The probability density function, fu(X), describing relative use of a habitat with environmental covariates, X, is typically defined using:
- (eqn 1)
where fa(X) gives the relative availability of a habitat composed of environmental covariates X, in the study area and the integral applies to all possible combinations of environmental conditions in environmental space E (i.e. the multi-dimensional space represented by environmental variables). This specification (eqn 1) can be seen as a weighted distribution (Patil & Rao 1978; Patil 2002; Lele & Keim 2006), where the organism samples the available distribution fa(X) with a probability proportional to w(X) in order to obtain the used distribution fu(X). Hence, in environmental space, w(X) is proportional to the ratio between habitat use and availability. w(X) is known as the habitat or resource selection function and is most often modelled as an exponential function of covariates, , where XT is the transpose of X (McDonald, Manly & Raley 1990; Lele & Keim 2006; Johnson et al. 2008). Although rarely explicitly stated, eqn 1 assumes that, conditional on w(X), changes in fa(X) lead to proportional changes in fu(X). Assuming the parameters in the habitat selection function are conditional on fa(X), and fa(X) is uniform in geographical space (G), the likelihood function for the entire dataset can be simplified to:
- (eqn 2)
where the integral of eqn 1, originally defined in environmental space, is replaced by the integral evaluated over all of geographical space; here A is the entire study area. This integral can be approximated by evaluation and averaging of w(X) at random or regular points in geographical space. This alternative specification of the likelihood function illustrates that the habitat selection model fitted in environmental space (see eqn 1), is equivalent to a model which quantifies use in geographical space as a function of the underlying environmental conditions (i.e. eqn 2 – Aarts, Fieberg & Matthiopoulos 2012). An advantage of specifying the likelihood according to eqn 1, however, is that it becomes readily apparent that the estimated habitat selection function w(X) is conditional on what is considered to be available to the organism [i.e. fa(X)]. As will be illustrated later (see e.g. Figs 1-4), the estimated w(X) may vary drastically as a function of absolute habitat availability, even though the animal uses the same movement rules to explore and exploit space.
Figure 1. (a) Fixed-time usage of two essential habitats (simulation 1), and (b) preferential selection of two substitutable habitats (simulation 2). Grey dots represent individual organisms. Dark and light grey areas represent habitat A and B, respectively. The availability of habitat A varies from top-left to bottom-right from 90% to 10%. In simulation 1, the total number of locations in each habitat is independent of absolute habitat availability, whereas in simulation 2, the number of locations in each habitat increases with the availability.
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Figure 2. Left (a) the estimated habitat selection coefficients under the fixed-time use of essential habitats (simulation 1) and right (b) the substitutable habitat simulation (simulation 2). When the disproportionality between usage and availability is modelled using traditional habitat selection functions (i.e. eqn 3), the estimated coefficients (○) decrease as a function of availability under the fixed-time simulation (a). When use increases proportionally with availability (b), the habitat selection coefficients (○), remain constant. If the contribution of availability is allowed to vary by estimating ω (eqn 4, solid line), the estimated habitat selection coefficients remain constant.
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The likelihood function in eqn 2 is equivalent to that of a conditional inhomogeneous Poisson process (CIPP – Cressie 1993) and identical to the likelihood used by MaxEnt when an exponential function is used to model w(X) (Phillips, Anderson & Schapire 2006; Aarts, Fieberg & Matthiopoulos 2012; Renner & Warton 2013). Alternatively, one can fit an Unconditional Inhomogeneous Poisson Process (UIPP) model [Cressie (1993), eqn. 8.5.16, page 655, and Aarts, Fieberg & Matthiopoulos (2012), eqn. 4]. In contrast to the CIPP, the UIPP estimates an intercept, which relates to the mean intensity. In the UIPP likelihood, the integral in the denominator of eqn 2 is exponentiated, and must still be evaluated numerically. In both the (CIPP) and UIPP, the (relative) intensity of the point process is given by w(X). The two approaches will result in equivalent slope parameters associated with environmental covariates [see Aarts, Fieberg & Matthiopoulos (2012), Appendix A], but the conditional approach cannot estimate an intercept parameter since it cancels from both the numerator and denominator in eqn 2 (Lele & Keim 2006). We fitted models using the unconditional likelihood; such models can easily be fit as a Poisson log-linear model in most statistical software packages using the following numerical trick (Baddeley & Turner 2000; Warton & Shepherd 2010): First, a regular grid with a spatial resolution similar to the environmental data is constructed, an availability point is located at the centre of each grid cell, and for each grid cell (i) the total number of used and available locations (ni) is calculated. Next, a Poisson log-linear model is fitted to the data, where the value of the response variable is set to 0 for each availability point, but to ni for each animal location, and 1/ni were specified as ‘prior weights’ for all points in the GLM. For more details see Baddeley & Turner (2000).