Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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2.3. LINEAR PREDICTION 15 useful in predicting short data records (Robinson and Treitel, 1980). Burg’s method first estimates the reflection coefficients by minimizing the forward and backward prediction errors. The accuracy of the method decreases for higher-order AR prediction models, and time series with longer data samples. Similarly to the Yule-Walker equations, the Levinson recursion determines prediction coefficients defined as the last autoregressive parameter estimates for each model order p (Marple, 1987). I follow Marple’s (1987) way to obtain the equations of Burg’s algorithm. and The forward and backward prediction errors are, respectively, given by: ε f n = x f p n + a i=1 f i xn−i , (2.22) ε b n = x b p n−p + a i=1 b ixn+i. (2.23) Burg’s method estimates prediction coefficients via a least squares solution. Mini- mizing the arithmetic mean of forward and backward prediction error power subject to recursion similar to equations (2.22) and (2.23) as follows: ρ fb p = 1 1 [ 2 N N n=p+1 |ε f p| 2 + 1 N N n=p+1 |ε b p| 2 ] . (2.24) where N denotes the length of recorded data. Note that only data that has been recorded is used in the summation. Therefore, ρ fb p is assumed as a single parameter, so that the prediction coefficient kp is a prediction coefficient. Prediction errors from order p − 1 are found and by setting the complex derivative of the equation (2.24) to zero
2.3. LINEAR PREDICTION 16 ∂ρ fb p ∂Rekp + i ∂ρfb p ∂Imkp = 0 (2.25) where Re and Im are real and imaginary parts, respectively, of the complex derivative. A least squares solution ensures a solution for prediction coefficients, kp as follows: kp = where ∗ is the complex conjugation. −2 N n=p+1 ε f p−1(n)εb∗ p−1(n − 1) Nn=p+1 |ε f p−1(n)| 2 + N n=p+1 |εb p−1(n − 1)| 2 (2.26) 2.3.4 Computing the AR coefficients without limiting the aperture The Yule-Walker and Burg algorithms are often used in signal processing algorithms for their computational efficiency at the time of computing prediction error coefficients. In what follows I will provide a method that I have introduced in order to solve for the coefficients of the linear prediction problem using only the data that are available. In other words, I will avoid creation of the correlation matrix and directly posed the problem as a least-squares problem. Consider a filter length p = 3 and a time series of length N = 7. Using equations (2.12) and (2.13) I can write the following equations:
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: 2.3. LINEAR PREDICTION 14 which can
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 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
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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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