independent component analysis versus principle component analysis
Independent component analysis P15
... by principal component analysis (PCA); see Sections 13.1.2 and 13.2.2 In noisy ICA, we also encounter a new problem: estimation of the noise-free realizations of the independent components (ICs) ... consider the ~ noisy independent components, given by si = si + ni , and rewrite the model as x = A~ s (15.3) We see that this is just the basic ICA model, with modified independent components What ... model This way we can estimate the mixing matrix and the noisy independent components The estimation of the original independent components from the noisy ones is an additional problem, though;...
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Independent component analysis P16
... ESTIMATION OF THE INDEPENDENT COMPONENTS Maximum likelihood estimation Many methods for estimating the mixing matrix use as subroutines methods that estimate the independent components for a known ... methods for reconstructing the independent components, assuming that we know the mixing matrix Let us denote by m the number of mixtures and by n the number of independent components Thus, the mixing ... small, the (t) are the realizations of the independent components, and C is an irrelevant constant The functions fi are the log-densities of the independent components Maximization of (16.7) with...
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Independent component analysis P17
... unknown realvalued m -component mixing function, and is an n-vector whose elements are the n unknown independent components Assume now for simplicity that the number of independent components n equals ... : : : xn into n independent components y1 : : : yn , giving a solution for the nonlinear ICA problem This construction also clearly shows that the decomposition in independent components is by ... quite nonlinear, because nonlinear factor analysis is able to explain the data with 10 components equally well as linear factor analysis (PCA) with 21 components Different numbers of hidden neurons...
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Independent component analysis P18
... means that simply finding a matrix so that the components of the vector C A V z(t) = Vx(t) (18.3) are white, is not enough to estimate the independent components This is because there is an infinity ... infinity of different matrices that give decorrelated components This is why in basic ICA, we have to use the nongaussian structure of the independent components, for example, by minimizing the higher-order ... time-lagged covariances only This is in contrast to ICA using higher-order information, where the independent components are allowed to have identical distributions Further work on using autocovariances...
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Independent component analysis P19
... the signal x(t) to be deconvolved Estimating just one independent component , we obtain the original deconvolved signal s(t) If several components are estimated, they correspond to translated ... one independent component, which x x 364 CONVOLUTIVE MIXTURES AND BLIND DECONVOLUTION is easier than estimating all of them In convolutive BSS, however, we often need to estimate all the independent ... maximum likelihood principle This principle was shown to be quite closely related to the maximization of the output entropy, which is often called the information maximization (infomax) principle; see...
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Independent component analysis P20
... by independent subspaces analysis, to be explained next 20.2.2 Independent subspace analysis Independent subspace analysis [204] is a simple model that models some dependencies between the components ... methods of independent subspaces or topographic ICA, on the other hand, we assume that we cannot really find independent components; instead we can find groups of independent components, or components ... that the components si are independent However, ICA is often applied on data sets, for example, on image data, in which the obtained estimates of the independent components are not very independent, ...
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Independent component analysis P21
... into independent components One can always obtain uncorrelated components, and this is what we obtain with FastICA In image feature extraction, however, one can clearly see that the ICA components ... subspace is relatively independent of the phase of the input This is in fact what the principle of invariant-feature subspaces, one of the inspirations for independent subspace analysis, is all about ... must note that ICA applied to image data usually gives one component representing the local mean image intensity, or the DC component This component normally has a distribution that is not sparse;...
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Independent component analysis P22
... the existence of statistically independent source signals, their instantaneous linear mixing at the sensors, and the stationarity of the mixing and the independent components (ICs) The independence ... concomitant sound Principal component analysis (PCA) has often been used to decompose signals of this kind, but as we have seen in Chapter 7, it cannot really separate independent signals In fact, ... response components For each component presented in Fig 22.4 a and Fig 22.4 b, left, top and, right side views of the corresponding field pattern are shown Note that the first principal component...
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Independent component analysis P23
... user’s subsequent transmitted symbols are assumed to be independent, these products are also independent for a given user i Denote the independent sources a1 m b1 m : : : aLm bKm by yi (m) i ... cost function can be separated, and the different independent components can be found one by one, by taking into account the previously estimated components, contained in the subspace spanned by ... minimization task becomes much simpler [344] Complexity minimization then reduces to principal component analysis of temporal correlation matrices This method is actually just another example of blind...
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Independent component analysis P24
... time-varying underlying factor or independent component sj (t) on the measured time series is approximately linear The assumption of having some underlying independent components in this specific application ... mean and unit variance), the independent components 444 OTHER APPLICATIONS 20 40 28 48 16 20 40 28 48 16 20 40 28 48 16 20 40 28 48 16 Fig 24.2 Four independent components or fundamental factors ... different orders in the classic AR prediction method for each independent component In reality, especially in real world time series analysis, the data are distorted by delays, noise, and nonlinearities...
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Tài liệu Bài 1: Introduction(Independent component analysis (ICA) doc
... guiding principle W Another principle that has been used for determining is independence: the components yi should be statistically independent This means that the value of any one of the components ... the source separation problem, the original signals were the independent components” of the data set 1.3 INDEPENDENT COMPONENT ANALYSIS 1.3.1 Definition We have now seen that the problem of blind ... which the components are statistically independent In practical situations, we cannot in general find a representation where the components are really independent, but we can at least find components...
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Tài liệu Bài 7: What is Independent Component Analysis? docx
... (Hardback); 0-471-22131-7 (Electronic) What is Independent Component Analysis? In this chapter, the basic concepts of independent component analysis (ICA) are defined We start by discussing a couple ... distribution Now let us mix these two independent components Let us take the following mixing matrix: A0 = 10 10 (7.13) 156 WHAT IS INDEPENDENT COMPONENT ANALYSIS? Fig 7.6 The joint distribution ... vertical axis: s2 158 WHAT IS INDEPENDENT COMPONENT ANALYSIS? Fig 7.9 The joint distribution of the observed mixtures x1 and x2 , obtained from supergaussian independent components Horizontal axis:...
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Tài liệu Independent Component Analysis - Chapter 14: Overview and Comparison of Basic ICA Methods pptx
... estimation principles There are, however, a couple of differences between the estimation principles as well Some principles (especially maximum nongaussianity) are able to estimate single independent components, ... the components All these principles are essentially equivalent or at least closely related The principle of maximum nongaussianity has the additional advantage of showing how to estimate the independent ... to estimate all the components at the same time Some objective functions use nonpolynomial functions based on the (assumed) probability density functions of the independent components, whereas...
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Tài liệu Independent Component Analysis - Chapter 16: ICA with Overcomplete Bases ppt
... ESTIMATION OF THE INDEPENDENT COMPONENTS Maximum likelihood estimation Many methods for estimating the mixing matrix use as subroutines methods that estimate the independent components for a known ... methods for reconstructing the independent components, assuming that we know the mixing matrix Let us denote by m the number of mixtures and by n the number of independent components Thus, the mixing ... small, the (t) are the realizations of the independent components, and C is an irrelevant constant The functions fi are the log-densities of the independent components Maximization of (16.7) with...
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Tài liệu Independent Component Analysis - Chapter 17: Nonlinear ICA ppt
... unknown realvalued m -component mixing function, and is an n-vector whose elements are the n unknown independent components Assume now for simplicity that the number of independent components n equals ... : : : xn into n independent components y1 : : : yn , giving a solution for the nonlinear ICA problem This construction also clearly shows that the decomposition in independent components is by ... quite nonlinear, because nonlinear factor analysis is able to explain the data with 10 components equally well as linear factor analysis (PCA) with 21 components Different numbers of hidden neurons...
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Tài liệu Independent Component Analysis - Chapter 18: Methods using Time Structure ppt
... means that simply finding a matrix so that the components of the vector C A V z(t) = Vx(t) (18.3) are white, is not enough to estimate the independent components This is because there is an infinity ... infinity of different matrices that give decorrelated components This is why in basic ICA, we have to use the nongaussian structure of the independent components, for example, by minimizing the higher-order ... time-lagged covariances only This is in contrast to ICA using higher-order information, where the independent components are allowed to have identical distributions Further work on using autocovariances...
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Tài liệu Independent Component Analysis - Chapter 19: Convolutive Mixtures and Blind Deconvolution pptx
... the signal x(t) to be deconvolved Estimating just one independent component , we obtain the original deconvolved signal s(t) If several components are estimated, they correspond to translated ... one independent component, which x x 364 CONVOLUTIVE MIXTURES AND BLIND DECONVOLUTION is easier than estimating all of them In convolutive BSS, however, we often need to estimate all the independent ... maximum likelihood principle This principle was shown to be quite closely related to the maximization of the output entropy, which is often called the information maximization (infomax) principle; see...
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Tài liệu Independent Component Analysis - Chapter 20: Other Extensions pptx
... by independent subspaces analysis, to be explained next 20.2.2 Independent subspace analysis Independent subspace analysis [204] is a simple model that models some dependencies between the components ... methods of independent subspaces or topographic ICA, on the other hand, we assume that we cannot really find independent components; instead we can find groups of independent components, or components ... that the components si are independent However, ICA is often applied on data sets, for example, on image data, in which the obtained estimates of the independent components are not very independent, ...
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Tài liệu Independent Component Analysis - Chapter 22: Brain Imaging Applications docx
... the existence of statistically independent source signals, their instantaneous linear mixing at the sensors, and the stationarity of the mixing and the independent components (ICs) The independence ... concomitant sound Principal component analysis (PCA) has often been used to decompose signals of this kind, but as we have seen in Chapter 7, it cannot really separate independent signals In fact, ... response components For each component presented in Fig 22.4 a and Fig 22.4 b, left, top and, right side views of the corresponding field pattern are shown Note that the first principal component...
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