Difference between Hausdorff and earth mover (EMD) distance

Question

I have two dataset that i want to compare. each dataset contain the weight of 10 different person measured for 3 different day.

I am interested in measuring the probabily that the two sample originate from the same population.

People seem to suggest doing a Kolmogorov-Smirnov test but i need a measurement.

I was thinking doing the EMD to compare the distribution for each day

EMD(dataset1.day1,dataset2.day1) + EMD(dataset1.day2,dataset2.day2) + EMD(dataset1.day3,dataset2.day3)

where dataset1.day1 is the histogram of the value for day1 in dataset 1...

But i could probably take each person as a 3d datapoint and do the EMD in 3d.

One other possibility was to do the Hausdorff distance but doing the average of the distance for each point instead of taking the maximum distance.

The two dataset have very different skewness so i was also considering using the Mann-Whitney-Wilcoxon_test.

What are the main difference between the two technique.

Answer 1

The intuitive difference between Hausdorff distance and EMD between sets A and B is:

• EMD tells you the total work required to move all A's mass onto B, under the optimal scheme for doing so.
• Hausdorff tells you the worst-case distance between an element of A and the nearest element of B. If you consider each point to have unit mass, then you can think of Hausdorff as telling you the worst-case amount of work required to move a single element of A onto some element of B, under the optimal scheme for doing so.

Your modification of Hausdorff would have the characterization:

• It tells you the average amount of work required to move each element of A onto some element of B, under the optimal scheme for doing so.

Of course, which one you want depends on your application...

Answer 2

There are two questions here: (1) how to determine the probability that the two samples are from the same distribution and (2) what kind of distance metric could be used to measure their overlap.

For the first, one simple way would be to determine the distribution of the first sample (perhaps it's multivariate normal?) and then calculate the posterior density of the second sample under the assumption of the distribution of the first. I like this approach because the interpretation is very straightforward.

For the second, I wouldn't do what you're suggesting with EMD unless you have some natural pairing of individuals in samples 1 and 2 (see @whuber's questions above). One common point between Hausdorff and EMD is that both let you specify an arbitrary distance metric for the points, e.g., euclidean or cosine, so you don't have to average the points (I'd go further and say you shouldn't if you use these methods). The downside is that your results will depend on your choice of the distance metric so you need some way of justifying your choice.

Because of the downside of the distance metric being arbitrary, I'd consider, instead, the Bhattacharyya distance or perhaps mutual information, provided you can make some information choice about what the distributions are.