Introduction to Density Estimation

Transcription

1 Introduction to Density Estimation

2 2 Goal Estimate population size N (= abundance) or density D of animal species* Problems: Many species occur at (very) low density over (very) large areas Many of these areas are hard (i.e. expensive) to access Many species not conspicuous (e.g. cetaceans spend almost all their time underwater) More traditional (visual) methods have shortcomings * D=N/A A=area

3 3 How to count them To estimate abundance/density we can: Count them all: implement a census Impossible for most situations Sample: Count part of the population and scale up Where to sample? Survey design question not our main focus here. How to sample are all animals in the sampled area detected? We focus on this.

4 4 Traditional survey methods Traditional methods use visual sightings Plot sampling Distance sampling (Buckland et al. 2001) Visual line transects Point transects Cue counting Capture-recapture (a.k.a. mark recapture) (Williams et al. 2002) Photo-ID Biopsy (Genetics-based)

5 5 Plot sampling Interest in A and survey a (typically A >> a) Key: randomness and replication

6 6 Plot sampling Coverage points survey units lines high a / A low

7 7 Design based inferences Based on (e.g.) simple random sampling: Covered area a Density = n/a Abundance = n Density = n/a N = n / coverage probability Survey area A

8 8 Model based inferences?? Based on a model (e.g. of longitude): Covered area a Density = n/a Abundance = n model model Survey area A

9 9 Example I You counted all the n animals in a given area of size a. Density in a is by definition D=n/a If a = 100km 2, n = 300, then D=3 animals/km 2 Now assume that we only detect a proportion p of the animals. Under that scenario D n ap If we knew p we could get D. Assume p = D 4 animals per km

10 10 But we do not know p! Frameworks Estimating Distance sampling Capture-recapture P Probability of detection D Density (or abundance) Underlying idea simple & intuitive (sometimes) complex implementation

11 11 Distance sampling Randomly allocate large number of line or point transects over study area

12 12 Distance sampling Collect distances to all detected animals Detected animal Missed animal

13 13 Distance sampling Use distances to model a detection function g(x), which represents the probability of detecting an animal given it is at a distance x from the transect Distance in meters Source: Williams an Thomas JCRM

14 14 Distance sampling Use detection function to estimate p

15 15 Distance sampling Use detection function to estimate p P=

16 16 Lines and points: estimating p Line transects Point transects and cue counting True distribution of animals Detection function, g(x) Observed distribution, f(x)

17 17 Distance sampling recap

18 18 Distance sampling Many transects, independent of animal distribution, allows us to assume known animal distribution: Line transects: uniform Point transects / traditional cue counting: triangular Many transects also allow: Better spread of transects Robust variance estimation

19 19 Distance sampling assumptions All animals at 0 distance are detected, g(0)=1 If animals on (or near) the point or line are missed, density estimators are biased low (by g(0)) Distances are measured without error If distances have errors, D can be (potentially severely) biased Observation process is instantaneous Slow animal movement (relative to platform speed) not a problem Responsive movement causes bias (towards observer over-estimation and visa-versa)

20 20 Capture-recapture Example: two visits, closed population: 1st visit, 100 animals captured and marked. 2nd visit, 90 animals are captured, of which 30 are marked. How can we estimate pop size N?

21 21 Capture-recapture II One intuitive approach: Proportion marked in the population = 100/N Proportion marked in the sample=30/90... and given a random sample and a closed population these are (on average) the same

22 22 Capture-recapture III In practice, n animals are captured over K sessions and we observe their capture histories session

23 23 Capture-recapture IVa What can you say about the values of p and N in this scenario? session High p, small N

24 24 Capture-recapture IVb What can you say about the values of p and N in this scenario? session High p, big N

25 25 Capture-recapture IVc What can you say about the values of p and N in this scenario? session Low p, (fairly) small N

26 26 Capture-recapture IVd What can you say about the values of p and N in this scenario? session Low p, (very) big N

27 27 Problem 1: N is ill defined Except in rare occasions (e.g. small island, closed population), the estimated N is valid for an unknown area (at best ad hoc procedures to get surveyed area) can t get D!

28 28 Problem 2: Unmodelled heterogeneity If some animals are much easier to see than others overestimate p understimate N session

29 29 Spatially explicit capture recapture (SECR) An animal s capture history tells us something (0,0,1,1,0,1,1,0,0,1) But it can tell us more: it has a spatial component usually ignored (0,0,8,12,0,11,7,0,0,12) Borchers 2012 (SECR for dummies) Borchers & Efford 2008 (tech foundations) Dawson & Efford 2009 (1st acoustic app) Example: 10 capture sessions 16 traps

30 30 SECR II SECR advantages over conventional capturerecapture: Reduced heterogenity, as much of it can come from the spatial component. less bias Can make inferences about the area surveyed Density is a parameter in the model (hence density estimates are obtained directly)

31 31 Standard vs. non-standard ways to estimate p Distance sampling, mark-recapture and (more recently) SECR standard methods in the literature In some passive acoustics situations, standard methods can t be used but variations can be, based on similar ideas See next lecture, plus 2 of the 4 case studies

32 32 Variance and Precision 100 animals per 1000 km2 it means something... Estimate Scenario Worst Nightmare Wildest Dream Likely real...but it does not mean much! 95% Confidence Interval ( ) (98 102) (75 135)

33 33 Measures of Precision Variance: s2 Standard Deviation: s Standard Error, SE = s/ n Coefficient of variation, CV = SE/estimate (often reported as a percentage, i.e. 100) Confidence Intervals are often useful Analytic (formula based) estimates of variance Resampling procedures (e.g., bootstrap)

34 34 Variance and Precision In terms of CV for density estimation, general rules of thumb are: CV (%) >50 Scenario Liar Dream Great Good Most Likely Nightmare

35 35 A very useful result: the delta method Given an estimator made up of a number of constants and random components, the delta method states that if the random components are independent, then we can estimate the CV of the estimator as CV ( Dˆ ) 2 CV (random components)

36 36 Indirect methods Want to estimate animal density But often do not detect individual animals themselves. Examples include: Some sign produced by the animal is detected (dung, nests, songs) When groups rather than individual animals are detected So, we can estimate sign or group density need multipliers to convert to animal density. In acoustics, we often deal with such indirect methods

37 37 Indirect methods example: cue counting Developed for visual shipboard surveys of big whales count whale blows (blow= cue ) Use distance sampling methods to get density of blows, Dc To get animal density, we need to divide by blow rate ( cue rate ) r, i.e., D = Dc /r Works just the same for animal sounds...

38 38 Multipliers Cue rate is an example of a multiplier blow production rate sound production rate Other examples that may be needed: average group size false positive detection rate... It is strongly desired that these are collected in the same time and place as the main survey. If not, are they representative? Possibly the main stumbling block with acoustic methods...

39 39 Many animals are hard to see

40 40 To be or not to be (seen) 1500 Depth (m) 0 Blainville s beaked whale Mesoplodon densirostris, Bahamas, ~4 hours 0 Time (hours) 4 Credit: Robin Baird

41 41 To be or not to be (seen) Depth (m) 0 Blainville s beaked whale Mesoplodon densirostris, Bahamas, ~80 hours 1500 ~ 85% of the time below 3m Time (hours) Credit: Robin Baird

42 42 Visual disadvantages (or acoustics advantages) Visual methods are: Problematic for animals that are submerged for long periods Restricted to daylight periods Dependent on good weather conditions Often susceptible to small sample sizes Impractical for continuous long-term monitoring

43 43 Density information in sound Information about number of animals Number of detections Information about probability of detection Sound intensity Ominidirectional source Distance to sound Sound characteristics (e.g. frequency) Distance to sound Multiple sensors detect same sound Indication of location

44 44 Now with sound: an intuitive density estimator Count n sounds in an area a, what is the estimated density of animals? n ˆ D acˆ hat ^ denotes an estimate C is a correction that accounts for many factors, including: (not only) Detection probability (but also) Sound production rate False positives Observation effort (recording time, etc)

45 45 Important issues not covered here Obtaining acoustic data Type of sensors Hardware Processing acoustic data Detection Classification (and eventually) Localization

46 46 QUESTIONS?

47 Estimating density using acoustic data

48 2 Goals of lecture Show how to apply/adapt the methods introduced in the previous lecture to passive acoustics Provide a roadmap to help choose which method to use when Note: The focus here is on estimating density from fixed passive acoustic sensors (although we ll mention active and towed) Examples presented are very broad-brush. Due to time constraints some important issues such as variance estimation are hardly mentioned. Will return to these issues in the case studies. Most examples come from our work so we can criticize them freely!

49 Roadmap Part 1 3

50 4 Roadmap Part 2 Fixed acoustic sensors

51 5 Type of acoustic survey Active methods allow animals to be detected when they are not vocalizing Ranging is often feasible Can use standard distance sampling methods But: Animals may respond to the sound (assumption violation) Possible welfare issues If you re working with deep-diving animals that may be missed at depth, then you ll need to account for this. Example Cox et al. (2011) on krill. Active acoustics will not be our focus here

52 What can you count? 6

53 7 Counting cues, counting groups If you can count individuals acoustically, then you can (potentially) estimate the density of individuals More commonly, however, one can only count groups or cues (or it may be better to do so) indirect methods Then, you can estimate the density of groups or cues. Need a multiplier to convert to density of individuals: group size or mean cue rate Need auxiliary data to get these Often the Achilles heel of indirect methods Better to get them from a representative sample taken at the time and place of the main survey For more on this, see the case studies

54 Type of sensor platform 8

55 9 Moving platforms Multi-sensor towed (or bow mounted) shipboard (/submarine/glider) array Obtain bearing to individuals (or groups) Multiple bearings give position Analogous to a standard line transect survey Examples: Hastie et al. 2003; Barlow and Taylor 2005; Lewis et al. 2007

56 10 Moving platforms - issues Detection not certain at zero distance Can be corrected for if you know vocalization pattern Inaccurate localization causes measurement error in perpendicular distances Usually not a major problem Methods exist for dealing with measurement error if you know the error distribution (Marques et al. 2004, Borchers et al. 2010)

57 11 Moving platforms issues (contd.) Unknown depth means horizontal perpendicular distance is unknown Only a problem for deep-diving species Ignoring problem overestimates distances underestimates density Can be incorporated if you know the distribution of depths for vocalizing animals and work in the vertical plane with the direct (radial) distances. In some cases, count groups rather than individuals Then need to get an estimate of mean group size (e.g., visually)

58 12 Moving platforms issues (contd.) Species mis-classification Can treat each species separately False positives need accounting for Often use manual analysis of a sample of data to ground truth an automated detector - need to be careful with sampling design (systematic random sample is best) Obtain estimate of proportion of detections that are false positives can use as a multiplier False negatives not a problem (if there are none on the trackline) A more coherent approach is to deal with mis- classification for all species together Caillat et al. (2013); Conn et al. (2013)

59 13 Moving platforms single sensor Many gliders only have a single sensor No bearings no range For possible solutions see later Image: Liquid robotics

60 14 Fixed sensors Advantages over towed systems: Often cheaper to deploy (although gliders) Can make use of existing systems Better temporal coverage Disadvantages over towed systems: Possibly poor spatial coverage Need to account for animal movement More difficult to do ranging (Note: floating sensors and gliders may be more like fixed sensors if they move slowly compared with animal speed)

61 15 Total count methods Count of things detected Area surveyed Multiplier(s) to convert number of things detected to number of individuals e.g., group size, if detections are groups Simple, no modelling required: good if you can do it, but often not feasible (e.g., needs lots of sensors)

62 16 Image: Diane Claridge Fig. from arvis/extended.html Thanks to D. Moretti. Example: Dive counting for beaked whales at AUTEC Moretti et al. (2010) Monitoring period: 10 days around time of a Navy exercise Identify time and approximate location of start of a group dive Assume certain detectability Assume can tell whether inside or out of survey area Assume no mis-classification

63 17 Example: Dive counting for beaked whales at AUTEC mean group size from separate visual surveys number of dive starts area monitored time spent monitoring mean dive rate taken from a sample of tagged whales Issues: s and r come from different time and small samples dive counting hard to automate groups diving close together

64 18 Example: Sperm whales at AUTEC See Ward et al. (2012) Snapshot total count of individuals Monitoring period: 28 archived recordings over 42 days. Screened recordings for possible sperm whale presence. Whales present on 8 recordings, making up pp=0.38 of the total monitoring period. Looked at the recordings possibly containing whales in more detail

65 19 Example: Sperm whales at AUTEC Took a systematic random sample of minute periods from recordings that possibly contain sperm whales Treated each sample as a snapshot. Used sophisticated processing algorithm to isolate direct-path click trains Localized animals and excluded those detected off the range (i.e., outside the study area) Used max number of overlapping click trains as an estimate of the number of diving animals Separately, used data from tagged animals to determine the proportion of 10-minute intervals animals spend clicking,

66 20 Example: Sperm whales at AUTEC number of individuals counted in the 50 samples area monitored number of snapshots (50) proportion of the whole time monitored where sperm whales were possibly present proportion of time whales vocalize in a 10 minute period Issues: Assumption that 10 mins is a snapshot Assumption we have a perfect count of vocalizing whales on range Assumption that tag data from another time period and (for some tags) place tell us pv

67 21 Distance sampling methods

68 22 Examples: (a) North pacific right whales; (b) fin whales North Pacific right whales: Marques et al. (2011) + Case Study 1 Fin whales: Harris et al. (2013) Examples of cue count method Treated as point transect cue count cues are specific calls (right whales: up call, fin whales: 20 Hz pulse). Cue rate required Image:

69 23 Example: Fin whales in the Atlantic Data from 24 Ocean Bottom Seismometers (OBSs) Each OBS has 3 seismometer channels - can get distances. 1 year of data

70 24 Example: Fin whales in the Atlantic Fitted detection function NB: Same data, scaled differently Assuming a triangular distribution of animals about the hydrophones,

71 25 Example: Fin whales in the Atlantic Cue count method (with allowance for mis-classification) number of cues (20 Hz calls) detected estimated proportion of false positives (from a manually processed sample) estimated cue rate area monitored time spent monitoring (summed over the k sensors) truncation distance estimated average detection probability of a 20 Hz call within the area monitored

72 26 Example: Fin whales in the Atlantic Benefits: 24 sensors assumption that true distribution of call distances is triangular is more plausible (compare to Case Study 1 ). Issues: Call rate for vocalizing fin whales is not known. Proportion of whales that do not vocalize is not known Therefore, density of animals not possible to estimate

73 Avoiding cues using tracks Alternatively, if you could track individuals within range of each sensor, could use a snapshot approach Assuming all individuals can be tracked at zero distance, you get: number of individuals detected area monitored estimated proportion of false positives number of snapshots estimated average detection probability of an individual at a snapshot moment (from standard detection function modelling) But: still only estimate density of calling animals need to know proportion of animals calling to estimate density of all fin whales in the study region. 27

74 28 Cue counting vs snapshots Cue counting pros Easy to identify cue Occurs at an instant so no need to worry about movement Cue counting cons Need cue rate multiplier Detection of cues may not be independent Snapshot pros No need for cue rate multiplier Snapshot cons Need to be able to count individuals What snapshot interval/spacing to use: arbitrary Need to be careful with variance estimation Ad hoc Would be better to have methods that explicitly incorporate animal movement under development (e.g., DiTraglia 2007)

75 29 Density without distances: (MR and) SECR

76 30 SECR for acoustic data SECR was designed for trapping studies; location where animals trapped tells you about the detection probability and the home range centre Can treat a sound like an animal: it starts from a single location ( home range centre ) and radiates out, being detected ( trapped ) at hydrophones ( traps ) (Efford et al. 2009, Dawson and Efford 2009) Only need one trapping occasion as the same sound can be detected at multiple hydrophones From Dawson and Efford 2009

77 31 Example: Minke whales at PMRF Image: Reefteach Case study 2 16 hydrophones at the Pacific Missile Range Facility (PMRF), off Kauaii, Hawaii Minke whale boings detected; TDOA and dominant frequency were used to associate calls across hydrophones Issue: still require a cue rate See also: Marques et al. (2012); Martin et al. (2013)

78 32 Beyond SECR SECR makes use of associations But in many cases you have more information about location of sound: Can often localize some sounds Relative received levels may contain information about relative distances. Ditto for frequency components Sometimes you have bearing information We are working on methods that use all of this information (Borchers 2012; Borchers et al. 2014)

79 33 Density without distances: p from auxiliary information This is a sub-optimal scenario, as you need to rely on auxiliary information not part of the main survey to get detection probability Just as with all multipliers, you need to be careful this information is applicable See Borchers (2002)

80 34 Example: Baltic harbour porpoises with auxiliary visual observations Kyhn (2010); Kyhn et al. (2012) Evaluation of concept: density estimation from T-PODs T-PODs are porpoise detectors record detection of porpoise clicks T-PODs were deployed at Fyns Hoved, Denmark close to shore, overlooked by visual observers Snapshot-based method: object counted is the number of 15s intervals where a porpoise is detected (assumes max 1 porpoise) Detection probability obtained by using visual observers to set up trials Estimator just like snapshot estimator, except here p will come from visual observer trials

81 35 Example: Baltic harbour porpoises Getting the p: Observers tracked porpoises visually. Assuming linear movement between surfacings path. Estimate true position every 15 seconds. Model probability of detection vs distance (logistic regression) Assuming triangular distribution of animals around hydrophone, can get average p Approach called a trapping point transect in the terrestrial literature (Buckland et al. 2006, Potts et al. 2012) Example detection function (data from Line Kyhn)

82 36 Example: Baltic harbour porpoises Issues: Assumes max 1 animal per snapshot Assumes triangular distribution of animals around the T-POD In practice, we wish to apply estimates of p to PODs placed throughout the Baltic (SAMBAH project). But intensive studies can only take place in limited places. See

83 37 Example: beaked whales at AUTEC with auxiliary tag data Marques et al. (2009) and Case Study 3 Cue-based method object counted is beaked whale clicks over 82 hydrophones for 6 days Detectability estimated from separate tagging experiment to set up trials Detection function estimated by logistic regression more complex than porpoise as covariates were used Fitted detection function

84 38 Example: Beaked whales via acoustic modelling Küsel et al. (2011); Helble et al. (2013) (a & b); see also Case study 4 (on blue whales) Use assumptions about source level combined with acoustic modelling of transmission loss and detector characterization to predict the detection function, and then estimate p. Animal location distribution Source level distribution Acoustic prop. model Ambient noise distribution Detector performance average prob of detection, p

85 39 Comments on acoustic modelling approach Advantage (relative to other auxiliary information methods): no expensive tagging/visual observations needed Disadvantages: answers only as good as the modelling! single sensor unlikely animals uniformly distributed around the instrument BUT collected data cannot estimate the distribution have to assume a uniform distribution. In general, our view is this should be a last resort!

86 40 Summary methods considered Towed acoustic line transects on individuals/groups Fixed sensors: Plot sampling on cues (beaked whale dive starts) and individuals (sperm whales) Point transects on cues (right whales) and individuals via snapshot (fin whales) SECR on cues (minke whales) Trapping point transect on individuals via snapshot (harbour porpoise) and cues (beaked whales) Cue counting with p estimated from acoustic modelling (beaked whales and blue whales)

87 We have not covered all possible methods for fixed sensors there are some approaches still to be explored.

88 42 Summary which method? Roadmap will (hopefully) help Not all methods on the map have been covered here Survey design: choosing a method to maximize reliability Minimize use of multipliers (e.g., use individuals rather than cues; certain detection rather than p) Measure multipliers as part of the main survey (e.g., get distances to estimate p). If not possible, use a good sampling design in same time and place as survey. If not possible, do this with any component you can (e.g., detector characterization)

89 43 Summary which method? Total count of individuals is best (but expensive) Distance sampling or SECR (or beyond SECR) on individuals is 2nd best. But often neither are practical usually we work on cues; often we need auxiliary info approaches to get p Nevertheless, use a good sampling design to obtain any multipliers you can, e.g.: detector performance (false positive rate, detection prob given SNR) ambient noise

90 44 QUESTIONS???