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[gnuastro-commits] master 0975bf8: Book: minor edits to the newly added
From: |
Mohammad Akhlaghi |
Subject: |
[gnuastro-commits] master 0975bf8: Book: minor edits to the newly added skewness section |
Date: |
Sat, 18 Dec 2021 11:22:11 -0500 (EST) |
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.
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