Multidimensional empirical mode decomposition: Difference between revisions

Revision as of 21:50, 26 May 2022

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==Ensemble empirical mode decomposition==

To improve the accuracy of measurements, the ensemble mean is a powerful approach, where data are collected by separate observations, each of which contains different noise over an ensemble of universe's. To generalize this ensemble idea, noise is introduced to the single data set, <math>x(t)</math>, as if separate observations were indeed being made as an analogue to a physical experiment that could be repeated many times. The added [[white noise]] is treated as the possible random noise that would be encountered in the measurement process. Under such conditions, the i th ‘artificial’ observation will be <math>x_i(t)=x(t)+w_i(t)</math>.

In the case of only one observation, one of the multiple-observation ensembles is mimicked by adding different copies of white noise, <math>w_i(t)</math>, to that single observation as given in the equation. Although adding noise may result in a smaller signal -to-noise ratio, the added white noise will provide a uniform reference scale distribution to facilitate EMD; therefore, the low signal-noise ratio doeswill not affect the decomposition method but actually enhances it by avoiding mode mixing. Based on this argument, an additional step is taken by arguing that adding white noise may help extract the true signals in the data, a method that is termed Ensemble Empirical Mode Decomposition (EEMD).

The EEMD consists of the following steps:

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</math><ref name=":5" />

where RX (1, i, j), RX (2, i, j), and RX (m, i, j) are the ''m'' sets of signal as stated (also here we use ''R'' to indicate row decomposing). The relation between these m 2D decomposed signals and the original signal is given as <math>X(i,j)=\sum_{ k \mathop =1}^mRX(k,i,j)</math>. <ref name=":5" />

The first row of the matrix RX (m, i, j) is the mth EMD component decomposed from the first row of the matrix X (i, j). The second row of the matrix RX (m, i, j) is the mth EMD component decomposed from the second row of the matrix X (i, j), and so on.

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where

: <math>K'_i=\frac{\sum_{i=1}^nK_iN_i}{N}. </math>&nbsp;&nbsp;<ref name=":7" />

Therefore, the compression rate of the spatial ___domain is as follows