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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 00:29:42 +0000

On 18/01/2007 19:46, K Randolph wrote:
...
Notice that the shin has the second lowest multiple. The sin and shin
together have a multiple still on the low side of the average of 1.784
(taken from the total row).

And sin has the highest multiple. You can't win that way. Also you can't put sin and shin together in these statistics by simple addition, so your added calculation is not valid - although in practice I accept that sin and shin taken together would fall in the middle of the range, if you accept sin-shin pairs as real homonyms.

Even though I disagree with the lexicographic method, the results seem
to imply even stronger than defining by action that the sin and shin
together were originally one letter. (Defining according to action
gives an observed frequency closer to the predicted numbers than to
the "observed" frequency from your received list.)

I would be interested in your statistics on the number of homonym for each letter defined "by action". I assume that you have these numbers as without them you could not observe a frequency.

To repeat my original observation, I noticed that sin and shin
together are observed to have a lower frequency of homonyms than
expected, especially lower than expected if they were originally two
phonemes. That, along with other clues, caused me to raise the
question, were they originally one phoneme with one letter to
represent that one phoneme?

Well, let us see the statistics on which you based your original observation.

My calculations suggest that the distribution of verb roots is essentially random over the space of possible three letter roots, with no special constraint on duplicates. The implication of this is that there would be no difference in the frequency of homonyms between simple letters and those which were originally multiple letters; and thus that this method of counting homonyms is not helpful in deciding whether sin and shin were originally separate.

A bit more analysis. Imagine verb roots to be distributed within a cube, with one dimension of the cube for each of the three root letters. The cube is subdivided into zones for each letter in each dimension, with the width of the zone corresponding to the overall probability of the root letter being that letter. My hypothesis is that the roots are distributed essentially randomly in this three dimensional space. On this hypothesis:

Let p(1,A) be the probability that the first root letter is A (standing for an unknown letter, as do B and C), and p(ABC) be the probability of the verb ABC. Let N be the total number of roots. Thus:

p(ABC) = p(1,A) * p(2,B) * p(3,C)

For any ABC, p(ABC) is rather small. The probable number of homonyms for the verb ABC comes out close to (1/2)N^2 * (p(ABC))^2; I can't calculate the exact figure but the difference from this is very small for small p(ABC) and large N.

Thus the expected total number of homonyms with A as the first root letter, H(1,A), is the sum over all B and C of the above:

H(1,A) = sum over all B and C of ((1/2)N^2 * p(1,A)^2 * p(2,B)^2 * p(2,C)^2)
= (1/2)N^2 * p(1,A)^2 * sum over all B of (p(2,B)^2) * sum over all C of (p(2,C)^2)

And of course similarly for H(2,B) and H(3,C).

H(1,A) is certainly proportional to the square of p(1,A) as stated before, as the other factors are independent of A. The variation from this observed square law for root letter resh may be because the probabilities with this letter are high enough that the approximations break down.

Plugging this into my spreadsheet (for sin and shin treated as distinct), I estimate a total of 285 homonyms, compared with 267 actually listed in my source. Adding together for each root letter (i.e. H(1,A) + H(2,A) + H(3,A)) I get the following figures, second column is estimates and third is observed:

letter 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
total 856 801

Most of these figures seem to be in remarkably close agreement considering how crude the original hypothesis was, that the verbs are distributed randomly in the three dimensional space with no mutual interference. As observed before, the number of homonyms for vav and resh seems to be significantly below the estimate, and this discrepancy is sufficient to account for the difference between the estimated and observed totals. The other observed figures are distributed equally on either side of the estimates, and the differences are little more than could be expected statistically. As the standard deviation is approximately the square root of the estimate, I can say note that the observations for 13 letters are within 1 standard deviation of the estimate (expected 68% of the time, I think), and another 6 are within 2 standard deviations (expected 95% within 2 standard deviations, I think, observed 19 out of 24). Resh and vav are both close to three standard deviations below estimate. Zayin is nearly four standard deviations above estimate and tsade nearly three standard deviations above estimate, but shin is nearly 2.5 standard deviations below estimate. This doesn't look like totally random behaviour, especially as for both zayin and shin (but not tsade) the discrepancy relates almost entirely to cases where this letter is the first root letter. But this deviation from randomness doesn't seem to be linked to sin and shin being separate letters or not.

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





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