kernel density is important
re #1: That seems to break for me, because (taking the Gaussian example) if I have two data points, centers -0.25 and 0.25 .. the maximum of the total area after calculating both kernels will be at x=0, which wasn't a zero-crossing of either Gaussian point in isolation.
- 2: I like this idea!
I've just now switched (experimentally) to using the Tricube kernel because I like it's shape: flattish in the center and trailing off to either side. I have it adjusted to slightly overhang the precise intersection width of each data point. Since it only exists from (-1,1), I've got some of your suggestion #2 built in, and turn skipping has pretty much ceased! We'll see how well this kernel compares, of course....
For #1 I did not mean the zero-crossing of any one point, I meant the zero-crossings of the sum of all the derivatives of the kernel density function. Of course, whether it's efficient to calculate those zeros or not all depends on what the kernel density function is (probably not practical for gaussian, trivial for triangular, as two extereme cases)
Hmm... tricube sounds like an interesting one, though that's quite a bit of multiplication it uses. I wonder if this is the sort of thing that would be worth doing a rough approximation of really. I mean... it probably wouldn't affect the results too much to do the kernel density as a piecewise "sum of rectangles" approximation, and it would be much faster.
