https://sccn.ucsd.edu/eeglab/download/eeglab_jnm03.pdf

EEGLAB: an open source toolbox for analysis of single-trial EEG dynamics including independent component analysis.

https://hal.archives-ouvertes.fr/hal-00417283/document

Independent component analysis, a new concept?

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

We present two classes of improved estimators for mutual information M(X,Y), from samples of random points distributed according to some joint probability density mu(x,y). In contrast to conventional estimators based on binnings, they are based on entropy estimates from k -nearest neighbor distances. This means that they are data efficient (with k=1 we resolve structures down to the smallest possible scales), adaptive (the resolution is higher where data are more numerous), and have minimal bias. Indeed, the bias of the underlying entropy estimates is mainly due to nonuniformity of the density at the smallest resolved scale, giving typically systematic errors which scale as functions of k/N for N points. Numerically, we find that both families become exact for independent distributions, i.e. the estimator M(X,Y) vanishes (up to statistical fluctuations) if mu(x,y)=mu(x)mu(y). This holds for all tested marginal distributions and for all dimensions of x and y. In addition, we give estimators for redundancies between more than two random variables. We compare our algorithms in detail with existing algorithms. Finally, we demonstrate the usefulness of our estimators for assessing the actual independence of components obtained from independent component analysis (ICA), for improving ICA, and for estimating the reliability of blind source separation.

Estimating mutual information.

Independent component models have gained increasing interest in various fields of applications in recent years. The basic independent component model is a semiparametric model assuming that a p-variate observed random vector is a linear transformation of an unobserved vector of p independent latent variables. This linear transformation is given by an unknown mixing matrix, and one of the main objectives of independent component analysis (ICA) is to estimate an unmixing matrix by means of which the latent variables can be recovered. In this article, we discuss the basic independent component model in detail, define the concepts and analysis tools carefully, and consider two families of ICA estimates. The statistical properties (consistency, asymptotic normality, efficiency, robustness) of the estimates can be analyzed and compared via the so called gain matrices. Some extensions of the basic independent component model, such as models with additive noise or models with dependent observations, are briefly discussed. The article ends with a short example.


Keywords:

blind source separation;
fastICA;
independent component model;
independent subspace analysis;
mixing matrix;
overcomplete ICA;
undercomplete ICA;
unmixing matrix

Independent Component Analysis

The paper considers an application of blind identification to beamforming. The key point is to use estimates of directional vectors rather than resort to their hypothesised value. By using estimates of the directional vectors obtained via blind identification, i.e. without knowing the array manifold, beamforming is made robust with respect to array deformations, distortion of the wave front, pointing errors etc., so that neither array calibration nor physical modelling is necessary. Rather suprisingly, ‘blind beamformers’ may outperform ‘informed beamformers’ in a plausible range of parameters, even when the array is perfectly known to the informed beamformer. The key assumption on which blind identification relies is the statistical independence of the sources, which is exploited using fourth-order cumulants. A computationally efficient technique is presented for the blind estimation of directional vectors, based on joint diagonalisation of fourth-order cumulant matrices; its implementation is described, and its performance is investigated by numerical experiments.

Blind beamforming for non-gaussian signals

Simultaneous diagonalization of several matrices can be implemented by a Jacobi-like technique. This note gives the required Jacobi angles in close form.

Jacobi Angles for Simultaneous Diagonalization

A brain-computer interface (BCI) is a communication system that allows to control a computer or any other device thanks to the brain activity. The BCI described in this paper is based on the P300 speller BCI paradigm introduced by Farwell and Donchin. An unsupervised algorithm is proposed to enhance P300 evoked potentials by estimating spatial filters; the raw EEG signals are then projected into the estimated signal subspace. Data recorded on three subjects were used to evaluate the proposed method. The results, which are presented using a Bayesian linear discriminant analysis classifier, show that the proposed method is efficient and accurate.

/pdf/xdawn-algorithm-to-enhance-evoked-potentials-application-to-yhpxnjoim5.pdf

xDAWN Algorithm to Enhance Evoked Potentials: Application to Brain–Computer Interface

A new algorithm for computing the nonorthogonal joint diagonalization of a set of matrices is proposed for independent component analysis and blind source separation applications. This algorithm is an extension of the Jacobi-like algorithm first proposed in the joint approximate diagonalization of eigenmatrices (JADE) method for orthogonal joint diagonalization. The improvement consists mainly in computing a mixing matrix of determinant one and columns of equal norm instead of an orthogonal mixing matrix. This target matrix is constructed iteratively by successive multiplications of not only Givens rotations but also hyperbolic rotations and diagonal matrices. The algorithm performance, evaluated on synthetic data, compares favorably with existing methods in terms of speed of convergence and complexity.

Nonorthogonal Joint Diagonalization by Combining Givens and Hyperbolic Rotations

The aim of this study was to compare eight methods for the estimation of the image-derived input function (IDIF) in [(18)F]-FDG positron emission tomography (PET) dynamic brain studies. The methods were tested on two digital phantoms and on four healthy volunteers. Image-derived input functions obtained with each method were compared with the reference input functions, that is, the activity in the carotid labels of the phantoms and arterial blood samples for the volunteers, in terms of visual inspection, areas under the curve, cerebral metabolic rates of glucose (CMRglc), and individual rate constants. Blood-sample-free methods provided less reliable results as compared with those obtained using the methods that require the use of blood samples. For some of the blood-sample-free methods, CMRglc estimations considerably improved when the IDIF was calibrated with a single blood sample. Only one of the methods tested in this study, and only in phantom studies, allowed a reliable calculation of the individual rate constants. For the estimation of CMRglc values using an IDIF in [(18)F]-FDG PET brain studies, a reliable absolute blood-sample-free procedure is not available yet.

https://journals.sagepub.com/doi/pdf/10.1038/jcbfm.2009.93

Antoine Souloumiac

Papers

Blind beamforming for non-gaussian signals

Jacobi Angles for Simultaneous Diagonalization

xDAWN Algorithm to Enhance Evoked Potentials: Application to Brain–Computer Interface

Nonorthogonal Joint Diagonalization by Combining Givens and Hyperbolic Rotations

Comparison of Eight Methods for the Estimation of the Image-Derived Input Function in Dynamic [18F]-FDG PET Human Brain Studies: