Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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3.2. NONLINEAR MODELING OF TIME SERIES 33 Equation (2.9) (linear prediction model) is a time-invariant linear system where I have replaced the output by the one-step ahead prediction of the input. Similarly, I can consider equation (3.12) and construct a time-variant nonlinear prediction operator with a nonlinear Volterra model of the form (order of p, q, r): xn = + + p ai xn−i i=1 q q bjk xn−jxn−k j=1 k=1 r r r clms xn−lxn−mxn−s l=1 m=1 s=1 + εn , (3.13) where ai, bjk, and clms are the linear, the nonlinear quadratic, and the nonlinear cubic impulse responses of the nonlinear system, respectively, also known as Volterra kernels. The first term on the right hand side of equation (3.13) represents the classical linear prediction problem with a prediction operator of order p. The second and third terms of equation (3.13) represent the expansion of the signal in terms of quadratic and cubic nonlinearities. The modeling error is given by εn. Note that equation (3.13) is the nonlinear AR formulation as an extension of linear AR modeling. The last expression can also be written in prediction error form εn = xn − − − p ai xn−i i=1 q q bjk xn−jxn−k j=1 k=1 r r r clms xn−lxn−mxn−s . (3.14) l=1 m=1 s=1
3.2. NONLINEAR MODELING OF TIME SERIES 34 White Noise Sequence et crrr bqq ap c b2q c222 c221 b22 c b c112 c111 b z + Σ Σ 2rr a 2 1rr 1q 11 1 − z a z xt AR Sequence Figure 3.3: Volterra AR diagram. Modified from Marple (1987) and Ulrych and Sacchi (2005). The coefficients of the linear term of the Volterra series are given by a1, · · · , ap, the coefficients of the quadratic term of the Volterra series are given by b11, · · · , bqq, and the coefficients of the cubic term of the Volterra series are given by c111, · · · , crrr. Equation (3.14) clearly states that I have more flexibility to model the deterministic part of the complex signal when the nonlinear terms are incorporated in the model. The third-order Volterra representation of the data requires the estimation of p + q 2 + r 3 coefficients: ai, i = 1, . . . , p, bjk, j, k = 1, . . . , q, and clms, l, m, s = 1, . . . , r. I will use symmetry properties of the Volterra series to reduce the number of coefficients of the quadratic contribution from q 2 to (q (q + 3)/2 − q) and the number of coefficients of the cubic contribution from r 3 to (r 2 + r!/(r − 3)!3!). Table 3.1 shows various filter lengths and the number of prediction coefficients using symmetry arguments of a Volterra series. It can be seen that the number of parameters increases dramatically with a small increment in the filter order. Again, I can use forward and backward prediction x f p n = a i=1 f q q i xn−i+ b j=1 k=1 f jk xn−jxn−k+ r r r l=1 m=1 s=1 c f lms xn−lxn−mxn−s + εn , (3.15)
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: 3.2. NONLINEAR MODELING OF TIME SER
Page 51 and 52: 3.2. NONLINEAR MODELING OF TIME SER
Page 53 and 54: 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
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5.4. REAL DATA EXAMPLES 84 Time [s]
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Page 103 and 104:
Page 105 and 106:
5.5. SUMMARY 90 5.5 Summary In this
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series in this thesis has been trun
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6.1. FUTURE DIRECTIONS 94 Noise red
Page 111 and 112:
BIBLIOGRAPHY 96 Canales, L. L. (198
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BIBLIOGRAPHY 98 Marple, S. L. (1980
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BIBLIOGRAPHY 100 Trad, D. (2003). I
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Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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