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Keywords:

  • Akaike Information Criterion;
  • annually varying parameter;
  • Chondrichthys;
  • data-poor;
  • detectability;
  • detection/non-detection data;
  • hierarchical model;
  • presence/absence;
  • stepwise model selection;
  • temporal variability

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information
  1. Many population dynamics models and common software packages, including for capture–mark–recapture, occupancy and catch-at-age models, are estimable using parameters that are either constant for all years or vary annually.
  2. We develop a spline approximation to time-varying parameters, which includes the constant or annually varying case as two extremes, where the Akaike Information Criterion is used to select an appropriate degree of smoothness, ranging from change in every year to no change at all. Simulation modelling is used to evaluate the performance of this method relative to constant, annually varying, or random-effect parameter estimates when applied to an occupancy model that simultaneously estimates abundance and detectability in multiple sampling strata. We also demonstrate this method by approximating time-varying abundance using 25 years of detection/non-detection data for 42 Chondrichthyes species off northern Australia.
  3. Simulation modelling indicates that the stepwise-spline approximation results in lower estimation errors for the proportion of abundance in each stratum, regardless of sample sizes, the underlying form for time-varying abundance and the inclusion of unmodelled process errors.
  4. Applied to the Chondrichthyes data from northern Australia, the spline approximation identifies temporal variability in abundance for 34 of 42 species, and an annually varying model is never selected.
  5. We recommend this stepwise-spline approximation in cases where a decision is necessary between constant or annually varying forms for an estimated parameter. Possible applications include occupancy, capture–mark–recapture, catch-at-age and index standardization models.

Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information

Population dynamics models are a fundamental tool in assessing and managing wild populations, including the effects of human activities such as harvesting. Modern population dynamics models approximate biological and sampling processes by specifying one or more process model equations describing the biological processes. Parameters are then estimated by minimizing the divergence between observed and predicted data, where the prediction data are based on an observation model describing the sampling process (Hilborn & Mangel 1997).

Many population dynamics models can be parameterized in terms of annually varying parameters (i.e. a separate value for each year) for biological or sampling processes. One example is capture–mark–recapture models (Thomson, Cooch & Conroy 2009), wherein the resighting of uniquely marked individuals is sufficiently informative to allow annually varying survival and detection rates to be estimated. Other examples include occupancy models (MacKenzie et al. 2005), which estimate detectability and abundance from detection/non-detection data, and integrated catch-at-age models (Maunder 2003), which estimate population status and productivity from a variety of data sources.

However, an annually varying parameter may not be parsimonious given the quantity and quality of data that are available. In such cases, common statistical packages – including m-surge (Choquet et al. 2004) for capture–mark–recapture models and ‘unmarked’ (Fiske & Chandler 2011) for occupancy models – frequently allow a parameter to be either estimated as annually varying or constant for all years. This decision will frequently impact study results and is often aided using common model selection tools such as the Akaike Information Criterion (AIC, Akaike 1974).

In this study, we propose an approach to time-varying parameters that includes the constant or annually varying forms as nested models. Specifically, we propose to approximate time-varying parameters using piecewise polynomials, also known as splines (Hastie & Tibshirani 1990; Hastie, Tibshirani & Friedman 2009), and use model selection tools such as the AIC to determine a parsimonious level of smoothness in time variability, at the same time as any other model selection decisions. Model selection has played a central role in ecological model building and inference for at least a decade (Buckland, Burnham & Augustin 1997; Burnham & Anderson 2002), so we believe that this approach will be intuitive to many ecologists. Additionally, model selection can also identify a model with time-invariant parameters, so this method can be used to identify whether time variability is supported for one or more parameters.

We demonstrate this method using an occupancy model and detection/non-detection data to estimate the proportion of population abundance in different spatial strata over time. We chose to use detection/non-detection data to demonstrate that it is possible to identify and account for time-varying effects even in relatively data-poor situations. Furthermore, we use occupancy modelling for this application to allow explicit estimation of detection probabilities rather than using other methods that make implicit, but strict, assumptions regarding spatially constant detectability. We use simulation modelling to contrast the stepwise-spline approximation with an occupancy model that either (i) assumes abundance is constant over time, (ii) estimates abundance in each stratum and year as a fixed effect or (iii) estimates annual changes in abundance as random effects. We demonstrate application of the stepwise-spline approximation using data for 42 Chondrichtyes species in northern Australia, while aggregating sampling data from 12 sampling methods over 25 years. These data had previously been used to assess impacts of fishing on bycatch species in a multi-species prawn trawl fishery (Zhou & Griffiths 2008), without considering the possibility of time-varying parameters.

Materials and methods

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information

Stepwise-spline approximation to time-varying parameters

To estimate a time-varying parameter ψt, we replace its time-constant version ψ with a fixed-knot spline approximation (eqn 1):

  • display math(eqn 1)

where dfψ is the number of estimated parameters, βi is the i-th estimated parameter, Si,t is the linear basis for the spline approximation (with ψt rows), and t is an index for year. For simplicity, we present a piecewise-linear (e.g. first-order) spline basis (eqn 2):

  • display math(eqn 2)

where inline image are the dfψ evenly spaced knots that join the spline components for ψt, and nyears in the number of years for which data are available. More details regarding this spline, and computation for alternative spline orders and forms, are given in the Supporting information. β1 represents the value for ψ1, whereas subsequent values of βi represent temporal changes in ψt. This spline approximation for time-varying abundance is appropriate for any number of time-varying parameters and allows biologically plausible bounds (or likelihood penalties) to be placed on annual changes in abundance for each stratum (e.g. large annual increases in total abundance could be heavily penalized for a slow-growing species).

In this approximation, the number of evenly spaced knots (i.e. where different piecewise components with equal width are joined) is selected using model selection tools such as the AIC. This is carried out in a stepwise algorithm starting with all parameters having one degree of freedom (i.e., constant parameters) as follows:

  1. For the first parameter (denoted here as ψt), define a set of spline degrees of freedom dfψ across which to search. In our application, this search set was defined as {−7, −4, −3, −2, −1, 0, +1, +2, +3, +4, +7}, where any spline degrees of freedom outside the possible degrees of freedom (e.g. >25 or <1) was not included (we present a spline for 25 years of data to correspond to the example presented later). Thus, when starting the search algorithm at dfψ = 1, the search algorithm searches across {1, 2, 3, 4, 7}, while if dfψ = 5, the search set would be {1, 2, 3, 4, 5, 6, 7, 8, 9, 14}. Whichever of these yielded the lowest AIC is then selected for dfψ.
  2. Repeat this process for all other time-varying parameters.
  3. Conduct model selection for all other parameters that have multiple possible forms.
  4. Repeat this search algorithm until the selected form for all parameters is unchanged for a full cycle of the search algorithm. This is then the selected model structure.

Occupancy modelling

Both abundance and the probability of non-detection can be estimated simultaneously using occupancy models (Royle & Nichols 2003; Zhou & Griffiths 2007) applied to detection/non-detection data. This non-detection or ‘false negative’ probability is also called detectability or catchability in some ecological models. When abundance is estimated in several strata that have different levels of fishing effort, the estimate of abundance in each stratum can be combined to provide a rough estimate of fishing mortality. This procedure for estimating fishing mortality using the proportion overlap between fishing effort and stock range is called the Sustainability Assessment for Fishing Effects (SAFE) method (Zhou & Griffiths 2008), and is appropriate in data-poor or bycatch fisheries when estimates (or conservative values) can be assigned to fishery catchability (e.g. the proportion of fishes within the nominal area swept for a given gear that are captured) and post-encounter mortality, when data on fishing effort in each stratum are available, and occupancy model assumptions (as detailed later) are approximately met (more details are given in the Supporting information).

Occupancy models are able to simultaneously estimate abundance and survey detectability by employing a repeated measures design (Royle & Nichols 2003), where multiple samples in the same grid cell and time period are assumed to sample a locally closed population (i.e. an unobserved, but constant, number of target individuals at that location). The repeated measures design requires several strong assumptions, including local closure (e.g. no immigration, emigration, birth and death) during the sampling season, an independent probability of encountering each local individual using the sampling gear, and no missing variables related to detection probability at each site (MacKenzie et al. 2005). Although one or more of these assumptions will often not be met (or cannot be validated) in any given application, these models may still be preferable to models that make the strong, implicit assumption that detectability is constant spatially (Mazerolle et al. 2007). In the application to marine species presented in this study, the assumption of a locally closed population is almost certainly not met, but an occupancy model is still useful both (i) because catchability may vary between strata, and ignoring this would bias estimates of fishing mortality in the SAFE method, and (ii) because derived model outputs related to ratios (e.g. the proportion of abundance within a given strata) may be less sensitive to violated assumptions than absolute values (see Marescot et al. 2011 for an example).

In this occupancy model, we allow the estimate of abundance in fished (Nr = 1,t) and unfished (Nr = 2,t) strata to vary by year t. Therefore, we model total abundance using the stepwise-spline approximation on a log-scale (eqn 3) and the proportion of total abundance in the fished strata using the stepwise-spline approximation on a logit scale (eqn 4).

  • display math(eqn 3)
  • display math(eqn 4)

where dfN, β(N) and S(N) are, respectively, the number of estimated parameters, the vector of estimated parameters, and the spline basis for total abundance Nt, and dfP, β(P) and S(P) are, respectively, the number of estimated parameters, the vector of estimated parameters, and the basis for the proportion of abundance in the fished strata Pt. Abundance in the fished strata is Nr = 1,t = Nt · Pt, and abundance in the unfished strata is Nr = 2,t = Nt · (1−Pt). Parameters for this occupancy model were estimated for the constant, annually varying and stepwise-spline models using a quasi-Newton algorithm and automatic differentiation as implemented by the ADMB software platform (Fournier et al. 2012). More model details can be found in the Supporting information.

We also compare the spline approximation with a random-effect occupancy model, where abundance in each year in the fished and unfished strata is treated as a separate random effect. Parameters for the random-effect occupancy model were estimated by maximum marginal likelihood, where the marginal likelihood was computed by integrating across abundance in each stratum and year while specifying a normal hyperdistribution for changes in the natural logarithm of total abundance (Nt) or the proportion of abundance in the fished strata (Pt). This model allows the variance of annual changes in the natural logarithm of total abundance (inline image) and the proportion of abundance in the fished strata (inline image) to be estimated, and these estimated variance parameters control the degree of smoothness for time variability in Nt and Pt in a way that is analogous to the degrees of freedom selected by AIC in the stepwise-spline approximation. The high dimensional integral necessary to calculate the marginal likelihood was evaluated using the Laplace approximation as implemented in the admb-re software platform (http://admb-project.org/) using separable functions (Skaug & Fournier 2006). The code for this and the stepwise-spline occupancy model, as well as all code necessary for replicating the simulation modelling evaluation, are available on the first author's website (https://sites.google.com/site/thorsonresearch/code/spline).

Simulation evaluation

Simulation modelling is used to explore the accuracy of estimates resulting from these occupancy models. Data sets with a known time series of abundance in each of two strata are generated and compared with estimates from occupancy models that make various assumptions about abundance. Specifically, we compare a model where stepwise AIC is used to select spline degrees of freedom with a ‘constant-abundance’ model that assumes abundance, and the proportion within each stratum is constant for all years (i.e. dfN = 1 and dfP = 1), an ‘annually varying’ model that separately estimates abundance in each year (i.e. dfN = 25 and dfP = 25) and the random-effect occupancy model. For this evaluation, we compare annual estimates of availability inline image for each model with its true value inline image.

Data are simulated for one survey design over nyears = 25 years, where the number of random samples each year (nsamples) varies among simulation scenarios to explore the impact of varying sample sizes. Total abundance is constant for all cases and years (Nt = 1000), while fish redistribute between fished and unfished areas, but the fished areas themselves do not change over time (Fig. 1).

image

Figure 1. True abundance (left column: ‘Abundance’; dotted line: abundance in unfished area; solid line: abundance in fished area) and spatial distribution in years 1 (‘Year 1’), 13 (‘Year 13’) and 25 (‘Year 25’) in either fished (left of black line) and unfished (right of black line) areas for one simulation, showing configurations where abundance is dome-shaped (top row), follows a mean-reverting random walk (middle row), or follows a mean-reverting random walk with additional process errors (bottom row).

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We look at a 3 × 3 cross (Table 1) of three annual sample sizes (L: nsamples = 10; M: nsamples = 40; H: nsamples = 100) and three patterns of temporal change in abundance between fished and unfished areas (1: dome-shaped; 2: random-walk; 3: random-walk with process errors). Data are generated as follows:

Table 1. Root-mean-squared error when estimating the proportion of population abundance that is fished in each year inline image for four forms of the occupancy model (i.e. stepwise selection of spline degrees of freedom, constant abundance, annually varying abundance and random-effect abundance) for a 3 × 3 cross of different sample sizes and abundance trends, as well as the average spline degrees of freedom for the stepwise-spline approximation for each scenario
ScenarioSample sizeAbundance and errorsRoot-mean-squared error for each modelDegrees of freedom for stepwise model
StepwiseConstantAnnualRandom effectTotal abundanceProportion fished
L-1nsamples = 10Dome-shaped0·1630·2520·2250·1711·5404·010
M-1nsamples = 40Dome-shaped0·0690·1860·1260·1291·4953·685
H-1nsamples = 100Dome-shaped0·0560·1770·0880·0941·7203·465
L-2nsamples = 10Random walk0·1670·2930·2960·2271·7054·205
M-2nsamples = 40Random walk0·1070·2380·1960·1771·9755·640
H-2nsamples = 100Random walk0·0940·2140·1590·1472·2806·610
L-3nsamples = 10Random walk with process error0·1940·2950·3170·2391·9754·080
M-3nsamples = 40Random walk with process error0·1110·2400·2030·1872·1655·255
H-3nsamples = 100Random walk with process error0·1050·2240·1680·1612·7156·100
  1. Abundance in fished and unfished strata for each year is simulated according to one of the three abundance forms (depicted in Fig. 1 and explained later).
  2. For each year, fished abundance is randomly assigned among 50 equally sized grid cells (with ac = 1 for all cells) that are designated as fished, and unfished abundance is similarly assigned among 50 equally sized unfished cells (other columns of Fig. 1).
  3. Sampling is conducted for each year, where survey locations in each year are randomly assigned among all cells. Each sampling observation samples the entire grid (i.e., γg/ac = 1) and has detectability dg = 0·1 in the fished strata and 0·05 in the unfished strata, and for each survey, the year is recorded as well as whether a survey captures at least one individual of the species or not.
  4. Simulated sampling data are used to estimate the parameters of the occupancy model with four alternative treatments for abundance: stepwise selection of spline degrees of freedom (i.e. dfN and dfP estimated), constant abundance (i.e. dfN = dfP = 1), annually varying abundance (i.e. dfN = dfP = 25) and treating abundance as a random effect. For each model, true and estimated availability are recorded for each year.
  5. This process is replicated 200 times for each of the nine configurations, and results from each of the four model types (stepwise, constant, annually varying and random-effect abundance) are evaluated by calculating the root-mean squared error (RMSE) for each simulation configuration.

Abundance in fished and unfished strata is simulated according to the following three scenarios:

Scenario 1 (Dome-shaped): Abundance redistributes from 20% in fished areas to 80% in fished areas after 12·5 years and then shifts back. This scenario was designed to demonstrate the systematic bias that is likely to occur when assuming constant abundance given a redistribution of abundance between strata.

Scenario 2 (Random-walk): Abundance redistributes according to a mean-reverting autoregressive random walk according to the following equations (eqns 5-7):

  • display math(eqn 5)
  • display math(eqn 6)
  • display math(eqn 7)

where ρt is the random-walk value for year t, which has correlation AR1 = 0·8 between adjacent years and standard deviation σρ = 0·5, W = 0·9 is the parameter governing the mean-reversion process (which is included to ensure that abundance does not end up entirely fished or unfished), and inline image is the proportion of abundance in the fished strata r = 1. This scenario was designed to demonstrate changes in accuracy and precision arising from the spline approximation in an ideal situation, that is, when occupancy model assumptions are met.

Scenario 3 (Random-walk with process errors): Abundance redistributes according to a mean-reverting correlated random walk (σρ = 0·5, AR1 = 0·8, W = 0·9, eqns 5-7), while there are additional process errors causing the probability of detecting each local individual in sample i, Di, to vary for each survey occasion i (i.e. violating occupancy model assumptions about repeated measures samples) according to a beta distribution (eqn 8):

  • display math(eqn 8)

where d is the average probability of detecting each individual in a given cell, and inline image = 0·05. This scenario was designed to illustrate the impact of non-independence of detection probabilities, which has been shown previously to affect the performance of occupancy models (Martin et al. 2011).

Case study application

As an example application of time-varying parameters in data-poor ecological situations, we estimate availability inline image where again r = 1 is the fished stratum, and r = 2 is the unfished stratum using survey data for 53 species of Chondrichthyans along the northern Australian coast. Fishing mortality for these species occurs due to bycatch in the Northern Prawn Fishery, which primarily targets two species of tiger prawns (Penaeus semisulcatus and P. esculentus) from August to November. This management region is divided into 0·1 latitude by 0·1 longitude (c. 120 km2) cells that are designated as being in the ‘fished’ stratum if they have experienced >3 days of fishing between 2007 and 2009 and in the ‘unfished’ stratum otherwise, leading to fished and unfished strata with approximate areas of 8700 and 827 200 nm2, respectively (for simplicity in this case study application, we ignore the division of fished and unfished areas into the five sampling strata used in Zhou & Griffiths 2008). Survey data for these species are available for 12 surveys operating intermittently between 1979 and 2003. This resulted in 22 estimated sampling gears after classifying each survey as a separate gear in fished and unfished strata (two surveys were only in the unfished strata). Aggregating data for all 22 gears results in a total of 6565 total observations, with occurrence probabilities for these Chondrichthyes species ranging from 0·01% to 23%. All samples within the same cell and the same year are treated as repeated measures. Further details of the case study may be found in Zhou & Griffiths (2008). We note that estimating differences in detectability between strata justifies the use of the occupancy modelling framework.

To analyse these data, we fit an occupancy model with AIC-selected degrees of freedom for total abundance (dfN) and proportion fished (dfP). We hypothesize that both total abundance and the proportion of abundance in each stratum has likely changed for many of these species during these 25 years. We note the number of species for which stepwise-AIC model selection leads to a model with time-varying abundance, and also compare the AIC in these instances with the AIC for a model that assumes abundance is constant for all years, as well as that for a model that estimates annually varying forms for abundance and the proportion fished. We do this to evaluate whether time-varying abundance is selected for all species, or simply for species with relatively large sample sizes.

Results

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information

Simulation modelling

The occupancy model using a stepwise-selected spline approximation for abundance had lower error than the constant, annually varying or random-effect models in all simulation configurations (Table 1, Fig. 2). As expected, the constant-abundance model showed relatively little improvement in RMSE with increasing sample size, because increased sample sizes improved estimation of detectability, but could not improve estimation of changes in abundance. The RMSE for the constant-abundance model given large sample sizes arose primarily from systematic errors due to approximating a time-varying process with a constant parameter. A plot of RMSE by year for the dome-shaped abundance model (Fig. 2) shows this systematic error for the constant-abundance model, which is positively biased in early and late years, and negatively biased in intermediate years. By contrast, the annually varying abundance model achieved the greatest reduction in RMSE with increased sample sizes, ranging from RMSE = 0·23–0·32 for nsamples = 10 to RMSE = 0·09–0·17 for nsamples = 100. The random-effect model had approximately equal or lower RMSE than either the annually varying or constant-abundance models because it allowed for time-varying abundance but avoided aberrantly large annual deviations in abundance due to shrinkage. The configurations where the random effects model performed slightly worse than the annually varying model were for dome-shaped abundance, because abundance changes in this case were not ‘exchangeable’ and violated the assumptions of the random-effect estimator. However, the random-effect model did not perform as well as the stepwise-spline approximation in any configuration. Exploratory analysis showed that this arose because the time variability in abundance was highly autocorrelated. This autocorrelation was not included in the simple implementation of random effects, but is easily approximated using the spline model.

image

Figure 2. Regions showing 80% simulation intervals (e.g. 10th and 90th percentile) for errors when estimating availability inline image for different data availability (columns: left: nsamples = 10; middle: nsamples = 40; right: nsamples = 100) and abundance trends (rows: top: dome-shaped; middle: mean-reverting random walk; bottom: mean-reverting random walk with additional process errors) for constant (green dotted line), annual (dark blue dashed line), stepwise-selected (red solid line) and random-effect (light blue) models.

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Overall, the relative ranking of models did not change among configurations. There was a small bias in availability estimates for all models due to a small bias in estimating the relative detectability between fished and unfished strata. This bias did not change appreciably when process errors were introduced in detectability for each sample (i.e. the ‘random walk with process errors’ scenario). This demonstrates that a spline approximation can decrease estimation errors, and an occupancy model may be appropriate for estimating a ratio statistic such as availability, even when occupancy model assumptions are violated. Finally, the time required to apply each model (Fig. 3) was similar among different abundance scenarios, increased with increasing sample sizes, and varied greatly among estimation models. The constant-abundance model was by far quickest to apply in all cases, followed by the annual-varying abundance model. The stepwise-spline model generally took 4–8 times less time (including the AIC stepwise selection process) than the random-effect model, even when the latter was implemented using partially separable random effects to speed convergence (Skaug & Fournier 2006).

image

Figure 3. Time (in seconds) required to estimate parameters for the constant (‘const.’), stepwise-AIC (‘step.’), annually varying (‘ann.’) and random-effect models (‘rand.’) for different data availability and abundance trends.

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Chondrichthys application

The Akaike Information Criterion identifies time variability in total abundance or the proportion in the fished stratum for 34 of the 42 Chondrichthys species (Table 2). Reductions in AIC arising from the spline approximation are extremely large (ΔAIC > 100) for 14 of these species, and a time-varying form is selected for a species with as few as 10 occurrences over 25 years (Ornate Eagle Ray, Aetomylaeus vespertilio). As one example, estimates for Cowtail Stingray, Pastinachus atrus (Fig. 4), after accounting for detectability that varies by gear and strata show that availability was low from 1980 through 1998, but has dramatically increased since then. By comparison, the constant-abundance occupancy model misses any indication that susceptibility of this species to fishing gear has increased over time.

image

Figure 4. Availability estimates inline image for Cowtail Stingray, Pastinachus atrus, for occupancy models with different dfN and dfP that were encountered during the AIC-stepwise model building algorithm (ΔAIC ≤ 3: solid; 3 < ΔAIC ≤ 6: dashed; 6 < ΔAIC ≤ 10: dotted) with the constant-abundance model included for comparison.

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Table 2. Stepwise-selected degrees of freedom for total abundance (dfN) and redistribution between fished and unfished areas (dfP) for each of 42 Chondrichthyan species, along with the difference in Akaike Information Criterion (AIC) between the AIC-selected model and the constant-abundance model (ΔAICconstant, dfN = dfP = 1) or the annually varying model (ΔAICannual, dfN = dfP = 25), the number of encounters (Nobs) from 6565 sampling occasions, and the number of gears that ever encountered the given species (Ngear), where numbers of estimated fixed effects are Ngear + dfN + dfP, Ngear + 2, Ngear + 50 for the AIC-selected model, the constant-abundance model and the annually varying model, respectively
Scientific nameCommon name N obs N gear dfNdfPΔAICconstantΔAICannual
Aetobatus ocellatus Whitespotted Eagle Ray30711084·4
Aetomylaeus vespertilio Ornate Eagle Ray10332303·5399·8
Aetomylaeus nichofii Banded Eagle Ray1401113 11·475·9
Neotrygon annotata Plain Maskray526829111·175·2
Anoxypristis cuspidata Narrow Sawfish8812210·496·1
Carcharhinus amboinensis Pigeye Shark5310110171·2
Carcharhinus brevipinna Spinner Shark39911091·9
Carcharhinus dussumieri Whitecheek Shark15101932470·6576
Carcharhinus fitzroyensis Creek Whaler33721172·292·7
Carcharhinus limbatus Common Blacktip Shark2591141114·997·1
Carcharhinus macloti Hardnose Shark1661211087·8
Carcharhinus sorrah Spot-tail Shark547166101562·93195·8
Carcharhinus tilstoni Australian Blacktip Shark114616192642·32401·2
Chiloscyllium punctatum Grey Carpetshark556724 53·4124·8
Dasyatis brevicaudata Smooth Stingray22561343·4439·1
Neotrygon kuhlii Bluespotted Maskray230817 72·2156·3
Neotrygon leylandi Painted Maskray5271141 13·181·2
Dasyatis thetidis Black Stingray254410·989·4
Eusphyra blochii Winghead Shark4812110211
Galeocerdo cuvier Tiger Shark58621230628·7
Gymnura australis Australian Butterfly Ray10851039358·172·7
Hemigaleus australiensis Weasel Shark6381213110·1261·6
Hemipristis elongata Fossil Shark541312088·4 78
Himantura fai Pink Whipray105110333·1
Himantura granulata Mangrove Whipray19316120·9212·3
Himantura toshi Brown Whipray13617 44·9139·1
Himantura astra Blackspotted Whipray120313210295·569·5
Himantura uarnak Reticulate Whipray120627136118·3
Himantura leoparda Leopard Whipray8882414·3 78
Nebrius ferrugineus Tawny Shark256136·197·4
Negaprion acutidens Lemon Shark71531166193·9
Pastinachus atrus Cowtail Stingray111715 37·690·4
Pristis microdon Freshwater Sawfish10611091·9
Pristis zijsron Green Sawfish146210·799·3
Rhina ancylostoma Shark Ray42611084·2
Glaucostegus typus Giant Shovelnose Ray153120·183·5
Rhizoprionodon acutus Milk Shark8172019 55·959·6
Rhizoprionodon taylori Australian Sharpnose Shark8312216·8142
Rhynchobatus australiae Whitespotted Guitarfish96512214117·959·8
Sphyrna lewini Scalloped Hammerhead2121652895·51004·2
Sphyrna mokarran Great Hammerhead10415112 29·578·4
Stegostoma fasciatum Zebra Shark1511025 51·449·5

Discussion

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information

Time-varying parameters in ecological models

Ecological models approximate complicated biological and sampling processes with simple equations and functional forms. However, the parameters that are used to approximate these processes will often vary over time. The process errors induced by these types of variability will often lead to qualitative differences in the predictions of ecological models (Clark 2003), and accounting for this variability remains a central challenge for ecological modelling.

The stepwise-spline approximation that we propose represents a family of time-varying forms: assuming a parameter is constant over time, estimating a smoothed form for a time-varying parameter, or estimating annually varying parameters. Similarly, a piecewise-constant (e.g. 0th-order spline) is equivalent to estimating a separate value for different blocks of time, as has been proposed for some time-varying parameters in fishery population dynamics models (Legault 2008). We show that model selection tools that are commonly used by ecologists can identify an appropriate degree of smoothness (at the same time as other model selection decisions). We additionally propose that AIC-selection of a model with time variability in a parameter indicates that the process represented by this parameter is likely to be changing over time. We note that the sensitivity and specificity of the stepwise approximation when identifying the presence or absence of time-varying processes requires additional testing. However, in cases where failing the account for time-varying processes is very risky (such as time-varying abundance for managed marine species), we believe that this method for identifying time variability is appropriate.

The spline approximation presented here requires several subjective decisions within any given study. B-splines (Hastie, Tibshirani & Friedman 2009) will be appropriate in cases where rapid convergence is necessary, because each spline component has local support and low covariance with other coefficients. By contrast, the parameters of the splines that we use will exhibit high covariance, but will allow logical bounds (or likelihood penalties) to be placed on estimated coefficients. Additionally, the order of the spline (e.g., piecewise-constant, piecewise-linear, piecewise-quadratic, etc.) must be chosen. This could in theory be selected using AIC or cross-validation, although the authors know of no study that has explored this, and many packages assume a cubic-spline (e.g., Wood 2006) by default. Researchers must also choose a criterion for selecting an appropriate number of knots, which could include AIC, the low sample size bias adjustment for AIC (AICc), the Schwarz Bayesian Information Criterion (BIC Schwarz 1978), generalized cross-validation (Wood 2006) or many others. We have used AIC for simplicity, but future research could explore the appropriate degree of penalty for selecting spline degrees of freedom in different applications. Future research could also evaluate spline models that select among many possible breakpoints instead of the evenly spaced knots used here. Breakpoint techniques such as this are conceptually simple, but would significantly decrease the speed of any model selection algorithm. Selection of the location of knots would presumably improve model performance when parameters change more rapidly at some times than others and is an important topic of future research.

Population dynamics studies will frequently select between the constant and annually varying form for parameters using the AIC (e.g., Cubaynes et al. 2011; Monk, Berkson & Rivalan 2011; Marescot et al. 2011), perhaps due to the ease with which a parameter can be specified as constant or annually varying within common software such as m-surge for capture–mark–recapture (Choquet et al. 2004) and ‘unmarked’ for occupancy modelling (Fiske & Chandler 2011). We propose that easy symbolic notation for the spline approximation, such as that used in the ‘mgcv’ package for generalized additive models (Wood 2006), would ease the incorporation of the spline approximation into applied studies if included in common software for population dynamics models.

In addition to the spline approximation presented here, time-varying processes can be treated as random effects or penalized coefficients. However, both of these methods have drawbacks and will not be feasible or appropriate for all modelling situations. Random effect estimation requires integration and is computationally demanding (de Valpine 2009). In our case, we compared a basic implementation of a random-effect model with the stepwise-spline approximation. We used the admb-re software to evaluate the necessary multivariate integral, which required some familiarity with the ‘partial separability’ of state-space random-effect coefficients (Skaug & Fournier 2006) to develop a parameterization that was computationally feasible, relatively fast and numerically stable. However, this random-effect model did not improve estimation performance, likely because time variability in our simulation experiment was highly autocorrelated and hence was more easily approximated using the spline method than random effect deviations. Alternatively, a time-varying parameter can be approximated using a smoothing spline (or variations thereon) with multiple fixed effects and specifying a penalty to constrain changes to be small or gradual, where the penalty itself is estimated using simple or generalized cross-validation (Wood 2006; Maunder & Harley 2011). However, simple cross-validation will frequently require numerous cycles of optimization to evaluate the fit for each model (e.g. 10 for 10-fold cross-validation), which may be prohibitive for large and complicated models. By contrast, generalized cross-validation is faster but requires estimating the effective degrees of freedom for a smoothed parameter (Meyer & Woodroofe 2000). This estimation is likely possible for many ecological models, but requires resolving several issues specific to the occupancy models presented here, that is, their hierarchical design and discrete-valued predictions. We therefore suggest that developing generalized cross-validation methods for occupancy models is an important area of future research.

Occupancy models

Occupancy models have been a hot research topic for at least a decade (MacKenzie et al. 2005) and can simultaneously estimate detectability and abundance using survey data (Royle & Nichols 2003). Occupancy models have been modified to account for false-positive detections (Royle & Link 2006), finite-population corrections (Royle & Kéry 2007), spatially varying detection probabilities (MacKenzie et al. 2002) and aggregation behaviours for target individuals (Zhou & Griffiths 2007). An occupancy model was chosen to demonstrate the stepwise-spline approximation to time-varying parameters because occupancy models are informative about both abundance and detectability, and hence time variability in one is unlikely to be confounded with the other. Detectability is one of the most difficult-to-estimate parameters in marine stock assessment modelling and is frequently confounded with other biological processes (Zhou et al. 2011), so the capacity to estimate it using survey data is a powerful reasons for applying occupancy modelling.

Occupancy models involve several strong assumptions, including locally closed populations, independent detection of individuals, and no missing variables related to detectability (MacKenzie et al. 2005). However, these models may still be preferable to implicit, but strong assumptions about spatially constant detectability, even in cases where occupancy model assumptions are violated or impossible to verify (Mazerolle et al. 2007). We have demonstrated that estimates of the proportion of abundance in a given spatial stratum are only mildly biased even when local detectability is distributed according to a beta-binomial model, thus responding to concerns raised by Martin et al. (2011). Marescot et al. (2011) similarly demonstrated robust results when using a different ratio of estimates, that is, the change in abundance over time. We hope that further research will continue to identify which model estimates (or derived quantities) are sensitive or robust to violations of occupancy model assumptions.

We propose that increased parsimony regarding time-varying parameters will increase the accuracy and interpretability of future occupancy model applications as well as for other common ecological models. However, future applications of the stepwise-spline approximation to time-varying parameters may require additional diagnostics related to confounded parameters (e.g. inspection of the estimated correlation among parameters) when used with models that cannot separately estimate parameters. The consequences of estimating time-varying parameters in models that are highly confounded are an important topic for future research.

Acknowledgements

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information

Support was provided primarily by the Commonwealth Scientific and Industrial Research Organization (CSIRO), with preliminary support provided by the University of Washington, the NMFS-Sea Grant Population Dynamics Fellowship (NA09OAR4170120), a NMFS groundfish project grant to the University of Washington, and the Joint Institute for the Study of the 353 Atmosphere and Ocean (JISAO) under NOAA Cooperative Agreement No. 354 NA17RJ1232. We thank M. Fuller and S. Griffiths for help obtaining and processing data, D. Smith, P. Powell and L. Wyld for logistical support, B. Venables and N. Klaer for editorial suggestions, as well as the efforts of personnel with the CSIRO Marine and Atmospheric Research and state agencies who originally collected the Chondrichthys data set. The manuscript was improved by comments from Julien Martin and an anonymous reviewer.

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  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information
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Supporting Information

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information

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