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Biorthogonal decomposition: Difference between revisions
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The combination chrono/topo at a given ''k'', ψ<sub>k</sub>(i) φ<sub>k</sub>(j), | The combination chrono/topo at a given ''k'', ψ<sub>k</sub>(i) φ<sub>k</sub>(j), | ||
is called a spatio-temporal 'mode' of the fluctuating system, and is constructed from the data matrix without any prejudice regarding the mode shape. | is called a spatio-temporal 'mode' of the fluctuating system, and is constructed from the data matrix without any prejudice regarding the mode shape. | ||
The λ<sub>k</sub> are the eigenvalues (sorted in decreasing order), where ''k=1,...,min(N,M)''. | The λ<sub>k</sub> are the eigenvalues (sorted in decreasing order), where ''k=1,...,min(N,M)'', and directly represent the square root of the fluctuation energy contained in the corresponding mode. | ||
This decomposition is achieved using a standard [[:Wikipedia:Singular value decomposition|Singular value decomposition]] of the data matrix ''Y(i,j)''. | This decomposition is achieved using a standard [[:Wikipedia:Singular value decomposition|Singular value decomposition]] of the data matrix ''Y(i,j)''. | ||
Thus, the oscillations of the spatiotemporal fluctuating field are represented by means of a very small number of spatio-temporal modes that are constructed from the data themselves, without prejudice regarding the mode shape. | Thus, the oscillations of the spatiotemporal fluctuating field are represented by means of a very small number of spatio-temporal modes that are constructed from the data themselves, without prejudice regarding the mode shape. | ||