Multidimensional empirical mode decomposition: Difference between revisions

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By construction, the EOF patterns and the principal components are independent. Two factors inhibit physical interpretation of EOFs: (i)The orthogonality constraint and (ii) the derived patterns may be ___domain dependent. Physical systems are not necessarily orthogonal and if the patterns depend on the region used they may not exist if the ___domain changes.

==== Spatial-temporal signal using Multimulti-dimensional ensemble empirical mode decomposition<ref name=":7" /> ====

Assume, we have a spatio-temporal data ''T''(''s'', ''t''), where ''s'' is spatial locations (not necessary one dimensional originally but needed to be rearranged into a single spatial dimension) from 1 to ''N'' and ''t'' temporal locations from 1 to ''M''.

Using PCA/EOF, one can express ''T ''(''s'', &nbsp;''t'') into <math>T(s,t)=\sum_{i=1}^K Y_i(t)V_i(t)</math><ref name=":7" />

where Yi''Y''_''i''(''t'') is the ith''i''th Principalprincipal Componentcomponent and Vi''V''_''i''(''t'') the ith''i''th [[Empiricalempirical orthogonal function]] (EOF) pattern and ''K'' is the smaller one of ''M'' and ''N''. PC and EOFs are often obtained by solving the eigen-valueeigenvalue/eigen-vectoreigenvector problem of either temporal co-variance matrix or spatial co-variance matrix based on which dimension is smaller. The variance explained by one pair of PCA/EOF is its corresponding eigenvalue divided by the sum of all eigenvalues of the co-variance matrix.

If the data subjected to PCA/EOF analysis is all white noise, all eigenvalues are theoretically equal and there is no preferred vector direction for the principal component in PCA/EOF space. To retain most of the information of the data, one needs to retain almost all the PC's and EOF's, making the size of PCA/EOF expression even larger than that of the original but If the original data contain only one spatial structure and oscillate with time, then the original data can be expressed as the product of one PC and one EOF, implying that the original data of large size can be expressed by small size data without losing information, i.e. highly compressible.