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Topic: The same four proportional weighting factors work for each
00/01/10/11 when 0.25 is subtracted from each !!!

Replies: 506   Last Post: Nov 20, 2012 9:21 PM

 Messages: [ Previous | Next ]
 Halitsky Posts: 600 Registered: 2/3/09
The relationship of UCP's to "u"
Posted: May 30, 2012 9:10 PM

You asked the right initial question ? how do the UCP?s relate to
?u?. That?s the first question that must be answered in order for you
to understand the first table in my last post.

So, let me begin with the necessary basics, which I?m going to provide
with no mention of biology whatsoever.

Imagine you work for an eccentric billionaire who for some reason
known only to him merely wants you to continuously set type for brief
announcements in the public notice section of a weird kind of
newspaper he publishes.

All these announcements are similar in that they must each consist of
a string of letters drawn from the 20-letter alphabet
{ACDEFGHIKLMNPQRSTVWY}, and for the sake of discussion, we?ll say that
each announcement must be 24 to 124 letters long.

Because of his eccentricity, your boss gives you announcements to
typeset in a weird code that only you are privy to.

In this code of his, each of the twenty letters {ACDEFGHIKLMNPQRSTVWY}
is DEGENERATELY represented by AT LEAST one of the 64 3-letter words
that can be formed from the four letter alphabet (t,c,a,g}, except for
the three words taa,tag, and tga. (These three triples do not
represent any of the letters {ACDEFGHIKLMNPQRSTVWY} because your boss
uses them to signify ?end of announcement?, i.e. when you see one of
them, you know that the announcement has ended.)

looks like this:

ttt F tct S tat Y tgt C
ttc F tcc S tac Y tgc C
tta L tca S taa - tga -
ttg L tcg S tag - tgg W

ctt L cct P cat H cgt R
ctc L ccc P cac H cgc R
cta L cca P caa Q cga R
ctg L ccg P cag Q cgg R

att I act T aat N agt S
atc I acc T aac N agc S
ata I aca T aaa K aga R
atg M acg T aag K agg R

gtt V gct A gat D ggg G
gtc V gcc A gac D ggc G
gta V gca A gaa E gga G
gtg V gcg A gag E ggg G

This code is ?degenerate? because with the exception of the letters W
and M (which are each only encoded by one 3-letter word), the
remaining 18 letters of the alphabet {ACDEFGHIKLMNPQRSTVWY} are
encoded by 2, 4, or 6 different 3-letter words. For example, the
letter F is encoded by the two 3-letter words ttt,ttc, the lettter V
is encoded by the four 3-letter words gcX (X=t/c/a/g)m and the letter
L is encoded by the six 3-letter words tta, ttg, ctX (X=t/c/a/g).

And at first glance, your boss seems to be providing you with encoded
announcements in which he?s chosen randomly among the 3-letter words
that can signify the same letter. For example, sometimes, he?ll
signify the letter N of an announcement with the code word aat, and
sometimes with the code word aac, and there doesn?t seem to be any
rhyme nor reason for the choices he makes.

But, because your work is so boring, you decide to pay attention to
the choices your boss is making, and you realize that they?re not
random. In particular, you?ve noticed that in the encoded
announcements he gives you to typeset, he uses the the two-word
sequences att ccc, att cca, and att cct to represent the two-letter
sequence IP far more frequently than he uses the other 9 possible two-
word sequences that can represent IP according to the code he?s given
you (see above.) And in fact, the following 63 2-word sequences all
seem to be over-represented in the encoded announcements he gives you
to typeset:

ttt ttc|FF
ttt tta|FL
ttg ctc|LL
ttt gcc|FA
tta aaa|LK
ctg ctc|LL
ctt tcc|LS
ctc acc|LT
ctc tac|LY
ctc ggc|LG
att ccc|IP
att cca|IP
att cct|IP
att gcc|IA
atg ctc|ML
gtg ctg|VL
gtt tcc|VS
gtc acc|VT
gtc agc|VS
gtc ggc|VG
tcg ctg|SL
tct cca|SP
ccg gtg|PV
ccc ggc|PG
acg ctg|TL
acg ccg|TP
gct tta|AL
gcg ctg|AL
gcg gtg|AV
gcg ccg|AP
gct cca|AP
gct aat|AN
gcc agc|AS
tat ccg|YP
cat ccg|HP
cag cga|QR
aaa aat|KN
aaa aag|KK
gat ttg|DL
gaa aat|EN
gaa aaa|EK
gag cga|ER
gag cgg|ER
tgg cga|WR
tgg cgg|WR
cgg gtg|RV
cgg tac|RY
cgc tgg|RW
cgg cga|RR
agt gga|SG
agg cga|RR
agg cgg|RR
aga aga|RR
ggt ttt|GF
ggg ctg|GL
ggg tat|GY
ggc tac|GY
ggt tat|GY
gga aaa|GK
ggc aag|GK
ggc tgg|GW
ggg cgg|GR
ggc agc|GS

For example, since there are 12 ways to represent the two letter
sequence IP in the code he?s given you:

att cct
att ccc
att cca
att ccg
atc cct
atc ccc
atc cca
atc ccg
ata cct
ata ccc
ata cca
ata ccg

each of the three two-word sequences att ccc, att cca, and att cct
should occur 1/12 of the time if your boss is choosing randomly,
whereas in the actual announcements he gives you to typeset, they each
occur significantly more than 1/12 of the time (significantly in the
formal statistical sense.)

So just to pass the time when you?re not actively typesetting, you
decide to determine the degree ?u? of over-representation of the above
63 two-word sequences in any given announcement that your boss has
given you to typeset.

And to do so, you use the following table of expected frequencies for
the 49 2-letter sequences that are encoded by the above 63 two-word
sequences:

AL 0.0833 (the above 63 2-word sequences contain 2 out of the 12 ways
to code AL)
AN 0.1250 (the above 63 2-word sequences contain 1 out of the 8 ways
to code AN)
AP 0.1250 (the above 63 2-word sequences contain 2 out of the 16 ways
to code AP)
AS 0.0417 ...
AV 0.0625
DL 0.0833
EK 0.2500
EN 0.2500
ER 0.1667
FA 0.1250
FF 0.2500
FL 0.0833
GF 0.1250
GK 0.2500
GL 0.0417
GR 0.0417
GS 0.0417
GW 0.2500
GY 0.3750
HP 0.1250
IA 0.0833
IP 0.2500
KK 0.2500
KN 0.2500
LG 0.0417
LK 0.0833
LL 0.0556
LS 0.0278
LT 0.0278
LY 0.0833
ML 0.1667
PG 0.0625
PV 0.0625
QR 0.0833
RR 0.1111
RV 0.0278
RW 0.1667
RY 0.0833
SG 0.0417
SL 0.0278
SP 0.0417
TL 0.0417
TP 0.0625
VG 0.0625
VL 0.0417
VS 0.0833
VT 0.0625
WR 0.3333
YP 0.1250

In particular, you use this table as follows.

Suppose your boss gives you an encoded announcement in which:

a) a 2-word sequence for AP appears just once and this 2-word sequence
is att ccc (one of the above 63 2-word sequences)

b) a 2-word sequence for AL appears just once and this 2-word sequence
is gcg ctg (one of the above 63 2-word sequences)

c) no other 2-word sequence from the above 63 occurs in the
announcement (equivalently, you will typeset an announcement in which
none of the above 49 2-letter sequences appears other than one AP and
one AL.)

Then the degree ?u? of average over-representation of the 63 2-word
sequences in the announcement is

1 / 0.8333 (for AL)

plus

1 / 0.25 (for AP)

equals

12 + 4 = 16

divided by 2 (

equals 8.

That is, on average, the 63 2-word sequences are represented in the
announcement 8 times more frequently than expected.

Let me stop there and see if I?ve made the relationship between UCP?s
and ?u? clear, and if not, what questions you have.

If you do now understand the ?expected frequencies? table discussed
above for S63, you should also understand why ?by definition? (as you
said), the expected frequencies for C711 are those above subtracted
from 1. If you don?t, then that will be the topic of my next post.

Date Subject Author
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