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## [gnuastro-commits] master 26c5e7b 1/5: Book: add magnitude quantile to t

 From: Mohammad Akhlaghi Subject: [gnuastro-commits] master 26c5e7b 1/5: Book: add magnitude quantile to the surface brightness limit section Date: Fri, 17 Dec 2021 21:53:26 -0500 (EST)

branch: master
Author: Sepideh Eskandarlou <sepideh.eskandarlou@gmail.com>

Book: add magnitude quantile to the surface brightness limit section

Until now, the definition of skewness was used to quantitatively compare
signal and noise distribution in surface brighness limit section.

Now, for quantifying the signal and noise distributions instead of skewness
we used from quantile of the mean. Because the distribution of the signal
and the noise are not Gaussian and we cant used the skewness, we should use
the quantile of the mean instead of skewness.
---
doc/gnuastro.texi | 109 +++++++++++++++++++++++++++++++++++++++++++++++++++---
1 file changed, 104 insertions(+), 5 deletions(-)

diff --git a/doc/gnuastro.texi b/doc/gnuastro.texi
index 7fa39e1..a565855 100644
--- a/doc/gnuastro.texi
+++ b/doc/gnuastro.texi
@@ -4683,19 +4683,118 @@ $astarithmetic r_detected.fits -hINPUT-NO-SKY set-in \ in det nan where -odet-masked.fits$ ds9 det-masked.fits
$aststatistics det-masked.fits + +------- + Number of elements: 918698 + Minimum: -0.113805 + Maximum: 0.130365 + Median: -0.00226983 + Mean: -0.0002118496391 + Standard deviation: 0.02569687481 +------- +Histogram: + | ** * + | * ** * * + | ** ** * * + | * ** ** ** * + | ** ** ** ** * ** + | ** ** ** ** * ** * + | * ** ** ** ** * ** ** + | ** ** ** ** ** * ** ** * + | ** ** ** ** ** ******* ** ** * + | ** ** ** ** ** ** ******* ** ** ** * + |******** ** ** ** ** ** **************** ** ** ** ** ** ** ** ** ** ** + |---------------------------------------------------------------------- + +@end example + +@noindent +From above histogram, we see taht the distribution of the noise is roughly symmetric. +Let us to see the signal distribution in the image. + +@example +$ aststatistics r_detected.fits -hINPUT-NO-SKY
+
+-------
+  Number of elements:                      3049472
+  Minimum:                                 -0.113805
+  Maximum:                                 159.25
+  Median:                                  0.0239832
+  Mean:                                    0.1056523001
+  Standard deviation:                      0.6981762756
+-------
+Histogram:
+ |*
+ |*
+ |*
+ |*
+ |*
+ |*
+ |*
+ |*
+ |*
+ |*
+ |******************************************* ***  ** ****  * *   *  * *
+ |----------------------------------------------------------------------
+
+@end example
+
+@noindent
+As you can see, the distribution is very elongated because the galaxy inside
the image is very bright.
+If you compare the above two distributions, you will see that the minimum
value of the image has not changed because we have not masked the minimum
values while the maximum value of the image has changed.
+If we compare the mean and median values of the signal distribution with the
mean and mean values of the noise distribution, we see how the mean and median
values of the noise distribution are close together, while these values are
very different in signal distribution.
+Now let's by using the @option{--lessthan} optin, limit the distribution of
the signal and make it similar to the noise distribution and then compare them
together.
+
+@example
+$aststatistics r_detected.fits -hINPUT-NO-SKY --lessthan=0.130365 + +------- + Number of elements: 2532028 + Minimum: -0.113805 + Maximum: 0.130354 + Median: 0.0135445 + Mean: 0.01720879614 + Standard deviation: 0.03591988828 +------- +Histogram: + | * * + | * * * ** + | * * * ** * + | ** ** * * ** ** + | ** ** * * ** ** * + | * ** ** **** ** ** ** + | ** ** ** **** ** ** ** * + | * ** ** ******* ** ** ** ** * + | * ** ** ** ******* ** ** ** ** ** ** + | * ** ** ** ************* ** ** ** ** ** ** ** * ** ** + |******** ** ** ** ** ******************* ** ** ** ** ***************** + |---------------------------------------------------------------------- @end example @noindent -From the ASCII histogram, we see that the distribution is roughly symmetric. -We can also quantify this by measuring the skewness (difference between mean and median, divided by the standard deviation): +If we compare the above signal distribution with the noise distribution. +We can see the noise distribution is completely symmetric, while the signal distribution in this range is asymmetric, especially in outer part. +This asymmetric is due to the effect of the signal. +Because we found and masked all those signals in the NoiseChisel, the noise distribution is completely symmetrical. + +@noindent +In @ref{Quantifying signal in a tile} we showed that when our distribution is skewed, the standard deviation is not defined at all, because the distribution is not Gaussian. +In scenarios like this, where our distribution is not Gaussian, we use quantile of the mean instead of skewness. +Now let's quantify these distribution by measuring the quantile of the mean: + +@example +$ aststatistics r_detected.fits -hINPUT-NO-SKY --quantofmean
+0.8105163158
+@end example

@example
-$aststatistics det-masked.fits --mean --median --std \ - | awk '@{print ($1-$2)/$3@}'
+0.5111848629
@end example

@noindent
-Showing that the mean is larger than the median by @mymath{0.08\sigma}, in
other words, as we saw in @ref{NoiseChisel optimization}, a very small residual
signal still remains in the undetected regions and it was up to you as an
exercise to improve it.
+Showing that in the signal distibution the mean is larger than the median by
@mymath{0.8\sigma}.
+While in noise distribution the mean is larger than the median by
@mymath{0.5\sigma}, in other words, as we saw in @ref{NoiseChisel
optimization}, a very small residual signal still remains in the undetected
regions and it was up to you as an exercise to improve it.
So let's continue with this value.
Now, we will use the masked image and the surface brightness limit equation in
@ref{Quantifying measurement limits} to measure the @mymath{3\sigma} surface
brightness limit over an area of @mymath{25 \rm{arcsec}^2}: