Geometric mean with logarithmically spaced data?

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This is partly a Matlab question, partly conceptual. I generated three vectors with logarithmically spaced numbers using logspace. If I want to average the first element in the three vectors, then separately the second element, and so on, should I take the arithmetic or geometric mean? If the latter, how do I do this?
  1 Comment
Stephen23
Stephen23 on 12 Jul 2022
Edited: Stephen23 on 12 Jul 2022
"If I want to average the first element in the three vectors, then separately the second element, and so on, should I take the arithmetic or geometric mean?"
You gave no information in your question that helps us to understand what you need those means to achieve or how you will use them. In the absence of an such information, are you looking to understand different types of mean?:

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Accepted Answer

DGM
DGM on 12 Jul 2022
Edited: DGM on 12 Jul 2022
"how do I do this?"
% three vectors:
A = logspace(0,2,10);
B = logspace(0,1.9,10);
C = logspace(0.2,2,10);
% put them in an array
allvecs = [A; B; C]
allvecs = 3×10
1.0000 1.6681 2.7826 4.6416 7.7426 12.9155 21.5443 35.9381 59.9484 100.0000 1.0000 1.6260 2.6438 4.2987 6.9895 11.3646 18.4785 30.0454 48.8527 79.4328 1.5849 2.5119 3.9811 6.3096 10.0000 15.8489 25.1189 39.8107 63.0957 100.0000
% take the arithmetic mean
avg = mean(allvecs,1)
avg = 1×10
1.1950 1.9353 3.1358 5.0833 8.2440 13.3764 21.7139 35.2647 57.2990 93.1443
% take the geometric mean
gmn = geomean(allvecs,1)
gmn = 1×10
1.1659 1.8957 3.0824 5.0119 8.1491 13.2502 21.5443 35.0303 56.9581 92.6119
  1 Comment
John D'Errico
John D'Errico on 12 Jul 2022
Edited: John D'Errico on 12 Jul 2022
I'm confused as to the question. Is it asking if you should use a geometric mean or an arithmetic mean? Surely that is your choice? @DGM tells you how to compute either, but there is no truly correct way to do something that only you know why you are doing it.
Another simple way to form a set of several geometric sequences is::
S = 2.^([0;0;0] + [1;2;3].*linspace(0,1,10))
S = 3×10
1.0000 1.0801 1.1665 1.2599 1.3608 1.4697 1.5874 1.7145 1.8517 2.0000 1.0000 1.1665 1.3608 1.5874 1.8517 2.1601 2.5198 2.9395 3.4290 4.0000 1.0000 1.2599 1.5874 2.0000 2.5198 3.1748 4.0000 5.0397 6.3496 8.0000
I'll let you figure out how I generated them..
Now, since I carefully chose the rate constants, a property of the geometric mean is that it will return the middle sequence.
geomean(S,1)
ans = 1×10
1.0000 1.1665 1.3608 1.5874 1.8517 2.1601 2.5198 2.9395 3.4290 4.0000
Do you see that the geometric mean here just reproduced the middle one?

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