Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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3.2. NONLINEAR MODELING OF TIME SERIES 39 where Linear, Quadratic, and Cubic denote the matrices of linear, quadratic, and cubic filter coefficients respectively: and ⎢ Quadratic = ⎢ ⎣ ⎡ ⎢ Cubic = ⎢ ⎣ ⎡ ⎢ Linear = ⎢ ⎣ ⎡ xl+1 xl+2 · · · xl+p . . .. xm xm · · · xm . . xn−1 xn−2 · · · xn−p x 2 l+1 2xl+1xl+2 2xl+1xl+3 · · · x 2 l+q . . .. x 2 m 2xmxm 2xmxm · · · x 2 m . . . . . . . . . . .. x 2 n−1 2xn−1xn−2 2xn−1xn−3 · · · x 2 n−q ⎤ ⎥ , (3.26) ⎥ ⎦ . . ⎤ ⎥ , (3.27) ⎥ ⎦ x 3 l+1 · · · 3x 2 l+1xl+2 3x 2 l+1xl+3 · · · 6xl+1xl+2xl+3 · · · x 3 l+r . . . . x 3 m · · · 3x 2 mxm 3x 2 mxm · · · 6xmxmxm · · · x 3 m . . . . x 3 n−1 · · · 3x 2 n−1xn−2 3x 2 n−1xn−3 · · · 6xn−1xn−2xn−3 · · · x 3 n−r . . . . . . .. . . ⎤ ⎥ . ⎥ ⎦ (3.28) The unknown vector m contains linear and nonlinear prediction coefficients orga- nized in lexicographic form in the dimension of P + Q(Q+3) 2 − Q + R2 + R! ) × 1. (R−3)!3!
3.3. 1-D SYNTHETIC AND REAL DATA EXAMPLES 40 It is clear that the problem is linear in the coefficients. One possible solution vector is given by the regularized least squares solution (damped least squares). Adopt- ing the least squares method with zero-order quadratic regularization leads to the solution of the filter coefficients: m = (A T A) + µI) −1 A T d . (3.29) For small systems (small order of parameters p, q, and r) one can use direct inversion methods to solve equation (3.29). For systems that involve long operators, I suggest the use of semi-iterative solvers like the method of conjugate gradients (Wang and Treitel, 1973). In our examples, however, I have adopted direct inversion methods. In general, f − x algorithms do not require the inversion of large systems of equations. In many cases this is a consequence of working with small spatial windows. 3.3 1-D Synthetic and Real Data Examples I have developed an algorithm to invert the coefficients of a third-order Volterra series. I focus on a 1-D synthetic time series which is generated with a real second- order Volterra system. Figures 3.4(a) and 3.5(a) show a 1-D input data for 100 samples. Figures 3.4(b) and 3.4(c) represent the predicted series modeled via the third- order Volterra series (with parameters p = 8, q = 8, and r = 8) and the associated modeling error, respectively. Figures 3.4(d) and 3.4(e) portray the predicted data using linear prediction theory and modeling error (p = 8, q = 0, and r = 0). It is
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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 53: 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]
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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
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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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