Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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2.3. LINEAR PREDICTION 11 prediction of the input signal. Mathematically this is equivalent to the following model for the signal xn: xn = p aixn−i + εn, i=1 = a1xn−1 + a2xn−2 + . . . apxn−p + εn (2.10) where p is the order of the prediction filter an, n = 1, . . . , p. I have also introduced a prediction error εn to account for the part of the signal that cannot be properly predicted. These models are of importance for the analysis of time series in both time and frequency domain. This model is also called an Autoregressive model of order p (AR(p)). One can immediately see that the latter entails a regression model but rather that being a regression on arbitrary basis functions, the regression uses past values of the data one is trying to predict. The latter is the reason behind the name Auto-regressive model. The coefficients of the system (AR coefficients) can be estimated using various methods. Marple (1987) provides a review of methods for solving the prediction problem. In particular, one can adopt a minimum error formulation (least squares approach) to find the coefficients that minimize the error function J given by: J = (xn − n p ai xn−i) 2 . (2.11) The filter ai, i = 1 . . . p that minimizes the cost function J can be used to predict the signal and to compute the so called AR-spectral estimator (Marple, 1987). i=1
2.3. LINEAR PREDICTION 12 In my thesis I have adopted the least squares approach to estimate the coefficients of linear and non-linear prediction systems but bear in mind that other method (at least for the linear prediction problem) are also plausible. In the following section I will discuss three methods to estimate the coefficients a(i), i = 1 . . . p from the data x(n): the Yule-Walker method, Burg’s algorithm, and the least squares approach. 2.3.1 Forward and backward prediction Off-line processing allows computation of forward and backward prediction oper- ators. In other words, I can use past samples of data to predict future samples and/or I can use future samples of data to predict past samples. This allows us to formulate a problem with two systems of equations. One for forward prediction, and, the other for backward prediction, x f p n = a i=1 f i xn−i + εn, (2.12) x b p n = a i=1 b ixn+i + εn . (2.13) 2.3.2 Estimating AR coefficients via Yule-Walker Equations I can write equation (2.6) as follows X(z) = E(z) , (2.14) A(z) and the autocorrelation in the z domain (Ulrych and Sacchi, 2005) is
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: 2.3. LINEAR PREDICTION 10 E(z) 1 X(
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
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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 61 and 62: 3.4. SUMMARY 46 3.4 Summary In this
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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
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4.4. NOISE REMOVAL AND VOLTERRA SER
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4.5. SYNTHETIC AND REAL DATA EXAMPL
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4.6. SUMMARY 74 4.6 Summary In this
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5.1. INTRODUCTION 76 ples remains a
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5.3. SYNTHETIC DATA EXAMPLES 78 Not
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5.3. SYNTHETIC DATA EXAMPLES 80 Tim
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5.3. SYNTHETIC DATA EXAMPLES 82 Tim
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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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