<p>You can do any mathematical operation to any number you have, but you should have well justifications for doing so. </p>

<p>For example, the original transformation from flux to magnitude using -2.5*log transformation is for compressing the range of value presentation (i.e., instead of 10^n to just n), and the minus sign is to make it easily implemented and interpreted because of the ancient brightness system ranking brighter for lower number. (I am not sure if there is any good reason for making -2.5 instead of just -1. haha).</p>

<p>Your transformation seems more like modelling reasons. There are a lot of cases in the history (including not only in physics but also other fields) that mathematical relations were ansatz, and purely phenomenological/observational rather than theorey driven. Some might be still used without underlying theory supporting, but we just keep using them because they provide good implications. Yours is one of the former case. </p>

<p>If you cannot make any sensible justification of why taking double-log to fluxes is something awesome, it is hard to make peer reviewers (who judge your works) to accept it. (This is for sure subjective to personal opinion). And by the meaning of sensible justification, since you are dealing with "physics," it means the connection from the physical process to the functional form.</p>

<p>So, the reason you should not take double-log might be because there is no physcial explanation of why doing so is physically meaningful.</p>

CC BY-SA 4.0
2018-08-16T20:10:07.510Z
I was in two minds about doing it in the first place, since it gives such a tiny improvement to the correlation coefficient. You’ve just convinced me to pursue another method.
Jim421616
CC BY-SA 4.0
2018-08-16T13:28:55.733Z
2018-08-16T13:28:55.733Z
Kornpob Bhirombhakdi
<p>You <em>can</em>, the question is whether you should.</p>

<p>Is there a theoretical reason why you think that magnitude should vary exponentially with redshift? If not then you may be falling into a fallacy.</p>

<p>You see with a polynomial with sufficiently many terms, I can get a curve that fits your data perfectly, <em>r</em>=1, Does this my polynomial is a better model? Probably not.  Just throwing functions at your data and choosing the one that gives the biggest <em>r</em> isn't mathematical modelling of data.</p>

<p>I would question why you are using magnitude at all? You seem to be finding that "close objects appear brighter" which isn't a new discovery.</p>

CC BY-SA 4.0
2018-08-15T21:41:22.123Z
Indeed, it isn't, but the point of the study is to find a tighter relationship between magnitude in a wavelength or a combination of magnitudes that has a high enough correlation coefficient that we can reliably use it to estimate the redshift, without having to do a spectroscopic analysis of the light. Photometry is easier than spectrometry. At the moment, the best correlation we get is about r=0.5
Jim421616
CC BY-SA 4.0
2018-08-15T07:14:56.830Z
2018-08-15T07:14:56.830Z
James K
<p>I'm investigating the relationship between the magnitude of quasars in various wavelengths, and their redshift $z$. I've found that if I take the $log$-$mod$-transform ($L_{mag}$) of the magnitudes, where</p>

<p>$L(y) = sgn(y) * log(|y| + 1)$</p>

<p>which I got from <a href="https://blogs.sas.com/content/iml/2014/07/14/log-transformation-of-pos-neg.html" rel="nofollow noreferrer">here</a> and <a href="http://juluribk.com/2016/03/30/dealing-with-plotting-negative-zero-and-positive-values-in-log-scale/" rel="nofollow noreferrer">here</a>, I can get a slightly tighter fit for my data, which is the aim of the study. Here's a plot for the W1 magnitude and $L(W1)$ as above, both against $log(z)$:</p>

<p><a href="https://i.sstatic.net/549t2.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/549t2.png" alt="enter image description here"></a></p>

<p>showing the regression coefficient for each.</p>

<p>My question, before I take this to my supervisor, is: <strong>is there any reason why I shouldn't be taking the $log$ of an already $log$ quantity?</strong></p>

CC BY-SA 4.0
2018-08-15T13:59:35.343Z
I'm voting to close this question as off-topic because This is a question about validity of statistical analysis methods, not astronomy
Carl Witthoft
CC BY-SA 4.0
2018-08-15T21:39:13.550Z
Can it be migrated to statistics?
Jim421616
CC BY-SA 4.0
2018-08-16T13:16:27.937Z
I don't see any relation to statistics for this question.
Kornpob Bhirombhakdi
<redshift><quasars><magnitude>
Can I take logarithms of magnitudes (which are already log)
Jim421616
# Can I take logarithms of magnitudes (which are already log)

(Asked by: username_0 on 2018-08-15)

I'm investigating the relationship between the magnitude of quasars in various wavelengths, and their redshift $z$. I've found that if I take the $log$-$mod$-transform ($L\_{mag}$) of the magnitudes, where

$L(y) = sgn(y) \* log(|y| + 1)$

which I got from [here](https://blogs.sas.com/content/iml/2014/07/14/log-transformation-of-pos-neg.html) and [here](http://juluribk.com/2016/03/30/dealing-with-plotting-negative-zero-and-positive-values-in-log-scale/), I can get a slightly tighter fit for my data, which is the aim of the study. Here's a plot for the W1 magnitude and $L(W1)$ as above, both against $log(z)$:

[![enter image description here](https://i.sstatic.net/549t2.png)](https://i.sstatic.net/549t2.png)

showing the regression coefficient for each.

My question, before I take this to my supervisor, is: **is there any reason why I shouldn't be taking the $log$ of an already $log$ quantity?**

**Tags:** redshift, quasars, magnitude
> **Comments:**
> **username_1**: I'm voting to close this question as off-topic because This is a question about validity of statistical analysis methods, not astronomy  
> **username_0**: Can it be migrated to statistics?  
> **username_2**: I don't see any relation to statistics for this question.  

---

## 2 answers

### Answer by username_2 on 2018-08-16. Score: 1 | Accepted answer

You can do any mathematical operation to any number you have, but you should have well justifications for doing so.

For example, the original transformation from flux to magnitude using -2.5\*log transformation is for compressing the range of value presentation (i.e., instead of 10^n to just n), and the minus sign is to make it easily implemented and interpreted because of the ancient brightness system ranking brighter for lower number. (I am not sure if there is any good reason for making -2.5 instead of just -1. haha).

Your transformation seems more like modelling reasons. There are a lot of cases in the history (including not only in physics but also other fields) that mathematical relations were ansatz, and purely phenomenological/observational rather than theorey driven. Some might be still used without underlying theory supporting, but we just keep using them because they provide good implications. Yours is one of the former case.

If you cannot make any sensible justification of why taking double-log to fluxes is something awesome, it is hard to make peer reviewers (who judge your works) to accept it. (This is for sure subjective to personal opinion). And by the meaning of sensible justification, since you are dealing with "physics," it means the connection from the physical process to the functional form.

So, the reason you should not take double-log might be because there is no physcial explanation of why doing so is physically meaningful.

> **Comments:**
> **username_0**: I was in two minds about doing it in the first place, since it gives such a tiny improvement to the correlation coefficient. You’ve just convinced me to pursue another method.  

---

### Answer by username_3 on 2018-08-15. Score: 1

You *can*, the question is whether you should.

Is there a theoretical reason why you think that magnitude should vary exponentially with redshift? If not then you may be falling into a fallacy.

You see with a polynomial with sufficiently many terms, I can get a curve that fits your data perfectly, *r*=1, Does this my polynomial is a better model? Probably not. Just throwing functions at your data and choosing the one that gives the biggest *r* isn't mathematical modelling of data.

I would question why you are using magnitude at all? You seem to be finding that "close objects appear brighter" which isn't a new discovery.

> **Comments:**
> **username_0**: Indeed, it isn't, but the point of the study is to find a tighter relationship between magnitude in a wavelength or a combination of magnitudes that has a high enough correlation coefficient that we can reliably use it to estimate the redshift, without having to do a spectroscopic analysis of the light. Photometry is easier than spectrometry. At the moment, the best correlation we get is about r=0.5  

---

