Bidirectional Mean Averaging and The Wave Matrix

    1 - Oscillation Data Set

      A = {X₁, X₂, X₃, ... , X_n-2, X_n-1, X_n}

       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 e^d) / (1 + e^d)
      U₃ = (U₂ + X₃ e^d) / (1 + e^d)
      ...
      U_n-2 = (U_n-3 + X_n-2 e^d) / (1 + e^d)
      U_n-1 = (U_n-2 + X_n-1 e^d) / (1 + e^d)
      U_n  = (U_n-1 + X_n e^d) / (1 + e^d)

      Up Mean Averaging Set -  U = {U₁, U₂, U₃, ... , U_n-2, U_n-1, U_n}

    3 - Down Mean Averaging

      D_n  = X_n
      D_n-1 = (D_n + X_n-1 e^d) / (1 + e^d)
      D_n-2 = (D_n-1 + X_n-2 e^d) / (1 + e^d)
      ...
      D₃ = (D₄ + X₃ e^d) / (1 + e^d)
      D₂ = (D₃ + X₂ e^d) / (1 + e^d)
      D₁ = (D₂ + X₁ e^d) / (1 + e^d)

      Down Mean Averaging Set -  D = {D₁, D₂, D₃, ... , D_n-2, D_n-1, D_n}

    4 - Bidirectional Mean Averaging

        B = (U + D) / 2

              ¯

      Y₁ = (U₁ + D₁) / 2
      Y₂ = (U₂ + D₂) / 2
      Y₃ = (U₃ + D₃) / 2
      ...
      Y_n-2 = (U_n-2 + D_n-2) / 2
      Y_n-1 = (U_n-1 + D_n-1) / 2
      Y_n  = (U_n + D_n) / 2

      Bidirectional Mean Averaging Set -  B_ma(A,d) = B = {Y₁, Y₂, Y₃, ... , Y_n-2, Y_n-1, Y_n}
          A -  oscillation data set.
          d -  degree of weighting data.
              -¥ £ d £ +¥, d is any real number.
              lim d ®  +¥, B_ma(A,d) = B = A
              lim d ®  -¥, B_ma(A,d) = B = (X₁ + X_n ) / 2

  5 - Iteration of Bidirectional Mean Averaging

      B₁ = B_ma(A,d)
      B₂ = B_ma(B₁,d)
      B₃ = B_ma(B₂,d)
      ...
      B_i-2 = B_ma(B_i-3,d)
      B_i-1 = B_ma(B_i-2,d)
      B_i  = B_ma(B_i-1,d)

      Iteration of Bidirectional Mean Averaging -  I_bma(A,d,i) = B_i

          A - oscillation data set.   
          d - degree of weighting.
          i - iteration of averaging.
            i ³ 1, i is any positive integer.

    6 - I_bma as Wave Data Set and Remainder Set of Wave Data Set

        Wave Data Set -  W = I_bma(A,d,i)

       R = A - W

            ¯

      R₁ = X₁ - Y₁
      R₂ = X₂ - Y₂
      R₃ = X₃ - Y₃
      ...
      R_n-2 = X_n-2 - Y_n-2
      R_n-1 = X_n-1 - Y_n-1
      R_n = X_n - Y_n

       R = {R₁, R₂, R₃, ... , R_n-2, R_n-1, R_n}

      Remainder Set of Wave Data Set -  R 

    7 - Iteration of Remainder Set, the Wave Matrix and Remainder Set of the Wave Matrix

      W₁ = I_bma(A,d,i) ,    R₁ = A - W₁
      W₂ = I_bma(R₁,d,i) ,    R₂ = R₁ - W₂
      W₃ = I_bma(R₂,d,i) ,    R₃ = R₂ - W₃
      ...
      W_j-2 = I_bma(R_j-3,d,i) ,    R_j-2 = R_j-3 - W_j-2
      W_j-1 = I_bma(R_j-2,d,i) ,    R_j-1 = R_j-2 - W_j-1
      W_j  = I_bma(R_j-1,d,i) ,    R_j = R_j-1 - W_j

      The Wave Matrix -  W_m(A,d,i,j) = {W₁, W₂, W₃, ... , W_j-2, W_j-1, W_j}
      Remainder Set of the Wave Matrix -  R_m(A,d,i,j) = R_j

         A - oscillation data set.
            A = W₁ + W₂ + W₃ + ... + W_j-2 + W_j-1 + W_j + R_j
        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 ® +¥, R_j = 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 I_bma(A,d,i)²) / n
                                  =  [a = 1 to a = n] Ö(S (Y_a)²) / n

        RMS of Remainder Data Set -  _rmsR =  Ö(S (R)²) / n 
                                      =  Ö(S (A - W)²) / n
                                      =  Ö(S (A - I_bma(A,d,i))²) / n
                                      =  [a = 1 to a = n] Ö(S (R_a)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 = I_bma(A,d₁,i) ,    R₁ = A - W₁ ,  d₁ à _rmsW₁ = _rmsR₁
      W₂ = I_bma(R₁,d₂,i) ,    R₂ = R₁ - W₂ ,  d₂ à _rmsW₂ = _rmsR₂
      W₃ = I_bma(R₂,d₃,i) ,    R₃ = R₂ - W₃ ,  d₃ à _rmsW₃ = _rmsR₃
      ...
      W_j-2 = I_bma(R_j-3,d_j-2,i) ,    R_j-2 = R_j-3 - W_j-2 ,  d_j-2 à _rmsW_j-2 = _rmsR_j-2
      W_j-1 = I_bma(R_j-2,d_j-1,i) ,    R_j-1 = R_j-2 - W_j-1 ,  d_j-1 à _rmsW_j-1 = _rmsR_j-1
      W_j = I_bma(R_j-1,d_j,i)  ,      R_j = R_j-1 - W_j  ,      d_j à _rmsW_j = _rmsR_j

      Degree of Weighting Data Set -  d = {d₁, d₂, d₃, ... , d_j-2, d_j-1, d_j}

      RMS Adjusted Wave Matrix -  W_m(A,d,i,j) = {W₁, W₂, W₃, ... , W_j-2, W_j-1, W_j}
      RMS Adjusted Remainder Set of the Wave Matrix -  R_m(A,d,i,j) = R_j