# Kernel Contours

The pros & cons of different analysis techniques are discussed in detail in the Review Of Home Range Analyses and for a more comprehensive recent review, see "A Manual for Wildlife Radio Tagging" (Kenward 2001) and Kenward et al. (2001).

## Introduction

Kernels and Harmonic means are both contour analyses which estimate density indices for locations at intersections of a matrix set across an animals range, and then interpolate contours between the values of that matrix.

(Worton (1989) noted that the Harmonic Mean estimator was one of a family of Kernel estimators and that the bivariate normal estimator (among others) could handle d=0 without special treatment. The bivariate normal kernel used by (Worton (1989) uses a negative exponential function of location distance from estimation points for estimating location density indices. As exp(0)=1, this function has no problem handling locations that coincide with estimation points. However, the function is also inherently more smoothing than Harmonic Mean contours. This makes it more appropriate than Harmonic Mean approaches for obtaining stable estimates of range size with small samples of locations, but less precise for estimating cores to be used in analyses of habitat content and sociality, especially when there are distant outliers.

Smoothing of kernel analyses involves estimation of a smoothing parameter (h), sometimes called the reference bandwidth or window width ((Seaman & Powell 1996). (Worton (1989) pointed out that his reference h might overestimate range areas by over-smoothing the distribution when ranges are strongly multimodal. He suggested the use of a fractional multiplier of h, and proposed estimating this smoothing factor by Least Squares Cross Validation (LSCV) of the mean integrated square error. See below for an explanation of how you can alter the smoothing parameter.

As for Harmonic Mean estimation, the contours can be based solely on the mean and variance of the density index distribution across locations, which is most appropriate for range size estimates, or fitted to selected proportions of the locations, which may better define cores for subsequent analyses of sociality and habitat. The matrix size can be varied from its default of 40x40, or frozen to prevent re-scaling. Matrix size does not influence the kernel estimates greatly. However, the expansive nature of outermost kernel contours can result after re-scaling in ranges covering only 50% of the matrix, so selection of matrices larger than the default 40x40 may help to obtain smooth contours across multimodal distributions.

The majority of options for kernel contours are exactly the same as those for harmonic mean contours except Kernel type and Smoothing which replaces whether to centre locations between matrix intersections.

## Locations Kc and statistics

The density index can be estimated at each location. This enables estimation of statistics for the density distribution, and also provides values for interpolating contours either to the locations or based on location density alone. The Locations, Kc & statistics option runs more quickly than other options in contour analyses because the routine does not have to calculate a matrix of location density values.

The statistics produced can give a useful insight into the structure of the range based on the distribution of the locations. A range statistics output file can be specified in the Output Files. The statistics include range spread and the density score at the central fix, with the dispersion, skewness and kurtosis of the density distribution (Spencer & Barrett 1984).

## Selected cores

This option allows you to examine range structure and to define core areas. By excluding low density areas the edges enclose areas most used by the animal. See the introduction to Location Analyses for more details.

You can choose one or more values for the percentage of locations or of location density to be included. Type them in ascending order, separated by either spaces or commas.

The display shows coordinates of the Kc (Kernel centre) and its distance from the focal site.

In the Output Files column you can specify a range areas and statistics output files. The estimates are in column format, suitable for spreadsheets. Each row has the 7 range variables, followed by X,Y coordinates for the range centre, followed by 5 range statistics followed by as many areas as there were core percentages. The statistics include the Kc coordinates, and the spread, dispersion, skew and kurtosis of the location distribution (Spencer & Barrett 1984).

## Cores at 5% intervals

This option provides plots which help to decide which locations are part of a core, and which are outliers.

You can choose to save both edge (polygon) and utilisation files the latter can be plotted in the main window. Density analyses such as this save edges from 20-99% (because 100% of the distribution cannot be estimated). The cores are saved at 5% intervals, a total of 17 sets.

The analysis produces a results screen which shows a map of the range edges, a table showing the area and % of total area of each % edge, and a graph of utilisation distribution.

Core ranges

Graphs of utilisation distribution can be used to assess (by eye) whether a sharp discontinuity in area, after the elimination of outlying locations, indicates a core range. Better estimation may be possible if the utilisation file is saved and plotted in the main window.

## Incremental area analysis

Incremental area analysis is used to answer the question "how many locations do I need to estimate a home range?" Starting with the first three locations (the minimum needed to estimate a polygon area without a boundary strip), the new area is estimated as each location is added. This permits the consecutive areas, which tend to increase initially as the animal is observed using different parts of its range, to be plotted against number of locations until there is evidence of stability, which indicates that adding further locations will not improve the home range estimate. The default is to plot the edge round all the locations that have been added, but it is also possible to choose a single, smaller core. The results are saved to a .inc file which can be examined using opened in the main window.

## Kernel Type

When Worton (1989) proposed the use of the bivariate normal kernel estimator, he noted that the smoothing parameter (h) can be adjusted in several ways. One approach is to vary h locally across the matrix, by weighting initial values to produce an "adaptive kernel".

#### Fixed kernels

These use the reference smoothing parameter (bandwidth), which is the standard deviation of x and y coordinates divided by the sixth root of the sample size.

Worton (1989) suggested weighting by the inverse of the initial density index, to emphasise the tail of the distribution. Subsequent reviews show this to overestimate range sizes (Worton 1995b, Seaman & Powell 1996).

#### Core weighting

An alternative is to core-weight the estimates, which tends to de-emphasise the tail of the density distribution. However, it is probably best to base estimates on the default, fixed value.

## Contours

#### Contours based on location density alone (default)

In this option, contour plots are based solely on the mean and variance for the distribution of density indices across the locations. The contours estimate the probability of including a particular proportion of locations and may include more or less than the proportion of locations actually recorded. As in ellipse analyses, the outermost contour is estimated to include 99% of the location distribution, because the location density at 100% would be infinitely small. This classic approach is most appropriate for estimating stable range sizes, typically to include 95% of the density distribution, for which reason it is the default in Ranges 6.

#### Contours fitted to locations

When contours are fitted to locations, the density index values at locations are ranked. Contours are then plotted to just include a given percentage of the locations. This approach is most analogous to the polygon approach. For example, it puts the 100% contour through the outermost location. There tend also to be irregular gaps between contours fitted to locations, which simplify detection of a core by inspection of utilisation plots and may define cores best for analyses of sociality and habitat content.

## Smoothing multiplier, hRef and LSCV

#### hRef

The reference smoothing parameter (hRef) is the standard deviation of rescaled x and y coordinates divided by the sixth root of the number of locations.

#### Fixed multiplier

As the reference smoothing parameter tends to overestimate range areas, it can be multiplied by a fractional value (Worton 1989), in which case the contour areas from bivariate normal kernels tend to approach those of Harmonic Mean contours. Ranges allows multipliers between 0.1 (which often produces tight contours round locations) and 2.0 (which smooths contours highly).

#### LSCV (Least Squares Cross Validation)

LSCV provides an objective way to find a multiple of hRef. There are a number of potential ways of implementing LSCV. In Ranges, the routine starts with a multiplier of 1.51 and works downwards in steps of 0.02 to 0.09. The default implementation [LSCV Inflection] stops if it reaches an inflection, at which a decreasing downward slope either becomes an upward slope (indicating a local minimum) or increases again in a downward direction (indicating that a local minimum would have been likely with a much smaller step size than 0.02). [Ranges also supports LSCV Local minimum (in which case inflections other than true minima are ignored) and LSCV Global minimum (in which case the minimum that gives tightest smoothing is used). These options are denoted by LSCVI, LSCVL and LSCVG respectively.]

If no minimum or inflexion is found, the reference value (*1) is used. Thus, a multiplier of 1 is an indication that the routine has failed ( the start and interval value used for the local minimum search means that it can find an optimal multiplier of 0.99 or 1.01 but not 1 itself ).

If you have chosen to use least squares cross validation the message "OPTIMISING THE SMOOTHING CONSTANT" appears while the programme is running and the value of the multiplier is displayed.

LSCV can be applied on a range-by-range basis, but that makes the size of individual ranges dependent not only on the area covered by the locations but also on how they are distributed within that area (Kenward 2001). An alternative approach is to use all values that are not 1 (the reference value) to estimate a median fraction for the multiplier, and then apply that multiple of hRef to all the ranges. The value of this multiple should be stated when describing the analysis. To do this first run an LSCV analysis with a single selected core, and ‘Output stats and areas file’ selected, the multipliers will be in the xhRef column, then repeat the analysis selecting [[#Fixed multiplier|Fixed Multiplier] and entering the median value you have calculated.

LSCV tends not to work well with fewer than 30 locations or for data with large resolution relative to range size (Seaman et al. 1999). If a minimum is found only for very few ranges in a set, it may be better to use the reference value (hRef * 1), or to adopt a value that another study has found to be appropriate for the species.

## Matrix size

#### Matrix, set no. of cells

Contouring is most detailed when there are small distances between intersections of the estimation matrix. The default in Ranges is a 40x40 matrix, because this gives rapid runs, little change in definition for larger matrices and comparability with RangesV. However, Ranges lacks the memory constraints of previous versions and matrix size can be increased up to 200.

#### Matrix, set size of cells

This new option allows the size of matrix cells, rather than the number, to be set by the user. This allows you to retain the same plotting resolution for home ranges of very different size (perhaps predator and prey), but may result in somewhat coarse plots for the smaller home range. If you set the size of cells such that the resulting matrix would be greater than the maximum of 200 cells, cell size will be set to create a 200 cell matrix and the warning ‘Matrix interval enlarged’ will be displayed at the end of the run. The size of the matrix cells is displayed in the ‘interval’ column in the statistics output file.

## Matrix rescaling

Contours tend to extend beyond the outermost locations, especially when based on location density alone. To plot such contours, the matrix is set to extend beyond the locations. In Ranges, the default is to set the locations to span the central 70% of the matrix. If an initial estimation of density at grid edges indicates that the outermost contour will still extend beyond the matrix, the proportion of the matrix spanned is decreased automatically in steps of 5% until a fit is likely. Re-scaling of the matrix is prevented by selecting the Freeze matrix option. This option is not available if ‘Matrix, set size of cells’ is selected.

Differences that have been noted in contour estimates between different software packages (Larkin & Halkin 1994) are likely to depend partly on aspects of the matrix, such as whether the quoted size includes or excludes a "contour-completion" boundary. For comparability between packages, estimation conditions must be set carefully. For example, if you want a 25x25 grid across the locations in Ranges, where the proportion of matrix spanning the locations is by default 70%, you should select a 36x36 grid and freeze it.