In this Chapter we introduced the Random Encounter Model developed by Rowcliffe et al. (2008). The main focus of the Chapter has been on the development of a maximum likelihood framework, which can later be extended, to estimate density of unmarked an- imals from camera trap data in REM.
We started by explaining the ideal gas model for which REM is based, and gave the derivation of REM by Rowcliffe et al. (2008). The ideal gas model is a model for col- lision rates or encounters, which depends on three things: the size and speed of the particles and their density (Yapp, 1956). REM uses this concept to determine the rate of encounter between animals and camera traps. The encounter rate in REM is a function of the dimensions of the camera, the density and an estimate of animal speed of movement, which is treated as a fixed value. In this Chapter we have illustrated the maximum likelihood REM formula using a small data set from Whispsnade Wild Animal Park, and the results were compared with the results from Rowcliffe et al. (2008).
Rowcliffe et al. (2008) used the total encounters divided by the total number of camera trapping days and multiplied by a constant term, which includes the average speed and the dimensions of the camera trap. This formula gives the equivalent of a maximum likelihood estimate with an underlying assumption of a Poisson model, which we have shown. To estimate the variance of the estimated density Rowcliffe et al. (2008) used bootstrapping by resampling camera locations with replacement and taking the variance of a large number of resampled density estimates. Our approach is different from Row- cliffe et al. (2008) in that we modelled the encounter data using a Poisson REM while assuming a fixed value of animal speed as in Rowcliffe et al. (2008). We have estimated the variance of estimated density using four methods: inverse of the negative Hessian matrix, which comes from the optim function for minimizing the negative log-likelihood in R; an adjusted variance method, which uses Taylor expansion to approximate the variance; bootstrap method, which involves resampling the animal speed data and en- counter data, and the direct standard error method.
Testing REM using the WWAP data Rowcliffe et al. (2008) found that the estimated densities for three of the four species (wallaby, water deer and muntjac) did not differ significantly from, and were within 22% of the census results. The underestimation of the mara species was a result of nonrandom placements of camera traps in areas where the mara frequented during the survey period. The estimates of the density from the maximum likelihood framework and the density from the census differ substantially since the model does not account for the variability in the density in heterogeneous habitats. Also, the current REM formula does not account for the variability in animal speed, which would have an effect on standard error estimates. Estimates of the stand- ard error based on the direct method and the Hessian of the log-likelihood function are the same, as expected, and are guaranteed to be valid asymptotically, as the sample size becomes infinite. However, the standard error estimates are biased since the sample size is small, as well as possible non-normality of the sample. The confidence intervals based on the Hessian are also narrow and do not capture the density from the census within an approximate confidence for all species. The standard error estimates based on the adjusted variance method and the bootstrap method are comparable, and the density from the census is captured within an approximate 95% confidence interval. The variability in the speed is huge, and the standard error estimates of the estimated density from the adjusted variance method is dominated by this uncertainty, which res- ulted in wider confidence intervals. The bootstrap method also allows for the variability in the speed as well as the variability in the encounter data. The advantage of using the adjusted variance method and the nonparametric bootstrap method for estimation precision is that they do not assume any distribution for the data. However, it is worth noting that the adjusted variance method and the direct method are limited in their use and cannot be used to estimate standard errors of the density for more complex models. For example, it may be possible to use the adjusted variance method for the Poisson REM with habitat, but it may not be easy to do so.
We have considered including habitat as a covariate in REM to determine whether it would have an effect on estimated density. We found that for the mara species the estimated mean density across habitats increased but the increase is nowhere near the
density from the census. For the muntjac and wallaby species the estimated mean dens- ity improved substantially when habitat is considered as it is closer to the density from the census, and the density from the census for the muntjac species is captured within a approximate 95% confidence interval based on the Hessian of the log-likelihood function and the bootstrap method. For the water deer species the mean density is overestimated and the bias is larger compared with the bias when habitat is not included in the model. The mean density from the census is also not captured within an approximate 95% con- fidence interval based on the estimates from the Hessian and the bootstrap method. It is worth noting that the adjusted variance method is limited in its flexibility. While it may be possible to estimate variance within and across habitats using the adjusted variance method, it may not be easy to do so.
The method adopted by Rowcliffe et al. (2008) for estimating density of individuals moving in groups, from camera trap data, is limited to the case that group sizes are small and counts of the number of individuals in each group is possible. There is an inherent difficulty with camera traps in that it is not possible to measure true group size. Also, if the group size is dependent on the detection distance, there would be dif- ficulties in obtaining an unbiased estimate of the expected group size. This dependence arises because large groups are more likely to be detected further away from the camera, while small groups might remain undetected. Buckland et al. (2001) suggest that this phenomenon would cause an overestimation of the expected group size because too few small groups are detected (that is, they are under-represented in the sample). Another complication is that large groups near the camera would be detected but it is possible that their centres would lie outside of the detection zone; for instance, all animals in the group might not be detected given the narrow width of the detection zone near the camera. According to Buckland et al. (2001), if the centre of the group is located inside the detection zone then the count of the size of the group must include all individuals in the group, even if some animals are beyond the detection zone; and if the centre of the group is outside the detection zone, then no observation is recorded. The difficulty with this in camera trapping is that even if groups detected have their centres within the detection zone the distance from the camera to the centre of the group is unobservable. Buckland et al. (2001) suggest that a possible approach would be to replace a group of
a given size by objects with the same size of the group at the same distance. In this case there would be no need to estimate mean group size. But the issue of violation of the independence assumption arises, invalidating analytical variance estimation and model selection procedures. However, if robust inference methods for variance estima- tion are adopted this difficulty can be avoided, but the issue of model selection remains. This approach would require exploration, and remains an avenue for future research in camera trapping analysis.
We have also shown that varying the time scale in aggregating the data from camera traps (that is over 1 day, 2 days or a week) has no effect on estimated density and its standard error. In this Chapter we have organized the encounters data in the form per camera per day to facilitate later modelling which would include covariates such as habitat and the random location of camera traps.
We have concluded that REM is relatively accurate but precision is dependent upon the method of variance estimation used. Therefore, we consider an integrated likelihood approach (iREM), which is comprehensive as it combines all of the data sets in one coherent framework, accounting for the variability in the observed data. This iREM approach is a more general and flexible approach in estimating density and its standard error. We could easily include covariates such as habitat or weather, and other inde- pendent variables required to estimate density into the model. iREM is discussed in Chapter 3. This approach corrects the precision, and allows for the accurate treatment of correlation in the estimators.
Chapter 3
integrated Random Encounter Model (iREM)
In this Chapter we develop an integrated likelihood method to estimate animal density. The integrated Random Encounter Model (iREM) builds on the Random Encounter Model (REM) discussed in Chapter 2. Rather than using a fixed estimate of animal speed as REM does, iREM simultaneously models the encounter data and animal speed data in one coherent framework. iREM utilizes an integrated population modelling (IPM) approach to estimate animal density, which is discussed in Section 3.1. The Chapter then continues with Section 3.2, which provides a description of the model. In Section 3.3 and Section 3.4 descriptions of the parametric distributions used to model animal speed of movement and the encounters, respectively, are given. Some species generally move around in pairs or family groups, so we also show how iREM can be extended to include group data in Section 3.2, and the model for the group data is given in Section 3.5. Examples of the likelihood function are given in Section 3.6. The performance of the models is tested via simulations in Section 3.7. To conclude the Chapter, Section 3.8 provides an illustration of the application of iREM using the Whipsnade Wild Animal Park (WWAP) data set.
3.1
Integrated Population Modelling (IPM)
Demographic and survey information at the population and individual levels are often simultaneously available when monitoring wildlife populations. The information col- lected is often analysed separately or in isolation, and separate results are presented.
For example, a survey may be designed to provide information on population abund- ance, while another survey may be designed to provide information about survival for a specific life stage. These surveys, however, may provide overlapping information about demographic processes or abundances; and analyses that utilize that overlap would be more powerful, and would provide more information, than multiple piecemeal analyses. One such approach which utilizes overlapping information is an integrated population model (IPM) (Newman et al., 2014, Chpt. 9; McCrea and Morgan, 2014, Chpt.12). Integrated population modelling provides a single, coherent analysis framework for a range of data sets collected from different surveys, all relating to the same species. By combining all sources of information in a single analysis, integrated population models simultaneously describe all the data, and consequently generally result in more precise parameter estimators (McCrea and Morgan, 2014, Chpt. 12). Early work on integrated population modelling was done by Fournier and Archibald (1982), who presented the idea for fishery data. Fournier and Archibald (1982) developed a flexible model to in- clude extra information regarding the aging procedure of a fishery. Since then there have been several developments in that area. In fisheries research, the approach is termed integrated analysis (McCrea and Morgan, 2014, Chpt. 12).
Besbeas et al. (2002) developed an integrated analysis of different types of census and demographic data on animals of the same species. They devised a state-space model forming a combined likelihood for census data and data on survival from ring-recovery, under the assumption that the data sets are independent. The likelihood is formed by means of the Kalman filter, using appropriate normal variables to approximate Poisson and binomial random variables. By maximizing the combined likelihood the paramet- ers estimated provided a simultaneous description of both data sets, and parameters such as productivity, which could not be estimated from the data sets separately were estimable under the combined likelihood framework. But to overcome a potential de- ficiency in combining likelihoods that are formed using specialist computer programs, which is an obstacle to the joint analysis, Besbeas et al. (2003) suggested a multivariate normal approximation, which was evaluated on data sets of two birds species, lapwings and herons, and which has been shown to be efficient and accurate. Extending this work Besbeas et al. (2005) adopted a multivariate normal approximation for the form
of the likelihood of the survival data, making use of parameter estimates and corres- ponding estimates of error obtained from analysing the survival data alone. In this case, the particular computer programs or packages need only be run once, to obtain maximum-likelihood estimates of the relevant parameters, and of their standard error and correlation. Schaub et al. (2007) argued that these integrated population mod- els, however, have been applied to species without the lack of demographic data, so Schaub et al. (2007) have demonstrated the flexibility of integrated population models to estimate demographic parameters from sparse data, with a relictual colony of greater horseshoe bats (Rhinolopbus f errumequinum). Schaub et al. (2007) applied a Bayesian integrated population modelling approach to the data and found that if the data were analysed separately, they would not have been able to estimate fecundity, the estimates of survival would have been less precise, and the estimate of population growth would have been biased.
As described above integrated population models have the advantage over the piece- meal approach of estimating parameters that are otherwise inestimable, and obtaining more precise parameter estimates. Cole and McCrea (2016) for example, demonstrated that aside from the natural advantages of improved precision of parameter estimates and reduced correlation, integrated population models have the additional advantage of making it possible to estimate some parameters that were not estimable from modelling the data individually.
There are some problems, however, with the use of integrated population models. For instance, as in the case of Besbeas et al. (2002), the specialist computer programs or packages in which the separate component likelihoods are constructed and combined would be an obstacle to the joint analysis. Also, integrated population models rely on the assumption that different data sets are independent, which is frequently violated in practice (Abadi et al., 2010). Besbeas et al. (2009) for example, showed that the danger of combining recovery information with dependent census data is increased root mean square errors compared with the case of combining with independent census data. But Abadi et al. (2010) used simulation methods to assess how the violation of the assump- tion of independence affects the statistical properties of the parameter estimators. They
found that this violation had only minor consequences on the precision and accuracy of the parameter estimates.
This thesis is particularly interested in estimating animal density and its variability correctly. The integrated Random Encounter Model (iREM) developed in this Chapter uses an integrated population modelling approach, which combines the speed data and the encounter data in a single framework. The iREM method is advantageous over piecemeal approaches as it accounts for the sampling variability of the estimator of animal speed of movement. iREM also allows for accurate treatment of precision and correlation in the estimators. The encounter data and animal speed data at WWAP were collected from separate sources, therefore, they are considered statistically independent, so their contributions to the likelihood could be multiplied. The next section describes the iREM method used to estimate animal density.