Bidirectional Mean Averaging and The Wave Matrix
1 - Oscillation Data Set
A = {X1, X2, X3, ... , Xn-2, Xn-1, Xn}
A - oscillation data set is any set of X values oscillating on the x-axis.
oscillating X values can be obtained by removing any regression components.
possible regression components are:
constant, linear, exponential, logarithmic, power, geometric, polynomial, etc.
2 - Up Mean Averaging
U1 = X1
U2 = (U1 + X2 ed) / (1 + ed)
U3 = (U2 + X3 ed) / (1 + ed)
...
Un-2 = (Un-3 + Xn-2 ed) / (1 + ed)
Un-1 = (Un-2 + Xn-1 ed) / (1 + ed)
Un = (Un-1 + Xn ed) / (1 + ed)
Up Mean Averaging Set - U = {U1, U2, U3, ... , Un-2, Un-1, Un}
3 - Down Mean Averaging
Dn = Xn
Dn-1 = (Dn + Xn-1 ed) / (1 + ed)
Dn-2 = (Dn-1 + Xn-2 ed) / (1 + ed)
...
D3 = (D4 + X3 ed) / (1 + ed)
D2 = (D3 + X2 ed) / (1 + ed)
D1 = (D2 + X1 ed) / (1 + ed)
Down Mean Averaging Set - D = {D1, D2, D3, ... , Dn-2, Dn-1, Dn}
4 - Bidirectional Mean Averaging
B = (U + D) / 2
¯
Y1 = (U1 + D1) / 2
Y2 = (U2 + D2) / 2
Y3 = (U3 + D3) / 2
...
Yn-2 = (Un-2 + Dn-2) / 2
Yn-1 = (Un-1 + Dn-1) / 2
Yn = (Un + Dn) / 2
Bidirectional Mean Averaging Set - Bma(A,d) = B = {Y1, Y2, Y3, ... , Yn-2, Yn-1, Yn}
A - oscillation data set.
d - degree of weighting data.
-¥ £ d £ +¥, d is any real number.
lim d ® +¥, Bma(A,d) = B = A
lim d ® -¥, Bma(A,d) = B = (X1 + Xn ) / 2
5 - Iteration of Bidirectional Mean Averaging
B1 = Bma(A,d)
B2 = Bma(B1,d)
B3 = Bma(B2,d)
...
Bi-2 = Bma(Bi-3,d)
Bi-1 = Bma(Bi-2,d)
Bi = Bma(Bi-1,d)
Iteration of Bidirectional Mean Averaging - Ibma(A,d,i) = Bi
A - oscillation data set.
d - degree of weighting.
i - iteration of averaging.
i ³ 1, i is any positive integer.
6 - Ibma as Wave Data Set and Remainder Set of Wave Data Set
Wave Data Set - W = Ibma(A,d,i)
R = A - W
¯
R1 = X1 - Y1
R2 = X2 - Y2
R3 = X3 - Y3
...
Rn-2 = Xn-2 - Yn-2
Rn-1 = Xn-1 - Yn-1
Rn = Xn - Yn
R = {R1, R2, R3, ... , Rn-2, Rn-1, Rn}
Remainder Set of Wave Data Set - R
7 - Iteration of Remainder Set, the Wave Matrix and Remainder Set of the Wave Matrix
W1 = Ibma(A,d,i) , R1 = A - W1
W2 = Ibma(R1,d,i) , R2 = R1 - W2
W3 = Ibma(R2,d,i) , R3 = R2 - W3
...
Wj-2 = Ibma(Rj-3,d,i) , Rj-2 = Rj-3 - Wj-2
Wj-1 = Ibma(Rj-2,d,i) , Rj-1 = Rj-2 - Wj-1
Wj = Ibma(Rj-1,d,i) , Rj = Rj-1 - Wj
The Wave Matrix - Wm(A,d,i,j) = {W1, W2, W3, ... , Wj-2, Wj-1, Wj}
Remainder Set of the Wave Matrix - Rm(A,d,i,j) = Rj
A - oscillation data set.
A = W1 + W2 + W3 + ... + Wj-2 + Wj-1 + Wj + Rj
d - degree of weighting.
i - iteration of averaging.
j - iteration of the wave data set.
j ³ 1, j is any positive integer.
lim j ® +¥, Rj = 0
8 - Determining d Values in the Wave Data Set through Root Mean Square (RMS) Equivalence and Degree of Weighting Set
RMS of Wave Data Set - rmsW = Ö(S (W)2) / n
= Ö(S Ibma(A,d,i)2) / n
= [a = 1 to a = n] Ö(S (Ya)2) / n
RMS of Remainder Data Set - rmsR = Ö(S (R)2) / n
= Ö(S (A - W)2) / n
= Ö(S (A - Ibma(A,d,i))2) / n
= [a = 1 to a = n] Ö(S (Ra)2) / n
Determined d Value - d à rmsW = rmsR
Read as: d is determined when rmsW is equal to rmsR.
the value of d is found through a feedback root find algorithm.
W1 = Ibma(A,d1,i) , R1 = A - W1 , d1 à rmsW1 = rmsR1
W2 = Ibma(R1,d2,i) , R2 = R1 - W2 , d2 à rmsW2 = rmsR2
W3 = Ibma(R2,d3,i) , R3 = R2 - W3 , d3 à rmsW3 = rmsR3
...
Wj-2 = Ibma(Rj-3,dj-2,i) , Rj-2 = Rj-3 - Wj-2 , dj-2 à rmsWj-2 = rmsRj-2
Wj-1 = Ibma(Rj-2,dj-1,i) , Rj-1 = Rj-2 - Wj-1 , dj-1 à rmsWj-1 = rmsRj-1
Wj = Ibma(Rj-1,dj,i) , Rj = Rj-1 - Wj , dj à rmsWj = rmsRj
Degree of Weighting Data Set - d = {d1, d2, d3, ... , dj-2, dj-1, dj}
RMS Adjusted Wave Matrix - Wm(A,d,i,j) = {W1, W2, W3, ... , Wj-2, Wj-1, Wj}
RMS Adjusted Remainder Set of the Wave Matrix - Rm(A,d,i,j) = Rj
My Self Picks are optimized to produce
the most number of wins with
the least amount of effort.
Order is a subset of Chaos.
Thank You,
Doug