Dinh Tuan Pham

Blind separation of mixture of independent sources through a quasi-maximum likelihood approach

We propose two methods for separating mixture of independent sources without any precise knowledge of their probability distribution. They are obtained by considering a maximum likelihood (ML) solution corresponding to some given distributions of the sources and relaxing this assumption afterward. The first method is specially adapted to temporally independent non-Gaussian sources and is based on the use of nonlinear separating functions. The second method is specially adapted to correlated sources with distinct spectra and is based on the use of linear separating filters. A theoretical analysis of the performance of the methods has been made. A simple procedure for optimally choosing the separating functions is proposed. Further, in the second method, a simple implementation based on the simultaneous diagonalization of two symmetric matrices is provided. Finally, some numerical and simulation results are given, illustrating the performance of the method and the good agreement between the experiments and the theory.

Bilinear markovian representation and bilinear models

An extension of the linear Markovian repsentation called the bilinear Markovian representation is introduced, and is shown to provide representations of all-diagonal bilinear time series models. Some properties of the bilinear Markovian representation are also given.

Markovian source separation

A maximum likelihood (ML) approach is used to separate the instantaneous mixtures of temporally correlated, independent sources with neither preliminary transformation nor a priori assumption about the probability distribution of the sources. A Markov model is used to represent the joint probability density of successive samples of each source. The joint probability density functions are estimated from the observations using a kernel method. For the special case of autoregressive models, the theoretical performance of the algorithm is computed and compared with the performance of second-order algorithms and i.i.d.-based separation algorithms.

Levinson-Durbin-type algorithms for continuous-time autoregressive models and applications

Different forms of Levinson-Durbin-type algorithms, which relate the coefficients of a continuous-time autoregressive model to the residual variances of certain regressions or their ratios, are derived. The algorithms provide parametrizations of the model by a finite set of positive numbers. They can be used for computing the covariance structure of the process, for testing the validity of such a structure, and for stability testing.

On the bias of the least squares estimator for the first order autoregressive process

The paper provides an exact formula for the bias of the parameter estimator of the first order autoregressive process and derives the asymptotic bias.

Exploiting source non stationary and coloration in blind source separation

A new method for blind source separation of instantaneous mixtures is developed. It exploits both the spectral and time diversity of the sources and is based on Gaussian mutual information. As a result, it uses only second order statistics and can be efficiently implemented through a joint diagonalization algorithm. Simulation results illustrate the good performance of the method.

Tests for Non-Correlation of Two Cointegrated Arma Time Series

In multivariate time series modelling, we are often led to investigate the existence of a relationship between two time series. Here, we generalize the procedure proposed by Haugh (1976) and extended by El Himdi and Roy (1997) for multivariate stationary ARMA time series to the case of cointegrated (or partially nonstationary) ARMA series. The main contribution consists in showing that, in the case of two uncorrelated cointegrated time series, an arbitrary vector of residual cross-correlation matrices asymptotically follows the same distribution as the corresponding vector of cross-correlation matrices between the two innovation series. The estimation method from which the residuals are obtained can be the conditional maximum likelihood method as discussed in Yap and Reinsel (1995) or some other which has the same convergence rate. From this result, it follows that the considered test statistics, which are based on residual cross-correlation matrices, asymptotically follow χ-super-2 distributions. The finite sample properties, under the null hypothesis, of the test statistics are studied by simulation. Copyright 2003 Blackwell Publishing Ltd.

Maximum likelihood estimation for a multivariate autoregressive model

The paper provides an analytical expression for the exact log likelihood function and its first derivatives for a multivariate autoregressive model. Based on these results, two algorithms for constructing the maximum likelihood estimate, using the Fishers scoring technique, are proposed. The estimated model is guaranteed to be stable. Simulation examples show that this algorithm has good convergence properties and the resulting maximum likelihood estimator could perform better than earlier methods, in cases where the record length is short and the autoregressive polynomial has roots near the unit circle. >

Efficient computation of autoregressive estimates through a sufficient statistic

It is shown that various time reversible methods, in particular, Burgs algorithm, for autoregressive model estimation may be performed through the use of a simple sufficient statistic. This provides more efficient computation of the estimators. >

Asymptotic normality of double-indexed linear permutation statistics

The paper provides sufficient conditions for the asymptotic normality of statistics of the form ΣaijbRiRj, whereaijandbijare real numbers andRiis a random permutation.