Biorthogonal decomposition: Difference between revisions

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The Biorthogonal Decomposition (BOD, also known as Proper Orthogonal Decomposition, POD<ref>P. Holmes, J.L. Lumley, and G. Berkooz, ''Turbulence, Coherent Structures, Dynamical Systems and Symmetry'', Cambridge University Press (1996) ISBN 0521634199</ref>) applies to the analysis of multipoint measurements
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:<math>Y(i,j)\,</math>
 
where ''i=1,...,N'' is a temporal index and ''j=1,...,M'' a spatial index (typically).
The time traces ''Y(i,j)'' for fixed ''j'' are usually sampled at a fixed rate (so ''t(i)'' is equidistant); however the measurement locations ''x(j)'' need not be ordered in any specific way.
 
The data are decomposed in a small set of linearly independent modes, determined from the structure of the data matrix ''Y'' itself, without prejudice regarding the mode shape.
 
== Description ==
 
The BOD decomposes the data matrix as follows:
 
:<math>Y(i,j) = \sum_k \lambda_k \psi_k(i) \phi_k(j),\,</math>
 
where &psi;<sub>k</sub> is a 'chrono' (a temporal function) and &phi;<sub>k</sub> a 'topo' (a spatial or detector-dependent function), such that the chronos and topos satisfy the following orthogonality relation
 
:<math>\sum_i{\psi_k(i)\psi_l(i)} = \sum_j{\phi_k(j)\phi_l(j)} = \delta_{kl}.\,</math>
 
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)'', 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)'':
 
:<math>Y = U S V^T.\,</math>
 
where ''S'' is a diagonal ''N&times;M'' matrix and ''S<sub>kk</sub>'' = &lambda;<sub>k</sub>, the first min(''N,M'') columns of ''U'' (''N&times;N'') are the chronos and the first min(''N,M'') columns of ''V'' (''M&times;M'') are the topos. <ref>[[:Wikipedia:MATLAB|MATLAB]] code: <code>[U,S,V] = svd(Y,'econ');</code></ref>
 
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.
<ref>[http://link.aip.org/link/?PHPAEN/1/3288/1 T. Dudok de Wit et al., ''The biorthogonal decomposition as a tool for investigating fluctuations in plasmas'', Phys. Plasmas '''1''' (1994) 3288]</ref>
 
A limitation of the technique is that it assumes space-time separability.
This is not always the most appropriate assumption:
e.g., travelling waves have a structure such as ''cos(kx-&omega;t)''; however, most propagating waves can still be recognised clearly by their distinct footprint in the biorthogonal modes (provided there are not too many): a travelling wave will produce a pair of modes with similar amplitude and a 90&deg; phase difference.
 
== Relation with signal covariance ==
 
Assuming the signals ''Y(i,j)'' have zero mean (their temporal average is zero, or &Sigma;<sub>i</sub> ''Y(i,j)'' = 0), their [[:Wikipedia:Covariance|covariance]] is defined as:
 
:<math>C(j_1,j_2) = \sum_i {Y(i,j_1)Y(i,j_2)},\!</math>
 
Substituting the above expansion of ''Y'' and using the orthogonality relations, one obtains:
 
:<math>C(j_1,j_2) = \sum_k {\lambda_k^2 \phi_k(j_1)\phi_k(j_2)}</math>
 
The technique is therefore ideally suited to perform cross covariance analyses of multipoint measurements.
 
By multiplying this expression for the covariance matrix ''C'' with the vector &phi;<sub>k</sub> it is easy to show that the topos &phi;<sub>k</sub> are the eigenvectors of the covariance matrix ''C'', and &lambda;<sub>k</sub><sup>2</sup> the corresponding eigenvalues.
 
== Physical interpretation ==
 
For linear systems, the biorthogonal modes converge to the linear eigenmodes of the system in the limit of large ''N''. <ref>[http://dx.doi.org/10.1006/jsvi.2001.3930 G. Kerschen and J. C. Golinval, ''Physical interpretation of the proper orthogonal modes using the Singular Value Decomposition'', Journal of Sound and Vibration '''249''', 5 (2002) 849]</ref>
 
== See also ==
 
* [[:Wikipedia:Principal component analysis|Principal component analysis]]
 
== References ==
<references />

Revision as of 03:28, 19 May 2011

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