Editing Proper orthogonal decomposition (section)

== Mathematical expression ==
The first idea behind the Proper Orthogonal Decomposition (POD), as it was originally formulated in the domain of fluid dynamics to analyze turbulences, is to decompose a random vector field '''''u(x, t)''''' into a set of deterministic spatial functions ''Φ<sub>k</sub>''(''x'') modulated by random time coefficients ''a<sub>k</sub>''(''t'') so that:

:<math>u(x,t)=\sum_{k=1}^\infty a_k (t) \phi_k(x)</math>
[[File:POD snapshots.png|thumb|POD snapshots]]
The first step is to sample the vector field over a period of time in what we call snapshots (as display in the image of the POD snapshots). This snapshot method<ref>{{Cite journal|last=Sirovich|first=Lawrence|date=1987-10-01|title=Turbulence and the dynamics of coherent structures. I. Coherent structures|journal=Quarterly of Applied Mathematics|volume=45|issue=3|pages=561–571|doi=10.1090/qam/910462|issn=0033-569X|doi-access=free}}</ref> is averaging the samples over the space dimension '''''n''''', and correlating them with each other along the time samples '''''p''''':

:<math>U = \begin{pmatrix} u(x_1,t_1) & \cdots &  u(x_n,t_1)\\ \vdots &  & \vdots \\ u(x_1,t_p) & \cdots &  u(x_n,t_p) \end{pmatrix}</math> with '''n''' spatial elements, and '''p''' time samples

The next step is to compute the [[covariance matrix]] C

:<math>C = \frac{1}{(p-1)} U^T U</math>[[File:Mor-diagram.png|thumb]]

We then compute the eigenvalues and eigenvectors of C and we order them from the largest eigenvalue to the smallest.

We obtain n eigenvalues λ1,...,λn and a set of n eigenvectors arranged as columns in an n × n matrix Φ:

: <math>\phi = \begin{pmatrix} \phi_{1,1} & \cdots & \phi_{1,n} \\ \vdots &  & \vdots \\ \phi_{n,1} & \cdots & \phi_{n,n} \end{pmatrix}</math>