# Ellipses

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).

## Contents

## Introduction

Ellipses are plotted using the original Jennrich & Turner (1969) approach. Assuming a bivariate normal distribution of the locations, their variance and covariance are used to estimate their density about major and minor axes centred on the arithmetic mean coordinates. The major axis is inclined at ? degrees to the horizontal. A measure of distribution asymmetry (JTasym) is the standard deviation of the major axis divided by that of the minor axis.

The Ellipses options menu is the same as for convex polygons, except that it is unnecessary to choose a range centre.

## 99% cores

Whereas convex polygon cores are based on shapes that include the required percentage of locations, ellipses enclose a proportion of the bivariate normal density distribution. They may therefore include more or less than the appropriate percentage of locations. The outermost ellipse is estimated to include 99% of the density distribution because the 100% ellipse would be at infinity.

If you choose to file data, you will be offered the option of filing edges and of creating another file containing the distances from a selected range centre to each location. The distances are in column format, suitable for input to a spreadsheet. Each row contains the distance preceded by the 7 range variables and followed by the ellipse statistics. This is a .csv file with column headers that can be double-clicked to open in Microsoft Excel or imported to an alternative spreadsheet.

## 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. N.B. if the range is multinuclear (i.e. has more than 1 core area) the home range is best described by cluster analyses or by contours. 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.

In the **Output Files** column you can specify a range areas and statistics output file. 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. Structure statistics include the ellipse centre coordinates, the radius of the major and minor axis for the maximum core size, the inclination, the x-variance, y-variance and covariance, the correlation coefficient and the asymmetry ratio. This is a .csv file with column headers that can be opened in Stats Viewer, double-clicked to open in Microsoft Excel or imported to an alternative spreadsheet.

## 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 cores are saved at 5% intervals, from 20-99% (because 100% of the distribution cannot be estimated), a total of 17 sets.

Utilisation files can be opened on the main window where the plot will be displayed. Note that ellipses have a smooth utilisation distribution plots (which are of little use) because they are based purely on a normal distribution.

## 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 consecutive area estimates have to be saved to an output file, so that the result can be examined in the main window.

The smoothing entailed in ellipse estimation leads to rapid achievement of an asymptote in incremental analyses, often with as few as 10 locations. However, if several successive locations add to a core area following a number of outliers, the estimated ellipse areas may fall after the initial increase.