Talk:Chromosome structure via Euclidean Distance Matrices

<pre>

%%% Ronan Fleming, E.coli molecule data

%%% -Jon Dattorro, August 9 2008

clear all

load ecoli

frame = 4; % 1 through 12

G = her49imfs12movfull(frame).cdata; % uint8

G = (double(G)-128)/128; % Gram matrix

N = size(G,1);

Vn = [-ones(1,N-1); speye(N-1)];

[evec evals flag] = eigs(Vn'*G*Vn, [], 20, 'LA');

if flag, disp('convergence problem'), return, end;

close all

Xs = [zeros(3,1) sqrt(real(evals(1:3,1:3)))*real(evec(:,1:3))']; % Projection of -Vn'D Vn on PSD cone rank 3

plot3(Xs(1,:), Xs(2,:), Xs(3,:), '.')

</pre>

== E.coli realization ==

[[Image:E.coli-4.jpg|thumb|right|560px|Test image E.coli]]

I regard the [http://www.convexoptimization.com/TOOLS/ecoli.mat autocorrelation data] you provided as a Gram matrix.

Then conversion to a Euclidean distance matrix (EDM) is straightforward - <br>Chapter 5.4.2 of [http://meboo.convexoptimization.com/BOOK/EuclideanDistanceMatrix.pdf Convex Optimization & Euclidean Distance Geometry].

The program calculates only the first 20 eigenvalues of an oblique projection of the EDM on a positive semidefinite (PSD) cone - <br>

Chapter 7.0.4 - 7.1 [http://meboo.convexoptimization.com/BOOK/ProximityProblems.pdf ''ibidem''].

You can see at runtime that there are many significant eigenvalues; which means, the Euclidean body (the molecule) lives in a space higher than dimension 3, assuming I have interpreted the E.coli data correctly.

To get a picture corresponding to physical reality, we obliquely project the EDM on the closest rank-3 subset of the boundary of that PSD cone; this means, precisely, we truncate eigenvalues.

It is unlikely that this picture is an accurate representation unless the number of eigenvalues of that projection approaches 3 prior to truncation.

Matlab Figures allow 3D rotation in real time, so you can get a good idea of the body's shape.

I include a low-resolution figure here (frame 4) for reference.