User:Jarle Pahr/SVD

Notes on Singular Value Decomposition (SVD):

See also http://en.wikipedia.org/wiki/Singular_value_decomposition

Eigenvectors and eigenvalues:

An eigenvector of a matrix is A is a non-zero vector [math]\displaystyle{ \overrightarrow v }[/math] that satisfies the equation

[math]\displaystyle{ A\overrightarrow v = \lambda \overrightarrow v }[/math]

Where [math]\displaystyle{ \lambda }[/math] is a scalar. [math]\displaystyle{ \lambda }[/math] is called an eigenvalue.

Singular values:

Any mxn matrix A can be factored as:

[math]\displaystyle{ A = U\sum {V^T} }[/math] where:

• [math]\displaystyle{ U }[/math] is an mXm orthogonal matrix where the columns are the eigenvectors of [math]\displaystyle{ AA^T }[/math]
• [math]\displaystyle{ V }[/math] is an nXn orthogonal matrix where the columns are the eigenvectors of [math]\displaystyle{ A^{T}A }[/math]
• [math]\displaystyle{ \sum{} }[/math] is an mXn diagonal matrix where the [math]\displaystyle{ r }[/math] first diagona elements are the square roots of the eigenvalues of [math]\displaystyle{ A^{T}A }[/math], also called the singular values of A. Singular values are always real and positive.

For any SVD, the following facts apply:

• The rank of a matrix A equals the number of singular values of A.
• The column space of A is spanned by the first r columns of U.
• The null space of A is spanned by the last n − r columns of V.
• The row space of A is spanned by the first r columns of V.
• The null space of AT is spanned by the last m − r columns of U.

The columns of [math]\displaystyle{ U }[/math] are called the left singular vectors, and the columns of [math]\displaystyle{ V }[/math] theright singular vectors.

SVD in NumPy

http://docs.scipy.org/doc/numpy/reference/generated/numpy.linalg.svd.html

To find the rank of a matrix A, assign u,s,vh = linalg.svd(A) and r = len(s)

Applications to metabolic networks

Given a metabolic network with stoichoimetric matrix [math]\displaystyle{ S }[/math], the change in metabolite concentrations [math]\displaystyle{ x }[/math] as function of the reaction rates [math]\displaystyle{ v }[/math] is given by the equation:

[math]\displaystyle{ Sv = \frac{{dx}}{{dt}} }[/math]

The singular value decomposition of S is

[math]\displaystyle{ S = U\sum {V^T} }[/math]

Which can be written as:

[math]\displaystyle{ SV = U\sum }[/math]

The above expression defines a series of independent equations on the form

[math]\displaystyle{ S{v_k} = {\sigma _k}{u_k} }[/math]

Where [math]\displaystyle{ v_k }[/math] are the column vectors of V (the right singular vectors), [math]\displaystyle{ {\sigma _k} }[/math] are the singular valus and [math]\displaystyle{ u_k }[/math] are the column vectors of U (the left singular vectors).

Applying the SVD to S, we get:

[math]\displaystyle{ {U^T}\frac{{dx}}{{dt}} = \sum {V^T}v }[/math]

The columns of U, called the left singular vectors and denoted [math]\displaystyle{ U_i }[/math] form linear combinations of the concentations variables. Similarly, columns of V, called the right singular vectors' and denoted [math]\displaystyle{ V_i }[/math] form linear combinations of the fluxes. Linear combinations of concentration variablesare called pools while linear combinations of fluxes are called pathways.

For each nonzero singular value there is a corresponding column [math]\displaystyle{ u_k }[/math] in U and a row [math]\displaystyle{ v_k^T }[/math] whose relationship is described by the equation

[math]\displaystyle{ \frac{{d(u_k^Tx)}}{{dt}} = {\sigma _k}(v_k^Tv),\,\,k = 1,...,r }[/math]

where the linar combination of metabolite concentrations

[math]\displaystyle{ u_k^Tx = {u_{k1}}{x_1} + {u_{k2}}{x_2} + ... + {u_{kr}}{x_m} }[/math]

is being driven by the linear combination of metabolic fluxes

[math]\displaystyle{ v_k^Tv = {v_{k1}}{v_1} + {v_{k2}}{v_2} + ... + {v_{kn}}{v_n} }[/math]

Each column [math]\displaystyle{ u_k }[/math] of U thus defines an eigen-reaction or systemic metabolic reaction which can be expressed as

[math]\displaystyle{ \sum_{\text{for } v_{kj \lt 0}} u_{ki}x_{i}\underset{{\sum\limits_{{\text{for }}{{\text{v}}_{kj \lt 0}}} {{v_{kj}}{v_j}} }}{\overset{{\sum\limits_{{\text{for }}{{\text{v}}_{kj \gt 0}}} {{v_{kj}}{v_j}} }}{\rightleftharpoons}} \sum_{\text{for } v_{kj \gt 0}} u_{ki}x_{i} }[/math]

The elements of [math]\displaystyle{ u_k }[/math] are called systemeic stoichiometric coefficients and the elements of [math]\displaystyle{ v_k }[/math] are called systemeic participation numbers.

Attention: The right singular vectors [math]\displaystyle{ v_k }[/math] should not be confused with the flux vector [math]\displaystyle{ v }[/math].

Links

http://www.uwlax.edu/faculty/will/svd/

math.stackexchange.com/questions/261801/how-can-you-explain-the-singular-value-decomposition-to-non-specialists

http://www.ams.org/samplings/feature-column/fcarc-svd

http://campar.in.tum.de/twiki/pub/Chair/TeachingWs05ComputerVision/3DCV_svd_000.pdf

http://langvillea.people.cofc.edu/DISSECTION-LAB/Emmie%27sLSI-SVDModule/p4module.html

Finding the null space of a matrix

Finding null space of matrix: http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi

https://github.com/amilsted/evoMPS/blob/master/evoMPS/nullspace.py

http://wiki.scipy.org/Cookbook/RankNullspace

http://pastebin.com/5ms5QEtn

A rational basis for the nullspace of a matrix with rational elements can be found using SymPy.

Links

http://mathbio.colorado.edu/mediawiki/index.php/MBW:Singular_value_decomposition_of_stoichiometric_matrices

Bibliography

Palsson, 2006. Systems Biology: Properties of reconstructed networks. Cambridge.

Famili & Palsson. Journal of theoretical biology 224 (2003) 87-96.