# Convex Polygons

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

Ranges estimates convex polygons by finding the most southwest location as the first corner, then seeking the least clockwise location (i.e. the next location on the outside of the range moving in a clockwise direction) as a second corner, and then working through each location that is least clockwise to the vector from the previous corner until reaching the most southwest location again. The 100% convex polygon contains all the recorded locations (Mohr 1947), but a core territory can be defined by plotting a convex polygon around a lesser proportion of the locations. A mononuclear peeled polygon, estimated by excluding a proportion of locations furthest from a peel centre is a simple index of the area most used by an animal (Kenward 1987).

After you provide an input filename, the analysis options menu will be displayed. This menu is similar in all other range analyses. It allows you to select particular ranges from a set, to select particular types of location (defined by LQVs) within a file, to select different types of display (including use of a background map) and to file range edges, areas and structure statistics for subsequent analyses.

## 100% cores

The 100% cores option gives a rapid estimation of a convex polygon around all the locations in a range. 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. This is a .csv file with column headers that can be double-clicked to open in Microsoft Excel or imported to an alternative spreadsheet. Each row contains the distance preceded by the 7 range variables and followed by the LQVs.

## Selected cores

This option allows you to examine range structure and to define core areas. By excluding outlying locations (in linkage analyses) or low density areas (in Ellipse or Contour analyses), 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 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. Structure statistics include mean, median and maximum distances from locations to the range centre, and the span of maximum distance between any two locations. This is a .csv file with column headers that can be 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. Linkage analyses save edges from 20-100%, density analyses generally from 20-99% (because 100% of the distribution cannot be estimated). The cores are saved at 5% intervals, a total of 17 sets.

Utilisation files can be opened on the main window where the plot will be displayed.

#### 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 consecutive area estimates have to be saved to an output file, so that the result can be examined using Range Use Plots.

Note that for polygons, adding locations will result in the same size range (if the new locations are within the area where the animal has already been recorded), or a bigger range (if the new location is outside those already recorded). However, contours may decrease in size as locations are added, because there is more certainty about where the distribution lies.

## Choose Peel Centre

To estimate cores that exclude some locations, it is necessary to choose a peel centre. The furthest locations from the peel centre are excluded first.

#### Focal site

The focal site is defined in the range label of the location file. If this option is chosen and no focal site is specified within the location file, a short warning of ‘no site’ will be displayed for that range. The routine will then continue analysis on other ranges within the file.

#### Harmonic mean centre (Hc)

The harmonic mean centre is the location where the inverse reciprocal mean distance to all the other fixes is a minimum (Spencer and Barrett 1984). This provides a more robust estimator than the simple arithmetic mean, which can be estimated in an area devoid of locations.

#### Kernel centre (Kc)

The kernel centre is the location at which the Gaussian kernel estimator indicates highest density (Worton 1989). This is frequently the same location as the harmonic mean centre.

#### Arithmetic mean centre (Ac)

The arithmetic mean of all the x and y coordinates defines the "centre of activity" sensu Hayne (1949). However, its position is strongly influenced by outlying locations, and it can lie in an area devoid of locations, especially in multinuclear ranges.

#### Recalculated Ac (RAc)

This tends to focus on the area of densest locations by recalculating the arithmetic mean position after excluding each furthest location. When running this analysis you will see the centre drift as more locations are excluded.