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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 | |||
The Biorthogonal Decomposition (BOD, also known as Proper Orthogonal Decomposition, POD | |||
: | :<math>Y(i,j)\,</math> | ||
where ''i=1,...,N'' is a temporal index and ''j=1,...,M'' a spatial index (typically). | where ''i=1,...,N'' is a temporal index and ''j=1,...,M'' a spatial index (typically). | ||
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The BOD decomposes the data matrix as follows: | The BOD decomposes the data matrix as follows: | ||
: | :<math>Y(i,j) = \sum_k \lambda_k \psi_k(i) \phi_k(j),\,</math> | ||
where & | where ψ<sub>k</sub> is a 'chrono' (a temporal function) and φ<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'', & | 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 & | 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)'': | ||
: | :<math>Y = U S V^T.\,</math> | ||
where ''S'' is a diagonal ''N& | where ''S'' is a diagonal ''N×M'' matrix and ''S<sub>kk</sub>'' = λ<sub>k</sub>, the first min(''N,M'') columns of ''U'' (''N×N'') are the chronos and the first min(''N,M'') columns of ''V'' (''M×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. | 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. | A limitation of the technique is that it assumes space-time separability. | ||
This is not always the most appropriate assumption: | This is not always the most appropriate assumption: | ||
e.g., travelling waves have a structure such as ''cos(kx-& | e.g., travelling waves have a structure such as ''cos(kx-ωt)''; 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° phase difference. | ||
== Relation with signal covariance == | == Relation with signal covariance == | ||
Assuming the signals ''Y(i,j)'' have zero mean (their temporal average is zero, or & | Assuming the signals ''Y(i,j)'' have zero mean (their temporal average is zero, or Σ<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: | 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. | 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 & | By multiplying this expression for the covariance matrix ''C'' with the vector φ<sub>k</sub> it is easy to show that the topos φ<sub>k</sub> are the eigenvectors of the covariance matrix ''C'', and λ<sub>k</sub><sup>2</sup> the corresponding eigenvalues. | ||
== See also == | == See also == | ||
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== References == | == References == | ||
<references /> | |||