Talk:Chromosome structure via Euclidean Distance Matrices

From Wikimization

Revision as of 22:56, 6 August 2008

%%% Ronan Fleming, E.coli molecule data 
%%% -Jon Dattorro, August 2008
clear all
load ecoli

frame = 12;                                         % 1 through 12
G = her49imfs12movfull(frame).cdata;                % uint8
G = (double(G)-128)/128;                            % Gram matrix
N = size(G,1);

D = diag(G)*ones(N,1)' + ones(N,1)*diag(G)' - 2*G;  % EDM D

clear her49imfs12movfull G;

Vn = [-ones(1,N-1); speye(N-1)];
VDV = (-Vn'*D*Vn)/2;

clear D Vn

[evec evals flag] = eigs(VDV, [], 20, 'LR');
if flag, disp('convergence problem'), return, end;

close all

Xs = sqrt(real(evals(1:3,1:3)))*real(evec(:,1:3))'; % Projection of -VDV on PSD cone rank 3
plot3(Xs(1,:), Xs(2,:), Xs(3,:), '.')

E.coli realization

I regard the autocorrelation data you provided as a Gram matrix.

Then the conversion to an EDM is straightforward - Chapter 5.4.2 of Convex Optimization & Distance Geometry.

The program calculates the first 20 eigenvalues of the projection of the EDM on the positive semidefinite (PSD) cone.

You can see that there are many significant eigenvalues; which means, the Euclidean body 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, you have to project on the PSD cone, rank 3 subset.