Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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3.2. NONLINEAR MODELING OF TIME SERIES 37 Symmetry arguments in equations (3.19, 3.20 and 3.21) are also valid for the backward prediction. The number of coefficients for the cubic part is reduced from r 3 to r 2 + r!/(r − 3)!3! and the total number of prediction coefficients is reduced form p + q 2 + r 3 to p + (q (q + 3)/2 − q) + r 2 + r!/(r − 3)!3!. and The form of the forward and backward prediction equations is now given by: x f n = a f 1xn−1 + a f 2xn−2 · · · + a f pxn−p + b f 11x 2 n−1 + 2b f 12xn−1xn−2 + 2b f 13xn−1xn−3 + · · · + b f qqx 2 n−q + c f 111x 3 n−1 + 3c f 112x 2 n−1xn−2 + · · · + 6c f 123xn−1xn−2xn−3 + · · · + c f rrrx 3 n−r + εn , (3.22) x b n = a b 1xn+1 + a b 2xn+2 · · · + a b pxn+p + b b 11x 2 n+1 + 2b b 12xn+1xn+2 + 2b b 31xn+1xn+3 + · · · + b b qqx 2 n+q + c b 111x 3 n+1 + 3c b 112x 2 n+1xn+2 + · · · + 6c b 123xn+1xn+2xn+3 + · · · + c b rrrx 3 n+r + εn . (3.23) As in the linear prediction problem, I will assume that the Volterra coefficients (linear, quadratic, and cubic) are obtained using actual observations. For example, assume p = 1, q = 2, c = 3, and a time series of N = 7 points:
3.2. NONLINEAR MODELING OF TIME SERIES 38 x1 = a1x2 + b11x 2 2 + 2b12x2x3 + b22x 2 3 + c111x 3 2 + · · · + 3c112x 2 2x3 + · · · + 6c123x2x3x4 + ε1 x2 = a1x3 + b11x 2 3 + 2b12x3x4 + b22x 2 4 + c111x 3 3 + · · · + 3c112x 2 3x4 + · · · + 6c123x3x4x5 + ε2 x3 = a1x4 + b11x 2 4 + 2b12x4x5 + b22x 2 5 + c111x 3 4 + · · · + 3c112x 2 4x5 + · · · + 6c123x4x5x6 + ε3 x4 = a1x5 + b11x 2 5 + 2b12x5x6 + b22x 2 6 + c111x 3 5 + · · · + 3c112x 2 5x6 + · · · + 6c123x5x6x7 + ε4 x4 = a1x3 + b11x 2 3 + 2b12x3x2 + b22x 2 2 + c111x 3 3 + · · · + 3c112x 2 3x2 + · · · + 6c123x3x2x1 + ε4 x5 = a1x4 + b11x 2 4 + 2b12x4x3 + b22x 2 3 + c111x 3 4 + · · · + 3c112x 2 4x3 + · · · + 6c123x4x3x2 + ε5 x6 = a1x5 + b11x 2 5 + 2b12x5x4 + b22x 2 4 + c111x 3 5 + · · · + 3c112x 2 5x4 + · · · + 6c123x5x4x3 + ε6 x7 = a1x6 + b11x Linear 2 6 + 2b12x6x5 + b22x 2 5 + c111x Quadratic 3 6 + · · · + 3c112x 2 6x5 + · · · + 6c123x6x5x4 +ε7 Cubic or in matrix form: ⎡ ⎤ ⎢ xl ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ xm ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ ⎥ ⎣ ⎦ xn ⎛ ⎞ ⎜ ⎝ d N × 1 ⎟ ⎠ = ⎡ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ Linear | Quadratic | Cubic ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎛ ⎜ ⎝ A N × (P + Q(Q+3) 2 − Q + R2 + R! (R−3)!3! ) ⎤ ⎞ ⎟ ⎠ × ⎡ ⎤ ⎢ a1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ap ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ b11 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ bqq ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ c111 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ ⎥ ⎣ ⎦ crrr m + ⎡ (3.24) ⎤ ⎢ εl ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ εl+1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ ⎥ ⎣ ⎦ εn ⎛ ⎞ ⎜ ⎝ ε N × 1 ⎟ ⎠ (3.25)
Page 1 and 2: Name of Author: Soner Bekleric Univ
Page 3 and 4: University of Alberta Faculty of Gr
Page 5 and 6: Abstract Linear filter theory has p
Page 7 and 8: Contents 1 Introduction 1 1.1 Appli
Page 9 and 10: List of Tables 3.1 Filter length an
Page 11 and 12: 4.4 (a) Prediction of Figure 4.2(b)
Page 13 and 14: List of symbols Symbol Name or desc
Page 15 and 16: List of abbreviations Abbreviation
Page 17 and 18: In applied seismology predictive de
Page 19 and 20: 1.1. APPLICATIONS 4 applied to vide
Page 21 and 22: 1.3. THESIS OUTLINE 6 • Chapter 4
Page 23 and 24: 2.2. LINEAR PROCESS 8 Finally, I pr
Page 25 and 26: 2.3. LINEAR PREDICTION 10 E(z) 1 X(
Page 27 and 28: 2.3. LINEAR PREDICTION 12 In my the
Page 29 and 30: 2.3. LINEAR PREDICTION 14 which can
Page 31 and 32: 2.3. LINEAR PREDICTION 16 ∂ρ fb
Page 33 and 34: 2.3. LINEAR PREDICTION 18 I have an
Page 35 and 36: 2.4. 1-D SYNTHETIC AND REAL DATA EX
Page 39 and 40: 2.6. SUMMARY 24 Relative PSD Relati
Page 41 and 42: Chapter 3 Nonlinear Prediction 3.1
Page 43 and 44: 3.1. NONLINEAR PROCESSES VIA THE VO
Page 45 and 46: 3.1. NONLINEAR PROCESSES VIA THE VO
Page 47 and 48: 3.2. NONLINEAR MODELING OF TIME SER
Page 49 and 50: 3.2. NONLINEAR MODELING OF TIME SER
Page 51: 3.2. NONLINEAR MODELING OF TIME SER
Page 61 and 62: 3.4. SUMMARY 46 3.4 Summary In this
Page 63 and 64: 4.1. LINEAR PREDICTION IN THE F −
Page 65 and 66: 4.2. ANALYSIS OF OPTIMUM FILTER LEN
Page 67 and 68: 4.2. ANALYSIS OF OPTIMUM FILTER LEN
Page 69 and 70: 4.3. NONLINEAR PREDICTION OF COMPLE
Page 73 and 74: RMSE 2 4.3. NONLINEAR PREDICTION OF
Page 77 and 78: 4.4. NOISE REMOVAL AND VOLTERRA SER
Page 79 and 80: 4.5. SYNTHETIC AND REAL DATA EXAMPL
Page 89 and 90: 4.6. SUMMARY 74 4.6 Summary In this
Page 91 and 92: 5.1. INTRODUCTION 76 ples remains a
Page 93 and 94: 5.3. SYNTHETIC DATA EXAMPLES 78 Not
Page 95 and 96: 5.3. SYNTHETIC DATA EXAMPLES 80 Tim
Page 97 and 98: 5.3. SYNTHETIC DATA EXAMPLES 82 Tim
Page 99 and 100: 5.4. REAL DATA EXAMPLES 84 Time [s]
Page 101 and 102: 5.4. REAL DATA EXAMPLES 86 Time [s]
Page 103 and 104:
5.4. REAL DATA EXAMPLES 88 Time [s]
Page 105 and 106:
5.5. SUMMARY 90 5.5 Summary In this
Page 107 and 108:
series in this thesis has been trun
Page 109 and 110:
6.1. FUTURE DIRECTIONS 94 Noise red
Page 111 and 112:
BIBLIOGRAPHY 96 Canales, L. L. (198
Page 113 and 114:
BIBLIOGRAPHY 98 Marple, S. L. (1980
Page 115 and 116:
BIBLIOGRAPHY 100 Trad, D. (2003). I
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Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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