Frédéric Vrins

Sensor Array and Electrode Selection for Non-invasive Fetal Electrocardiogram Extraction by Independent Component Analysis

Recently, non-invasive techniques to measure the fetal electrocardiogram (FECG) signal have given very promising results. However, the important question of the number and the location of the external sensors has been often discarded. In this paper, an electrode-array approach is proposed; it is combined with a sensor selection algorithm using a mutual information criterion. The sensor selection algorithm is run in parallel to an independent component analysis of the selected signals. The aim of this method is to make a real time extraction of the FECG possible. The results are shown on simulated biomedical signals.

SWM : a class of convex contrasts for source separation

We derive a class of contrasts for blind source separation (BSS) to separate bounded sources (or more generally, finite sources), based on support width measures (SWM) of the marginal output distributions. These contrasts are shown to have no spurious local maxima, i.e., all the local maxima are relevant from the source separation point of view; they all correspond to non-mixing BSS solutions so that a gradient-ascent method can be used.

Local minima of information-theoretic criteria in blind source separation

Recent simulation results have indicated that spurious minima in information-theoretic criteria with an orthogonality constraint for blind source separation may exist. Nevertheless, those results involve approximations (e.g., density estimation), so that they do not constitute an absolute proof. In this letter, the problem is tackled from a theoretical point of view. An example is provided for which it is rigorously proved that spurious minima can exist in both mutual information and negentropy optima. The proof is based on a Taylor expansion of the entropy.

A Minimum-Range Approach to Blind Extraction of Bounded Sources

In spite of the numerous approaches that have been derived for solving the independent component analysis (ICA) problem, it is still interesting to develop new methods when, among other reasons, specific a priori knowledge may help to further improve the separation performances. In this paper, the minimum-range approach to blind extraction of bounded source is investigated. The relationship with other existing well-known criteria is established. It is proved that the minimum-range approach is a contrast, and that the criterion is discriminant in the sense that it is free of spurious maxima. The practical issues are also discussed, and a range measure estimation is proposed based on the order statistics. An algorithm for contrast maximization over the group of special orthogonal matrices is proposed. Simulation results illustrate the performances of the algorithm when using the proposed range estimation criterion

Mixing and Non-Mixing Local Minima of the Entropy Contrast for Blind Source Separation

In this paper, both non-mixing and mixing local minima of the entropy are analyzed from the viewpoint of blind source separation (BSS); they correspond respectively to acceptable and spurious solutions of the BSS problem. The contribution of this work is twofold. First, a Taylor development is used to show that the exact output entropy cost function has a non-mixing minimum when this output is proportional to any of the non-Gaussian sources, and not only when the output is proportional to the lowest entropic source. Second, in order to prove that mixing entropy minima exist when the source densities are strongly multimodal, an entropy approximator is proposed. The latter has the major advantage that an error bound can be provided. Even if this approximator (and the associated bound) is used here in the BSS context, it can be applied for estimating the entropy of any random variable with multimodal density

Improving independent component analysis performances by variable selection

Blind source separation (BSS) consists in recovering unobserved signals from observed mixtures of them. In most cases the whole set of mixtures is used for the separation, possibly after a dimension reduction by PCA. This paper aims to show that in many applications the quality of the separation can be improved by first selecting a subset of some mixtures among the available ones, possibly by an information content criterion, and performing PCA and BSS afterwards. The benefit of this procedure is shown on simulated electrocardiographic data by extracting the fetal electrocardiogram signal from mixtures recorded on the abdomen of a pregnant woman.

Electrode Selection for Noninvasive Fetal Electrocardiogram Extraction using Mutual Information Criteria

Blind source separation (BSS) techniques have revealed to be promising approaches for the noninvasive extraction of fetal cardiac signals from maternal abdominal recordings. From previous studies, it is now believed that a carefully selected array of electrodes well-placed over the abdomen of a pregnant woman contains the required ‘information’ for BSS, to extract the complete fetal components. Based on this idea, previous works have involved array recording systems and sensor selection strategies based on the Mutual Information (MI) criterion. In this paper the previous works have been extended, by considering the 3-dimensional aspects of the cardiac electrical activity. The proposed method has been tested on simulated and real maternal abdominal recordings. The results show that the new sensor selection strategy together with the MI criterion, can be effectively used to select the channels containing the most ‘information’ concerning the fetal ECG components from an array of 72 recordings. The method is hence believed to be useful for the selection of the most informative channels in online applications, considering the different fetal positions and movements.

Information theoretic versus cumulant-based contrasts for multimodal source separation

Recently, several authors have emphasized the existence of spurious maxima in usual contrast functions for source separation (e.g., the likelihood and the mutual information) when several sources have multimodal distributions. The aim of this letter is to compare the information theoretic contrasts to cumulant-based ones from the robustness to spurious maxima point of view. Even if all of them tend to measure, in some way, the same quantity, which is the output independence (or equivalently, the output non-Gaussianity), it is shown that in the case of a mixture involving two sources, the kurtosis-based contrast functions are more robust than the information theoretic ones when the source distributions are multimodal.

Double T Copula Pricing of Structured Credit Products: Practical Aspects of a Trustworthy Implementation

In spite of its simplicity, the popular One Factor Gaussian Copula model remains the market standard for the valuation of CDO tranches and n-th to default. It suffers however from well-know weaknesses, mainly due to the tail behavior of the Normal distribution (namely: the tails are too light, and there is no tail dependence, whatever is the copula correlation). Alternative models have been proposed, among those is the double t copula, which does not share the Gaussian copula drawbacks while not being much more complex. In spite of its nice features, this framework suffers from some technical problems related to its implementation. Without a lot of care, this technique could easily lead to inconsistent results. In our opinion, these difficulties have prevent practitioners to really turn that theoretically sounding model into a workable pricing tool. This paper aims at filling this gap by giving routes toward a reliable implementation of the double t copula framework, throwing away the drawbacks of this framework compared to the Gaussian one. The first purpose of this letter is to show that the implementation issues related to the double t model actually reduce to the estimation of integrals with respect to some student-t distributions. The second part of this note presents an efficient numerical method to perform this tedious task.

Conic martingales from Stochastic integrals

In this paper we introduce the concept of conic martingales}. This class refers to stochastic processes having the martingale property, but that evolve within given (possibly time-dependent) boundaries. We first review some results about the martingale property of solution to driftless stochastic differential equations. We then provide a simple way to construct and handle such processes. Specific attention is paid to martingales in