Mathematics in Independent Component Analysis

More documents

Recommendations

Info

270 Chapter 20. Signal Processing 86(3):603-623, 2006 2.2.1 Independent component analysis Independent component analysis (ICA) describes the task of (here: linearly) transforming a given multivariate random vector such that its transform is stochastically independent. In our setting the random vector is given by N realizations, and ICA is applied to solve the BSS problem (1), where S is assumed to be independent. As in all BSS problems, a key issue lies in the question of identifiability of the model, and it can be shown that A (and hence S) is already uniquely — except for column permutation and scaling — determined by X if S contains at most one Gaussian and is square integrable [26,27]. This enables us to apply ICA to the BSS problem and to recover the original sources. The idea of ICA was first expressed by Jutten and Hérault [28], while the term ICA was later coined by Comon [26]. In contrast to principal component analysis (PCA), ICA uses higher-order statistics to fully separate the data. Typical algorithms are based on contrasts such as minimum mutual information, maximum entropy, diagonal cumulants or non-Gaussianity. For more details on ICA we refer to the two available excellent text-books [1,2]. In the following we will use the so-called JADE (joint approximate diagonalization of eigenmatrices) algorithm, which identifies the sources using the fact that due to independence, the diagonal cumulants of the sources vanish. Furthermore, fixing two indices of the 4th order cumulants, it is easy to see that such cumulant matrices of the mixtures are diagonalized by A. After pre-processing by PCA, we can assume that A is orthogonal. Then diagonalization of one mixture cumulant matrix already yields A, given that its eigenvalues are pairwise different. However, in practice this is not always the case; furthermore, estimation errors could result in a bad estimate of the cumulant matrix and hence of A. Therefore, joint diagonalization of a whole set of cumulant matrices yields an improved estimate of A. Algorithms for actually performing joint diagonalization include gradient descent on the sum of off-diagonal terms, iterative construction of A by Givens rotation in two coordinates [29], an iterative two-step recovery of A [30] or — more recently — a linear least-squares algorithm for diagonalization [31], where the latter two algorithms can also search for non-orthogonal matrices A. One method we use for analyzing the s-EMG signals is JADE-based ICA. Confirming results from [32] we show that ICA can indeed extract the underlying sources. In the case of s-EMG signals, all sources are strongly super-Gaussian and can therefore safely be assumed to be non-Gaussian, so identifiability holds. However, due to the nonnegativity of A, the scaling indeterminacy is reduced to multiplication with a positive scalar in each column. If we additionally use the common assumption of unit variance of the sources, this 9
Chapter 20. Signal Processing 86(3):603-623, 2006 271 already eliminates the scaling indeterminacy. In order to use an ordinary ICA algorithm, we simply have to add a ’postprocessing’ stage: to guarantee a nonnegative matrix, column signs are flipped to have only (or as many as possible) nonnegative column entries. Also note that statistical independence meaning that the multivariate probability densities factorize is not related to the synchrony in the firing of MUs [33] — otherwise overlapping MUs could not be separated. We finally want to remark that ICA can also be interpreted as sparse signal decomposition method in the case of super-Gaussian sources. This follows from the fact that a good and often-used contrast for ICA is given by maximization of non-Gaussianity [34] — this can be approximately derived from the fact that due to the central limit theorem, a mixture of independent sources tends to be more Gaussian than the sources, so the process can be inverted by maximizing non-Gaussianity. In our setting the sources are very sparse, hence strongly non-Gaussian. An ICA decomposition is therefore closely related to a decomposition into parts of maximal sparseness — at least if sparseness is measured using kurtosis. 2.2.2 (Sparse) nonnegative matrix factorization In contrast to other matrix factorization models such as PCA, ICA or SCA, nonnegative matrix factorization (NMF) strictly requires both matrices A and S to have nonnegative entries, which means that the data can be described using only additive components. Such a constraint has many physical realizations and applications, for instance in object decomposition [5]. If additionally some sparseness constraints are put on A and S, we speak of sparse NMF, see [14] for more details. Typically, NMF is performed using a least-squares (Euclidean) contrast E(A,S) = �X − AS� 2 , (2) which is to be minimized. This optimization problem, albeit convex in each variable separately, is not convex in both at the same time and hence direct estimation is not possible. Paatero [35] minimizes (2) using a gradient algorithm, whereas Lee and Seung [36] develop a multiplicative update rule increasing algorithm performance considerably. Although NMF has recently gained popularity due to its simplicity and power in various applications, its solutions frequently fail to exhibit the desired sparse object decomposition. Therefore, Hoyer [14] proposes a modification of the NMF model to include sparseness. However, a simple modification of the cost function (2) could yield undesirable local minima, so instead he chooses to minimize (2) under the constraint of fixed sparseness of both A and S. Here, 10
Page 1 and 2:
Statistical machine learning of bio
Page 3:
to Jakob
Page 6 and 7:
vi Preface
Page 8 and 9:
viii CONTENTS 3 Signal Processing 8
Page 11 and 12:
Chapter 1 Statistical machine learn
Page 13 and 14:
1.1. Introduction 5 auditory cortex
Page 15 and 16:
1.2. Uniqueness issues in independe
Page 17 and 18:
1.2. Uniqueness issues in independe
Page 19 and 20:
S U M M A R Y T his �le contains
Page 21 and 22:
1.3. Dependent component analysis 1
Page 23 and 24:
Table 1.1: BSS algorithms based on
Page 25 and 26:
Page 27 and 28:
Page 29 and 30:
Page 31 and 32:
Page 33 and 34:
1.4. Sparseness 25 1.4 Sparseness O
Page 35 and 36:
1.4. Sparseness 27 Theorem 1.4.3 (S
Page 37 and 38:
1.4. Sparseness 29 R 3 R 3 A BSRA R
Page 39 and 40:
1.4. Sparseness 31 Sparse projectio
Page 41 and 42:
1.5. Machine learning for data prep
Page 43 and 44:
Page 45 and 46:
Page 47 and 48:
Page 49 and 50:
Page 51 and 52:
1.6. Applications to biomedical dat
Page 53 and 54:
Page 55 and 56:
Page 57 and 58:
Page 59 and 60:
1.7. Outlook 51 noise data signal i
Page 61:
Part II Papers 53
Page 64 and 65:
56 Chapter 2. Neural Computation 16
Page 66 and 67:
Page 68 and 69:
Page 70 and 71:
Page 72 and 73:
Page 74 and 75:
Page 76 and 77:
Page 78 and 79:
Page 80 and 81:
Page 82 and 83:
Page 84 and 85:
Page 86 and 87:
Page 88 and 89:
Page 90 and 91:
82 Chapter 3. Signal Processing 84(
Page 92 and 93:
Page 94 and 95:
Page 96 and 97:
Page 98 and 99:
90 Chapter 4. Neurocomputing 64:223
Page 100 and 101:
Page 102 and 103:
Page 104 and 105:
Page 106 and 107:
Page 108 and 109:
Page 110 and 111:
Page 112 and 113:
104 Chapter 5. IEICE TF E87-A(9):23
Page 114 and 115:
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
114 Chapter 6. LNCS 3195:726-733, 2
Page 124 and 125:
116 Chapter 6. LNCS 3195:726-733, 2
Page 126 and 127:
118 Chapter 6. LNCS 3195:726-733, 2
Page 128 and 129:
120 Chapter 6. LNCS 3195:726-733, 2
Page 130 and 131:
122 Chapter 6. LNCS 3195:726-733, 2
Page 132 and 133:
124 Chapter 7. Proc. ISCAS 2005, pa
Page 134 and 135:
Page 136 and 137:
Page 138 and 139:
130 Chapter 8. Proc. NIPS 2006 Towa
Page 140 and 141:
132 Chapter 8. Proc. NIPS 2006 1.3
Page 142 and 143:
134 Chapter 8. Proc. NIPS 2006 stru
Page 144 and 145:
136 Chapter 8. Proc. NIPS 2006 5.5
Page 146 and 147:
138 Chapter 8. Proc. NIPS 2006
Page 148 and 149:
140 Chapter 9. Neurocomputing (in p
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
Page 156 and 157:
Page 158 and 159:
Page 160 and 161:
Page 162 and 163:
Page 164 and 165:
156 Chapter 10. IEEE TNN 16(4):992-
Page 166 and 167:
Page 168 and 169:
Page 170 and 171:
162 Chapter 11. EURASIP JASP, 2007
Page 172 and 173:
Page 174 and 175:
Page 176 and 177:
Page 178 and 179:
Page 180 and 181:
Page 182 and 183:
Page 184 and 185:
176 Chapter 12. LNCS 3195:718-725,
Page 186 and 187:
178 Chapter 12. LNCS 3195:718-725,
Page 188 and 189:
180 Chapter 12. LNCS 3195:718-725,
Page 190 and 191:
182 Chapter 12. LNCS 3195:718-725,
Page 192 and 193:
184 Chapter 12. LNCS 3195:718-725,
Page 194 and 195:
186 Chapter 13. Proc. EUSIPCO 2005
Page 196 and 197:
Page 198 and 199:
Page 200 and 201:
192 Chapter 14. Proc. ICASSP 2006 S
Page 202 and 203:
194 Chapter 14. Proc. ICASSP 2006 A
Page 204 and 205:
196 Chapter 14. Proc. ICASSP 2006
Page 206 and 207:
198 Chapter 15. Neurocomputing, 69:
Page 208 and 209:
Page 210 and 211:
Page 212 and 213:
Page 214 and 215:
Page 216 and 217:
Page 218 and 219:
Page 220 and 221:
Page 222 and 223:
Page 224 and 225:
Page 226 and 227:
Page 228 and 229: 220 Chapter 15. Neurocomputing, 69:
Page 238 and 239: 230 Chapter 16. Proc. ICA 2006, pag
Page 248 and 249: 240 Chapter 17. IEEE SPL 13(2):96-9
Page 254 and 255: 246 Chapter 18. Proc. EUSIPCO 2006
Page 260 and 261: 252 Chapter 19. LNCS 3195:977-984,
Page 270 and 271: 262 Chapter 20. Signal Processing 8
Page 300 and 301: 292 Chapter 21. Proc. BIOMED 2005,
Page 306 and 307: 298 BIBLIOGRAPHY Barber, C., Dobkin
Page 308 and 309: 300 BIBLIOGRAPHY Georgiev, P. (2001
Page 310 and 311: 302 BIBLIOGRAPHY Keck, I., Theis, F
Page 312 and 313: 304 BIBLIOGRAPHY Schießl, I., Sch
Page 314 and 315: 306 BIBLIOGRAPHY Theis, F. and Inou
show all

Mathematics in Independent Component Analysis

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?