# How to quantify the similarity between three sets of complex numbers? [closed]

I have multiple groups of measurements, each containing three sets of complex numbers (impedances of the same thing measured under three conditions). The Nyquist plots belows shows two of such groups.

As can be seen in the Nyquist plots, there is a larger dispersion between the top group then the bottom. I'm looking to generate a single value that represents this dispersion for each individual group, i.e. a number for the top plot and another for the bottom that would tell me the top one is more dispersed than the bottom.

I tried to calcualte the deviation from the mean at each point, and can take the average of all points in the set as a representation of the difference - which is essentially the mean absolute deviation. But I came up with this on my own without much statistics background and was wondering if it there are more established ways to go about this.

I'm looking for a single value because I want to later calculate the correlation between the amount of variations and other factors.

Thank you in advance for any insights!

Here are the Nyquist plots of two groups of data, each containing three sets of complex numbers:

• Could you show the plot for clarity? Nov 9, 2023 at 19:30
• @ShawnHemelstrand I've added some pics, help they make more sense! Nov 9, 2023 at 22:27
• I've edited the question to explain the two groups and three sets, hopefully that makes a bit more sense and this question can get reopened. Apologize for the confusion! Nov 10, 2023 at 14:16
• Hm. Right now, it sounds like you are only interested in the difference between your two plots regardless of the colors. Is that so? Then you could simply take the sum or mean of all pairwise distances in the two plots and compare them. That said, the overlap of your points suggests that you not only have three groups of general points, but that triplets of points - one from each group - actually belong together. Is that so, and do you want or need to take this into account? Nov 10, 2023 at 15:55

"Three sets of complex numbers" very much sounds like you have a predefined clustering of 2-dimensional points into three clusters. Which in turn suggests the silhouette score, which is a very common KPI in clustering and essentially compares how "similar" any given point is to points within its cluster, compared to points in other clusters. This is at least well accepted and understood in the clustering community, so there may be some name recognition there.

• Do you mean I can treat the two groups of data (see plots added to question) as two clusters and calculate average inter & intra distances between any points to all other points in each cluster (that's how I understood what the Silhouette score is)? Could you clarify how that is related to the dispersion among each group? Nov 9, 2023 at 23:00
• Yes, that is what I am suggesting. (However, your question and your plots show three groups, while your comment says "two"? Also, you have two separate plots, so this does start to look a little more complicated...) In the calculation, the $a(i)$ for point $i$ captures the dispersion within the cluster in which point $i$ lies, while $b(i)$ captures the distance of point $i$ to any other cluster. I am not entirely sure this is what you want or need, especially now with the pictures added to your question... Nov 10, 2023 at 7:45
• I have updated the question to explain the two groups & three sets. Do you mind having a look at it and see if it still works? Thank you! Nov 10, 2023 at 14:18