# 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.

## 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]

3. MATLAB code: [U,S,V] = svd(Y,'econ');