Danilo P. Mandic

Multivariate empirical mode decomposition

Despite empirical mode decomposition (EMD) becoming a de facto standard for time-frequency analysis of nonlinear and non-stationary signals, its multivariate extensions are only emerging; yet, they are a prerequisite for direct multichannel data analysis. An important step in this direction is the computation of the local mean, as the concept of local extrema is not well defined for multivariate signals. To this end, we propose to use real-valued projections along multiple directions on hyperspheres (n-spheres) in order to calculate the envelopes and the local mean of multivariate signals, leading to multivariate extension of EMD. To generate a suitable set of direction vectors, unit hyperspheres (n-spheres) are sampled based on both uniform angular sampling methods and quasi-Monte Carlo-based low-discrepancy sequences. The potential of the proposed algorithm to find common oscillatory modes within multivariate data is demonstrated by simulations performed on both hexavariate synthetic and real-world human motion signals.

Tensor Decompositions for Signal Processing Applications: From two-way to multiway component analysis

The widespread use of multisensor technology and the emergence of big data sets have highlighted the limitations of standard flat-view matrix models and the necessity to move toward more versatile data analysis tools. We show that higher-order tensors (i.e., multiway arrays) enable such a fundamental paradigm shift toward models that are essentially polynomial, the uniqueness of which, unlike the matrix methods, is guaranteed under very mild and natural conditions. Benefiting from the power of multilinear algebra as their mathematical backbone, data analysis techniques using tensor decompositions are shown to have great flexibility in the choice of constraints which match data properties and extract more general latent components in the data than matrix-based methods.

Complex Empirical Mode Decomposition

A method for the empirical mode decomposition (EMD) of complex-valued data is proposed. This is achieved based on the filter bank interpretation of the EMD mapping and by making use of the relationship between the positive and negative frequency component of the Fourier spectrum. The so-generated intrinsic mode functions (IMFs) are complex-valued, which facilitates the extension of the standard EMD to the complex domain. The analysis is supported by simulations on both synthetic and real-world complex-valued signals

Filter Bank Property of Multivariate Empirical Mode Decomposition

The multivariate empirical mode decomposition (MEMD) algorithm has been recently proposed in order to make empirical mode decomposition (EMD) suitable for processing of multichannel signals. To shed further light on its performance, we analyze the behavior of MEMD in the presence of white Gaussian noise. It is found that, similarly to EMD, MEMD also essentially acts as a dyadic filter bank on each channel of the multivariate input signal. However, unlike EMD, MEMD better aligns the corresponding intrinsic mode functions (IMFs) from different channels across the same frequency range which is crucial for real world applications. A noise-assisted MEMD (N-A MEMD) method is next proposed to help resolve the mode mixing problem in the existing EMD algorithms. Simulations on both synthetic signals and on artifact removal from real world electroencephalogram (EEG) support the analysis.

Biometrics from Brain Electrical Activity: A Machine Learning Approach

The potential of brain electrical activity generated as a response to a visual stimulus is examined in the context of the identification of individuals. Specifically, a framework for the visual evoked potential (VEP)-based biometrics is established, whereby energy features of the gamma band within VEP signals were of particular interest. A rigorous analysis is conducted which unifies and extends results from our previous studies, in particular, with respect to 1) increased bandwidth, 2) spatial averaging, 3) more robust power spectrum features, and 4) improved classification accuracy. Simulation results on a large group of subject support the analysis

The Quaternion LMS Algorithm for Adaptive Filtering of Hypercomplex Processes

The quaternion least mean square (QLMS) algorithm is introduced for adaptive filtering of three- and four-dimensional processes, such as those observed in atmospheric modeling (wind, vector fields). These processes exhibit complex nonlinear dynamics and coupling between the dimensions, which make their component-wise processing by multiple univariate LMS, bivariate complex LMS (CLMS), or multichannel LMS (MLMS) algorithms inadequate. The QLMS accounts for these problems naturally, as it is derived directly in the quaternion domain. The analysis shows that QLMS operates inherently based on the so called ldquoaugmentedrdquo statistics, that is, both the covariance E{ xx H} and pseudocovariance E{ xx T} of the tap input vector x are taken into account. In addition, the operation in the quaternion domain facilitates fusion of heterogeneous data sources, for instance, the three vector dimensions of the wind field and air temperature. Simulations on both benchmark and real world data support the approach.

A generalized normalized gradient descent algorithm

A generalized normalized gradient descent (GNGD) algorithm for linear finite-impulse response (FIR) adaptive filters is introduced. The GNGD represents an extension of the normalized least mean square (NLMS) algorithm by means of an additional gradient adaptive term in the denominator of the learning rate of NLMS. This way, GNGD adapts its learning rate according to the dynamics of the input signal, with the additional adaptive term compensating for the simplifications in the derivation of NLMS. The performance of GNGD is bounded from below by the performance of the NLMS, whereas it converges in environments where NLMS diverges. The GNGD is shown to be robust to significant variations of initial values of its parameters. Simulations in the prediction setting support the analysis.

Empirical Mode Decomposition-Based Time-Frequency Analysis of Multivariate Signals: The Power of Adaptive Data Analysis

This article addresses data-driven time-frequency (T-F) analysis of multivariate signals, which is achieved through the empirical mode decomposition (EMD) algorithm and its noise assisted and multivariate extensions, the ensemble EMD (EEMD) and multivariate EMD (MEMD). Unlike standard approaches that project data onto predefined basis functions (harmonic, wavelet) thus coloring the representation and blurring the interpretation, the bases for EMD are derived from the data and can be nonlinear and nonstationary. For multivariate data, we show how the MEMD aligns intrinsic joint rotational modes across the intermittent, drifting, and noisy data channels, facilitating advanced synchrony and data fusion analyses. Simulations using real-world case studies illuminate several practical aspects, such as the role of noise in T-F localization, dealing with unbalanced multichannel data, and nonuniform sampling for computational efficiency.

Empirical Mode Decomposition for Trivariate Signals

An extension of empirical mode decomposition (EMD) is proposed in order to make it suitable for operation on trivariate signals. Estimation of local mean envelope of the input signal, a critical step in EMD, is performed by taking projections along multiple directions in three-dimensional spaces using the rotation property of quaternions. The proposed algorithm thus extracts rotating components embedded within the signal and performs accurate time-frequency analysis, via the Hilbert-Huang transform. Simulations on synthetic trivariate point processes and real-world three-dimensional signals support the analysis.

Multiscale Image Fusion Using Complex Extensions of EMD

Empirical mode decomposition (EMD) is a fully data driven technique for decomposing signals into their natural scale components. However the problem of uniqueness, caused by the empirical nature of the algorithm and its sensitivity to changes in parameters, makes it difficult to perform fusion of data from multiple and heterogeneous sources. A solution to this problem is proposed using recent complex extensions of EMD which guarantees the same number of decomposition levels, that is the uniqueness of the scales. The methodology is used to address multifocus image fusion, whereby two or more partially defocused images are combined in automatic fashion so as to create an all in focus image.