# Biorthogonal decomposition

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.

## Contents

## Description

The BOD decomposes the data matrix as follows:

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

where ψ_{k} is a 'chrono' (a temporal function) and φ_{k} 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*, ψ_{k}(i) φ_{k}(j),
is called a spatio-temporal 'mode' of the fluctuating system, and is constructed from the data matrix itself.
The λ_{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)*:

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

where *S* is a diagonal *N×M* matrix and *S _{kk}* = λ

_{k}, 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 Σ_{i} *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 φ_{k} it is easy to show that the topos φ_{k} are the eigenvectors of the covariance matrix *C*, and λ_{k}^{2} 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

- ↑ P. Holmes, J.L. Lumley, and G. Berkooz,
*Turbulence, Coherent Structures, Dynamical Systems and Symmetry*, Cambridge University Press (1996) ISBN 0521634199 - ↑ N. Aubry, R. Guyonnet and R. Lima,
*Spatiotemporal analysis of complex signals: Theory and applications*, J. Statistical Physics**64**, 3-4 (1991) 683 - ↑ MATLAB code:
`[U,S,V] = svd(Y,'econ');`

- ↑ T. Dudok de Wit et al.,
*The biorthogonal decomposition as a tool for investigating fluctuations in plasmas*, Phys. Plasmas**1**(1994) 3288 - ↑ 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 - ↑ 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