Nonuniformly sampled trivariate empirical mode decomposition

Abstract

Multichannel data-driven time-frequency algorithms, such as the multivariate empirical mode decomposition (MEMD), have emerged as important tools in the analysis of inter-channel dependencies that arise in multivariate data. Such methods employ uniform projection schemes on hyper spheres in order to estimate the local mean, thus requiring dense but underutilised sampling when processing unbal anced data channels. To this end, we propose a nonuniform projection scheme that adapts to the second order statistics of trivariate data; this provides the estimation of the local mean in the case of power imbalances and correlations between the channels. The algorithm is particularly useful for gener ating a low number of direction vectors within MEMD. Its performance is illustrated on synthetic and real-world data.

Multi-Fault Diagnosis of Rolling Bearings via Adaptive Projection Intrinsically Transformed Multivariate Empirical Mode Decomposition and High Order Singular Value Decomposition

“…The detailed algorithm of MEMD has been elaborated in the previous work of authors [ 27 ]. To elaborate the principle of APIT-MEMD, non-uniformly sampled trivariate empirical mode decomposition (NS-TEMD) is described here as an illustration [ 44 ]. The NS-TEMD algorithm is basically about improving the standard TEMD [ 21 ] by relocating the direction vectors adopting conventional uniform sampling method to new locations on an ellipsoid.…”

Adaptive-projection intrinsically transformed multivariate empirical mode decomposition in cooperative brain–computer interface applications

“…The number of direction vectors for MEMD and APIT-MEMD was varied from 32 to 256 (moderate to very high), following the suggestion in [20] that the number of direction vectors should be considerably greater than the dimensionality of the signals in order to extract meaningful IMFs. In our own experience, trivariate signals require more than 16 projection vectors for the NS-TEMD to capture the direction of highest power imbalances [31]. The α value for the APIT-MEMD was 1 due to the power imbalances.…”

“…The NS-TEMD algorithm [31] enhances the performance of the conventional TEMD by relocating the direction vectors, pre-generated using the conventional uniform sampling scheme (figure 1a), to their new positions on an ellipsoid (figure 1b), with the directions and relative powers determined by all the eigenvectors and eigenvalues of the trivariate covariance matrix of the signal. In the case of multivariate signals, relocating the pre-generated n-dimensional uniform projection vectors to their new positions on an n-dimensional ellipsoid using the above scheme is an exceptionally complicated task, not least because the direction of global highest curvature of the original input signal may not always align with the direction of local first principal component of each sifting input, resulting in a suboptimal estimate of the local mean.…”

“…Another algorithm-dynamically sampled BEMD [30]-employs Menger curvature to quantify local signal dynamics of bivariate signals to generate projection vectors which align with the highest local dynamics. For trivariate signals, non-uniformly sampled TEMD (NS-TEMD) [31] globally identifies a set of non-uniformly sampled projections by performing eigendecomposition of the covariance of a signal, subsequently constructing an ellipsoid of non-uniform direction vectors which matches signal dynamics.…”

Augmented EMD for complex‐valued univariate signals

“…One of the main research streams on the EMD is to extend its capability to deal with bivariate (including complex-valued univariate) [6][7][8][9][10], trivariate [11,12] or even multivariate [13][14][15][16][17][18] signals. For the case of bivariate signals, several methods have been recently proposed, namely complex EMD (CEMD) [6], rotation-invariant EMD (RI-EMD) [7], bivariate EMD (BEMD) [8], non-uniformly sampled BEMD (NS-BEMD) [9] and dynamically sampled BEMD (DS-BEMD) [10].…”