Simulated data
Mosaic areas are already centred on actual range sizes when sample sizes are very small (five points per circle: Fig. 5A). Convex hulls consistently underestimate by a large margin, as expected. Less intuitively, the remaining two methods consistently overestimate. Based on the r 2 values (caption of Fig. 5), hypervolumes and mosaic areas are similarly precise. Thus, the issue is accuracy instead of precision.
Twenty data points per trial (Fig. 5B) is still a very low figure because it has long been recommended that at least 50 data points should be used to fix home ranges (Seaman et al. 1999). Here, the mosaic area values are still the only ones centred on the line of unity. Specifically, the median of ratios taken against known values is 0.95. The other three methods all fail. The 95% KDE and hypervolume estimates are still too high, with median ratios of 1.97 and 1.54. As expected, convex hull areas are biased in the opposite direction, with a median ratio of 0.60. The best one could say for these three methods is that their biases do not reverse as sample size increases.
Note that 95% KDEs are no more accurate than anything else when the sample size is five (caption of Fig. 5A) and are not very close to mosaic areas (r 2 = 0.8568 for KDEs vs. mosaic areas). These facts call 95% KDEs into question: they have no particular justification (Powell & Mitchell 2012), they are too high (Fig. 5), and they are not highly replicable using the best method discussed in this paper.
Spatial clustering of the data (Fig. 5C) biases the mosaic area values only weakly (median estimate:known area ratio 0.80), causes convex hull areas to fall short almost by the entire 50% that is possible (ratio 0.52), and also lowers the values for 95% KDEs and hypervolumes. However, they are still overestimates (1.41 and 1.27).
Mosaic areas also can handle a variety of range shapes even when only 10 points are sampled (Fig. 6). Median ratios of estimated to known areas are not far from one for most shapes: circles (1.00), squares (1.03), rectangles (1.17), and three-quarter rectangles (1.19). Results are worse for pairs of squares (2.06) and particularly rings (2.14). The first figure is philosophically problematic because it is hard to say whether two nearby clumps really should be considered separate shapes. If not, then 2.06 may be a reasonable compromise. With respect to rings, each one excludes half the area of the enclosing circle, so the approximate 2:1 ratio means that the method essentially treats rings as circles at this very low sampling level (if not at high levels: Figs. 1B, D). By contrast, ring areas are dramatically overestimated by 95% KDEs (6.12) and hypervolumes (4.75). These patterns are not illustrated because the ratios speak for themselves (and to save space). Again, shape solidity is a widespread assumption that is important for some methods, but not so much for the new one.
In general, the high performance of mosaic area estimation given this broad array of shapes is perhaps not too surprising because the underlying logic assumes that any shape can be covered adequately and accurately by a series of circuits connecting points, which stands to reason. The surprise is that reasonable results can be obtained with very small data sets.