Compressed Sensing (CS) has emerged as an alternate method to acquire high dimensional signals effectively by exploiting the sparsity assumption. However, owing to non-sparse and non-stationary nature, it is extremely difficult to process Electroencephalograph (EEG) signals using CS paradigm. The success of Bayesian algorithms in recovering non-sparse signals has triggered the research in CS based models for neurophysiological signal processing. In this paper, we address the problem of Temporal Modeling of EEG Signals using Block Sparse Variational Bayes (SVB) Framework. Temporal correlation of EEG signals is modeled blockwise using normal variance scale mixtures parameterized via some random and deterministic parameters. Variational inference is exploited to infer the random parameters and Expectation Maximization (EM) is used to obtain the estimate of deterministic parameters. To validate the framework, we present experimental results for benchmark State Visual Evoked Potential (SSVEP) dataset with 40-target Brain-Computer Interface (BCI) speller using two frequency recognition algorithms viz. Canonical Correlation Analysis (CCA) and L1-regularized Multiway CCA. Results show that the proposed temporal model is highly useful in processing SSVEP-EEG signals irrespective of the recognition algorithms used.

Temporal Modeling of EEG Signals using Block Sparse Variational Bayes Framework

The Hammerstein model is a cascade composition of a static memoryless nonlinear function followed by a linear time-invariant dynamical subsystem, which is capable of modeling a wide range of nonlinear dynamical systems. Model structural parameter selection (including the model order and the nonlinearity order) and sparse representation of the static nonlinear function are two issues that receive increasing interests in Hammerstein system identification. In this paper, a novel Bayesian sparse multiple kernel-based identification method (BSMKM) for multiple-input single-output (MISO) Hammerstein system is proposed to handle those issues, where the basis-function model and the finite impulse response model are used to represent the nonlinear part and the linear part respectively. Firstly, in order to jointly realize the model parameter estimation, the sparse representation of static nonlinear function (the nonlinearity order selection can also be realized indirectly) and the model order selection of linear dynamical system, a hierarchical prior distribution is constructed based on Gaussian scale mixture model and sparse multiple kernel, which can characterize both inter-group sparsity and intra-group correlation structure. Then, a full Bayesian method based on variational Bayesian inference is proposed to estimate all unknown model parameters, including finite impulse response coefficients, hyperparameters and noise variance. Finally, the performance of the proposed BSMKM identification method is evaluated by numerical experiments using both simulation and real data. 

Identification of MISO Hammerstein system using sparse multiple kernel-based hierarchical mixture prior and variational Bayesian inference.

We present a fast update rule for variational block-sparse Bayesian learning (SBL) methods. Using a variational Bayesian framework, we show how repeated updates of probability density functions (PDFs) of the prior variances and weights can be expressed as a nonlinear first-order recurrence from one estimate of the parameters of the proxy PDFs to the next. Specifically, the recurrent relation turns out to be a strictly increasing rational function for many commonly used prior PDFs of the variances, such as Jeffrey's prior. Hence, the fixed points of this recurrent relation can be obtained by solving for the roots of a polynomial. This scheme allows to check for convergence/divergence of individual prior variances in a single step. Thereby, the the computational complexity of the variational block-SBL algorithm is reduced and the convergence speed is improved by two orders of magnitude in our simulations. Furthermore, the solution allows insights into the sparsity of the estimators obtained by choosing different priors.

Fast Variational Block-Sparse Bayesian Learning

We present some details of Bayesian block sparse modeling using hierarchical prior having deterministic and random parameters when entries within the blocks are correlated. In particular, the effect of the threshold to prune out variance parameters of algorithms corresponding to several choices of marginals, viz. multivariate Jeffery prior, multivariate Laplace distribution and multivariate Student’s t distribution, is discussed. We also provide details of experiments with Electroencephalograph (EEG) data which shed some light on the possible applicability of the proposed Sparse Variational Bayes framework.

Some Comments on Variational Bayes Block Sparse Modeling with Correlated Entries

This paper addresses the problem of Bayesian Block Sparse Modeling when coefficients within the blocks are correlated. In contrast to the current hierarchical methods which do not exploit correlation structure within the blocks, we propose a three level hierarchical estimation framework. It employs heavy-tailed priors for block sparse modeling and variational inference for Bayesian estimation. This paper also describes the relationship between proposed framework and some of the existing Block Sparse Bayesian Learning (SBL) methods and show that these SBL methods can be viewed as its special cases. Extensive experimental results for synthetic signals are provided, demonstrating the superior performance of the proposed framework in terms of failure rate, relative reconstruction error, to name a few. We also demonstrate the applicability of this framework in telemonitoring of Fetal Electrocardiogram.

Variational Bayes Block Sparse Modeling with Correlated Entries

Bayesian Sparse Signal Recovery (SSR) for Multiple Measurement Vectors, when elements of each row of solution matrix are correlated, is addressed in the paper. We propose a standard linear Gaussian observation model and a three-level hierarchical estimation framework, based on Gaussian Scale Mixture (GSM) model with some random and deterministic parameters, to model each row of the unknown solution matrix. This hierarchical model induces heavy-tailed marginal distribution over each row which encompasses several choices of distributions viz. Laplace distribution, Student's t distribution and Jeffery prior. Automatic Relevance Determination (ARD) phenomenon introduces sparsity in the model. It is interesting to see that Block Sparse Bayesian Learning framework is a special case of the proposed framework when induced marginal is Jeffrey prior. Experimental results for synthetic signals are provided to demonstrate its effectiveness. We also explore the possibility of using Multiple Measurement Vectors to model Dynamic Hand Posture Database which consists of sequence of temporally correlated hand posture sequence. It can be seen that by exploiting temporal correlation information present in the successive image samples, the proposed framework can reconstruct the data with less linear random measurements with high fidelity.

Shruti Sharma

Papers

Variational Bayes Block Sparse Modeling with Correlated Entries

Temporal Modeling of EEG Signals using Block Sparse Variational Bayes Framework

Sparse Signal Recovery for Multiple Measurement Vectors with Temporally Correlated Entries: A Bayesian Perspective

Some Comments on Variational Bayes Block Sparse Modeling with Correlated Entries