How to interpret this formula? Sn=C*mediani{medianj|xi−xj|} where C = 1.1926

harrison

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Sn=C*mediani{medianj|xi−xj|} where C = 1.1926
If we have a time series {1,3,5,6} (small example)
My interpretation is that we would first subtract the values and get absolute values like so:
1-3 = 2
5-6 = 1
And then find the median of those values:
Median of 2,1 = 1.5
But then find the median of 1.5?
I do not understand at all.
Can someone please explain how I would use the above equation with the values I gave for input?
 
Sn=C*mediani{medianj|xi−xj|} where C = 1.1926
If we have a time series {1,3,5,6} (small example)
My interpretation is that we would first subtract the values and get absolute values like so:
1-3 = 2
5-6 = 1
And then find the median of those values:
Median of 2,1 = 1.5
But then find the median of 1.5?
I do not understand at all.
Can someone please explain how I would use the above equation with the values I gave for input?
I have no idea what you are saying. What does median i even mean? Similarly with median j. What is the relevance of n? Is S a sequence or a sum?

If I want the differences between the elements in your example, I get 6 - 5, 5 - 3, 3 - 1, 6 - 3,
5 - 1, 6 - 1 or 1, 2, 2, 3, 4, 5, which gives a median of 2.5, not 1.5.

I suggest that you give the problem from your book exactly and completely.
 
Sn=C*mediani{medianj|xi−xj|} where C = 1.1926
If we have a time series {1,3,5,6} (small example)
My interpretation is that we would first subtract the values and get absolute values like so:
1-3 = 2
5-6 = 1
And then find the median of those values:
Median of 2,1 = 1.5
But then find the median of 1.5?
I do not understand at all.
Can someone please explain how I would use the above equation with the values I gave for input?

Please tell us the context of the question, including whatever you know about the purpose of the calculation. I tried searching for the notation, and found that your expression is much like the MAD (probably Median Absolute Difference) that I found defined on some sites. I assume that n, i, and j are subscripts, and there is a set of (n?) data values x_i.

Once you confirm the context, and I have time (which I don't now) I can show you how I would interpret it.
 
I'll still want to see your context, but I'll tell you more about what I see.

First, your formula, as I understand it, is
\(\displaystyle S_n=C*median_i(median_j|x_i−x_j|)\)

What this would mean is that you first take one value of i and find the value of \(\displaystyle |x_i−x_j|\) for each value of j (that is, the absolute difference of your chosen value and every value in the data set (including the one your chose) and find the median of those n values. Repeat this for every value of i, so you will get a list of n medians; take the median of all those. Finally, you multiply by the given value of C.

For your data, {1,3,5,6}, the work would look like this:
i=1: {|1-1|, |1-3|, |1-5|, |1-6|} = {0, 2, 4, 5}, median = 3
i=2: {|3-1|, |3-3|, |3-5|, |3-6|} = {2, 0, 2, 3}, median = 2
i=3: {|5-1|, |5-3|, |5-5|, |5-6|} = {4, 2, 0, 1}, median = 1.5
i=4: {|6-1|, |6-3|, |6-5|, |6-6|} = {5, 3, 1, 0}, median = 2
median of {3, 2, 1.5, 2} = 2
1.1926 * 2 = 2.3852.

But I don't find that formula anywhere other than in your question here and on Stack Exchange. What I do find in various places is Median Absolute Deviation, which is

\(\displaystyle MAD = median( |X\)\(\displaystyle _i − median(X)| )\)

or, as I find it elsewhere looking more like yours, e.g. here,

\(\displaystyle MAD = b median_i( |x_i − \)\(\displaystyle median_j(x_j)| )\)

That would be calculated differently; but the "b" in the latter, which is called "k" in the Wikipedia page, is a constant that depends on the type of distribution, which converts MAD into an approximation of the standard deviation, which would be called s, so this may well be what your formula is supposed to be.

If I calculate the MAD for your example (using your constant), I get this:

median({1, 3, 5, 6}) = 4
median({|1-4|,
|3-4|, |5-4|, |6-4|}) = median(3, 1, 1, 2) = 1.5
1.1926 * 1.5 = 1.7889

I'm very curious about what your problem really is about.
 
First off, a huge thank you to Dr. Peterson. Your explanation was detailed and I fully understand the equation now.

Sorry for the late reply, busy past couple of days and although I did read your response yesterday I did not have time to reply.

The equation can be found here: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.393.7947&rep=rep1&type=pdf

I had been doing research on shot length distributions and stumbled upon that formula. It was simply out of curiosity that I wanted to learn how to calculate that formula.

There is another formula Qn which I don't know how to calculate either. But, they both have the same problem as median absolute deviation which is the 50% breakdown level where if 50% of the values are repeated then MAD = 0.
 
Last edited:
First off, a huge thank you to Dr. Peterson. Your explanation was detailed and I fully understand the equation now.

Sorry for the late reply, busy past couple of days and although I did read your response yesterday I did not have time to reply.

The equation can be found here: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.393.7947&rep=rep1&type=pdf

I had been doing research on shot length distributions and stumbled upon that formula. It was simply out of curiosity that I wanted to learn how to calculate that formula.

There is another formula Qn which I don't know how to calculate either. But, they both have the same problem as median absolute deviation which is the 50% breakdown level where if 50% of the values are repeated then MAD = 0.

I had to look up "order statistic" in Wikipedia to confirm the notation used for Q_n, which is what I guessed it might be. Your reference says

\(\displaystyle Q_n=c_{Qn}\times 2.2219\{|X_i-X_j|;i<j\}_{(k)}\)

What the subscript (k) means is "the kth element of the set, when they are arranged in increasing order". So you would calculate k as described following the formula; calculate all the differences between different elements of the set (only once for each pair, because it requires i<j); and put them in order, then count.
 
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