Context: I want to compare the sample probability distributions (PDFs) of two datasets (generated from a dynamical system). These datasets depend on a set of parameters, and I want a concise way to evaluate the distance between the two PDFs over several different parameter regimes, ideally by a single number. For a fixed parameter regiume, my two sample PDFs are given by the vectors $x$ and $y$, where $x_i$ is the relative frequency of samples which lie in the $i$th bin.

One method I've seen is the Kolmogorov-Smirnov statistic, which is the maximum vertical distance between the cumulative distribution functions of the two datasets. This would work for my purposes, but I'm starting to think that the chi-squared distance will be better (at the very least I had heard of it). It is given by: $d(x,y) = \frac{1}{2}\sum_i \frac{(x_i-y_i)^2}{x_i+y_i}$. But this doesn't seem to make sense in the case where $x_i=y_i=0$, which seems like a fairly reasonable occurrence.

Any recommendations?

Context: I want to compare the sample probability distributions (PDFs) of two datasets (generated from a dynamical system). These datasets depend on a set of parameters, and I want a concise way to evaluate the distance between the two PDFs over several different parameter regimes, ideally by a single number. For a fixed parameter regiume, my two sample PDFs are given by the vectors $x$ and $y$, where $x_i$ is the relative frequency of samples which lie in the $i$th bin.

One method I've seen is the Kolmogorov-Smirnov statistic, which is the maximum vertical distance between the cumulative distribution functions of the two datasets. This would work for my purposes, but I'm starting to think that the chi-squared distance will be better (at the very least I had heard of it). It is given by: $d(x,y) = \frac{1}{2}\sum_i \frac{(x_i-y_i)^2}{x_i+y_i}$. But this doesn't seem to make sense in the case where $x_i=y_i=0$, which seems like a fairly reasonable occurrence.

Any recommendations?

Edit: My first instinct was just to take the $L^2$ ($l^2$) norm of the difference of the two PDFs (sample PDFs). But then I thought the $L^1$ norm might be more suitable for a probability distribution. After looking a little further, I stumbled on the K-S and Chi-Squared distances.