Been there, done that. You are not telling me anything that I do not already know.
My EXCEL program calculates indices for any number or decks and any count, balanced or unbalanced, but I concentered on the infinite deck approximation.
And if I used two counts, I used Excel Solver to find the values of k1 and k2, for example, KO + k1*(5m7c) + k2*(AA89mTc) that maximized the absolute value of the CC between the tag values of the derived count an the EoR from BJA which EoR were calculated to four or five digits using combinatorial analysis.
I developed the LSL technique ETFAN was using Proportional Deflection (PD). ETFAN vefifid that my Excel file was correct. We tested each on several examples and LSL and PD produced identical results, adding more credibility that I did things correctly. Also indices generated from my Excel program for the HL for example, corresponded to the HL indices published. So another indication that things were done correctly. And also my formulas for playing strategy changes make logical sense also.
As another level of proof that I have done things correctly, every time I gave Gronbog a new set of strategy changes with values of "k" and a new index, the SCORE improved, every single time. For HL with AA78mTc I gave Gronbog some very crazy and insignificant soft doubling indices with new values of "k" and indices. And the SCORE improved, very slightly as expected, but it did improve. If i had made an error, the simluations would have shown in and the SCORE woudl not have increased. But the SCORE increased EVERY time I gave Gronbog new indices and values of "k".
I also tested my LSL linear combination of counts tecnhique against examples that Griffin used in his Therooy of Blackjack with his complicaed correlation coefficient matrix and I got the same results. My calcluations are CORRECT.
I have a book published in 2016 called Blackjack KO with Table of Critical Running Counts which I am redoing along with redoing KO with 5m7c and AA89mTc to make them easier to read. But in that book I covered what you mentioned above and much, much more including more formulas and relationships between infinite deck indices that I derived.
So here is how I described what I did in a footnote in my book Blackjack KO with Table of Critical Running Counts.
The multiple linear regression method used in this paper consists of defining a new count, X2 = X0 + k*(X1), and finding the value of ‘k’ that maximizes the absolute value of the infinite deck correlation coefficient between the tag values of X2 and Y = Effects of Removal, EoR. m = Slope(EoR, X2) = Slope(Y, X2) is the same whether X2 is balanced or not, that is, it is not necessary to convert X2 to a balanced count before calculating the infinite deck index. Once the value of ‘k’ is determined so that X2 is now defined, then the slope of the least squares line (LSL) between Y = EoR and X2 is calculated, i.e., m = slope of LSL = AAC (Average Advantage Change) in EV for X2 increasing by one and for 51 cards remaining. This is because Y = EoR is given for a single deck and so EoR(c) = EoR for card ‘c’ removed from a single deck = EV(51 cards remaining after removing card ‘c’) – EV(full deck) so 51 cards are remaining. See Effects of Removal Definition exhibit under Derivations in the Appendix. But AAC in EV for 52 cards remaining is required, i.e., AACpTCp = AAC per True Count point is required. So AACpTCp is calculated as AACpTCp = (51/52)*(AAC) = (51/52)*(slope). Then infinite deck index is calculated as Idx = FDHA / AACpTCp. Griffin used a slightly different (and more complicated and abstract, in my opinion) procedure where he constructs a 3 x 3 correlation coefficient matrix of X0, X1 and Y. The multiple linear regression method has intuitive appeal since as ‘k’ varies, X0 + k*(X1) can be seen to represent a series of various derived counts to choose from, each with its own correlation coefficient with EoR. The value of ‘k’ giving the best count is the value of ‘k’ that maximizes the absolute value of the correlation coefficient of the tag values of the derived count, X2, and EoR for the given playing strategy situation under consideration. See the exhibits under Calculation of KO Index for Standing on hard 15 v T in the Appendix for an example of these calculations. I used this technique to find the value of ‘k’ in KO + k*(5m7c) that maximized the absolute value of the infinite deck correlation coefficient between the tag values of KO + k*(5m7c) counts for each given situation’s EoR. Once the value of ‘k’ had been determined, then a LSL was fitted to the KO + k*(5m7c) count and the EoR and the index calculated. As a check, the proportional displacement (PD) technique was also used to calculate the index of KO + k*(5m7c) count. Both LSL and PD techniques produced identical indices.
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