I use the geometric mean (GM) as a scale factor for data normalization. To avoid the $0$ cancellation effect with positive values, I use the simple offset GM: $$ \hat{x} = \sqrt[n]{\prod\limits_{i=1}^n \left( x_i +1\right) } -1\,.$$ For a scale factor, using $|x_i| $ could suffice. However, one may want a scale sign related to the most represented values.

  1. Is there a trick to compute the geometric mean of negative numbers, or a proxy (a good appproximation)?
  2. Could it even be extended to complex numbers? After all, there exists a complex form for the arithmetic-geometric mean.
  3. Have them been used already in statistics?
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    $\begingroup$ You are struggling because the GM makes no sense for non-positive numbers. Instead of searching for meaningless extensions of it (such as complex values), have you thought of using a different estimator of scale? $\endgroup$ – whuber Apr 14 '16 at 15:43
  • $\begingroup$ Your comment belies a more fundamental underlying question. Evidently you have a need to estimate a scale and you also have some way to evaluate how "effective" this estimate is. Why, then, don't you ask that question? Tell us why you wish to estimate a scale and how you will compare the effectiveness of estimators. Then we could propose useful solutions. Wouldn't that be much better than having us criticize the obvious limitations of the GM and proposing some kind of ad hoc fix? $\endgroup$ – whuber Apr 14 '16 at 15:52
  • $\begingroup$ @whuber yes I did. I am collecting various normalization methods, because I am not sure of the most effective. One could for instance use $\exp{x_i}$ and apply the logarithm afterward, but I have not found traces of such methods. After all, for a long time, $x^2= -1$ made no sense $\endgroup$ – Laurent Duval Apr 14 '16 at 15:56
  • $\begingroup$ @whuber For a long time I believe the arithmetic-geometric mean only worked for positive real numbers, and discovered it existed for complex numbers while searching for this question. I am dealing with novel data, where people remove zero and negative values for GM normalization. I am trying first to understand the rationale behind that, and ways around, since I have little additional information. I wish I can ask a more interesting question soon, when I have more solid foundations for it. $\endgroup$ – Laurent Duval Apr 14 '16 at 16:03
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    $\begingroup$ It's not a question of interpreting complex numbers: people have known how to do that for centuries and there are no difficulties, either conceptual of computational, with it. For the statistical analysis of real-valued data, however, introducing complex numbers in an effort to estimate scale does appear to be meaningless in that it gives us no useful information that could not otherwise be supplied more simply by sticking with real numbers. $\endgroup$ – whuber Apr 14 '16 at 16:06

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