Proper Orthogonal Decomposition?

1. Jul 23, 2014

joshmccraney

Just as the title says, what is a POD? I've tried reading papers but I feel I am missing something. Does anyone have a good, intuitive understanding of this? Let me know if I've accidentally posted in the wrong section.
Thanks!

2. Jul 24, 2014

WWGD

Decomposition of what type of object ?

3. Jul 24, 2014

D H

Staff Emeritus
Principal orthogonal decomposition is just another name for the singular value decomposition, aka principal components analysis, aka the Karhunen–Loève transform, aka the Hoteling transform, aka factor analysis, and probably other names as well.

This concept has so many names because it is so extremely useful in so many different domains. Different people have developed several different schemes (hence the many names) for attacking these problems, but ultimately they're all pretty much the same thing.

4. Jul 25, 2014

joshmccraney

WWGD: the object would be of wind velocity (its turbulent).

DH: do you mind telling me (or referring me to a source) about what you mean by "all the same thing"? the fact that i don't understand you makes me feel i am missing something pretty big.

thanks both!

5. Jul 25, 2014

D H

Staff Emeritus
Berkooz, Gal, Philip Holmes, and John L. Lumley (1993) "The proper orthogonal decomposition in the analysis of turbulent flows." Annual review of fluid mechanics 25:1 539-575 provides an overview on POD toward exactly the problem you asked about. It also points out the similarity between POD, the KL transform, and principal components analysis. For some reason, they miss the singular value decomposition, which lies at the heart of all of these techniques.

Another tutorial on the POD, Chatterjee, Anindya (2000) "An introduction to the proper orthogonal decomposition." Current science 78:7 808-817, does point out the similarity between the POD and principal component analysis, the Karhunen–Loéve transform, and the singular value decomposition, most particularly the latter.

Shlens, Jonathon (2014). "A tutorial on principal component analysis." arXiv preprint arXiv:1404.1100 provides a nice tutorial on principal component analysis and talks about the intimate relationship between PCA and the singular value decomposition.

In all cases, the root of the technique involves decomposing some matrix $A$ as $A=UVW^\mathsf{T}$ (or $A=UVW^\ast$ if $A$ is complex). where the matrices $U$ and $W$ are orthogonal (unitary if $A$ is complex) and the matrix $V$ is a positive semidefinite diagonal matrix (the diagonal elements are real and non-negative). This decomposition always exists. The diagonal elements of $V$ contains the "singular values" of $A$, hence the name of the technique. The decomposition is performed such that the largest singular value is in $v_{1,1}$, the second largest in $v_{2,2}$, and so on.

The power of the technique lies in the fact that most of the singular values (and the left and right eigenvectors associated with them) are oftentimes zero or are very small compared to the largest few singular values. This provides a natural means for vastly reducing the dimensionality of a problem, from 1000x1000 (or more) to just a few dimensions.

Last edited: Jul 25, 2014
6. Jul 28, 2014

joshmccraney

Ohhhhh okay, so this is what is is! Makes sense now. (sorry for not posting this in the linear algebra spot, which i think is where it belongs?)

thanks for going into so much detail. i'll definitely follow up more! you've given me a good base; thanks so much!