In Manhattan and elsewhere, the streets and avenues are not symmetric: the avenues are much farther apart than are the streets. This means that W 53rd St. & 8th Ave is much farther from W 53rd & 7th than it is from W 52nd & 8th. A distance metric that treats all dimensions as equal would be off by a factor of about 2.5. A weighted distance metric that knew of this difference would produce distance values - and hence clusters - that more closely matched the real world.
Generalizing this to n-d, the new distance metric might look like this: distance = sum(abs(p2[i] - p1[i]) * s[i] ) where S = a vector of (positive) scale factors. Would this be an appropriate new clustering feature? Jeff
