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Rachel Fewster Department of Statistics, University of Auckland Variance estimation for systematic designs in spatial surveys

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Method of estimating density of objects in a survey region. Line transect sampling

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D # detections per unit area = p

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D = p Line transect sampling Density, D

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Estimate the variance of the ratio by the Delta method: “squared CVs add” D # detections per unit area = p ENCOUNTER RATE easy ENCOUNTER RATE VARIANCE: Largest and most difficult component Usually >70% of total variance

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Encounter rate estimates mean detections per unit line length Encounter Rate and its variance

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Inferential framework: which Var(n/L)? Animals from spatial p.d.f. Select lines Detect animals Variance is defined over conceptual survey repeats Find n/L

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Inferential framework: which Var(n/L)? Animals from spatial p.d.f. Select lines Detect animals Variance is defined over conceptual survey repeats Find n/L Gained value of n/L from first survey

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Same animals, new positions Second survey: Inferential framework: which Var(n/L)?

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Select new lines Same animals, new positions Detect new animals Find new n/L Inferential framework: which Var(n/L)? Second survey:

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Select new lines Same animals, new positions Detect new animals Find new n/L Inferential framework: which Var(n/L)? Gained value of n/L from second survey Overall, gives var(n/L) across the repeated surveys This is our ENCOUNTER RATE VARIANCE.

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To estimate a variance, use repeated observations with the same variance Random-line estimator: makes no assumptions about the unknown distribution of objects; How to estimate Var(n/L)?

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To estimate a variance, use repeated observations with the same variance Random-line estimator: makes no assumptions about the unknown distribution of objects; random variables are IID with respect to the design. How to estimate Var(n/L)?

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Systematic Survey Designs Surveys usually use SYSTEMATIC transect lines, instead of random lines. Grid has random start-point

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Systematic lines give LOWER VARIANCE than random lines in trended populations But the variance is HARD TO ESTIMATE A systematic sample has NO REPETITION: it is a sample of size 1!

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Variance for systematic designs There is no general design-unbiased variance estimator for data from a single systematic sample Approaches to systematic variance estimation are: 1.Ignore the problem and use estimators for random lines 2.Use some form of post-stratification 3.Model the autocorrelation in the systematic sample Approach used to date

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Variance for systematic designs There is no general design-unbiased variance estimator for data from a single systematic sample Approaches to systematic variance estimation are: 1.Ignore the problem and use estimators for random lines 2.Use some form of post-stratification 3.Model the autocorrelation in the systematic sample Approach in Fewster et al, Biometrics, 2009

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But the stratified estimators are still biased sometimes – e.g. high sampling fraction, or population clustering Stratified variance estimators: results Can we do better…?

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Variance for systematic designs There is no general design-unbiased variance estimator for data from a single systematic sample Approaches to systematic variance estimation are: 1.Ignore the problem and use estimators for random lines 2.Use some form of post-stratification 3.Model the autocorrelation in the systematic sample

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Historical Note Many estimators for systematic designs originated in social statistics – discrete surveys Correlation will clearly exist in responses of neighbours, but modelling the correlation is hard!

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But space is continuous! As a strip changes position very slightly...... it still covers many of the same objects.

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But space is continuous! As a strip changes position very slightly...... it still covers many of the same objects. Idea: 1.Divide the region into hundreds of tiny ‘striplets’ 2.Allow the number of objects available in each striplet to be random variables X 1, X 2, …, X J 3.The number of objects available in any full strip is the sum of the objects in the constituent striplets

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1.Divide the region into hundreds of tiny ‘striplets’ 2.Number of objects available in striplets 1, 2, …, J is X 1, X 2, …, X J 3. Number of objects available in any full strip is the sum of the objects in the constituent striplets. Expected number of objects per striplet Random number of objects per striplet, X 1, X 2, …, X J ~ Multinomial Striplet #objects available striplet position 0 1 2 3 4

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1.Divide the region into hundreds of tiny ‘striplets’ 2.Number of objects available in striplets 1, 2, …, J is X 1, X 2, …, X J 3. Number of objects available in any full strip is the sum of the objects in the constituent striplets. Striplet #objects available striplet position 0 1 2 3 4 Full strip at this position: 10 objects Full strip at next position: 7 objects Full strip at next position: 8 objects... etc

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Recap: We want the variance in the encounter rate, n/L, over: 1.Moving grid; 2.Moving objects; 3.Detections Account for: 1.Large-scale trends 2.Small-scale noise

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1. Trends in object density across the region Observed number of detections per unit search area #detections / unit area Points correspond to observed transects Fit a GAM to give a fitted object density for any search strip in the region x-coordinate

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#detections / unit area x-coordinate 1. Trends in object density across the region Fit a GAM to give a fitted object density for any search strip in the region For any striplet j, we now have an expected number of objects available, j

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Expected number of objects per striplet, j Striplet #objects available striplet position 0 1 2 3 4 Account for: 1.Large-scale trends

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Striplet #objects available striplet position 0 1 2 3 4 Account for: 2. Small-scale noise Random number of objects per striplet, X 1, X 2, …, X J ~ Multinomial(N, j /N) Striplet idea means we correctly model the autocorrelation between systematic grids

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Striplet #objects available striplet position 0 1 2 3 4 Account for: 2. Small-scale noise

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Recap: We want the variance in the encounter rate, n/L, over: 1.Moving grid; 2.Moving objects; 3.Detections Variance in number of objects available is taken care of (1 & 2) Variance in detections is Binomial given #objects available (1 & 2)

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Law of Total Variance: b is the grid placement: Mean and variance of #detections, n, given grid placement, is all that’s needed.

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Striplet variance estimator:

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Simulation Results: 3 habitat types but no clustering Clustering included

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Simulation Results: Red lines give correct answers

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Simulation Results: Ignoring the systematic design: appalling performance!

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Simulation Results: Post-stratification: improvement but still clear bias

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Simulation Results: Striplet method: huge improvement!

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Spotted Hyena in the Serengeti

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Short grass plains: prey herds congregate in wet season Long grass plains: unattractive in wet season

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Spotted Hyena in the Serengeti Wet season: non-territorial ‘commuters’ (n=186) Dry season: territorial residents (n=53)

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Wet season: highly clustered. cv(n/L) is: -17% ignoring systematic design -14% using poststratification -7% using striplets! Overall cv(D) is: -20% ignoring systematic design -17% using poststratification -11% using striplets The estimator matters!

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Dry season: not clustered; small n cv(n/L) is: -15% ignoring systematic design -12% using poststratification -13% using striplets Overall cv(D) is: -23% ignoring systematic design -20% using poststratification -21% using striplets Not much difference

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In Revision, Biometrics

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1. For a systematic design, variance estimators based on random lines are not adequate for trended or clustered populations 2. Post-stratification improves estimation for trended pops, but far from perfect 3. New ‘striplet’ method huge improvement in all line/strip situations trialled to date Variance can be highly overestimated Conclusions

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Striplet variance estimator: B is the number of possible grids, in discrete approximation j is fitted #objects in striplet j g j (b) is fitted P(detection) in striplet j

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Williams & Thomas, JCRM 2008 Application: British Columbia multi- species marine survey Select species with greatest and least trends in encounter rate for illustration

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Greatest trend: Dall’s Porpoise Highest encounter rates on short lines Worst case!

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Least trend: floating plastic garbage No trend in encounter rate with line length

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Results Dall’s Porpoise: previous reported CV=31% Stratified methods: reported CV=19% Estimated CV=31% using Poisson-based estimator with no adjustment for systematic lines Estimated CV=19% using design-based estimator with post-stratification and overlapping strata

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Results Floating garbage: previous reported CV=15% Stratified methods: reported CV=14% For untrended population, there is little difference in the different estimators

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But space is continuous! As a strip changes position very slightly...... it still covers many of the same objects.

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But space is continuous! As a strip changes position very slightly...... it still covers many of the same objects. Idea: 1.Divide the region into hundreds of tiny ‘striplets’ 2.Allow the number of objects available in each striplet to be random variables X 1, X 2, …, X J 3.The number of objects available in any full strip is the sum of the objects in the constituent striplets

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1.Divide the region into hundreds of tiny ‘striplets’ 2.Number of objects available in striplets 1, 2, …, J is X 1, X 2, …, X J 3. Number of objects available in any full strip is the sum of the objects in the constituent striplets. Striplet #objects available striplet position 0 1 2 3 4 Expected number of objects per striplet Random number of objects per striplet, X 1, X 2, …, X J ~ Multinomial

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1.Divide the region into hundreds of tiny ‘striplets’ 2.Number of objects available in striplets 1, 2, …, J is X 1, X 2, …, X J 3. Number of objects available in any full strip is the sum of the objects in the constituent striplets. Full strip at this position: 10 objects Full strip at next position: 7 objects Full strip at next position: 8 objects... etc Striplet #objects available striplet position 0 1 2 3 4

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1. Trends in object density across the region Observed number of detections per unit search area #detections / unit area Points correspond to observed transects Fit a GAM to give a fitted object density for any search strip in the region x-coordinate

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1. Trends in object density across the region #detections / unit area Fit a GAM to give a fitted object density for any search strip in the region x-coordinate For any new grid placement, we now have an expected number of objects available for that grid

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