Biorthogonal decomposition

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The Biorthogonal Decomposition (BOD, also known as Proper Orthogonal Decomposition, POD[1]) applies to the analysis of multipoint measurements

$ Y(i,j)\, $

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 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:

$ Y(i,j) = \sum_k \lambda_k \psi_k(i) \phi_k(j),\, $

where ψk is a 'chrono' (a temporal function) and φk a 'topo' (a spatial or detector-dependent function)[2], such that the chronos and topos satisfy the following orthogonality relation

$ \sum_i{\psi_k(i)\psi_l(i)} = \sum_j{\phi_k(j)\phi_l(j)} = \delta_{kl}.\, $

The combination chrono/topo at a given k, ψk(i) φk(j), is called a spatio-temporal 'mode' of the fluctuating system, and is constructed from the data matrix itself. The λk 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):

$ Y = U S V^T.\, $

where S is a diagonal N×M matrix and Skk = λk, 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. [3]

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. [4]

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-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

Assuming the signals Y(i,j) have zero mean (their temporal average is zero, or Σi Y(i,j) = 0), their covariance is defined as:

$ C(j_1,j_2) = \sum_i {Y(i,j_1)Y(i,j_2)},\! $

Substituting the above expansion of Y and using the orthogonality relations, one obtains:

$ C(j_1,j_2) = \sum_k {\lambda_k^2 \phi_k(j_1)\phi_k(j_2)} $

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 φk it is easy to show that the topos φk are the eigenvectors of the covariance matrix C, and λk2 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. [5] The biorthogonal decomposition is also highly sensitive to globally correlated oscillations. Recently, this property has been exploited to detect Zonal Flows. [6]

See also

References

  1. P. Holmes, J.L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press (1996) ISBN 0521634199
  2. N. Aubry, R. Guyonnet and R. Lima, Spatiotemporal analysis of complex signals: Theory and applications, J. Statistical Physics 64, 3-4 (1991) 683
  3. MATLAB code: [U,S,V] = svd(Y,'econ');
  4. T. Dudok de Wit et al., The biorthogonal decomposition as a tool for investigating fluctuations in plasmas, Phys. Plasmas 1 (1994) 3288
  5. G. Kerschen and J. C. Golinval, Physical interpretation of the proper orthogonal modes using the Singular Value Decomposition, Journal of Sound and Vibration 249, 5 (2002) 849
  6. 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, Plasma Phys. Control. Fusion 57, 2 (2015) 025005