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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<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 | ||
:<math>Y(i,j)\,</math> | :<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). | ||
The time traces ''Y(i,j)'' for fixed ''j'' are usually sampled at a fixed rate; however the measurement locations ''x(j)'' need not be ordered in any specific way. | 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: | The BOD decomposes the data matrix as follows: | ||
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:<math>Y(i,j) = \sum_k \lambda_k \psi_k(i) \phi_k(j),\,</math> | :<math>Y(i,j) = \sum_k \lambda_k \psi_k(i) \phi_k(j),\,</math> | ||
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 | where ψ<sub>k</sub> is a 'chrono' (a temporal function) and φ<sub>k</sub> a 'topo' (a spatial or detector-dependent function)<ref>N. Aubry, R. Guyonnet and R. Lima, ''Spatiotemporal analysis of complex signals: Theory and applications'', [[doi:10.1007/BF01048312|J. Statistical Physics '''64''', 3-4 (1991) 683]]</ref>, 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> | :<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'', 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 | is called a spatio-temporal 'mode' of the fluctuating system, and is constructed from the data matrix itself. | ||
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. | 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×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> | <ref>T. Dudok de Wit et al., ''The biorthogonal decomposition as a tool for investigating fluctuations in plasmas'', [[doi:10.1063/1.870481| 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. | ||
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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. | 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 | == Relation with signal covariance == | ||
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> | :<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 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. | |||
== Physical interpretation == | |||
For linear systems, the biorthogonal modes converge to the linear eigenmodes of the system in the limit of large ''N''. <ref>G. Kerschen and J. C. Golinval, ''Physical interpretation of the proper orthogonal modes using the Singular Value Decomposition'', [[doi:10.1006/jsvi.2001.3930|Journal of Sound and Vibration '''249''', 5 (2002) 849]]</ref> | |||
The biorthogonal decomposition is also highly sensitive to globally correlated oscillations. | |||
Recently, this property has been exploited to detect Zonal Flows. | |||
<ref>B.Ph. van Milligen, E. Sánchez, A. Alonso, M.A. Pedrosa, C. Hidalgo, A. Martín de Aguilera, A. López Fraguas, ''The use of the Biorthogonal Decomposition for the identification of zonal flows at TJ-II'', [[doi:10.1088/0741-3335/57/2/025005|Plasma Phys. Control. Fusion '''57''', 2 (2015) 025005]]</ref> | |||
== See also == | == See also == | ||