Hypervector Analysis of the Voynich Manuscript

[ReneZ]

Darrin,

Another thing that for the moment worries me in the description of methodology (You are not allowed to view links. Register or Login to view.) is the assumption of these particular vectors fulfilling the requirement of angle close to 90 deg (cos(angle) = ~0) for two different random vectors. In order for this to be true, the dot product of the two vectors must be near 0. The only way I can see for these to accomplish that is if the mean vector component value of at least one of the random vectors is 0. Therefore, two random vectors composed of random 0 or 1 will not work because their mean component value will then be approx. 0.5 if they are truly random. The dot product of the two vectors will be 0.25 times the number of components (dimensions) according to my calculations (resulting in a lower angle, around 60 degrees).
However, random vectors composed of either -1 or 1 randomly will work (each random vector component centered around 0 with equal distance). Perhaps this is what is meant?

Keep in mind that the components of the hypervectors are bits. They can only take two values.
Prof. Kanerva uses the values -1 and +1 while Darrin uses 0 and 1.

Addition and multiplication are defined in the same space.

I wonder, therefore, if there is a bug in Darrin's approach (which may turn out to have no impact).
The bit-wise multiplication based on "-1" and "+1" is straightforward:

-1 x -1 = 1
-1 x 1  = -1
 1 x -1 = -1
 1 x 1  =  1

Translated to the use of 0 and 1, this gives:

0 x 0 = 1
0 x 1 = 0
1 x 0 = 0
1 x 1 = 1

This means that multiplication is the same as the logical 'equivalent' function.
However, in Darrin's example he uses the opposite (which is the logical 'exclusive or' function).

Then, at the end, when all tri-grams are added, Darrin does this in the 'natural numbers' space, and then converts the result to 0 or 1 depending on whether it is more or less than 'half'.

However, it may well be that prof. Kanerva intended a bitwise addition, which is the same as 'exclusive or'. This result can be obtained from the 'natural numbers' sum by checking whether the number is even or odd.

On this last point I am not certain what prof. Kanerva meant.