There is a 2D continuous distribution from an actual physical process which I want to model mathematically. I have 40k samples. You can see a histogram here and here.

This can be summarized as 4 properties, listed in order of prevalence. Further down the list needs more samples to be apparent. I use $f(x,y)$ to denote the probability density function.

  1. It is symmetric: $f(x,y)=f(-x,y)=f(y,x)$
  2. It is centered on $(0,0)$: $|(x_1,y_1)|<|(x_2,y_2)|\implies f(x_1,y_1)>f(x_2,y_2)$
  3. Near the axes is more likely
  4. Near the diagonals is slightly more likely

I can't come up with a clean mathematical description for axes/diagonal, but in words, for a fixed $|(x,y)|$, the global maximum $A_1=f(x,y)$ is where $x=0$ or $y=0$, there is a local maximum $A_2$ where $x=y$ or $x=-y$, and compared to the global minimum $B$, $A_2-B$ is much less than $A_1-B$.

Properties #1 and #2 are immediately obvious, and #3 shows up with a small number of samples. The diagonals being more likely is only barely visible with 40k samples. I don't think other lines will be more probable, since the axes and diagonals are the special ones. If it does follow a half angle subdividing pattern or something, I'd think each $A_i-B$ will be exponentially smaller.

Since this is modelling a physical process, there is a bound, but it is high enough to not matter. We can assume it's unbounded.

So, what distribution has these properties?

Someone might say if it's a physical process, why not analyze that to find the distribution? That would require understanding the physical process, and it's too complicated for that. It's hard to even know all the variables. The result is a 2D point for each measurement, and that's easy to analyze. It is indeed only an approximation, but in practice an accurate approximation is good enough.

It is possible that the diagonals is just a fluke since the difference is so small. Perhaps the histogram bins were biased in a way that made diagonals look slightly more probable. Or the measurements had too large of a granularity. I'll have to go back and collect more samples to confirm.


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