2,1 A 1 p 2,2 η 2 1,A,B 1 ε 2 A,B + ε 1 A,A η2 0,A,B γ2 A,B B B p 3,1 p3,1 occupied in t=2 absent in t=2 98

99 The model as a multi-state The joint habitat/occupancy dynamics model can be seen (and analysed) as a multi-state model E.g.: Habitat types A & B; present or absent 4 states (joint habitat/occupancy) Habitat Occupancy State 1 A Present State 2 A Absent State 3 B Present State 4 B Absent If habitat type can only be suitable or unsuitable (where probability of occupancy = 0), the model simplifies to 3 states 99

100 Replicated count data

101 Replicated count data Surveys in which counts of individuals are collected Survey 1 Survey 2 Survey 3... Survey Κ Site Site Site Site s Could collapse data to detection/non-detection and apply occupancy models But, why would we throw away that information? 101

102 Replicated count data Occupancy model: detections d ij Bernoulli(p ij ) p ij probability of detection the species at site i during survey j N-mixture model: counts n ij Binomial(N i, r ij ) N i number of individuals at sampling site i r ij probability of detection of one individual at site i during survey j Assumptions include: all individuals are detected with the same probability each individual detected no more than once in each survey number of individuals in the site constant across the season (closure) Royle (2004) 102

103 Replicated count data N i is not known and is modelled as a random variable with a given distribution (e.g. Poisson) i.e. discrete mixture over site-abundance Same idea as in the Royle-Nichols model, but here with counts rather than detection/non-detection data Royle (2004) 103

104 Replicated count data e.g. likelihood h i = 302 Pr(h i = 302) = Pr(n i1 = 3 N i )Pr(n i2 = 0 N i )Pr(n i3 = 2 N i )Pr(N i ) N i N i n ij p ij n ij 1 p ij N i n ij e λ λ N i N i! Assumption that can only detect individuals once Pr n ij N i = 0 if n ij > N i i.e. at least max(n ij ) individuals in site i Royle (2004) 104

105 Replicated count data Estimates the parameters of the assumed underlying abundance distribution Need to be careful with the model assumptions, and the interpretation of abundance If individuals can be detected more than once per replicate, the binomial distribution is not suitable to describe the counts Particularly relevant when surveying for signs Can use a Poisson distribution instead (Guillera-Arroita, in press) n ij Poisson(N i ρ ij ) with ρ ij detection rate of one individual 105

106 Continuous sampling protocols

107 Discrete vs. continuous sampling protocols So far we have assumed discrete sampling protocols E.g. separate visits to each site at different points in time E.g. visits to discrete subunits within each site But in some cases data are collected in a continuous fashion E.g. camera-trap operating during a period of time E.g. detection data collected along transects 107

108 Discrete vs. continuous sampling protocols These data could be discretized using a given segment length and fitted with the models developed for discrete protocols But segment length is often arbitrary and the discretization can lead to (unnecessary) loss of data A more natural description of the data given by point processes 108

109 Discrete vs. continuous sampling protocols Point processes model random events in time (or space) Used in a wide variety of fields e.g. to model the arrival of customers in a queue e.g. to model the earthquake occurrences Also useful to describe detections of species (or individuals) in a period of time, or along a transect. 109

110 Discrete vs. continuous sampling protocols E.g. description of species detections at occupied sites: Discrete Continuous MacKenzie et al (2002) Independent Bernoulli trials Poisson process Guillera-Arroita et al (2011) Hines et al (2010) Bernoulli trials with Markovian dependence 2-Markov modulated Poisson process Guillera-Arroita et al (2011) Poisson process 2-MMPP 110

111 Wrap up

112 Further resources Occupancy Estimation and Modeling, MacKenzie et al. (2006) 112

113 Further resources Program MARK. A gentle introduction Evan Cooch & Gary White (Eds.) Mark-recapture, but has a lot of general advice on modelling & statistical issues 113

114 Further resources phi-dot website: Support forum (but read the manual first!) Not only occupancy, also CMR etc Course & conference announcements Donovan, T. M. and J. Hines Exercises in occupancy modeling and estimation. Exercises using Excel, PRESENCE, MARK 114

115 Final remarks Occupancy modelling offers a powerful and flexible framework for monitoring and ecological inference, including species distribution modelling Often more cost-effective than other techniques Many exciting developments going on (keep tuned!) BUT keep in mind:... your objectives (why you want to estimate occupancy)... model assumptions (for meaningful estimates)... survey design (to optimise your use of resources) 115