# Harmonic Mean 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

Harmonic means and kernels 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.

In Harmonic means, density indices are estimated as reciprocal functions of distance (1/r) to all the locations at matrix intersections. The very small value of 1/r for distant locations gives them a very small contribution to the Harmonic mean function (S1/r)/n, so the function emphasises local density. However, a location that occurs at the point of estimation makes an infinite contribution. To avoid this problem, the original implementation (Chapman & Dixon 1980) set r=1 (and hence 1/r=1) for r<1.

This solution creates a dependence of the contours, and hence the area estimates, on the scale used for the location coordinates relative to the matrix size used for contouring. In Ranges the calculations are performed in units of tracking resolution. As long as the size of each matrix cell is not greater than half the tracking resolution, locations will never be more than 1 matrix interval from an intersection and many 1/r values are 1. This smooths the distribution across the matrix and produces acceptable contouring. However, if matrix cells are much larger than half the tracking resolution (due to a small matrix, or small tracking resolution), harmonic mean values tend to be very small at intersections unless there is a nearby location. This results in an inadequately smoothed distribution and tends to produce contours as rings round locations.

This problem of scale dependence can be solved in two ways. One is to enlarge the number of matrix cells so that intervals are no more than half of the tracking resolution. The large matrices required are practical in the memories of modern computers, but estimation is slow. It is critical for tracking resolution to be registered correctly if this option is chosen. If the matrix is too small relative to the tracking resolution you will be presented with the warning ‘Matrix interval enlarged’ at the end of the run (more details in Location centring). The second approach is to treat all locations as if are at the centre of a square formed between the 4 nearest matrix intersections. The pros and cons of this location centring are discussed below in more detail. However, the large matrices (and hence slow runs) required for robust analyses should not deter use of Harmonic Mean estimates. The use of reciprocal distances gives less dependence on distant locations than with bivariate normal kernels, and thus a better fit of contours to range cores as well as less sensitivity to outlying locations.

## Locations Hc 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, Hc & 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 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 Hc (Harmonic mean location) 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 Hc 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.

#### 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 .

## 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.

#### 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

## Location centring

#### Unmodified locations

Relatively invariable estimations can be obtained in Harmonic Mean contouring if the intervals between intersection of the density matrix are no more than half of the minimum distance between locations. Analyses in Ranges are based on units of tracking resolution, which effectively relates scale to range size. In this way, a species with ranges that span up to 10km and tracking resolution of 100m has the same number of resolution units across the largest ranges (100) as a species tracked with 1m resolution over ranges that span 100m.

As the analysis with unmodified locations is the more robust treatment of locations (Kenward 2001), this is now the default for Ranges. The default matrix for the option is set to 150x150. If this results in any range having less than 2 matrix cells per unit of tracking resolution the warning : Matrix interval enlarged will be displayed ( and the resulting contours are likely to be under-smoothed, fitting very tightly to the locations). If this happens it is advisable (a) to check that the resolution value is appropriate (was the technique for a large animal really accurate to 1m?) and, if so, (b) to run that range with an appropriately large matrix.

The maximum recommended range spans (max. distance in either the N-S or E-W directions) for a grid size of 150 are 210m, 2.1 km and 21 km for tracking resolutions of 1m, 10m and 100m respectively. ( Maximum recommended range span = 0.7 * matrix size * tracking resolution * 2, the 0.7 allows for the fact that contours may extend beyond the span of the locations ).

(Note that the same warning message is also given if the size of matrix cells is set such that the resulting matrix would be greater than the maximum of 200 cells).

#### Centred in matrix squares

The approach of Spencer & Barrett (1984) was to treat all locations as if they were centred between the 4 nearest matrix intersections, such that 1/r values became functions of the intersection intervals. This removes the scaling problem, but instead makes smoothing decrease with increase in matrix size, such that the area estimated for the same range on a 100x100 matrix can be less than half that on a 40x40 matrix.

## 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. However, Ranges lacks the memory constraints of previous versions and matrix size can be increased up to a maximum of 200. Larger matrices than 40x40 are advisable, and are set by default, for some Harmonic Mean analyses. See Location centring for dependency on matrix size of some harmonic mean analyses.

#### 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 (Note that the same warning is displayed if the chosen matrix has less than 2 cells per unit of tracking resolution - see Location centring for details). 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.

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.