Thursday, October 13, 2011

Angular Separation as the Distance Correlation

One can use other numbers besides powers of 2 for the number system of the random-number generator. Here 54 = 625 is the number of objects that are chosen from. A sample of 9001 numbers was generated. The generator seems to work better with larger numbers which is why 54 was used. The resulting numbers can be converted into their base 5 representation and the four resulting base 5 numbers could be used to represent a part of a sample drawn from 5 objects.

Instead of the autocorrelation of the string of numbers with shifted versions of itself, a dot product was taken and the resulting anglar separation was used as the distance correlation. The "demons" are still present and occur at the midpoint of the sample.

The average projection of the unit vectors of one member of the sample onto another is 3/4 so the average normal would be about √7/4. This corresponds to an angular separation of 41.41°.

As can be seen, the distribution of the angular separations agrees fairly well with the expected mean. One might expect some variation from it if the initial (reference) "direction" is atypical and therefore an outlier.

The simplest definition of a distance correlation for generator strings of the same length would probably be the angle between their unit vectors or,

dcorr(e1,e2) = acos(e1·e2).

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