Karjalainen, Pasi A. Regularization and Bayesian methods for ...

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z h H θ ˆθ(z) ˆθ ˜θ C(θ, ˆθ) B(ˆθ) B(ˆθ|z) ˆθ B trace (A) W l J R (H) N (H) ψ diag (λ 1 ,...,λ p ) L α S S ⊥ dim(S) K S rank(A) q −1 A(q) S(ω) Γ Λ U,V S K t E {x} Vector of observations z = (z 1 ,...,z n ) T Model for observation Linear model for observation Parameter vector Estimator of parameter vector θ based on observations z Estimate of parameter vector θ Estimation error Cost function Bayes cost Conditional Bayes cost Bayesian estimate Trace of matrix A Weighting matrix Least squares index Jacobian Range of H Nullspace of H Basis vector Diagonal matrix Regularization matrix Regularization parameter Subspace Orthogonal complement of S Dimension of S Matrix of basis vectors of subspace S Rank of matrix A Time delay operator Polynomial of operator q −1 Spectrum Matrix of eigenvectors Diagonal matrix of eigenvalues Matrices of left and right singular vectors, respectively Diagonal matrix of singular values Kalman gain The empirical expectation, sample mean
CONTENTS 1 Introduction 15 2 Estimation theory 19 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Probability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 Bayesian estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4 Maximum likelihood estimation . . . . . . . . . . . . . . . . . . . . 23 2.5 Bayes cost method . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.6 Mean square estimation . . . . . . . . . . . . . . . . . . . . . . . . 25 2.7 Maximum a posteriori estimation . . . . . . . . . . . . . . . . . . . 27 2.8 Linear minimum mean square estimator . . . . . . . . . . . . . . . 28 2.9 Minimum mean square estimator for Gaussian variables . . . . . . 29 2.10 Mean square estimation with observation model . . . . . . . . . . . 31 2.11 Gauss–Markov estimate . . . . . . . . . . . . . . . . . . . . . . . . 32 2.12 Least squares estimation . . . . . . . . . . . . . . . . . . . . . . . . 33 2.13 Comparison of ML, MAP and MS estimates . . . . . . . . . . . . . 36 2.14 Selection of the basis vectors . . . . . . . . . . . . . . . . . . . . . 39 2.15 Modeling of prior information in Bayesian estimation . . . . . . . . 40 2.16 Recursive mean square estimation . . . . . . . . . . . . . . . . . . 41 2.17 Time-varying linear regression . . . . . . . . . . . . . . . . . . . . . 45 2.18 Properties of the MS estimator . . . . . . . . . . . . . . . . . . . . 47 3 Regularization theory 49 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2 Tikhonov regularization . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3 Principal component based regularization . . . . . . . . . . . . . . 52 3.4 Subspace regularization . . . . . . . . . . . . . . . . . . . . . . . . 54
Page 1: Karjalainen, Pasi A. Regularization
Page 5: Acknowledgements This work was carr
Page 10 and 11: 3.5 Selection of the regularization
Page 13 and 14: CHAPTER I Introduction Evoked poten
Page 15 and 16: 17 underlying parameters can usuall
Page 17 and 18: CHAPTER II Estimation theory In thi
Page 19 and 20: 2.2. Probability theory 21 and vect
Page 21 and 22: 2.3. Bayesian estimation 23 2.3 Bay
Page 23 and 24: 2.6. Mean square estimation 25 is t
Page 25 and 26: 2.7. Maximum a posteriori estimatio
Page 27 and 28: 2.9. Minimum mean square estimator
Page 29 and 30: 2.10. Mean square estimation with o
Page 31 and 32: 2.12. Least squares estimation 33 a
Page 33 and 34: 2.12. Least squares estimation 35 w
Page 35 and 36: 2.13. Comparison of ML, MAP and MS
Page 37 and 38: 2.14. Selection of the basis vector
Page 39 and 40: 2.16. Recursive mean square estimat
Page 41 and 42: 2.16. Recursive mean square estimat
Page 43 and 44: 2.17. Time-varying linear regressio
Page 45 and 46: 2.18. Properties of the MS estimato
Page 47 and 48: CHAPTER III Regularization theory I
Page 49 and 50: 3.2. Tikhonov regularization 51 and
Page 51 and 52: 3.3. Principal component based regu
Page 53 and 54: 3.5. Selection of the regularizatio
Page 55 and 56: CHAPTER IV Time series models In th
Page 57 and 58: 4.2. Stationary time series models
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4.3. Time-dependent time series mod
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4.5. Wiener filtering as estimation
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4.5. Wiener filtering as estimation
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5.2. Component based simulation of
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Page 69 and 70:
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5.3. Principal component based simu
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5.4. Discussion 75 5.4 Discussion T
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6.2. Ensemble analysis 77 • ensem
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6.2. Ensemble analysis 79 H = (I|
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6.2. Ensemble analysis 81 and thus
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6.2. Ensemble analysis 83 The prope
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6.3. Single trial estimation 85 and
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6.3. Single trial estimation 87 not
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6.3. Single trial estimation 89 dis
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6.4. Miscellaneous topics 91 severa
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6.5. Dynamical estimation of evoked
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6.5. Dynamical estimation of evoked
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6.6. Discussion 97 potentials. Some
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7.1. A systematic method for single
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7.3. The data sets 101 8. Select P
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7.4. Case study 1 103 process is us
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7.4. Case study 1 105 −0.2 −0.1
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7.4. Case study 1 107 110 105 100 9
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7.5. Case study 2 109 7.5 Case stud
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7.5. Case study 2 111 110 105 100 9
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7.5. Case study 2 113 -50 25 20 0 1
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7.6. Case study 3 115 −20 −20 0
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7.6. Case study 3 117 -20 15 -10 10
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7.6. Case study 3 119 −50 25 20 0
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7.7. Case study 4 121 selected so t
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7.8. Discussion 123 form such a met
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125 We also proposed a systematic m
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REFERENCES [1] J.J. Ackmann, P.P. E
Page 127 and 128:
References 129 auditory evoked pote
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References 131 Surveys on Mathemati
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References 133 nal on Biomedical En
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References 135 [155] P.G. Park and
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References 137 Neurophysiology, 92:
Page 137:
References 139 [230] N. Wiener. The
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