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Biorthogonal decomposition: Difference between revisions
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Biorthogonal decomposition (view source)
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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 BOD decomposes the data matrix as follows: | The BOD decomposes the data matrix as follows: | ||
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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 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 columns of ''U'' (''N×N'') are the chronos and the columns of ''V'' (''M×M'') are the topos. | |||
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> | <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> | ||
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:<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 correlation analyses of multipoint measurements. | |||
By multiplying this expression for the correlation 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 correlation matrix ''C'', and λ<sub>k</sub><sup>2</sup> the corresponding eigenvalues. | |||
== See also == | == See also == | ||