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Re: I had a sneaking suspicion you were going to insist on mapping to [0,1].
Posted:
Nov 19, 2012 3:35 AM
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On Nov 16, 6:24 pm, djh <halitsk...@att.net> wrote:
> I suspected there was going to be a need to map into [0,1], but didn?t
> want to raise the question until I knew whether you were going to
> dismiss the suggestion as unworkable for other reasons.
>
> Regarding c, here are the two salient points:
>
> a) we ignore strings in which c is 0;
>
> b) I can guarantee we will never examine a message segment containing
> more than 253 codons, and therefore 252 dicodons (assuming we allow
> overlapping dicodons, which we do);
>
> So, c can go from 1 to 252, where 252 is reached if every dicodon
> c(i)c(i+1) for 1<=i<251 is in the set of dicodons with which we?re
> working.
>
> Regarding u, suppose that:
>
> c) our dicodon set contains the L codon ctc and the S codon agt;
>
> d) in our string S of 252 overlapping dicodons c1...c252, c(i) is
> always ctc and c(i+1) is always agt for i odd.
>
> Then inasmuch as there are 6 L codons and 6 S codons, there are 6*6
> different dicodons signifying LS and 6*6 different dicodons signifying
> SL. And therefore, we would expect the LS dicodon ctcagt to occur
> 126/36 = 3.5 times in our string, and the SL dicodon agtctc to occur
> 126/36 = 3.5 times in our string. But since we insisted on (d) above,
> the LS dicodon ctcagt actually occurs 126 times in our string, and the
> SL dicodon agtctc actually occurs 126 times in our string.
>
> So ezch of these dicodons occurs 126/3.5 times more than expected, and
> therefore u for our string is 36.
>
> And inasmuch as no dicodon for any dipeptide has an expectancy LESS
> than 1/36, we can rightfully say that:
>
> e) the lowest possible u is 1/126 = .0079
>
> f) the highest possible u is 126/3.5 = 36.
>
> Assuming you agree on all of the above, I?m afraid you?ll have to tell
> me how to map each of
>
> e: [221.735, 308.65]
> c: [1,252]
> u: [.0079.,36]
>
> into [0,1].
>
> (Not being snippy here ... just confessing my usual feckless ignorance
> on this particular matter.)
>
> By the way, permit me to correct an egregious typo in my previous post
> regarding the orthogonal projection.
>
> This:
>
>> Orthogonal projection of this coordinate system onto the plane Peuc
>> thru (1,0,0), (0,1,0), (0,0,1) will take:
>>
>> e,0,0 into the point Pe = ( (1/sqrt6)e, (-sqrt2/2)e )
>> 0,c,0 into the point Pc = ( (-sqrt(2/3))e, 0 )
>> 0,0,u into the point Pu = ( (1/sqrt6)e, (+sqrt2/2)e )
>
> should of course have been:
>
>> Orthogonal projection of this coordinate system onto the plane Peuc
>> thru (1,0,0), (0,1,0), (0,0,1) will take:
>>
>> e,0,0 into the point Pe = ( (1/sqrt6)e, (-sqrt2/2)e )
>> 0,c,0 into the point Pc = ( (-sqrt(2/3))c, 0 )
>> 0,0,u into the point Pu = ( (1/sqrt6)u, (+sqrt2/2)u )
>
> (I assume you simply recognized and ?read-thru? this error, but I
> wanted to correct it for the record nonetheless.)
1. What you are suggesting is equivalent to rotating the axes using
the orthonormal transformation
[ a b -c ]
T = [ a -2b 0 ],
[ a b c ]
where a = -sqrt(1/3), b = sqrt(1/6), c = sqrt(1/2).
You discard the first column of T. Everything below holds
with or without the first column.
You have
[ Pe ] [ e 0 0 ]
[ Pc ] = [ 0 c 0 ] . T
[ Pu ] [ 0 0 u ]
where . denotes matrix multiplication (Mathematica notation).
The centroid of those three points is [e c u].T/3, and the average
centroid of any set of such points is [ebar cbar ubar].T/3, where
[ebar cbar ubar] is the centroid of the points in terms of the
original coordinates.
The same thing happens if you look at only pairs (e,c),(e,u),(c,u),
except the divisor 3 changes to a 2.
So all you're suggesting is to look at some linear transformations
of the centroids of the points at each L. That can never tell you
anything about the within-L relations or if/how they change with L.
2. When I asked how you were mapping e,c,u into [0,1], I had misread
your post as saying that you had done that. It isn't necessary. What
is necessary is that the values in each column of T imply a linear
combination that makes sense. Mapping x into [0,1] by
x' = (x - x_min) / (x_max - x_min)
does not necessarily make x' comparable to
y' = (y - y_min) / (y_max] - y_min)
for some other variable y.
The same holds for more complicated transformations. In general, any
transformation that is purely numerically driven, ignoring what the
numbers represent, is suspect.
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