Measuring Spatial Diffusion

In a previous article, I introduced a method that can quickly calculate spatial entropy, by turning distances in a dataset into a distribution over [0,1], that then of course has an entropy. This however does not vary with scale, in that if you multiply the entire dataset by a constant, the measure of entropy doesn’t change. Perhaps this is useful for some tasks, though it plainly does not capture the fact that two datasets could have the same proportional distances, but different absolute distances. If you want to measure spatial diffusion on absolute basis, then I believe the following could be a useful measure, that also has units of bits:

\bar{H} = \sum_{\forall i,j} \log(||x_i - x_j||).

Read literally, you take the logarithm of every pair of distances within the dataset, which will of course vary as a function of those distances. As a result, if you scale a dataset up or down, the value of \bar{H} will change as a function of that scale. In a previous note, I showed that we can associate any length with an amount of information given by the logarithm of that length, and so we can fairly interpret \bar{H} as having units of bits.


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