## Introduction

Home range is the fundamental measure of space use by animals. It is important in determining habitat preferences (Aebischer, Robertson & Kenward 1993), carrying capacities, aspects of species-extinction susceptibility (Woodroffe & Ginsberg 2000; Brashares 2003) and underpins several ecological theories including allometric scaling correlations (e.g. Mace, Harvey & Clutton-Brock 1982; Carbone & Gittleman 2002).

Advances in computing power have allowed ecologists to use increasingly sophisticated methods to estimate home ranges, including contouring methods for estimating complex probability density distributions (Dixon & Chapman 1980; Worton 1989). Contouring methods have considerable advantages over other popular home-range estimation methods such as the minimum convex polygon. They accommodate multiple centres of activity, do not rely on outlying points to anchor their corners and are less influenced by distant points, thereby excluding unused areas and leading to more accurate depictions of space use (Fig. 1).

Kernel density estimation (KDE) is widely viewed as the most reliable contouring method in ecology (Powell 2000; Kernohan *et al*. 2001). It was adapted to home-range analysis by Worton (1989) from a technique for estimating distributions from small samples (Silverman 1986). KDE creates isopleths of intensity of utilization by calculating the mean influence of data points at grid intersections. Each isopleth contains a fixed percentage (e.g. 95%) of the utilization density suggestive of the amount of time that the animal spends in the contour. A critical component of this calculation is the distance over which a data point influences the grid intersections; this value is the smoothing factor, or *h*. The larger the value of *h*, the larger and the less detailed the final home-range estimate (Silverman 1986; Worton 1989) (Fig. 1b–d). Conversely, small values of *h* reveal more of the internal structure of a home range but undersmooth in the outer density isopleths, leading to smaller estimates and creating discontinuous ‘islands’ of utilization (Fig. 1b). As KDE is sensitive to different values of *h*, the size and shape of the home-range estimates are dependent upon the methods used to calculate *h* (Silverman 1986; Wand & Jones 1995). This raises the possibility that variation in *h* may introduce systematic variation into home-range estimation that may complicate or invalidate some inter- and intrastudy comparisons.

The two preferred methods of calculating *h* in home-range analysis are the reference smoothing factor (eqn 1) and least-squares cross-validation (LSCV) (eqn 2).

For eqn 1, the reference smoothing parameter function: *n* is the number of locations and σ is the standard deviation of the *x* coordinates, with *y* coordinates transformed throughout the calculations to have the same standard deviation (Worton 1989).

The least-squares cross validation function is shown in eqn 2:

where *d _{ij}* is the distance between the

*i*th and

*j*th points and

*h*is a value of the smoothing parameter examined.

LSCV allows *h* to be chosen so as to minimize the squared distance between the fitted surface and the target surface, integrated over the area. It creates an estimate of this by a formula (eqn 2) derived from the difference between the predicted value at each data point based on a surface fitted using all the data and on one fitted after excluding the data point. This estimate of the error is then minimized by varying the bandwidth (Silverman 1986).

Previous studies have recommended KDE as a reliable home-range estimator and LSCV as the best available method of calculating *h* while concluding that *h*_{ref} oversmoothes (Worton 1995; Seaman & Powell 1996; Seaman *et al*. 1999; Powell 2000). However, these tests have been based on computer simulation of animal locations, not on field data.

Worton (1995) expanded on analyses by Boulanger & White (1990) using simulated data to test the performance of home-range estimators. Worton concluded that kernels were more reliable and accurate than the harmonic mean method approved by Boulanger & White but cautioned that the choice of smoothing factor had a profound effect on the bias observed in the final estimates.

Seaman *et al*. (1999) expanded on Worton's analysis. They used simulated data of between 10 and 200 points from more complicated distributions to mimic animal movements more closely and to test the influence of sample size and different methods of choosing *h* on kernel home-range estimates. The precision of KDE improved to an asymptote of 5–20% bias as sample size increased to 50 data points for simple distributions and 200 with complex distributions. They concluded that *h*-values chosen using LSCV produced the most reliable estimations of the distributions, giving the lowest frequency of poor estimates when compared with *h*_{ref} at sample sizes between 20 and 200 points.

Despite concerns over the superiority of kernels (Robertson *et al*. 1998), LSCV fixed kernels are viewed as applicable in all but a few specific situations (Blundell, Maeir & Debevec 2001). Perhaps worryingly for advocates of LSCV KDE, the method's performance has been reviewed more critically by statisticians (Sain, Baggerly & Scott 1994; Wand & Jones 1995; Jones, Marron & Sheather 1996). They point out that LSCV may underestimate the value of *h* appropriate for a distribution and that variation in values of *h* chosen by LSCV (*h*_{lscv}) may be considerable compared to methods such as the ‘solve-the-equation plug-in’ which have as yet not been adapted to home-range analysis (Wand & Jones 1995; Jones *et al*. 1996; Kernohan *et al*. 2001).

Girard *et al*. (2002) used GPS data from moose *Alces alces* to test LSCV KDE. Comparing kernels made from fewer locations to those estimated from using the majority of the data, they concluded that up to 300 locations were required for home-range estimates to become accurate and that accuracy improved up to sample sizes as large as 850. As such they advocate the use of GPS telemetry to acquire adequate sample sizes for optimum accuracy.

We extended these tests to a different species and to individuals with markedly different home-range use patterns in order to explore the relationship of sampling intensity with home-range size and stability in more detail. We used four large data sets (> 3000 points) spanning 9–12 months, collected from lions *Panthera leo* (Linneaus) with global positioning system (GPS) collars.