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I have plotted some experimental data of mine, and these data points fall into the following distributions:

enter image description here

enter image description here

So, these are fairly non-trivial looking distributions. I would like to figure out methods to quantify how these distributions differ. Perhaps a Kullback-Leibler divergence?

What other methods could I use to do this? There's also a question of how to deal with differing levels of sparsity/different sample sizes.

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  • $\begingroup$ A qq-plot is always a useful comparison. $\endgroup$ – Alex R. Mar 29 '17 at 18:45
  • $\begingroup$ @AlexR. How would one do this for these distributions? $\endgroup$ – ShanZhengYang Mar 29 '17 at 18:49
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You cannot treat your $Q$ distribution as it is treated canonically in the definition. A bit of statistical manipulation to your data will be necessary. Your options are 1. Subsample according to a uniform increment $P$ so that it can align with $Q$ or 2. you can interpolate pairs of points in $Q$ so that a subdivision can be instantiated to extrapolate more data points.

Depending on your application either option may be more or less advisable. Eg. if your data is dense with many change points, then go with option 2. and if the data is smooth go with the first option. You will have to make certain assumptions that are reasonable to use KL.

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