Biorthogonal decomposition

The Biorthogonal Decomposition (BOD, also known as Proper Orthogonal Decomposition, POD) 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 ψ_k is a 'chrono' (a temporal function) and φ_k a 'topo' (a spatial or detector-dependent function), 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):

$Y=USV^{T}.\,$

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.

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.

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_k² 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.

See also

• Principal component analysis