## Introduction

Analyses of animal resource selection functions (RSF) using data collected from relocations of individuals via remote telemetry devices have become commonplace. Technological advances, however, have produced statistical challenges in analysing such highly autocorrelated data. There have been several recent proposals for making statistical inference for RSFs from autocorrelated data. Most of these proposals have been variations of the weighted distribution (Patil 2002) approach (McDonald, Manly & Raley 1990; Lele & Keim 2006; Christ, Ver Hoef & Zimmerman 2008; Johnson *et al*. 2008; Forester, Im & Rathouz 2009). However, see Hooten *et al*. (2010) and Hanks *et al*. (2011) for alternatives to the traditional weighted distribution. Johnson *et al*. (2008) illustrates that the weighted distribution approach generalizes the notion of resource selection, typically defined as habitat use relative to the proportion of habitat available, by considering observed spatial locations as ‘habitat’ with availability governed by a movement model and the previous location of the animal. See Aarts, Fieberg & Matthiopoulos (2013) for additional exploration of the notion of availability in RSF studies. The weighted distribution approach makes inference on selection by modelling animal use as a function of a movement model and an RSF. There are three drawbacks to the weighted distribution approach, however. First, these methods can be computationally challenging due to an intractable normalizing constant. Second, there is a dearth of software available to animal ecologists for fitting these models. Finally, selection can only be inferred at the level of individual movement. The weighted distribution models are designed to model selection of the animal's next location given the current location. There is no straightforward way to aggregate over time and model selection relative to the entire study area while simultaneously accounting for temporal autocorrelation. This may be desirable as often location times are of no biological interest because they are a function of the sampling rate programmed into the telemetry device by the researcher. Throughout the remainder of the paper, we will refer to aggregation over time to mean ignoring the times of observation and modelling only the spatial location of the points.

Spatial point processes have recently been proposed for general analysis of presence–absence data (Warton & Shepherd 2010; Aarts *et al*. 2012). Warton & Shepherd (2010) proposed the spatial Poisson process as a solution to the ‘pseudo-absence problem’ that also appears in resource selection studies. The pseudo-absence problem arises due to the fact that one can often determine which spatial locations an animal has used, but it may be impossible to know which locations were *available* for the animal to use and were passed over for the selected locations. Early resource selection methods recognized this fact and used a researcher-selected sample of locations (the *availability sample*) to compare with known used locations (see chapter 5 of Manly *et al*. 2002). Controversy concerning the interpretation of the estimated parameters ensued due to the fact that inference was based on how the researcher and animal each selected their locations (Keating & Cherry 2004; Johnson *et al*. 2006; Beyer *et al*. 2010). Warton & Shepherd (2010) and Aarts *et al*. (2012) illustrated, however, that availability samples could be interpreted as quadrature points for approximating integrals within RSF likelihoods. Therefore, there is no case–control interpretation for the RSF with respect to use and availability, and we are free to select our availability points deterministically to best approximate the point process likelihood. While the point process likelihood presented by Warton & Shepherd (2010) and Aarts *et al*. (2012) eliminates the availability sampling problem for analysis of resource selection based on a landscape snapshot of use, telemetry data can contain substantial temporal autocorrelation that is not accounted for in spatial point process models previously used. Therefore, we propose the use of *space–time* point process models to account for this high autocorrelation.

By using space–time point process models, we take a conceptually different approach to analysing animal telemetry data for making RSF inference. Instead of considering the locations to be essentially a bivariate time series, as is the case with most movement-based weighted distribution models, we consider the telemetry data to be a realization of a space–time point process. The space–time point process models event occurrences (e.g. animal locations) in three dimensions: latitude, longitude and time. Under the space–time point process paradigm, the times of the relocations are also considered to be random rather than fixed. We view this as a more realistic model as there is often a random component to when telemetry locations are received. Even when devices are programed to record (and transmit) locations at specified times, there are usually some locations that will not be received by chance. These random components may be due to hypothesized sources such as animal state (e.g. sleeping) or habitat occupied (e.g. dense forest); however, this information is often not known to the researcher, thus a random process model of some kind is appropriate. Another benefit of using space–time models for telemetry data is that there is a theoretical basis for aggregating data over time to examine study area level selection. In fact, we will show that space–time point process models are a generalization of the weighted distribution models that Johnson *et al*. (2008) propose. Thus, even starting with a different motivation, we end up back at essentially the same place. However, by adding models for the arrival of location times, we can overcome the three main drawbacks of the weighted distribution RSF telemetry models: computational burden, lack of available software, and inability to aggregate over time.

We begin our discourse with a review of the weighted distribution approach and transition to development of space–time point process models and demonstrate that by modelling random times of location, we obtain a space–time point process model from the weighted distribution model. Following this, using the marginal intensity of the locations in a space–time point process, we will develop some methods for mitigating temporal dependence in locations when using space-only Poisson process models for RSF inference (Warton & Shepherd 2010; Aarts *et al*. 2012). For both versions we provide numerical approximations that allow model fitting with standard or readily available statistical software. Finally, we demonstrate space–time and space-only RSF inference by performing each analysis on a data set of northern fur seal (*Callorhinus ursinus*) telemetry locations in the Bering Sea off the coast of Alaska. In the fur seal analysis, we illustrate that the basic models developed earlier in the paper can be readily extended to accommodate more complex movement such as central place foraging.