Biblical Hebrew Forum

[Peter Kirk <peter AT qaya.org>]

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/