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Consider an annular region of space. We divide this annulus into four equal regions and we take some measurements in each of the four sectors.

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Imagine we are measuring something like a particle's energy which enters a given sector. The quantities are all strictly greater than zero. Measurements from one sector have weak to no correlation to other sectors so we just assume independence between sectors. The problem is that the number of measurements between sectors vary by orders of magnitude. One sector can return thousands of measurements, another can return hundreds, the third may return only tens, and the fourth may have even fewer. Furthermore, the measurements themselves in a given sector vary across orders of magnitude. For example, A may have a thousand measurements and their max and min might be 4 orders of magnitude apart.

Main question: What is a reasonable way to estimate a reasonable measure of central tendency for the entire annulus?

The measurements within a sector seem to be log-normally distributed. Here are some typical distributions for all four sectors.

enter image description here

The logs are all base ten here. The panels on the left show the distributions of the actual data in all four sectors. The right panels show the distribution of the data after I take the log for all four sectors. Within each bin, I am convinced that the "best" measure of central tendency is the median (in all of my cases, the median and the geometric mean are almost identical...the mean is always at like 75th percentile or higher). But then the question is, how can I get a measure of central tendency for the entire annulus? I can

  • take the average of the four medians
  • take the weighted mean of the four medians where the weights are inversely proportional to the number of measurements in a bin. So A gets the lowest weight and D gets the highest weight. This "makes sense" because I don't want A to influence the entire annulus too much. After all it is only a quarter of the entire annulus, just as valuable as D.
  • take the weighted mean of the four medians where the weights are directly proportional to the number of measurements in a bin. So A gets the highest weight and D gets the lower weights. This "makes sense" because A has the most number of measurements. It is the "cleanest" data set with the best statistics so it is the "best" representation of what is actually happening in the entire annulus.
  • merge all four datasets together and then take the median. This "makes sense" because the actual population in the entire annulus is highly skewed so median is a good measure for the entire annulus.

Which one of these is "best"? This is brand new stuff so we don't have much theory available. We have no idea what it is supposed to be theoretically. The end goal is just descriptive statistics as well as plugging the "average" energy into an equation and reporting numbers. At the end of the day, for now, we just want to be able to say, "The typical energy in the given annulus under these circumstances is ________ which plugging into equation 1 gives us Billy's constant as ________."

Assume that experiments are expensive and extremely time consuming and cannot be repeated so we have the data that we have. Does this have a name? Are there any books or papers that someone can recommend which we read, use, and cite? That'll be perfect.

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  • $\begingroup$ Are you aggregating your sector measurements from something less discrete? (For example, do you have a dense series of sensors distributed around the ring and then you're binning these sensors into sectors? Or are the sectors large sensors? Or do you have a single sensor in the center of each sector?) $\endgroup$ – Wayne Mar 22 '16 at 14:19
  • $\begingroup$ @Wayne Actually there are two detectors 180 degrees apart so they are always in opposite sectors. So at any one instant, each detector takes one measurement but the detectors are constantly moving, going around in circles with their radial distance changing. Both detectors cover the entire annulus. After all of the data is collected, we split up the annulus in four sectors (as shown), classify the data, and then make statements like "in A, the energy is usually _____ and in B, the energy is usually _______". $\endgroup$ – Fixed Point Mar 22 '16 at 20:42
  • $\begingroup$ @Wayne We collected and looked at the data first and then split up the annulus into sectors. This is also based on theory where the theory predicts that A would tend to have the highest values out of all four sectors. B and D would be the same but lower than A. C would tend to have the smallest values. This is why we divided up into sectors the way we did. And the measurements indeed follow this pattern most of the time. Now the question is, how to combine the sector statistics back into annular statistics? $\endgroup$ – Fixed Point Mar 22 '16 at 20:47
  • $\begingroup$ So you actually have continuous data, of sorts. The sectors are bins, and binning throws away information. I would think that if you want to make statements about typical energy in the ring, you should use the raw data rather than the binned data. Otherwise, you are throwing away information when you bin and then expecting to somehow get it back by combining bins. $\endgroup$ – Wayne Mar 22 '16 at 20:47
  • $\begingroup$ @Wayne Yes you are right. We have four bins. The data is kinda continuous with a constant cadence in time. How should we use the raw data? Any suggestions? $\endgroup$ – Fixed Point Mar 22 '16 at 20:49
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Two things that come to mind:

  1. It makes sense to bin readings by sector for statements about the sectors and their differences. It also makes sense to bin readings by state if there are different states (regimes, etc) of the system that need to be distinguished. But if you want to speak about the ring as a whole, binning to sectors and then trying to aggregate these bin values is coarsening your data for no reason. (Again, you may still have to consider readings during different states separately to make physical sense.)

  2. I keep thinking of the time factor, which is tied in to the rotation speed of your sensors and their measurement rate, as compared to any state changes in your system. That is: is your sensor essentially randomly sampling the entire ring during reasonably stable states or not?

If your sensors are sampling well, I'd think that the median of all of your samples, within each state, would be what you're looking for. I'm not a physicist, though, so hopefully actual experimentalists will jump in with comments or answers.

(I'm imagining that your sensors are essentially integrating the energy in the ring via random sampling.)

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