Figure 1.
The actual points used in the analysis, selected at random within boundaries defined in the methods to conform with the specified isopleth rules, are plotted here in the upper row for data sets A, B, and C. For each set, the 20% isopleth surrounds the densest aggregation of points that appear as relatively black areas in each of the plots. UDs constructed using the fixed kernel least-squares cross-validation method for these data are illustrations in the lower row (sizes have been adjusted to provide visual correspondence—where precise estimates of the fits are given in Table 1).
Figure 2.
Kruger National Park, showing the location of the four collared buffalo used in the empirical data test of the study. The Satara and Lower Sabie regions are shown as insets 1 and 2, respectively.
Table 1.
Total Error, with Type I and Type II Errors in parentheses for manufactured data sets A–C, as a percentage of total home range size, is listed for estimates obtained using the three LoCoH methods (100% isopleths and optimal—that is error minimizing—values k*, r* and a*) and the Gaussian kernel (GK) method (95%, 99% and optimal isopleths).
Figure 3.
Type I (dotted line), Type II (dashed line) and Total Error (solid line) (percentages) associated with the construction of 100% and 20% isopleths are plotted for the k-LoCoH, r-LoCoH, and a-LoCoH methods as a function of the parameters, k, r and a respectively for the three data sets (A, B, and C).
Figure 4.
The effect of sample size on the optimal (i.e. error minimizing) value of parameters, k, r and a and total errors associated with the construction of the 100% isopleth using the k-LoCoH (solid line), r-LoCoH (dashed line), and a-LoCoH (dotted line) methods respectively for the three data sets (A, B, and C). Mean and standard error for fifteen randomly generated datasets for each sample size are plotted.
Table 2.
Comparison of our heuristic rules for choosing initial parameter values k 1, r 1 and a 1 and optimal parameter values k*, r* and a* for the manufactured data.
Figure 5.
Illustrations of UDs constructed for data set A using k-LoCoH, r-LoCoH, and a-LoCoH methods with half, actual, and twice the optimal k, r and a parameter values. The darkest to lightest areas represent ascending decile areas from the 10th to 100th percentile isopleths.
Figure 6.
Illustrations of UDs constructed for data set B using k-LoCoH, r-LoCoH, and a-LoCoH methods with half, actual, and twice the optimal k, r and a parameter values. The darkest to lightest areas represent ascending decile areas from the 10th to 100th percentile isopleths.
Figure 7.
Illustrations of UDs constructed for data set B using k-LoCoH, r-LoCoH, and a-LoCoH methods with half, actual, and twice the optimal k, r and a parameter values. The darkest to lightest areas represent ascending decile areas from the 10th to 100th percentile isopleths.
Figure 8.
Comparisons of UD constructions using an a-LoCoH estimators where the value of the parameter is â obtained using the MSHC method (see text for details), and a parametric kernel, where the smoothing parameter h is calculated using the ad-hoc method of Silverman (1986). Panels: a. collar T07 and b. collar T15, both in the Satara Region; and c. collar T13 and d. collar T16. both in the Lower Sabie Region. Black circles are GPS collar locations and the hatched shape is the exclosure in a. and b. and the ridge area in c. and d. The left figure of each panel shows the 100% isopleth in light grey and the 95% isopleth in dark grey, using the a-LoCoH method. The right figure of each panel shows the 100% kernel in light grey and the 95% parametric kernel in dark grey.
Table 3.
Comparison of the areas in km2 estimated for the four buffalo GPS collar sets of data (n points) by the 95% and 100% isopleths for nonparametric (LoCoH) and parametric kernela methods.