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b-hebrew - Re: [b-hebrew] Peter Kirk's homonym percentage table

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  • From: Peter Kirk <peter AT qaya.org>
  • To: K Randolph <kwrandolph AT gmail.com>
  • Cc: B Hebrew <b-hebrew AT lists.ibiblio.org>
  • Subject: Re: [b-hebrew] Peter Kirk's homonym percentage table
  • Date: Fri, 19 Jan 2007 17:17:02 +0000

On 19/01/2007 14:10, K Randolph wrote:
Peter:

Could you do one letter by hand, showing how you came to your figures?
I don't understand your explanation, and I was unable to replicate
your results from your last letter, as far as I understood it.

This request is merely what I use to verify any formula that I
develop. I try it at least once using real numbers by hand aided by a
calculator. In this case I ask not that you solve for one letter, just
that you set up for one letter so I can see where the numbers go.
Thanks.

I'm not quite sure what you want, so I'll offer you this, the figures for all the letters, not just one, split into three blocks (the third you have already)

Verbs 1st 2nd 3rd total Homonyms 1st 2nd 3rd total
) 77 48 53 178 ) 13 8 7 28
B 78 97 78 253 B 18 16 12 46
G 58 49 28 135 G 7 3 2 12
D 57 62 75 194 D 11 9 4 24
H 31 42 226 299 H 5 9 54 68
H. 0 0 8 8 H. 0 0 0 0
W 0 175 1 176 W 0 35 0 35
Z 50 35 19 104 Z 13 4 2 19
X 135 70 86 291 X 32 9 14 55
+ 30 37 42 109 + 2 3 5 10
Y 84 34 0 118 Y 11 2 0 13
K 59 52 48 159 K 5 5 7 17
L 41 135 136 312 L 7 31 29 67
M 74 79 88 241 M 13 15 12 40
N 137 62 72 271 N 21 11 9 41
S 56 44 40 140 S 9 5 4 18
( 123 84 73 280 ( 31 16 9 56
P 84 78 86 248 P 11 8 16 35
C 61 63 55 179 C 12 12 9 33
Q 75 60 84 219 Q 11 3 11 25
R 101 170 210 481 R 11 44 53 108
& 49 18 10 77 & 6 4 0 10
$ 125 64 70 259 $ 15 8 7 30
T 30 57 27 114 T 3 7 1 11
Tot 1615 1615 1615 4845 267 267 267 801

p(1,A) p(2,B) p(3,C) p(1,A)^2 p(2,B)^2 p(3,C)^2 (1/2)N^2 H(1,A) H(2,B) H(3,C) letter
0.0477 0.0297 0.0328 0.0023 0.0009 0.0011 1304112.5 12 4 4 )
0.0483 0.0601 0.0483 0.0023 0.0036 0.0023 1304112.5 12 18 9 B
0.0359 0.0303 0.0173 0.0013 0.0009 0.0003 1304112.5 7 5 1 G
0.0353 0.0384 0.0464 0.0012 0.0015 0.0022 1304112.5 7 7 9 D
0.0192 0.0260 0.1399 0.0004 0.0007 0.0196 1304112.5 2 3 79 H
0.0000 0.0000 0.0050 0.0000 0.0000 0.0000 1304112.5 0 0 0 H.
0.0000 0.1084 0.0006 0.0000 0.0117 0.0000 1304112.5 0 59 0 W
0.0310 0.0217 0.0118 0.0010 0.0005 0.0001 1304112.5 5 2 1 Z
0.0836 0.0433 0.0533 0.0070 0.0019 0.0028 1304112.5 37 9 11 X
0.0186 0.0229 0.0260 0.0003 0.0005 0.0007 1304112.5 2 3 3 +
0.0520 0.0211 0.0000 0.0027 0.0004 0.0000 1304112.5 14 2 0 Y
0.0365 0.0322 0.0297 0.0013 0.0010 0.0009 1304112.5 7 5 4 K
0.0254 0.0836 0.0842 0.0006 0.0070 0.0071 1304112.5 3 35 29 L
0.0458 0.0489 0.0545 0.0021 0.0024 0.0030 1304112.5 11 12 12 M
0.0848 0.0384 0.0446 0.0072 0.0015 0.0020 1304112.5 38 7 8 N
0.0347 0.0272 0.0248 0.0012 0.0007 0.0006 1304112.5 6 4 2 S
0.0762 0.0520 0.0452 0.0058 0.0027 0.0020 1304112.5 31 13 8 (
0.0520 0.0483 0.0533 0.0027 0.0023 0.0028 1304112.5 14 12 11 P
0.0378 0.0390 0.0341 0.0014 0.0015 0.0012 1304112.5 8 8 5 C
0.0464 0.0372 0.0520 0.0022 0.0014 0.0027 1304112.5 11 7 11 Q
0.0625 0.1053 0.1300 0.0039 0.0111 0.0169 1304112.5 21 55 68 R
0.0303 0.0111 0.0062 0.0009 0.0001 0.0000 1304112.5 5 1 0 &
0.0774 0.0396 0.0433 0.0060 0.0016 0.0019 1304112.5 32 8 8 $
0.0186 0.0353 0.0167 0.0003 0.0012 0.0003 1304112.5 2 6 1 T
1.0000 1.0000 1.0000 0.0542 0.0572 0.0705 285 285 285

est obs
) 21 28
B 40 46
G 13 12
D 23 24
H 85 68
H. 0 0
W 59 35
Z 8 19
X 58 55
+ 7 10
Y 16 13
K 16 17
L 67 67
M 35 40
N 53 41
S 13 18
( 52 56
P 37 35
C 20 33
Q 29 25
R 144 108
& 6 10
$ 47 30
T 9 11
856 801

The formula for H(1,A) etc is as given in my previous post: (1/2)N^2 * p(1,A)^2 * sum of (p(2,B)^2) column * sum of (p(3,C)^2) column.

I noted also that for resh the estimated homonym probability is so high, in fact nearly one in three, that the assumptions I was making break down. My first estimate for the probable number of homonyms for each root is (1/2)N^2 * (p(ABC))^2, or (to simplify) (Np)^2/2. But in fact there are higher order terms here, to adjust for cases for three or more homonyms, and I think the full formula comes to something like:

(Np)^2/2 - (Np)^3/6 + (Np)^4/24 - ...

Now if for a particular root the first estimate is 1/2, that implies that (Np) = 1, and so the second term in the series is 1/6, the third is 1/24 etc etc., and the sum is just above 1/3. In fact for the root nun-vav-resh (one of the most probable), p(ABC) = (137/1615)*(175/1615)*(210/1615) = 0.001195, and (Np) = 1.93. With a value of (Np) this high my approximations break down completely. As this kind of breakdown is most common with resh and medial vav, and as the expected effect is that the estimated number of homonyms should be significantly reduced, it is not surprising that for vav and resh the estimates proved to be significantly higher than expected.

What all of this shows me is that the observed distribution of homonyms is very close to what would be expected from a random distribution of verbs. That tells us something about how the roots are distributed. It also, it seems to me, tells me that the people who detected homonyms among these verbs are on the right track, for they are consistently detecting the statistically expected number of homonyms.

PS Sorry if we are boring others on the list with this.

--
Peter Kirk
E-mail: peter AT qaya.org
Blog: http://speakertruth.blogspot.com/
Website: http://www.qaya.org/





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