Comparison between multiple curves/probability distributions I have several sets of numerical data, and we'll call those sets $i=1...n$. Each set describes a normalized probability mass distributions $y_{i,j}$ such that
$$\sum_j y_{i,j} = 1$$
These $y_{i,j}$ are realizations of a physical process, e.g. "distribution of the heights of sunflowers after 10 days in X soil". However, the distributions themselves are not sampled and have no sample size associated with them. They are not histograms.
Plotting these distributions on top of each other reveals that they fall into two distinct groups, and that the differences within the groups are small relative to the differences between the two groups. We'll call these "winning" and "losing" groups, arbitrarily.
What is a good way to statistically describe these groups? Given a new set, how can I classify it as a "winning" set or a "losing" set?
I've been thinking about the Jensen Shannon distance, as its symmetrical and works for these distributions, but then I'm left with a bunch of pairwise distances that I'm not sure how to continue to process.
 A: Once you have your pairwise distances (another possibility is the Earth Mover's Distance, also known as the Wasserstein metric), you can cluster your sets. The simplest clustering algorithm would be the k-means one, and if you already know you have two clusters, you can set $k=2$ and run pretty much any implementation that uses a matrix of pairwise distances. (An implementation that starts out from points in $n$-dimensional space and calculates Euclidean distances between the points internally won't be useful for you, because your data "points" are your sets, and you already have the distances.) You will end up with automatically generated labels on your sets. (Now is probably a good point to plot them, e.g., all with one label in one plot, all with the other ones in a parallel plot - the differences should be visible.)
Now, given a new set, calculate the distances to all existing (and labeled) sets, and use a k-nearest neighbor algorithm for classification. That is, pick the $k$ nearest (labeled!) sets and assign the majority label among these to your new set. Unfortunately, the parameter $k$ here has the same name as the $k$ in k-means, but it should be set to a different value - typically an odd one (e.g., $k=3$ or $k=5$) to avoid ties in the voting.
This would also work if you have more than two groups in your sets: just use a higher $k$ in your clustering algorithm. You can also experiment with other clustering approaches than k-means.
