Pierre-Olivier Amblard

On the use of stochastic resonance in sine detection

This paper deals with the use of stochastic resonance (SR) for detection purposes. The nonlinear physical phenomenon of SR generally occurs in dynamical bistable systems excited by a noisy sine: such systems are able to force cooperation between sine and noise such that the noise amplifies the sine. Because of this non-intuitive effect, the use of SR can be envisaged to detect small amplitude sines corrupted by additive noise. In this paper we recall some basics of detection and then show why SR can be used in sine detection context. After recalling some basics of SR in discrete time, we show how to use SR in a detection scheme.

On directed information theory and Granger causality graphs

Directed information theory deals with communication channels with feedback. When applied to networks, a natural extension based on causal conditioning is needed. We show here that measures built from directed information theory in networks can be used to assess Granger causality graphs of stochastic processes. We show that directed information theory includes measures such as the transfer entropy, and that it is the adequate information theoretic framework needed for neuroscience applications, such as connectivity inference problems.

Statistics for complex variables and signals—Part I: variables

Abstract This paper is devoted to the study of higher-order statistics for complex random variables. We introduce a general framework allowing the direct manipulation of complex quantities: the separation between the real and the imaginary parts of a variable is avoided. We give the rules to integrate and derive probability density functions and characteristic functions, so that calculations may be carried out. In the case of multidimensional variables, we use the natural framework of tensors. The study of complex variables leads to the extension of the notion of complex circular random variables already known in the Gaussian case.

Stochastic resonance in locally optimal detectors

The aim of the paper is to show that the nonlinear effect known as stochastic resonance, which corresponds to the improvement of the processing of information by noise, occurs naturally in some detection problems. We illustrate this by studying the problem of detecting a small amplitude sinusoid in non-Gaussian noise. We show that in some cases, the nonlinearity that appears in locally optimal detectors can be viewed as a stochastic resonator. If the parameters of the locally optimal detector (LOD) are not well tuned, the performance can be improved by the addition of noise.

Wavelet packets and de-noising based on higher-order-statistics for transient detection

Abstract In this paper, we present a detector of transient acoustic signals that combines two powerful detection tools: a local wavelet analysis and higher-order statistical properties of the signals. The use of both techniques makes detection possible in low signal-to-noise ratio conditions, when other means of detection are no longer sufficient. The proposed algorithm uses the adapted wavelet packet transform. It leads to a partition of the signal which is ‘optimal’ according to a criterion that tests the Gaussian nature of the frequency bands. To get a time dependent detection curve, we perform a de-noising procedure on the wavelet coefficients: The Gaussian coefficients are set to zero. We then apply a classical method of detection on the time reconstructed de-noised signal. We study the performance of the detector in terms of experimental ROC curves. We show that the detector performs better than decompositions using other classical splitting criteria. In the last part, we present an application of the algorithm on real flow recordings of nuclear plant pipings. The detector indicates the presence of a missing body in the piping at some instants not seen with a classical energy detector.

Stochastic resonance in discrete time nonlinear AR(1) models

This paper deals with stochastic resonance. This nonlinear physical phenomenon generally occurs in bistable systems excited by random input noise plus a sinusoid. Through its internal dynamics, such a system forces cooperation between the input noise and the input sine: provided the existence of fine tuning between the power noise and the dynamics, the system reacts periodically at the frequency of the sine. Of particular interest is the fact that the local output signal-to-noise ratio presents a maximum when plotted against the input noise power; the system resounds stochastically. Continuous-time systems have already been studied. We study the ability of intrinsically discrete-time systems [general nonlinear AR(1) models] to produce stochastic resonance. It is then suggested that such discrete systems can be used in signal processing.

The Relation between Granger Causality and Directed Information Theory: A Review

This report reviews the conceptual and theoretical links between Granger causality and directed information theory. We begin with a short historical tour of Granger causality, concentrating on its closeness to information theory. The definitions of Granger causality based on prediction are recalled, and the importance of the observation set is discussed. We present the definitions based on conditional independence. The notion of instantaneous coupling is included in the definitions. The concept of Granger causality graphs is discussed. We present directed information theory from the perspective of studies of causal influences between stochastic processes. Causal conditioning appears to be the cornerstone for the relation between information theory and Granger causality. In the bivariate case, the fundamental measure is the directed information, which decomposes as the sum of the transfer entropies and a term quantifying instantaneous coupling. We show the decomposition of the mutual information into the sums of the transfer entropies and the instantaneous coupling measure, a relation known for the linear Gaussian case. We study the multivariate case, showing that the useful decomposition is blurred by instantaneous coupling. The links are further developed by studying how measures based on directed information theory naturally emerge from Granger causality inference frameworks as hypothesis testing.

Phase tracking: what do we gain from optimality? particle filtering versus phase-locked loops

This paper studies the problem of tracking a Brownian phase with linear drift observed to within one digital modulation and one additive white Gaussian noise. This problem is of great importance as it models the problem of carrier synchronization in digital communications. The ultimate performances achievable for this problem are evaluated and are compared to the performances of three solutions of the problem. The optimal filter cannot be explicitly calculated and one goal of the paper is to implement it using recent sequential Monte-Carlo techniques known as particle filtering. This approach is compared to more traditional loops such as the Costas loop and the decision feedback loop. Moreover, since the phase has a linear drift, the loops considered are second-order loops. To make fair comparisons, we exploit all the known information to put the loops in their best configurations (optimal step sizes of the loops). We show that asymptotically, the loops and the particle filter are equivalent in terms of mean square error. However, using Monte-Carlo simulations we show that the particle filter outperforms the loops when considering the mean acquisition time (convergence rate), and we argue that the particle filter is also better than the loops when dealing with the important problem of mean time between cycle slips.

Statistics for complex variables and signals—Part II: signals

Abstract In this paper, we study the higher-order statistics of complex stationary signals. In a first part, we define precisely the multicorrelations and multispectra of complex signals. Different properties of these tools are put in evidence for certain classes of signals, such as analytic, band limited or circular signals. We show for example that band limited complex signals are circular up to a certain order. We then give the definition for the practical case of discrete time and discrete frequency processes. Finally, some extension of linear filtering relations are provided.

Stochastic discrete scale invariance

A definition of stochastic discrete scale invariance (DSI) is proposed and its properties studied. It is shown how the Lamperti (1962) transformation, which transforms stationarity in self-similarity, is also a means to connect processes deviating from stationarity and processes which are not exactly scale invariant: in particular we interpret DSI as the image of cyclostationarity. This theoretical result is employed to introduce a multiplicative spectral representation of DSI processes based on the Mellin transform, and preliminary remarks are given about estimation issues.