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I'm a student who studies machine learning.

When I studied this paper(https://dl.acm.org/doi/pdf/10.1145/335191.335388),

I saw the following formula and explanation.

$$(direct_{max} - direct_{min})/direct_{mean} = (indirect_{max} - indirect_{min})/indirect_{mean}$$

The $direct_{mean}$ is the mean value of $direct_{max}$ and $direct_{min}$.

And $indirect_{mean}$ is also the mean value of $indirect_{max}$ and $indirect_{min}$.

In this case, that paper said the fluctuate amount is the same between direct and indirect.

But I don't understand one thing.

Is there any specific meaning of the below formula?

$$2* (value_{max} - value_{min})/(value_{max} + value_{min})$$

I don't know why it has the meaning as the amount of fluctuating.

Thank you for reading my question.

beef stew
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  • To a crude approximation, you can think of $\frac{x_{max}+x_{min}}{2}$ as a kind of "average", so you're looking at the range divided by the "average", which is a kind of normalized way to describe the amount of fluctuation. – Ian Feb 22 '21 at 18:14
  • It looks like an amplitude $\Delta=v_{\text{max}}-v_{\text{min}}$ in relation to some sort of mean $\bar v = \frac{v_{\text{max}}+v_{\text{min}}}{2}$. – Gaff Feb 22 '21 at 18:19
  • @lan Oh, you mean that I can think of that formula as the very, very rough normalization concept such as min-max normalization? – beef stew Feb 22 '21 at 18:20
  • As far as I know it is called relative range (keyword). See this link. – callculus42 Feb 22 '21 at 18:23

1 Answers1

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You can rewrite the formula as follows:

$$ \frac{2*(\text{Max}-\text{Min})}{\text{Max}+\text{Min}} $$ $$ = \frac{\text{Max}-\text{Min}}{\frac{\text{Max}+\text{Min}}{2}} $$ $$ = \frac{\text{Range}}{\text{AVG(Min, Max)}} $$

The numerator is the range, and the denominator is the average of the max and min. You can think of that numerator as a measure of spread, and the denominator as a measure of central tendency. So, this ratio is similar to other dimensionless/normalized metrics that measure noise/signal, like $\frac{\sigma}{\mu}$ or $\frac{\text{IQR}}{\text{Median}}$.

EDIT: As noted in comments on the original question, this is a rough normalized measure of spread unless we know a lot more about the distribution. If the distribution of observed values is approximately symmetric and unimodal, then that average is likely close to the mean or median. But, if the distribution is skewed or multimodal, then the average of the min and max can be wildly different from most actual observations.

Amaan M
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