Delta-epsilon is at it's best when the linear neighborhoods are very small. One case where this occurs is in the linear congruential random-number generator found in Matlab; delta-epsilon is easily able to detect the deterministic nature of these random numbers with only about 1000 data points. This is because of the quite small size of the locally linear regions in the 1-dimensional map that describes the random number generator.
» ts = rand(1000,1); » [pre,post] = getimage(ts,1); » [delta,epsilon] = deltaeps(pre,post); » plot(delta,epsilon,'xb') » axis([0 0.0001 0 1])
Points are not scattered over the graph as would be expected for random numbers. Instead, they fall on a precise grid corresponding to the sawtooth nature of the map employed in the random number generator. Near delta=0, there are either very small or very large values of epsilon; these correspond to nearby points either being on the same teeth of the map, or on neighboring teeth separated by a discontinuity of height 1.
The success of delta-epsilon in this case is, I think, spectacular, but of little practical significance. Although the smallest distances between 1000 points scattered on the unit interval is very small, the same would not be true, for instance, of points scattered on the unit square. Thus, using a two-dimensional embedding would in this case completely destroy the results.
Similarly, even a small amount of noise in the data (e.g., one part in 16000) would eliminate the pattern seen here. Since real-world data are typically noise and of dimension > 1, the success seen here is unlikely to be repeated with field or experimental data.