[gnuastro-commits] master 26c5e7b 1/5: Book: add magnitude quantile to the surface brightness limit section

[Mohammad Akhlaghi]

branch: master
commit 26c5e7b28c7626c1280ad8c666bfcb7988af0f51
Author: Sepideh Eskandarlou <sepideh.eskandarlou@gmail.com>
Commit: Mohammad Akhlaghi <mohammad@akhlaghi.org>

    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)/$3@}'
+$ aststatistics det-masked.fits --quantofmean
+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}: