Spatial modelling etc

Transcription

1 Spatial modelling etc David L Miller & Mark V Bravington International Whaling Commission Scientific Committee 2017

2 Why are we here/why did we do this? Stratified Horvitz-Thompson is workhorse of many abundance estimates How is H-T going to fail? When do we need to use spatial models? What are Bad surveys? Report

3 Overview Today: 1. what's wrong with H-T? 2. Spatial models overview 3. What can go wrong with spatial models? 4. Testing designs in R 5. Model checking for spatial models Tomorrow: 1. What we missed, what is hard 2. Try out tester on your data 3. Other methods/software, future work 4. Guidelines

4 Practicalities Try not to talk for more than an hour without a break We both have funny accents, yell if you don't understand! There is maths don't worry

5 This is not a distance sampling course! This material usually takes 4 days+ to teach This will not prepare you to analyse spatial data BUT you can do this in St Andrews this summer! creem2.st-andrews.ac.uk

6 Why are we interested in spatially-explicit estimation?

7 Inferential aims

8 Part I

9 Horvitz-Thompson estimation: the good, the bad and the ugly

10 Horvitz-Thompson-like estimators Rescale the (flat) density and extrapolate N = study area n covered area i=1 s i p i s i p i are group/cluster sizes is the detection probability (from distance sampling)

11 Variance of H-T Multiple sources of randomness in H-T equation: p i - detectability n s - dealt with as, encounter rate - group size n/l

12 Hidden in this formula is a simple assumption Probability of sampling every point in the study area is equal Is this true? Sometimes. If (and only if) the design is randomised

13 Many faces of randomisation

14 What does this randomisation give us? Coverage probability H-T estimator assumes even coverage (or you can estimate) Otherwise not really valid

15 Estimating coverage We can estimate coverage of a non-uniform design! In Distance! Example from BC, Canada in this paper:

16 Estimating coverage

17 A complex survey plan Thomas, Williams and Sandilands (2007) Different areas require different strategies Zig-zags, parallel lines, census Analysis in Distance

18 Sideline: alternative terminology A design is an algorithm for laying down samplers in the survey area A realization (from that algorithm) is called a survey plan Len Thomas 2004)

19 H-T estimation again Can't estimate w/ H-T w/o coverage Fixed designs violate assumptions Some animals have P(included) = 0 Deteriorate pooling robustness property What can we do?

20 More on variance Encounter rate variance n j / l j n/l Within-transect variation can be bad e.g., N-S transect, N-S density gradient

21 Stratification If we suspect density change can stratify! Pre or post hoc (spatial and non-spatial)

22 I am going to stop talking very soon

23 Summary H-T is a spatial model (sort of) Violated an assumption if no randomness Hard to assess how bad this is Fewster et al (2009) and Fewster (2011) give variance approaches

24 Part II

25 Spatial models

26 Spatial models of distance sampling data Collect spatially referenced data Why not make spatially-explicit models? Go beyond stratified estimates Relate environmental covariates to counts

27 This is the rosy picture talk

28 We'll talk about the grim reality later

29 Example data in this talk

30 Sperm whales off the US east coast Hang out near canyons, eat squid Surveys in 2004, US east coast Combination of data from 2 NOAA cruises Thanks to Debi Palka, Lance Garrison for data. Jason Roberts for data prep.

31 Example data

32 Model formulation Pure spatial, pure environmental, mixed? May have some prior knowledge Biology/ecology What are drivers of distribution? Inferential aim Abundance Ecology

33 Density surface models Hedley and Buckland (2004) Miller et al. (2013)

34 Ignoring group size (more on that tomorrow)

35 Physeter catodon by Noah Schlottman

36 How do we model that?

37 SPOILER ALERT: your model is probably just a very fancy GLM

38 Generalised additive models (in 1 slide) Taking the previous example ( n j ) = A j p j exp β 0 + s ( ) [ k z kj ] k n j area of segment some count distribution probability of detection in segment (inverse) link function model terms

39 What about those s thingys?

40 Covariates space, time, environmental (remotely sensed?) data

41

42 Modelling smooths 1-dimension: not much difference 2D more tricky edge effects tricky boundaries more tomorrow Now going to do some maths (ignore at will)

43 How do we build them? Functions made of other, simpler functions Basis functions, b k Estimate β k s(x) = (x) K k=1 β kb k

44 Straight lines vs. interpolation Want a line that is close to all the data Don't want interpolation we know there is error Balance between interpolation and generality

45 How wiggly is a function?

46 Making wigglyness matter Fit needs to be penalised Something like: 2 2 s(x) x 2 R ( ) dx β T Sβ (Can always re-write this in the form ) Estimate the β k terms but penalise objective closeness to data + penalty (REML/ML)

47 Smoothing parameter

48 Sideline: GAMs are Bayesian models Generally: penalties are improper prior precision matrices (nullspace gives improper priors) Using shrinkage smoothers: proper priors empirical Bayes interpretation

49 Beyond univariate smooths? Can build (anisotropic) tensor product terms Take 2 or more univariate terms Thin plate regression splines allow multivariate terms (isotropic)

50 Spatial smoothing Can just smooth in space Valid abundance estimation technique Useful for EDA for env. cov. models (hard day 2!) Not good for extrapolations Basis choice can matter!

51 Why GAMs are cool... Fancy smooths (cyclic, boundaries, ) Fancy responses (exp family and beyond!) Random effects (by equivalence) Markov random fields Correlation structures See Wood (2006/2017) for a handy intro

52 Let's fit a model library(dsm) # environmental covariates dsm_env_tw <- dsm(count~s(depth) + s(npp) + s(sst), ddf.obj=df_hr, segment.data=segs, observation.data=obs, family=tw()) # space dsm_xy_tw <- dsm(count~s(x, y), ddf.obj=df_hr, segment.data=segs, observation.data=obs, family=tw()) dsm is based on mgcv by Simon Wood

53 Simple! Done?

54 NO

55 More on model checking later...

56 Predictions/abundance estimates Grid of covariates must be available Predict within survey area Extrapolate outside (with caution) Working on a grid of cells Plot is s(x,y) + s(depth) Add up to get abundance

57 Estimating variance Uncertainty from: detection function parameters spatial model Need to propagate uncertainty! Methods in dsm Bravington, Hedley & Miller (in prep)

58 Plotting uncertainty Maps of coefficient of variation CV for given stratum (better) Visualisation is hard

59 Communicating uncertainty Are animations a good way to do this? Simulate from posterior parameter distribution β N( β, Σ ) Some features (e.g. shelf, N-S gradient) stick out

60 I am going to stop talking very soon

61 Summary Build models in stages (detection function + GAM) Counts are functions of covariates Pure spatial models Environmental covariate models Mix?! Fit/check using dsm Most of the theory is resolved, applications are hard

62 Part III

63 H-T or spatial or give up?

64 Spatial models can help Spatial modelling can give ubiased abundance ests even with uneven coverage limits to extrapolation V. even coverage => HT? Evenness subtle, detectability effect e.g., weather bad in east

65 Weather or distribution? Weather has a big effect on detectability Need to record during survey Disambiguate between distribution/detectability Potential confounding can be BAD

66 Visibility during POWER 2014 Thanks to Hiroto Murase and co for this data!

67 Covariates can make a big difference!

68 Other stuff

69 Spatial modelling won't solve all yr problems Design issues Ludicrous extrapolation Survey plan not robust to weather issues Non-uniform distribution wrt sampler Migration Spatial models alone can't solve these issues

70 Spatial modelling won't solve all yr problems Violations of survey procedure Following animals Responsive movement Guarding the trackline Group size estimation Spatial models alone can't solve these issues

71 Spatial modelling won't solve all yr problems Detection functions Not enough observations Uncertain species ID Group size Spatial models alone can't solve these issues

72 @kitabet

73 Should everything be spatial? Do you have enough observations? If they do look good (even coverage, etc) Is it worth re-analysing from H-T? Point estimates similar? Variance may well be different?

74 I am going to stop talking very soon

75 Summary Spatial models don't solve all problems Complex models can lead to complex issues Recording weather conditions is important You can always give up!

76 Part IV

77 Testing designs

78 What can we do? Take a survey and simulate Is H-T robust? How do different spatial models compare? Only thinking about total abundance & CV

79 Software ltdesigntester github.com/dill/ltdesigntester (based on DSsim by Laura Marshall, CREEM) Setup simulations, test what can be done Most of the work needs to be done in GIS Survey shapefiles, covariates etc Import to R, runs models, shows output

80 Setting up a survey simulation

81 Density Grid in polygon of study area Either specify simple gradient or use other tools to make complex density Density as grid

82 Design Generate using GIS/Distance Export to shapefile

83 Detection function Functional form (halfnormal, hazard-rate) Parameters (scale, shape) Truncation (Covariates via multiple functions, more later)

84 Specification to simulation Generate multiple realizations Analyse each with a many models Different spatial, H-T Compare results

85 Test models Spatial smoothers thin plate spline, bs="tp" (Wood, 2003) thin plate spline with shrinkage, bs="ts" (Marra et al., 2011) Duchon spline, bs="ds", m=c(1, 0.5) (Miller et al., 2014) tensor of thin plate spline (w/ and w/o rotated covariates) Stratified estimates Horvitz-Thompson (w/ and w/o covariates) stratified Horvitz-Thompson (w/ and w/o covariates)

86 Comparing performance

87 Important caveats No model checking Dependent on good detection specs No group size model No g(0) or availability

88 Quick example code library(ltdesigntester) # setup a simulation my_sim <- build_sim(design_path="path/to/shp", dsurf=density_surface_matrix, n_grid_x=dsurf_dim_x, n_grid_y=dsurf_dim_y, n_pop=true_n, df=detection_function_specs, region="path/to/shp") # run it! res <- do_sim(nsim=number_of_sims, scenario=my_sim, pred_dat=prediction_data_frame,...)

89 We made a big deal about weather earlier... We can add covariates too (a wee bit unwieldy at the moment) Build multiple detection functions/sims in list() Covariates vary according to: logit function E-W (can set pars, 2 state) set values in segment data (already observed)

90 I am going to stop talking very soon

91 Summary We can test multiple (simple) scenarios Assumption of simple gradients Models likely won't work for difficult stuff if they don't work for simple things What will work/what won't Simple summary plots Better than the rest good

92 Part V

93 Model checking for DSMs

94 Model checking Count distribution Basis complexity Model (term) selection Sensitivity Observed vs. expected Cross-validation (replicability)

95 (Plus all the usual stuff for detection functions!)

96 Count distributions Response is a count Often, it's mostly zero Aggregations occur at scales smaller than spatial model Want response distribution that deals with that Could mess-up variance if ignored Linked to segmenting Flexible mean-variance relationship

97 Negative binomial Var (count) = (count) + κ(count) 2 Estimate κ Is quadratic relationship a strong assumption? Similar to Poisson: Var (count) = (count)

98 Tweedie distribution Var (count) = ϕ(count) q Common distributions are sub-cases: q = 1 q = 2 q = 3 Gaussian Poisson Gamma inverse- We are interested in 1 < q < 2 (here q = 1.2, 1.3,, 1.9 )

99 Basis complexity Before: s(x) = (x) K k=1 β kb k How big should k be? Big enough Penalty takes care of the rest?gam.check gives useful output (also residual checks etc)

100 gam.check text output gam.check(dsm_env_tw) Method: REML Optimizer: outer newton full convergence after 8 iterations. Gradient range [ e-08, e-08] (score & scale ). Hessian positive definite, eigenvalue range [ , ]. Model rank = 28 / 28 Basis dimension (k) checking results. Low p-value (k-index<1) may indicate that k is too low, especially if edf is close to k'. k' edf k-index p-value s(depth) s(npp) s(sst)

101 Tobler's first law of geography Everything is related to everything else, but near things are more related than distant things Tobler (1970)

102 Implications of Tobler's law

103 What can we do about this? Careful inclusion of terms Test for sensitivity (lots of models) Fit models using robust criteria (REML) Test for concurvity (mgcv::concurvity, dsm::vis.concurvity)

104 Term selection p (approximate) values (Marra & Wood, 2012) path dependence issues shrinkage methods (Marra & Wood, 2011) ecological-level term selection which biomass measure? include spatial smooth or not?

105 Observed vs. expected Diagnostic compare observed vs. expected counts Compare for different covariate/aggregations In next dsm, obs_exp() does this Going back to those rough POWER models > obs_exp(b, "beaufort") Observed Expected > obs_exp(b_nc, "beaufort") Observed Expected

106 Cross-validation How well does the model reproduce what we saw? Leave out one area, re-fit model, predict to new data Wenger & Olden (2012) have good spatial examples

107 Cross-validation example

108 Cross-validation example

109 I am going to stop talking very soon

110 2 (or more)-stage models Not cool (statistically), but Multi-stage models are handy! Understand and check each part Split your modelling efforts amongst people

111 Conclusions This methodology is general Bears, birds, beer cans, Loch Ness monsters Models are flexible! Linear things, smooth things, random effect things (and more) If you know GLMs, you can get started with DSMs Mature theoretical basis, still lots to do Active user community, active software development

112 Resources distancesampling.org/r/ distancesampling.org/workshops/duke-spatial-2015/

113 Thanks! Slides w/ references available at converged.yt

114 References Fewster, R.M., Buckland, S.T., Burnham, K.P., Borchers, D.L., Jupp, P.E., Laake, J.L., et al. (2009) Estimating the Encounter Rate Variance in Distance Sampling. Biometrics, 65, Fewster, R. M. (2011), Variance Estimation for Systematic Designs in Spatial Surveys. Biometrics, 67: Hedley, S. L., & Buckland, S. T. (2004). Spatial models for line transect sampling. Journal of Agricultural, Biological, and Environmental Statistics, 9(2). Marques, T. A., Thomas, L., Fancy, S. G., & Buckland, S. T. (2007). Improving estimates of bird density using multiple-covariate distance sampling. The Auk, 124(4). Marra, G., & Wood, S. N. (2011). Practical variable selection for generalized additive models. Computational Statistics and Data Analysis, 55(7). Marra, G., & Wood, S. N. (2012). Coverage Properties of Confidence Intervals for Generalized Additive Model Components. Scandinavian Journal of Statistics, 39(1). Wenger, S.J. and Olden, J.D. (2012) Assessing transferability of ecological models: an underappreciated aspect of statistical validation. Methods in Ecology and Evolution, 3,

115 Handy awkward question answers

116 Don't throw away your residuals!

117 gam.check

118 rqgam.check (Dunn and Smyth, 1996)

119 Penalty matrix For each b k calculate the penalty Penalty is a function of λ β T Sβ β S calculated once smoothing parameter ( ) dictates influence λ

120 How wiggly are things? We can set basis complexity or size ( ) Maximum wigglyness Smooths have effective degrees of freedom (EDF) k EDF < k Set k large enough

121 Let's talk about detectability

122 Detectability

123 Distance sampling Fit to the histogram Model: P [animal detected animal at distance y] = g(y; θ) Calculate the average probability of detection: w 1 p = g(y; θ )dy w 0

124 Distance sampling (extensions) Covariates that affect detectability (Marques et al, 2007) g(0) < 1 Perception bias ( ) (Burt et al, 2014) Availability bias (Borchers et al, 2013) Detection function formulations (Miller and Thomas, 2015) Measurement error (Marques, 2004)

125 That's not really how the ocean works...

126 Availability

127 We can only see whales at the surface What proportion of the time are they there? Acoustics Tags (DTAGs etc) Behavioural studies p Fixed correction to? Model via fancy Markov models (Borchers et al, 2013) Picture from University of St Andrews Library Special Collections