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This document provides an overview of secr 4.3, an R package for spatially explicit capture–recapture analysis (SECR). It includes some background on SECR, an outline of the package, and a more detailed description of how models are implemented. See secr-tutorial.pdf for an introductory tutorial. For details of how to use secr see the help pages and vignettes.

NOTE: secr was extensively re-written between versions 3.2 and 4.0, but the interface remained unchanged. For many datasets version 4 is significantly faster. Other changes are described in secr-version4.pdf.

Two add-on packages extend the capability of secr and are documented separately. secrlinear enables the estimation of linear density (e.g., animals per km) for populations in linear habitats such as stream networks (secrlinear-vignette.pdf). secrdesign enables the assessment of alternative study designs by Monte Carlo simulation; scenarios may differ in detector (trap) layout, sampling intensity, and other characteristics (secrdesign-vignette.pdf).

Spatial open-population capture–recapture models are implemented in the R package openCR (Efford and Schofield 2019). Other open-population packages due to Ben Augustine and Richard Glennie are available on GitHub (https://github.com/benaug/OpenPopSCR; https://github.com/r-glennie/openpopscr).

# Introduction to SECR

Spatially explicit capture–recapture (SECR) is a set of methods for modelling animal capture–recapture data collected with an array of 'detectors'. The methods are used primarily to estimate population density, but they also have advantages over non-spatial methods when the goal is to estimate population size (Efford and Fewster 2013). SECR methods overcome edge effects that are problematic in conventional capture–recapture estimation of animal populations (Otis et al. 1978). Detectors may be live-capture traps, with animals uniquely tagged, sticky traps or snags that passively sample hair, from which individuals are distinguished by their microsatellite DNA, or cameras that take photographs from which individuals are recognized by their natural marks. The concept of a detector extends to areas (polygons) or transects that are searched for animals or their sign.

The primary data for SECR are (i) the locations of the detectors, and (ii) detections of known individuals on one or more sampling occasions (i.e. their detection histories). The generic terms 'detector' and 'detections' cover several possibilities (see 'Detector types' below); we use them interchangeably with the more specific and familiar terms 'traps' and 'captures'. Table 1 gives a concrete example of trapping data (the structure differs for detectors that are not traps).

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Table 1. Some spatially explicit detection data. Each entry (e.g., A9) records the detector at which a known animal (ID) was observed at each sample time (occasion). '.' indicates no detection. Each detector has known x-y coordinates. Formats for data input are described in secr-datainput.pdf.

              Occasion
ID      1     2     3     4     5
----- ----- ----- ----- ----- -----
1     A9     .     .     .     .
2    A12   A12     .     .     .
3      .     .    C6    B5     .
4      .     .    G3     .    F3
etc.


In SECR, a spatial model of the population and a spatial model of the detection process are fitted to the spatial detection histories. The resulting estimates of population density are unbiased by edge effects and incomplete detection (other sources of bias may remain). Inverse prediction (IP SECR) and maximum likelihood (ML SECR) are alternative methods for fitting the spatial detection model (Efford 2004, Borchers and Efford 2008). Of these, ML SECR is the more flexible, with a caveat for data from single-catch traps. Data augmentation and Markov chain Monte Carlo (MCMC) methods have also been used for SECR (Royle and Young 2008, Royle et al. 2009, Singh et al. 2010, Royle and Gardner 2011, Royle et al. 2014), but this approach is much slower than ML SECR; it is not considered here.

# State and observation models

Like other statistical methods for estimating animal abundance (Borchers et al. 2002), SECR combines a state model and an observation model. The state model describes the distribution of animal home ranges in the landscape, and the observation model (a spatial detection model) relates the probability of detecting an individual at a particular detector to the distance of the detector from a central point in each animal's home range. The distances are not observed directly (usually we don't know the range centres), so conventional distance sampling methods do not apply.

## Distribution of home-range centres

The distribution of range centres in the population (Borchers and Efford 2008) will usually be treated as a homogeneous Poisson point process (Fig. 1a). Density (= intensity) is the sole parameter of a homogeneous Poisson process. An inhomogeneous Poisson distribution may also be fitted; this provides a means to evaluate the effects of habitat variables on density.

## Detection functions

A detection model describes the decline in detection probability with distance ($$d$$) from the home-range centre (Fig. 1b). The probability $$g(d)$$ is for the 'ideal' case of just one animal and one detector; the actual probability may differ (see discussion of 'traps' under Detector Types).

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Fig. 1. (a) Hypothetical Poisson distribution of range centres near an array of detectors. Each dot represents one individual. SECR estimates the density of this distribution. (b) Alternative detection functions. The halfnormal is defined by $$g(d) = g_0\exp\left(\frac{-d^2}{2\sigma^2}\right)$$ and the exponential by $$g(d) = g_0\exp\left(-\frac{d}{\sigma}\right)$$. See ?detectfn for more (also the list in Appendix 1).

## Detector types

The properties of detectors are an important part of the SECR observation model (Table 2). Inside secr, data are tagged with a detector type to ensure they are printed, plotted and analysed appropriately.

Some common detectors (camera 'traps' and hair snags for DNA) do not capture animals, but merely record that an animal has visited a site. These 'proximity' detectors can be considered to act independently of each other. With proximity detectors, each animal $$\times$$ occasion 'cell' of a detection history potentially contains several positive records. In the simplest case each cell contains a binary vector coding presence or absence at each detector (for such binary proximity detectors each observation has a Bernoulli distribution). A 'count' detector is a generalised proximity detector in which the data are vectors of counts, one per detector. Models for 'count' data will specify a distribution for the counts via the 'binomN' argument of secr.fit (binomN = 0 indicates Poisson; binomN > 1 indicates binomial with size = binomN; binomN = 1 indicates binomial with size given by the 'usage' attribute for the detector and occasion).

Detectors that are true traps do not act independently because capture of an animal in one trap prevents it being caught in another trap until it is released. Traps expose animals to competing risks of capture. The per-trap probability of capture may be adjusted for the competing risk from other traps by using an additive hazard model (Borchers and Efford 2008). However, if the detectors are traps that catch only one animal at a time then there is a further level of competition – between animals for traps. Multi-catch and single-catch traps therefore represent distinct detector types. No general adjustment has been found for the per-trap probability of capture in the single-catch case (it's an open research question), and there is strictly no known maximum likelihood estimator. However, density estimates using the multi-catch likelihood for single-catch data appear only slightly biased (Efford, Borchers and Byrom 2009).

Polygon and transect detectors are for binary or count detection data (e.g., number of detections per animal per polygon per occasion) supplemented with the x-y coordinates of each detection. When a study uses multiple search areas or multiple transects, detections may be either independent or dependent (e.g., maximum one per animal per polygon per occasion) as with traps. The dependent or 'exclusive' type is indicated by the suffix 'X'; in this case the counts are necessarily binary. Using the 'polygonX' or 'transectX' detector type ensures that a competing-risk model is fitted.

Acoustic 'signal strength' detectors produce a binary detection vector supplemented by measurements of signal strength, as from an array of microphones.

There is limited support in secr for the analysis of locational data from telemetry ('telemetry' detector type). Telemetry data are used to augment capture–recapture data (see addTelemetry and secr-telemetry.pdf).

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Table 2. Detector types in secr

Detector Description
single traps that catch one animal at a time
multi traps that may catch more than one animal at a time
proximity records presence at a point without restricting movement
count proximity detector allowing >1 detection per animal per time
capped proximity detector with maximum one animal at a time
polygon counts from searching one or more areas
transect counts from searching one or more transects
polygonX binary data from mutually exclusive areas
transectX binary data from mutually exclusive transects
signal detections and signal strengths at multiple microphones

# Origins and outline of the package secr

The program DENSITY (Efford et al. 2004, Efford 2012) provides a graphical interface to SECR methods that has been accepted by many biologists. However, DENSITY has significant drawbacks: it requires the Windows operating system, its algorithms are not always transparent or well-documented, it fits only homogeneous Poisson models, and it omits recent advances in SECR.

The R package secr was written to address these weaknesses and allow for further development. It implements almost all the methods described by Borchers and Efford (2008), Efford et al. (2009), Efford (2011), Efford and Fewster (2013), Efford et al. (2013) and Efford and Mowat (2014). secr 4.3 uses external C++ code via package Rcpp for computationally intensive operations (Eddelbuettel and Francois 2011); Multi-threading on multiple CPUs with RcppParallel (Allaire et al. 2019) gives major speed gains. Appendix 1 compares the features of DENSITY and secr. The most important functions of secr are listed in Appendix 2.

# How secr works

secr defines a set of R classes[1] and methods for data from detector arrays and models fitted to those data.

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Table 3. Essential classes in secr.

Class Data
traps locations of detectors; detector type ('proximity', 'multi', etc.)
capthist spatial detection histories, including a 'traps' object
mask raster map of habitat near the detectors
secr fitted SECR model

[1]:Technically, these are S3 classes. A 'class' in R specifies a particular type of data object and the functions (methods) by which it is manipulated (computed, printed, plotted etc). See the R documentation for further explanation.

To perform an SECR analysis you explicitly or implicitly construct each of these objects in turn. Fig. 2 indicates the relationships among the classes.

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Fig. 2. Essentials of the secr package.

• Each object class (shaded box) comes with methods to display and manipulate the data it contains (e.g. print, summary, plot, rbind, subset)[2].
• The function read.capthist forms a 'traps' object from the detector layout data and saves it as an attribute, along with capture data read from another file, in a 'capthist' object.
• By default, a habitat mask is generated automatically by secr.fit using a specified buffer around the detectors (traps). The function make.mask gives greater control over this step.
• Any of the objects input to secr.fit (traps, capthist, mask) may include a dataframe of covariates saved as an attribute. Covariate names may be used in model formulae; the covariates method is used to extract or replace covariates.
• Fitted secr models may be manipulated with the methods shown on the right and others listed in Appendix 3.

[2]:Text in this font refers to R objects that are documented in online help for the secr package, or in base R.

## Input

Data input is covered in the separate document secr-datainput.pdf. One option is to use text files in the formats used by DENSITY; these accommodate most types of data. Two files are required, one of detector (trap) locations and one of the detections (captures) themselves; the function read.capthist reads both files and constructs a capthist object. It is also possible to construct the capthist object in two stages, first making a traps object (with read.traps) and a captures dataframe, and then combining these with make.capthist. This more general route may be needed for unusual datasets.

## Output

The output from the function secr.fit is an object of class secr. This is an R list with many components. Assigning the output to a named object saves both the fit and the data for further manipulation. Typing the name at the R prompt invokes print.secr which formats the key results. These include the dataframe of estimates from the predict method for secr objects. Functions are provided for further computations on secr objects (e.g., AIC model selection, model averaging, profile-likelihood confidence intervals, and likelihood-ratio tests). Many of these are listed in Appendix 3.

One system of units is used throughout secr. Distances are in metres and areas are in hectares (ha). The unit of density for 2-dimensional habitat is animals per hectare. 1 ha = 10000 m2^ = 0.01 km2. To convert density to animals per km2, multiply by 100. Density in linear habitats (see package secrlinear) is expressed in animals per km.

# Documentation

The primary documentation for secr is in the help pages that accompany the package. Help for a function is obtained in the usual way by typing a question mark at the R prompt, followed by the function name. Note the 'Index' link at the bottom of each help page – you will probably need to scroll down to find it. The index may also be accessed with help(package = secr).

The consolidated help pages are in the file secr-manual.pdf. Searching this text is a powerful way to locate a function for a particular task.

Other documentation, in the form of pdf vignettes built with knitr, will be added from time to time. The 'User guides…' link in the package help index lists available files. The vignettes in Table 4 may be found on the Density website.

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Table 4. Vignettes for secr 4.3.

Vignette Topic
secr-overview.pdf introduction (this document)
secr-datainput.pdf data formats and input functions
secr-version4.pdf what's new in secr 4.x
secr-tutorial.pdf tutorial using Burnham and Cushwa snowshoe hare data
secr-models.pdf general description of models in secr
secr-troubleshooting.pdf problems with secr.fit, including speed issues
secr-densitysurfaces.pdf modelling density surfaces
secr-finitemixtures.pdf mixture models for individual heterogeneity
secr-markresight.pdf mark–resight models
secr-multisession.pdf data from multiple independent sessions
secr-noneuclidean.pdf non-Euclidean distance models
secr-parameterisations.pdf alternative parameterisations of detection
secr-polygondetectors.pdf using polygon and transect detector types
secr-sound.pdf analysing data from microphone arrays
secr-telemetry.pdf analysing combined telemetry and capture–recapture data
secr-varyingeffort.pdf variable effort (usage) in SECR models

The web page https://www.otago.ac.nz/density/ should be checked for news of bug fixes and new releases. New versions will be posted on CRAN, but there may be a delay of a few days. Help may be sought at phidot; see also the FAQ there for DENSITY and secr. Another forum intended for both software issues and wider discussion is secrgroup. For information on changes in each version, type at the R prompt:

news (package = "secr")


# Defining models with the 'model' argument of secr.fit

By default, the parameters of SECR models are assumed to be constant. We specify more interesting, and often better-fitting, models with the 'model' argument of secr.fit. Here 'models' relates to variation in the parameters that may be explained by known factors and covariates. The explanation in secr-models.pdf may help. If you just want to know how to use models, read on.

Models are defined symbolically in secr using R formula notation. A separate linear predictor is used for each core parameter. Core parameters are 'real' parameters in the terminology of MARK, and secr uses that term because it will be familiar to biologists.

Three real parameters are commonly modelled in secr 4.3; these are denoted 'D' (for density), 'g0' (or 'lambda0') and 'sigma'. Only the last two real parameters, which jointly define the model for detection probability as a function of location, can be estimated directly when the model is fitted by maximizing the conditional likelihood (CL = TRUE in secr.fit). D is then a derived parameter that is computed from an secr object with the function derived or one of its siblings (derivedCluster etc.).

Here is a simple example of the model argument in use:

secr.fit(captdata, model = g0~t)


The real parameter g0 is no longer constant, but takes a unique value on each sampling occasion (t).

Other 'real' parameters appear in particular contexts. 'z' is a shape parameter that is used only when the detection function has three parameters (annular halfnormal, cumulative gamma, hazard-rate etc. – see ?detectfn). Some detection functions primarily model 'exposure' or the cumulative hazard of detection, rather than the probability of detection; these use the real parameter 'lambda0' in place of 'g0' (see ?detectfn). 'lambda0' is also used with count detectors. A further 'real' parameter is the mixing proportion 'pmix', used in finite mixture models and hybrid mixture models (see ?hcov).

Sometimes it is illuminating and efficient to parameterise the detection function using a function of the primary 'real' parameters described above. This gives rise to the surrogate 'real' parameters a0 and sigmak; see the vignette secr-parameterisations.pdf for details and references.

Detection parameters and density parameters are modelled separately, as we now describe.

## Detection parameters

Effects on parameters of detection probability are specified via R formulae. The variable names used in formulae are either names for standard effects (Table 5) or the names of user-supplied covariates. Effects 'b', 'B', 'bk', and 'Bk' refer to individuals whereas 'k' and 'K' refer only to sites. Groups ('g') are used only in models fitted by maximizing the full likelihood; for conditional likelihood models use a factor covariate to achieve the same effect. See also the later section on modelling sex differences.

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Table 5. Automatically generated predictor variables used in detection models

Variable Description Notes
g group individual covariates listed in secr.fit argument 'groups'
t time factor one level for each occasion
T time trend linear trend over occasions on link scale
b learned response step change after first detection
B transient response depends on detection at preceding occasion (Markovian response)
bk animal x site response site-specific step change
Bk animal x site response site-specific transient response
k site learned response site effectiveness changes once any animal caught
K site transient response site effectiveness depends on preceding occasion
session session factor one level for each session
Session session trend linear trend on link scale
h2 2-class mixture finite mixture model with 2 latent classes
ts marking vs sighting two levels (marking and sighting occasions)

Any name in a formula that is not a variable in Table 5 is assumed to refer to a user-supplied covariate. secr.fit looks for user-supplied covariates in data frames embedded in the 'capthist' argument, or supplied in the 'timecov' and 'sessioncov' arguments, or named with the 'timevaryingcov' attribute of a traps object, using the first match (Table 6).

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Table 6. Types of user-provided covariate for in detection models. The names of columns in the respective dataframes, and names of components in the 'timevaryingcov' attribute, may be used in model formulae

Covariate type Data source Notes
Individual covariates(capthist) conditional likelihood only
Time timecov argument
Detector covariates(traps(capthist))
Detector x Time covariates(traps(capthist)) see ?timevaryingcov
Session sessioncov argument

The formula for any detection parameter (e.g., g0, lambda0 or sigma) may be constant ($$\sim$$ 1, the default) or some combination of terms in standard R formula notation (see ?formula). For example, g0 $$\sim$$ b + T specifies a model with a learned response and a linear time trend in g0; the effects are additive on the link scale. See Table 7 for other examples.

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Table 7. Some examples of the 'model' argument in secr.fit

Formula Effect
g0 $$\sim$$ 1 g0 is constant across animals, occasions and detectors
g0 $$\sim$$ b learned response affects g0
list(g0 $$\sim$$ b, sigma $$\sim$$ b) learned response affects both g0 and sigma
g0 $$\sim$$ h2 2-class finite mixture for heterogeneity in g0
g0 $$\sim$$ b + T learned response in g0 combined with trend over occasions
sigma $$\sim$$ g detection scale sigma differs between groups
sigma $$\sim$$ g*T group-specific trend in sigma
D $$\sim$$ cover density varies with 'cover', a variable in covariates(mask)
list(D $$\sim$$ g, g0 $$\sim$$ g) both density and g0 differ between groups
D $$\sim$$ session session-specific density

For other effects, the design matrix for detection parameters may also be provided manually in the argument dframe of secr.fit. This feature is untested.

## Inhomogeneous density models

The SECR log likelihood is evaluated by summing values at points on a 'habitat mask' (the 'mask' argument of secr.fit). Each point in a habitat mask represents a grid cell of potentially occupied habitat (their combined area may be almost any shape). The full design matrix for density (D) has one row for each point in the mask. As for the detection submodels, the design matrix has one column for the intercept (constant) term and one for each predictor.

Predictors may be based on Cartesian coordinates (e.g. 'x' for an east-west trend), a continuous habitat variable (e.g. vegetation cover) or a categorical (factor) habitat variable. Predictors must be known for all points in the mask (non-habitat excluded). The variables 'x' and 'y' are the coordinates of the habitat mask and are automatic, as are 'x2', 'y2', and 'xy'. Other spatial covariates should be named columns in the 'covariates' attribute of the habitat mask.

Regression splines are particularly effective for modelling spatial trend. For these and general guidance on fitting and displaying density surfaces, see the vignette secr-densitysurfaces.pdf.

# Model fitting and estimation

Models are fitted in secr.fit by numerically maximizing the likelihood. The likelihood involves integration over the unknown locations of the animals' range centres. This is achieved in practice by summation over points in the habitat mask, which has some implications for the user. Computation may be slow, especially if there are many points in the mask, and estimates may be sensitive to the particular choice of mask (either explicitly in make.mask or implicitly via the 'buffer' argument).

The default maximization algorithm is Newton-Raphson in the function stats::nlm. By default, all reported variances, covariances, standard errors and confidence limits are asymptotic and based on a numerical estimate of the information matrix. The Newton-Raphson algorithm is fast, but it sometimes fails to compute the information matrix correctly, causing some standard errors to be set to NA; see the 'method' argument of secr.fit for alternatives. Use confint.secr for profile likelihood intervals and sim.secr for parametric bootstrap intervals (both are slow).

We have already introduced the idea of a habitat mask. The SECR likelihood is evaluated by summing values at points on a mask; each point represents a grid cell of potentially occupied habitat. Masks may be constructed by placing a buffer of arbitrary width around the detectors, possibly excluding known non-habitat. How wide should the buffer be? The general answer is 'Wide enough not to cause bias in estimated densities'. This depends on the scale of movement of the animal, and on the chosen detection function. For specifics, see the separate vignette on habitat masks secr-habitatmasks.pdf and the help for 'mask' and the various mask-related functions (make.mask, mask.check, suggest.buffer, and esa.plot). Heavy-tailed detection functions such as the hazard-rate and lognormal can be problematic because they require an unreasonably large buffer for stable density estimates.

# Miscellaneous topics

## Modelling sex differences {#modellingsex}

There are many ways to model sex differences in secr. Here we sketch some possibilities, in order of usefulness (your mileage may vary).

1. Fit a hybrid mixture model as described in the online help (?hcov). This accommodates occasional missing values and estimates the sex ratio (pmix).

2. Use conditional likelihood (CL = TRUE) and include a categorical (factor) covariate in model formulae (e.g., g0 $$\sim$$ sex). To get sex-specific densities then specify groups = "sex" in derived.

3. Use full likelihood (CL = FALSE) and separate data for the two sexes as different sessions (most easily, by coding 'female' or 'male' in the first column of the capture file read with read.capthist). Then include a group term 'session' in relevant model formulae (e.g., g0 $$\sim$$ session).

4. Use full likelihood (CL = FALSE), define groups = "sex" or similar, and include a group term 'g' in relevant formulae (e.g., g0 $$\sim$$ g).

'CL' and 'groups' are arguments of secr.fit. Possibilities 1–4 should not be mixed for comparing AIC. Sex differences in home-range size (and hence sigma) may be mitigated by compensatory variation in g0 or lambda0 (Efford and Mowat 2014).

## Varying effort

The probability of observing an individual at a particular detector may depend directly on a known quantity such as how long the detector was exposed on a particular occasion. In the extreme, a detector may not have been operated. The terms 'effort' and 'usage' are used here interchangeably for variation in the duration of exposure and similar known effects. Usage is an attribute of the detectors in a traps object (a traps x occasions matrix); it may be entered with the detector coordinates in a trap layout file or added later (see ?usage). Models fitted to data including a usage attribute will adjust automatically for varying usage across detectors and occasions. Usage may take any non-negative value. This simplifies the modelling of data aggregated over varying numbers of occasions or nearby sites.

See the separate document secr-varyingeffort.pdf and Efford et al. (2013) for more.

## Mark–resight

Mark–resight data include sampling occasions on which previously marked animals were recorded, but new animals were not distinguished from each other. secr 4.3 provides a suite of spatial models for these data, as documented in secr-markresight.pdf. Two general classes of spatial mark–resight model are included: those in which the marking process is modelled (we call these spatial capture–mark–resight models), and those in which the process is not modelled and pre-marked animals are assumed to follow some distribution (e.g., uniform across a known area) (sighting-only or mark–resight models). Mark–resight models in secr 4.3 discard some spatial information in the unmarked sightings – information that is used in the models of Chandler and Royle (2013) and Sollmann et al. (2013). This results in some (probably small) loss of precision, and requires an adjustment for overdispersion to ensure confidence intervals have good coverage properties. The vignette secr-markresight.pdf should be consulted.

## Detector clusters

For surveying large areas it is efficient to use groups of detectors: within a group the detectors are close enough that animals may be re-detected at multiple points, while groups of detectors may be distributed across a region according to a probability design to sample possible spatial variation in density. secr allows for detector groups with the 'cluster' data structure. This is an attribute of a traps object that records which detectors belong to which cluster[3].

[3]:Clusters are assumed to share the same geometry (number of detectors, within-cluster spacing etc.).

Functions are provided to generate detector arrays with a clustered structure (trap.builder, make.systematic), to extract or replace the cluster attribute (clusterID), to compute the geometric centres and numbers of detections per cluster (cluster.centres, cluster.counts), etc.

A lacework design (Efford in prep.) is an alternative to a clustered design that is suitable when the region is intermediate size. Lacework designs may be constructed with make.lacework.

Data from a large, clustered design may often be analysed more quickly if the 'capthist' object is first collapsed into one using the geometry of a single cluster (the object retains a memory of the number of individuals from each original cluster in the attribute 'n.mash'). Use the function mash for this. Functions derived, derivedMash and the method predict.secr use 'n.mash' to adjust their output density, SE, and confidence limits.

## Parallel processing

On processors with multiple cores it is possible to speed up computation by using cores in parallel. In secr 4.3 this happens automatically in secr.fit and a few other functions using the multi-threading paradigm of RcppParallel (Allaire et al. 2019). The number of threads may be set directly with the function setNumThreads, or via the 'ncores' argument of several functions.

Earlier versions of secr relied on parallel processing with the parallel package (invoked when the argument ncores was set greater than 1). The benefits of that form of parallel processing were variable (considerable with simulations in sim.secr and ip.secr, but otherwise unimpressive).

The functions par.secr.fit, par.region.N and par.derived allowed models to be fitted or analysed in parallel, one per core. The greater speed of secr.fit in secr 4.3 makes par.secr.fit redundant. All three functions now appear to run faster with ncores=1 than with ncores>1. They are retained because it is sometimes useful to process several models at once.

## Regression splines

The standard models for 'real' parameters in secr are linear on the link scale, much like a generalised linear model. Semi-parametric 'regression spline' smooths provide more flexibility. These are implemented in secr using a method suggested by Borchers and Kidney (2014): Simon Wood's R package mgcv is used to parse s() and te() terms in model formulae and construct basis functions that are used like linear covariates within secr. Any 'real' parameter may be modelled with regression splines (D, lambda0, sigma, noneuc etc.). For details see the help page (?smooths) and the documentation for mgcv.

## Non-Euclidean distances

'Distance' in SECR models usually, and by default, means the Euclidean distance $$d = \sqrt{(x_1-x_2)^2 + (y_1 - y_2)^2}$$. The observation model can be customised by replacing the Euclidean distance with one that 'warps' space in some ecologically meaningful way. There are innumerable ways to do this. Royle et al. (2013) envisioned an 'ecological distance' that is a function of landscape covariates. Redefining distance is a way to model spatial variation in the size of home ranges, and hence the spatial scale of movement $$\sigma$$; Efford et al. (2016) use this to model inverse covariation between density and home range size. Distances measured along a linear habitat network such as a river system are also non-Euclidean (see package secrlinear).

secr provides general tools for specifying and modelling non-Euclidean distance, via the secr.fit details component 'userdist'. This may be a user-specified function or a pre-computed matrix. See secr-noneuclidean.pdf for a full explanation and examples.

# References

Allaire, J. J., Francois, R., Ushey, K., Vandenbrouck, G., Geelnard, M. and Intel (2019) RcppParallel: Parallel Programming Tools for 'Rcpp'. R package version 4.4.4. https://CRAN.R-project.org/package=RcppParallel.

Borchers, D. L., Buckland, S. T. and Zucchini, W. (2002) Estimating animal abundance: closed populations. Springer, London.

Borchers, D. L. and Efford, M. G. (2008) Spatially explicit maximum likelihood methods for capture–recapture studies. Biometrics 64, 377–385.

Borchers, D. L. and Fewster, R. M. (2016) Spatial capture–recapture models. Statistical Science 31, 219–232.

Borchers, D. L. and Kidney, D. (2014) Flexible density surface estimation for spatially explicit capture–recapture surveys. Technical Report, University of St Andrews.

Chandler, R. B. and Royle, J. A. (2013) Spatially explicit models for inference about density in unmarked or partially marked populations. Annals of Applied Statistics 7, 936–954.

Cooch, E. and White, G. (eds) (2014) Program MARK: A Gentle Introduction. 13th edition. Available online at http://www.phidot.org/software/mark/docs/book/.

Eddelbuettel, D. and Francois, R. (2011) Rcpp: Seamless R and C++ Integration. Journal of Statistical Software 40(8), 1–18. https://www.jstatsoft.org/v40/i08/.

Efford, M. G. (2004) Density estimation in live-trapping studies. Oikos 106, 598–610.

Efford, M. G. (2011) Estimation of population density by spatially explicit capture–recapture analysis of data from area searches. Ecology 92, 2202–2207.

Efford, M. G. (2012) DENSITY 5.0: software for spatially explicit capture–recapture. Department of Mathematics and Statistics, University of Otago, Dunedin, New Zealand https://www.otago.ac.nz/density.

Efford, M. G., Borchers D. L. and Byrom, A. E. (2009) Density estimation by spatially explicit capture–recapture: likelihood-based methods. In: D. L. Thomson, E. G. Cooch, M. J. Conroy (eds) Modeling Demographic Processes in Marked Populations. Springer. Pp 255–269.

Efford, M. G., Borchers D. L. and Mowat, G. (2013) Varying effort in capture–recapture studies. Methods in Ecology and Evolution 4, 629–636.

Efford, M. G., Dawson, D. K. and Borchers, D. L. (2009) Population density estimated from locations of individuals on a passive detector array. Ecology 90, 2676–2682.

Efford, M. G., Dawson, D. K., Jhala, Y. V. and Qureshi, Q. (2016) Density-dependent home-range size revealed by spatially explicit capture-recapture. Ecography 39, 676–688.

Efford, M. G. and Fewster, R. M. (2013) Estimating population size by spatially explicit capture–recapture. Oikos 122, 918–928.

Efford, M. G. and Mowat, G. (2014) Compensatory heterogeneity in spatially explicit capture–recapture data. Ecology 95, 1341–1348.

Huggins, R. M. (1989) On the statistical analysis of capture experiments. Biometrika 76, 133–140.

Laake, J. and Rexstad E. (2014) Appendix C. RMark - an alternative approach to building linear models in MARK. In: Cooch, E. and White, G. (eds) Program MARK: A Gentle Introduction. 13th edition. http://www.phidot.org/software/mark/docs/book/.

Lebreton, J.-D., Burnham, K. P., Clobert, J., and Anderson, D. R. (1992) Modeling survival and testing biological hypotheses using marked animals: a unified approach with case studies. Ecological Monographs 62, 67–118.

Otis, D. L., Burnham, K. P., White, G. C. and Anderson, D. R. (1978) Statistical inference from capture data on closed animal populations. Wildlife Monographs 62.

Royle, J. A., Chandler, R. B., Gazenski, K. D. and Graves, T. A. (2013) Spatial capture–recapture models for jointly estimating population density and landscape connectivity. Ecology 94 287–294.

Royle, J. A., Chandler, R. B., Sollmann, R. and Gardner, B. (2014) Spatial capture–recapture. Academic Press.

Royle, J. A. and Gardner, B. (2011) Hierarchical spatial capture–recapture models for estimating density from trapping arrays. In: A.F. O'Connell, J.D. Nichols and K.U. Karanth (eds) Camera Traps in Animal Ecology: Methods and Analyses. Springer, Tokyo. Pp. 163–190.

Royle, J. A., Nichols, J. D., Karanth, K. U. and Gopalaswamy, A. M. (2009). A hierarchical model for estimating density in camera-trap studies. Journal of Applied Ecology 46, 118–127.

Royle, J. A. and Young, K. V. (2008) A hierarchical model for spatial capture–recapture data. Ecology 89, 2281–2289.

Sollmann, R., Gardner, B., Parsons, A. W., Stocking, J. J., McClintock, B. T., Simons, T. R., Pollock, K. H. and O'Connell, A. F. (2013) A spatial mark–resight model augmented with telemetry data. Ecology 94, 553–559.

Stanley, T. R. and Burnham, K. P. (1999) A closure test for time-specific capture–recapture data. Environmental and Ecological Statistics 6, 197–209.

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# Appendix 1. Software feature comparisons {#appendix1}

• full implementation

◦ incomplete or inferior implementation.

Feature DENSITY 5.0 secr 4.3
General
Graphical interface
Inverse prediction (IP SECR)
Maximum likelihood estimation (ML SECR)
Non-spatial closed-population estimators
Simulation of spatial sampling
Build detector arrays
Control of random number generator
Closure tests
Import or export DENSITY text files
Convert BUGS data
Clustered detector layouts
Mash data from clustered layouts
Upload coordinates to GPS (uses GPSBabel)
Multi-threaded C++ for fast model fitting
ML secr
Density models (inhomogeneous 2-D Poisson)
Regional population size (region.N)
Varying effort (detector usage)
Fixed parameters
Parametric bootstrap
Between-session models
Profile likelihood confidence intervals
Mixture models for individual heterogeneity
Confidence ellipses
Formula-based model notation
Plot density models
Groups (e.g. males & females)
Score tests for model selection
Model averaging
Plot likelihood surface
Empirical variance from replicate units
Suggested buffer width
Contours of detection probability
Compute pdf for individual's range centre
Time-varying detector covariates
Hybrid mixture models (hcov)
Compensation (a0 parameterization)
Density-dependent sigma (sigmak parameterization)
Variance-only mode (method = 'none')
Combined telemetry-detection models
Regression splines
Non-Euclidean distance
Mark–resight
Detector types
Single-catch trapa^
Multi-catch trap
Proximity
Signal strength (acoustic)
Count
Capped
Polygon
Transect
Polygon (exclusive)
Transect (exclusive)
Telemetry
Detection functions
Halfnormal
Hazard rateb^
Exponential
Compound halfnormal
Uniforma^
w-exponential
Annular halfnormal
Binary signal strength
Signal strength
Signal strength spherical
Cumulative lognormalb^
Cumulative gamma
Hazard halfnormal
Hazard hazard rateb^
Hazard exponential
Hazard annular halfnormal
Hazard cumulative gamma
Hazard variable power

a. Not fitted by ML secr

b. Not recommended because of heavy tail

\pagebreak

# Appendix 2. Core functions of secr {#appendix2}

These are the core functions of secr 4.3 – the ones that you are most likely to use. S3 methods are marked with an asterisk.

Function Purpose
AIC* model selection, model weights
covariates extract or replace covariates of traps, capthist or mask
derived* compute density from conditional likelihood models
make.mask construct habitat mask (= mesh)
plot* plot capthist, traps or mask
read.capthist input captures and trap layout from Density format, one call
predict* compute 'real' parameters for arbitrary levels of predictor variables
predictDsurface evaluate density surface at each point of a mask
region.N* compute expected and realised population size in specified region
secr.fit maximum likelihood fit; result is a fitted 'secr' object
summary* summarise capthist, traps or mask
traps extract or replace traps object in capthist

\pagebreak

# Appendix 3. Classified index to secr functions {#appendix3}

Here is an index of secr functions classified by use (some minor functions are omitted). S3 methods are marked with an asterisk.

Function Purpose
Manipulate data objects
addCovariates add spatial covariates to 'traps' or 'mask'
as.mask coerce 'traps' object to 'mask' for specialised plotting
deleteMaskPoints edit 'mask'
discretize rasterize area-search capthist data
head* first rows of 'capthist', 'traps' or 'mask'
join combine sessions of multi-session 'capthist' object
make.grid construct detector array
make.capthist form 'capthist' from 'traps' and detection data
make.mask construct habitat mask (mesh)
make.systematic construct clustered random systematic design
make.lacework construct lacework random systematic design
MS.capthist combine 'capthist' objects into one multi-session 'capthist'
plot* plot 'capthist', 'traps' or 'mask'
plotMaskEdge draw line around mask cells
randomHabitat generates habitat mask with random landscape
rbind* append 'capthist', 'traps', 'popn' or 'mask' objects
read.capthist input captures and trap layout from Density format, one call
read.traps input detector locations from text file
reduce* aggregate detectors or occasions; change detector type
sim.capthist simulate capture histories
sightingPlot bubble plot of sightings in capthist object
snip split transect(s) into equal sections
split* split a single-session capthist or mask by various criteria
subset* filter 'capthist', 'traps' or 'mask'
summary* summarise 'capthist', 'traps' or 'mask'
tail* last rows of 'capthist', 'traps' or 'mask'
trap.builder construct various complex designs
verify* check 'capthist', 'traps' or 'mask' for internal consistency
Attributes of traps object
clusterID cluster identifier
clustertrap detector number within cluster
covariates* detector-level covariates
detector* detector type ('multi', 'proximity' etc.)
markocc vector distinguishing marking and sighting occasions
polyID* polygon or transect identifier
timevaryingcov name time-varying covariate(s)
usage* occasion- and detector-specific effort
Attributes of capthist object
addSightings add sighting data to a 'proximity', 'count' or 'polygon' object
addTelemetry add telemetry data to a 'proximity' or 'count' object
covariates* individual-level covariates, including grouping factors
session* session identifier(s)
signalmatrix sound x microphone table
telemetryxy coordinates of telemetry fixes
Tm counts of marked animals that were not identified
traps* embedded traps object(s)
Tu counts of unmarked animals
Data for each detection
alive TRUE/FALSE
animalID individual ID
clusterID cluster identifier
clustertrap detector number within cluster
noise noise (signal detectors)
occasion occasion
signal signal strength (signal detectors)
signalframe whole signal \
trap detector
xy detection coordinates (polygon and transect detectors)
Fit SECR model(s)
ip.secr fit simple SECR model by simulation \
par.secr.fit parallel secr.fit() (several models, using multiple cores)
secr.fit maximum likelihood fit; result is a fitted secr object
Operate on fitted model(s)
AIC* model selection, model weights
coef* 'beta' parameters
collate tabulate estimates from several models
confint* profile likelihood confidence intervals
CVa, CVa0 CV of individual detection from fitted mixture model
derived* density from conditional likelihood models
deviance* model deviance
df.residual* degrees of freedom for deviance
derivednj variance from replicated sampling units
derivedCluster variance from replicated sampling units
derivedExternal variance from replicated sampling units
ellipse.secr confidence ellipses for estimated parameters
fxi.secr probability density of home-range centre
LLsurface* compute likelihood surface and plot contours
logLik* log-likelihood of fitted model
LR.test likelihood-ratio test of two models
model.average combine estimates using AIC or AICc weights
par.derived parallel derived()
par.region.N parallel region.N()
plot* plot detection functions with confidence bands
predict* 'real' parameters for arbitrary levels of predictor variables
predictDsurface* evaluate density surface at each point of a mask
region.N* expected and realised population size in specified region
RSE extract precision (relative SE) of 'real' parameter estimates
score.test model selection with score statistic using observed information
secr.test Monte Carlo goodness-of-fit tests
simulate* generate realisations of fitted model
sim.secr parametric bootstrap
vcov* variance-covariance matrix of 'beta' or 'real' parameters
esa.plot cumulative plot esa or $$\hat D$$ vs buffer width
mask.check likelihood or estimates vs. buffer width and spacing
suggest.buffer find buffer width to keep bias within bounds
Specialised graphics
buffer.contour concave and convex boundary strips
fx.total summed pdfs of home-range centre pdfs (use with plot.Dsurface)
fxi.contour contour plot of home-range centre pdf(s)
pdot.contour contour plot of detection probability
strip.legend add colour legend to existing plot
Convert or export data
RMarkInput convert 'capthist' to dataframe for RMark
write.capthist export 'capthist' as text files for DENSITY
write.DA convert 'capthist' for analysis in WinBUGS
writeGPS upload coordinates to GPS using GPSBabel
Miscellaneous
ARL asymptotic range length
autoini generate starting values of D, g0 and sigma for secr.fit
clone replicate points to emulate overdispersion
closure.test closure tests of Otis et al. (1978) and Stanley \
closedN closed population size by various conventional estimators
counts summary data from 'capthist' object
CV coefficient of variation
dbar mean distance between capture locations
distancetotrap from an arbitrary set of points
edist Euclidean distance
MMDM mean maximum distance moved
moves distances between capture locations
nearesttrap from an arbitrary set of points
nedist Non-Euclidean distance
pdot location-specific net probability of detection
PG proportion of telemetry fixes in given polygons
pmixProfileLL profile likelihood as function of mixing proportion
RPSV 'root pooled spatial variance', a simple measure of home-range size
setNumThreads sets environment variable RCPP_PARALLEL_NUM_THREADS

\pagebreak

# Appendix 4. Datasets {#appendix4}

See each help page for details e.g., ?deermouse. Code for model fitting is in Appendix 2 of secr-version4.pdf.

deermouse

Peromyscus maniculatus Live-trapping data of V. H. Reid published as a CAPTURE example by Otis et al. (1978) Wildlife Monographs 62

hornedlizard

Repeated searches of a quadrat in Arizona for flat-tailed horned lizards Phrynosoma mcallii (Royle & Young Ecology 89, 2281–2289)

housemouse

Mus musculus live-trapping data of H. N. Coulombe published as a CAPTURE example by Otis et al. (1978) Wildlife Monographs 62

ovenbird

Multi-year mist-netting study of ovenbirds Seiurus aurocapilla at a site in Maryland, USA.

ovensong

Acoustic detections of ovenbirds (Dawson & Efford Journal of Applied Ecology 46, 1201–1209)

OVpossum

Brushtail possum Trichosurus vulpecula live trapping in the Orongorongo Valley, Wellington, New Zealand 1996–1997 (Efford and Cowan In: The Biology of Australian Possums and Gliders Goldingay and Jackson eds. Pp. 471–483).

possum

Brushtail possum Trichosurus vulpecula live trapping at Waitarere, North Island, New Zealand April 2002 (Efford et al. 2005 Wildlife Society Bulletin 33, 731–738)

secrdemo

Simulated data 'captdata', and some fitted models

skink

Multi-session lizard (Oligosoma infrapunctatum and O. lineoocellatum) pitfall trapping data from Lake Station, Upper Buller Valley, South Island, New Zealand (M. G. Efford, B. W. Thomas and N. J. Spencer unpublished).

stoatDNA

Stoat Mustela erminea hair tube DNA data from Matakitaki Valley, South Island, New Zealand (Efford, Borchers and Byrom 2009).