Multidimensional empirical mode decomposition: Difference between revisions

Revision as of 02:34, 30 July 2022

Revision as of 19:16, 16 August 2022

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=== Pseudo-bi-dimensional empirical mode decomposition<ref name=":5" /> ===

It should be pointed out herenoted that the “pseudo-BEMD” method is not limited to one-spatial dimension; rather, it can be applied to data of any number of spatial-temporal dimensions. Since the spatial structure is essentially determined by timescales of the variability of a physical quantity at each ___location and the decomposition is completely based on the characteristics of individual time series at each spatial ___location, there is no assumption of spatial coherent structures of this physical quantity. When a coherent spatial structure emerges, it better reflects the physical processes that drive the evolution of the physical quantity on the timescale of each component. Therefore, we expect this method to have significant applications in spatial-temporal data analysis.

To design a pseudo-BEMD algorithm the key step is to translate the algorithm of the 1D [[Hilbert huang transform|EMD]] into a Bi-dimensional Empirical Mode Decomposition (BEMD) and further extend the algorithm to three or more dimensions which is similar to the BEMD by extending the procedure on successive dimensions. For a 3D data cube of <math>i \times j \times k</math> elements, the pseudo-BEMD will yield detailed 3D components of <math>m \times n \times q</math> where <math>m</math>, <math>n</math> and <math>q</math> are the number of the IMFs decomposed from each dimension having <math>i</math>, <math>j</math>, and <math>k</math> elements, respectively.

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\begin{bmatrix}x(1,1) & x(1,2) & \cdots & x(1,j) \\x(2,1) & x(2,2) & \cdots & x(1,j) \\ \vdots & \vdots & & \vdots \\x(i,1) & x(i,2) & \cdots & x(i,j) \end{bmatrix}

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

At first we perform EMD in one direction of ''X''(''i'', ''j''), Row wise for instance, to decompose the data of each row into m components, then to collect the components of the same level of m from the result of each row decomposition to make a 2D decomposed signal at that level of m. Therefore, m set of 2D spatial data are obtained

: <math>

\begin{align}