Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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3.3. 1-D SYNTHETIC AND REAL DATA EXAMPLES 41 clear that quadratic and cubic terms are needed to capture the strong variability observed in the time series. In Figure 3.5(c) I portray the contribution due to the linear terms of a third- order Volterra series to predicted the data in Figure 3.5(b) due to the linear terms of third order Volterra series. Note that this contribution is negligible and shows that linear terms can not model the highly nonstationary part of signal. Figures 3.5(d) and 3.5(e) illustrate the parts of the prediction associated with quadratic (q = 8) and cubic (r = 8) terms in the third-order Volterra series. Figure 3.5(f) shows the contribution of nonlinear terms associated with quadratic and cubic terms (q = 8 and r = 8). It is clear that nonlinear quadratic and cubic terms are required to properly model the full aperture. A real data example corresponding to the so called Arctic oscillation time series (AO)-a time series from 1950 to 1999 of sea level pressures-is used to characterize the long term variability of nonseasonal sea level oscillations (Thomson, 2004). Figures 3.6(a) and 3.7(a) show the nonlinear AO data for the period from 1950 to 1999. The data consist of 104 samples (3 observations per year-January, February - March). Figures 3.6(b) and 3.6(c) illustrate predicted AO values for a Volterra system consisting of linear and nonlinear terms (p = 10, q = 10, and r = 10) and associated error, respectively. Figures 3.6(d) and 3.6(e) represent our attempt to model the data with a linear prediction filter (p = 10) and corresponding error. Again, it is clear that the dynamics of the time series is better captured by the third-order Volterra system. Figure 3.7(b) is the prediction using a third-order Volterra series with p = 10, q = 10, and r = 10. In Figure 3.7(c) I portray the part of the prediction attributed
3.3. 1-D SYNTHETIC AND REAL DATA EXAMPLES 42 Samples −4 −2 0 2 0 50 100 (a) Amplitude Samples Samples −4 −2 0 2 0 50 100 (b) (c) Amplitude −4 −2 0 2 0 50 100 (d) (e) Amplitude Samples Samples −4 −2 0 2 0 50 100 Amplitude −4 −2 0 2 0 50 100 (e) (f) Amplitude Figure 3.4: 1-D synthetic data for comparison of prediction between linear prediction theory and third order Volterra series. (a) Original data. (b) Prediction using a third-order Volterra series (p = 8, q = 8, and r = 8). (c) The error between the original data and the third-order Volterra prediction. (d) Prediction using the first order Volterra series, which is equivalent to linear prediction (p = 8). (e) The error between the original data and linear prediction. to the linear kernel (p = 10). Figures 3.7(d) and 3.7(e) show the parts of the prediction associated with quadratic (q = 10) and cubic (r = 10) terms in the third-order Volterra series. In addition, Figure 3.7(f) illustrates the part of the
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Name of Author: Soner Bekleric Univ
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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
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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 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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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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