Biblical Hebrew Forum

[Peter Kirk <peter AT qaya.org>]

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/