Generalized Correlation Formula

From Stanford Wong's BJ21

Posted by MathProf on 6 Jun 1998, 6:57 a.m.

T-Hopper asked in a thread below about generalize correlations. I believe that this is the formula he wanted, and thought it was worth a new thread.

Let us say we have several counts A1,A2,A3, ... and we wish to combine these into one count, by adding them together. They may be unbalanced and correlated with each other. We want to compute the correlation of the combined system, with respect to some variable B (like Betting Advantage, for example.)

First define for any count A, with tag values X1,X2,X,3...X13

    VA = [ Sum(Xi^2)-Sum(Xi)^2 ] /N

Where N is the sum of your weights. Ordinarily, N will be 13. However if you decided to distinguish suits (like count Red 7s different than Black 7s, or counting the 5 of Spades as a special card) you should use an N of 26 or even 52. You should use the same system of weights (N) for all the counts that you are combining.

    NA = square root (VA)

Note that VA has the same form as Variance. Also note that if the count is balanced, it corresponds to Griffin's Sum of Squares.

Now the correlation we will have will be a fraction. The Numerator will be

   Num = Sum {corr[i] * Ni} = corr(1)*NA1 + corr(2)*NA2 + ...  

Here Ni is the Ni for the count Ai, and corr(i) is the correlation of the ith count with the key Black Jack variable B.

The denominator is a bit more difficult. It is the square root of the sum of two different types of terms.

  Denom = sqrt (D1 + D2)  


   where  D1 = Sum {NAi^2 } = Sum{VAi} = VA1 + VA2 + VA3 + ... 

          D2 = Sum {NAi *Naj * corr(i,j) } , taken over pairs i,j: i<j

Here corr(i,j) is the correlation between count i and count j. Note that D2 is 0 when all these counts are uncorrelated. As an example, if you had three counts A1,A2,A3, then D1 and D2 would be

   D1 =  NA1*NA1 +  NA2*NA2 + NA3*NA3 = V1 + V2 + V3 

   D2 = 2 * NA1 *NA2 * corr(A1,A2) + 2 * N1 *N3 * corr(1,3) 

           + 2 * N2 *N3 * corr(2,3) 

T-Hopper: please let me know if you have any trouble applying this formula.