Initial results on an MMSE precoding and equalisation approach to MIMO PLC channels

This paper addresses some initial experiments using polynomial matrix decompositions to construct MMSE precoders and equalisers for MIMO power line communications (PLC) channels. The proposed scheme is based on a Wiener formulation based on polynomial matrices, and recent results to design and implement such systems with polynomial matrix tools. Applied to the MIMO PLC channel, the strong spectral dynamics of the PLC system together with the long impulse responses contained in the MIMO system result in problems, such that diagonlisation and spectral majorisation is mostly achieved in bands of high energy, while low-energy bands can resist any diagonalisation efforts. We introduce the subband approach in order to deal with this problem. A representative example using a simulated MIMO PLC channel is presented

A numerically efficient implementation of the expectation maximization algorithm for state space models

Peer reviewedPostprin

Graphical Modelling of Multivariate Time Series

This thesis mainly works on the parametric graphical modelling of multivariate time series. The idea of graphical model is that each missing edge in the graph corresponds to a zero partial coherence between a pair of component processes. A vector autoregressive process (VAR) together with its associated partial correlation graph defines a graphical interaction (GI) model. The current estimation methodologies are few and lacking of details when fitting GI models. Given a realization of the VAR process, we seek to determine its graph via the GI model; we proceed by assuming each possible graph and a range of possible autoregressive orders, carrying out the estimation, and then using model-selection criteria AIC and/or BIC to select amongst the graphs and orders. We firstly consider a purely time domain approach by maximizing the conditional maximum likelihood function with zero constraints; this non-convex problem is made convex by a ‘relaxation’ step, and solved via convex optimization. The solution is exact with high probability (and would be always exact if a certain covariance matrix was block-Toeplitz). Alternatively we look at an iterative algorithm switching between time and frequency domains. It updates the spectral estimates using equations that incorporate information from the graph, and then solving the multivariate Yule-Walker equations to estimate the VAR process parameters. We show that both methods work very well on simulated data from GI models. The methods are then applied on real EEG data recorded from Schizophrenia patients, who suffer from abnormalities of brain connectivity. Though the pretreatment has been carried out to remove improper information, the raw methods do not provide any interpretive results. Some essential modification is made in the iterative algorithm by spectral up-weighting which solves the instability problem of spectral inversion efficiently. Equivalently in convex optimization method, adding noise seems also to work but interpretation of eigenvalues (small/large) is less clear. Both methods essentially delivered the same results via GI models; encouragingly the results are consistent from a completely different method based on nonparametric/multiple hypothesis testing

Locally Stationary Functional Time Series

The literature on time series of functional data has focused on processes of which the probabilistic law is either constant over time or constant up to its second-order structure. Especially for long stretches of data it is desirable to be able to weaken this assumption. This paper introduces a framework that will enable meaningful statistical inference of functional data of which the dynamics change over time. We put forward the concept of local stationarity in the functional setting and establish a class of processes that have a functional time-varying spectral representation. Subsequently, we derive conditions that allow for fundamental results from nonstationary multivariate time series to carry over to the function space. In particular, time-varying functional ARMA processes are investigated and shown to be functional locally stationary according to the proposed definition. As a side-result, we establish a Cram\'er representation for an important class of weakly stationary functional processes. Important in our context is the notion of a time-varying spectral density operator of which the properties are studied and uniqueness is derived. Finally, we provide a consistent nonparametric estimator of this operator and show it is asymptotically Gaussian using a weaker tightness criterion than what is usually deemed necessary

Tensor Analysis and Fusion of Multimodal Brain Images

Current high-throughput data acquisition technologies probe dynamical systems with different imaging modalities, generating massive data sets at different spatial and temporal resolutions posing challenging problems in multimodal data fusion. A case in point is the attempt to parse out the brain structures and networks that underpin human cognitive processes by analysis of different neuroimaging modalities (functional MRI, EEG, NIRS etc.). We emphasize that the multimodal, multi-scale nature of neuroimaging data is well reflected by a multi-way (tensor) structure where the underlying processes can be summarized by a relatively small number of components or "atoms". We introduce Markov-Penrose diagrams - an integration of Bayesian DAG and tensor network notation in order to analyze these models. These diagrams not only clarify matrix and tensor EEG and fMRI time/frequency analysis and inverse problems, but also help understand multimodal fusion via Multiway Partial Least Squares and Coupled Matrix-Tensor Factorization. We show here, for the first time, that Granger causal analysis of brain networks is a tensor regression problem, thus allowing the atomic decomposition of brain networks. Analysis of EEG and fMRI recordings shows the potential of the methods and suggests their use in other scientific domains.Comment: 23 pages, 15 figures, submitted to Proceedings of the IEE