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.