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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> | ||
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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)'': | ||
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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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== Physical interpretation == | == Physical interpretation == | ||
For linear systems, the biorthogonal modes converge to the linear eigenmodes of the system in the limit of large ''N''. <ref> | 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 == | ||