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 BOD decomposes the data matrix as follows:

$Y(i,j)=\sum _k{\lambda }_k{\psi }_k(i){\varphi }_k(j),$

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

$\sum _i{\psi }_k(i){\psi }_l(i)=\sum _j{\varphi }_k(j){\varphi }_l(j)={\delta }_{kl}.$

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 without any prejudice regarding the mode shape. 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):

$Y=US{V}^T.$

where S is a diagonal N×M matrix and S_kk = λ_k, the columns of U (N×N) are the chronos and the 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 correlation

The correlation between signals j₁ and j₂ 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:

$C(j_1,j_2)=\sum _k{\lambda }_k^2{\varphi }_k(j_1){\varphi }_k(j_2)$

The technique is therefore ideally suited to perform cross correlation analyses of multipoint measurements.

By multiplying this expression for the correlation matrix C with the vector φ_k it is easy to show that the topos φ_k are the eigenvectors of the correlation matrix C, and λ_k² the corresponding eigenvalues.

See also

• Principal component analysis