The use of the model class developed in this chapter is demonstrated through its application to detection data from surveys of three different species: B. acutorostrata, N. annamen- sis, and A. lightfooti. These surveys are described below, and are also accompanied by descriptions of the analyses and a relevant simulation study, along with the corresponding results.
2.4.1 Common minke whale
Two independent observers aboard a plane surveyed the same area of ocean and recorded sightings of B. acutorostrata surfacings. For each detection, the perpendicular distance from the plane to the sighted whale was estimated using a declinometer. These data were collected during the 2001 North Atlantic Sightings Survey; see Pike, Paxton, Gunnlaugsson, and Vikingsson (2009) for further details.
Data like these are typically analysed using an MRDS model, and in this case distances must be assumed to be exact. An issue here is that when the observers sighted the same animal, their estimated distances routinely differed to some degree. This is typically rec- onciled by simply averaging the differences and assuming this mean difference is the exact, true difference. Here, an SECR model is fitted to these data using the auxiliary distance information. This allows for the observed distances to be modelled, and estimation of their associated measurement error.
In total, 70 whales were detected by at least one of the two observers. There was strong evidence to suggest that one observer was far more proficient in whale detection, and so a separate detection function was estimated for each3. An SECR model—estimating distance estimation error—provided a whale density estimate of 1.72 whales per hectare. An MRDS model—taking the distance estimates to be error free—provided a whale density estimate of 1.61 whales per hectare. Both estimates were associated with similar standard errors.
One particularly interesting feature of the SECR analysis is that the detectors (i.e., the observers) are considered to have the same location, and—for traditional SECR models— the detectors must be spatially discrete in order to estimate the detection function. Here it is shown that this is not a requirement for SECR, so long as spatial information is available from another source—in this case the estimated distances.
In order to ascertain the effect of modelling the error in the estimation of distances, a 3
This analysis was carried out in an outdated version of admbsecr. At the time of writing, admbsecr
no longer supports detector-specific detection function estimation. This may be reimplemented in the near future.
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Difference from true density (%)
SECR with
distances MRDS SECR withbearings SECR SECR withSS, TOA with SSSECR with TOASECR SECR Model type
B. acutorostrata N. annamensis A. lightfooti
Figure 2.5 Boxplots showing estimatedB. acutorostrata,N. annamensis, and A. lightfooti densi- ties from the simulated data sets for (i) the SECR and MRDS models (forB. acutorostrata), and (ii) the SECR models that do and do not incorporate auxiliary observed data (for N. annamensis
andA. lightfooti). Estimates are shown as percentage differences from the underlying density used
to generate the data.
simulation study was carried out. In total, 500 data sets were simulated, using parameter values similar to those estimated from the real data. Each was analysed using both an SECR model and an MRDS model (averaging the estimated distances for individuals detected by both observers). Both SECR and MRDS whale density estimators showed slight positive bias; however, the former had lower variance, and therefore also a lower mean-squared error (MSE; Figure 2.5, Table 2.1). The bias is suspected to be small-sample bias, as six parameters (whale density, distance measurement error, and two detection-function parameters for each observer) were estimated from detections of only approximately 70 animals, on average.
2.4.2 Northern yellow-cheeked gibbon
In 2010, an acoustic survey ofN. annamensis was conducted by Conservation International in the jungles of northeastern Cambodia. Three observers were stationed in a line, 500 m apart, and recorded detections of groups of calling gibbons. For each detection, the observer in question recorded an estimated bearing to the gibbon group. See Kidney et al. (in submission) for further details.
Table 2.1 Estimated biases, variances, and root mean square errors from the whale and gibbon simulation studies. All are given as percentages of the true underlying animal density used to generate the data.
Species Model type Bias (%) SD (%) RMSE (%)
B. acutorostrata SECR with distances 8.23 27.07 28.29
MRDS 10.79 30.80 32.63
N. annamensis SECR with bearings 1.56 25.88 25.93
SECR 4.29 86.76 86.86
A. lightfooti SECR with SS, TOA 0.14 6.15 6.16
SECR with SS 0.15 6.76 6.76
SECR with TOA −0.02 6.75 6.75
SECR −6.93 7.39 10.13
formation, resulting in an estimated gibbon group density of 0.319 groups km−2 (with a standard error of 0.074). A model that ignores the estimated bearings generates an estimate of 0.8290 groups km−2 (with a substantially larger standard error of 0.367).
A simulation study was carried out to investigate the effect of the incorporation of estimated bearings on the animal density estimator. As before, 500 data sets were simu- lated, with parameter values set at values similar to those estimated from the real data. Introducing the use of estimated bearings considerably reduces estimator variance (Figure 2.5, Table 2.1), and this due to the additional spatial information they provide (Figure 2.6). See Kidney et al. (in submission) for a rigorous analysis of these data, and further related methodological developments particular to the estimation of gibbon group density via SECR.
2.4.3 Cape Peninsula moss frog
See Section 3.5.1 for a detailed description of the A. lightfooti survey data. Briefly, six microphones were placed within a montane seepage at various sites, and male advertisement calls were recorded. The microphones were connected to a central audio recorder, allowing for the collection of relative TOAs of each detected call across the microphone array. Signal strengths were also observed for each detection. See Chapter 3 for further details.
Developing further methodology for the analysis of data from these surveys provides motivation for large portions of Chapter 3, where a rigorous treatment of the analysis of data from such a survey can also be found. Results from models fitted to these observed data are therefore not shown here, but this particular application is nevertheless mentioned
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With estimated bearings Without estimated bearings
Figure 2.6 Estimated conditional PDFs of a detected group’s location—both with and without incorporation of its estimated bearings. Arrows show the estimated bearings from from observers that detected the individual. It is estimated that the individual is located within the outermost contour of each set with a probability of 0.8, and is located within the innermost contour of each set with a probability of 0.1.
in order to further highlight the flexibility of the model class discussed.
Once more, a simulation study was carried out to investigate the effect of the use of the auxiliary information—in this case, the signal strength and TOA data. Incorporation of these data again was shown to reduce estimator variance (Figure 2.5, Table 2.1).
The models that did not incorporate the signal strength data fitted a half-normal detec- tion function; however, the data were generated using the signal strength detection function. This therefore constitutes a model misspecification, and likely is the reason for the observed bias in the SECR model that did not use any auxiliary data. It is interesting that this bias was virtually eliminated following the incorporation of TOA data.
A separate simulation study (not shown here for brevity) indicated that this bias was also eliminated by the use of the threshold detection function when neither signal strength nor TOA information were available.