Biorthogonal decomposition: Difference between revisions

Line 16:

The combination chrono/topo at a given ''k'', ψk(i) φk(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.

The λk are the eigenvalues (sorted in decreasing order), where ''k=1,...,min(N,M)''.

The λk 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)''.

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.

@@ Line 16: / Line 16: @@
 The combination chrono/topo at a given ''k'', &psi;<sub>k</sub>(i) &phi;<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.
-The &lambda;<sub>k</sub> are the eigenvalues (sorted in decreasing order), where ''k=1,...,min(N,M)''.
+The &lambda;<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)''.
 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.