[gnuastro-commits] master 0975bf8: Book: minor edits to the newly added skewness section

[Mohammad Akhlaghi]

branch: master
commit 0975bf8ec45aa057abb871b3238f0a0e7d56ea72
Author: Mohammad Akhlaghi <mohammad@akhlaghi.org>
Commit: Mohammad Akhlaghi <mohammad@akhlaghi.org>

    Book: minor edits to the newly added skewness section
    
    Until now, there were a few minor editorial mistakes in the text of the
    newly added section of the third tutorial.
    
    With this commit, after a re-read, I found and fixed them.
---
 doc/gnuastro.texi | 20 +++++++++++---------
 1 file changed, 11 insertions(+), 9 deletions(-)

diff --git a/doc/gnuastro.texi b/doc/gnuastro.texi
index e45c879..ae1d970 100644
--- a/doc/gnuastro.texi
+++ b/doc/gnuastro.texi
@@ -4677,7 +4677,7 @@ In the next section (@ref{Image surface brightness 
limit}), we will use this to
 However, to better understand NoiseChisel and also, the image surface 
brightness limit, understanding the skewness caused by signal, and how to 
measure it properly are very important.
 Therefore now that we have separated signal from noise, let's pause for a 
moment and look into skewness, how signal creates it, and find the best way to 
measure it.
 
-Let's start masking all the detected pixels and having a look at the noise 
distribution with the @command{astarithmetic} and @command{aststatistics} 
commands below (while visually inspecting the masked image with @command{ds9} 
in the middle).
+Let's start masking all the detected pixels found at the end of the previous 
section (@ref{NoiseChisel optimization}) and having a look at the noise 
distribution with Gnuastro's Arithmetic and Statistics programs as shown below 
(while visually inspecting the masked image with DS9 in the middle).
 
 @example
 $ astarithmetic r_detected.fits -hINPUT-NO-SKY set-in \
@@ -4720,11 +4720,12 @@ Histogram:
 @noindent
 @cindex Skewness
 This histogram shows a roughly symmetric noise distribution, so let's have a 
look at its skewness.
-The most commonly used definition of skewness (also known as ``Pearson's first 
skewness coefficient'') compares the difference between the mean and median, in 
untis of the standard deviation (STD):
+The most commonly used definition of skewness is known as the ``Pearson's 
first skewness coefficient''.
+It measures the difference between the mean and median, in untis of the 
standard deviation (STD):
 
 @dispmath{\rm{Skewness}\equiv\frac{(\rm{mean}-\rm{median})}{\rm{STD}}}
 
-The logic behind this definition is simple: as more signal is added (skewness 
is increased) and the mean shifts the positive faster than the median, so their 
distance should increase.
+The logic behind this definition is simple: as more signal is added to the 
same pixels that originally only have raw noise (skewness is increased), the 
mean shifts to the positive faster than the median, so the distance between the 
mean and median should increase.
 Let's measure the skewness (as defined above) over the image without any 
signal.
 Its very easy with Gnuastro's Statistics program (and piping the output to 
AWK):
 
@@ -4735,7 +4736,7 @@ $ aststatistics det-masked.fits --mean --median --std \
 @end verbatim
 
 @noindent
-We see that the mean and median are only @mymath{0.08\sigma} away from each 
other (which is very close)!
+We see that the mean and median are only @mymath{0.08\sigma} (rounded) away 
from each other (which is very close)!
 All pixels with significant signal are masked, so this is expected, and 
everything is fine.
 Now, let's check the pixel distribution of the sky-subtracted input (where 
pixels with significant signal remain, and are not masked):
 
@@ -4820,8 +4821,8 @@ $ aststatistics r_detected.fits --mean --median --std \
 
 The difference between the mean and median is now approximately 
@mymath{0.12\sigma}.
 This is larger than the skewness of the masked image (which was approximately 
@mymath{0.08\sigma}).
-At a glance (only to the quantified numbers), it seems that there is not much 
difference and the two distributions.
-However, visually looking at the non-masked image, or the ASCII histogram, you 
would expect the quantified skewness to be much larger than that of the masked 
image, but hasn't happened!
+At a glance (only looking at the numbers), it seems that there is not much 
difference between the two distributions.
+However, visually looking at the non-masked image, or the ASCII histogram, you 
would expect the quantified skewness to be much larger than that of the masked 
image, but that hasn't happened!
 Why is that?
 
 The reason is that the presence of signal doesn't only shift the mean and 
median, it @emph{also} increases the standard deviation!
@@ -4837,13 +4838,14 @@ We therefore need a better unit or scale to quantify 
the distance between the me
 A unit that is less affected by skewness or outliers.
 One solution that we have found to be very useful is the quantile units or 
quantile scale.
 The quantile scale is defined by first sorting the dataset (which has 
@mymath{N} elements).
-If we want the quantile of a value in a distribution, we first find the 
nearest data element to @mymath{V} in the sorted dataset (let's assume its the 
@mymath{i}-th element after sorting).
-The quantile of V is then defined as @mymath{i/N} (which will have a value 
between 0 and 1).
+If we want the quantile of a value @mymath{V} in a distribution, we first find 
the nearest data element to @mymath{V} in the sorted dataset.
+Let's assume the nearest element is the @mymath{i}-th element, counting from 
0, after sorting.
+The quantile of V in that distribution is then defined as @mymath{i/(N-1)} 
(which will have a value between 0 and 1).
 
 The quantile of the median is obvious from its definition: 0.5.
 This is because the median is defined to be the middle element of the 
distribution after sorting.
 We can therefore define skewness as the quantile of the mean (@mymath{q_m}).
-If @mymath{q_m\sim0.5} (the median), then we know the distribution is 
symmetric (possibly Gaussian, but the functional form is irrelevant here).
+If @mymath{q_m\sim0.5} (the median), then the distribution (of signal blended 
in noise) is symmetric (possibly Gaussian, but the functional form is 
irrelevant here).
 A larger value for @mymath{|q_m-0.5|} quantifies a more skewed the 
distribution.
 Furthermore, a @mymath{q_m>0.5} signifies a positive skewness, while 
@mymath{q_m<0.5} signifies a negative skewness.