Biorthogonal decomposition

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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

<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 BOD decomposes the data matrix as follows:

<math>Y(i,j) = \sum_k \lambda_k \psi_k(i) \phi_k(j),\,</math>

where &psi;_k is a 'chrono' (a temporal function) and &phi;_k 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;_k(i) &phi;_k(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;_k 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 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_kk = &lambda;_k, 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>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>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;_i Y(i,j) = 0), their 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;_k it is easy to show that the topos &phi;_k are the eigenvectors of the covariance matrix C, and &lambda;_k² the corresponding eigenvalues.

See also

• Principal component analysis

References

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