Talk:Chromosome structure via Euclidean Distance Matrices
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<pre> | <pre> | ||
%%% Ronan Fleming, E.coli molecule data | %%% Ronan Fleming, E.coli molecule data | ||
| - | %%% -Jon Dattorro, August 2008 | + | %%% -Jon Dattorro, August 9 2008 |
clear all | clear all | ||
load ecoli | load ecoli | ||
| Line 9: | Line 9: | ||
G = (double(G)-128)/128; % Gram matrix | G = (double(G)-128)/128; % Gram matrix | ||
N = size(G,1); | 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)]; | Vn = [-ones(1,N-1); speye(N-1)]; | ||
| - | + | [evec evals flag] = eigs(Vn'*G*Vn, [], 20, 'LA'); | |
| - | + | ||
| - | + | ||
| - | + | ||
| - | [evec evals flag] = eigs( | + | |
if flag, disp('convergence problem'), return, end; | if flag, disp('convergence problem'), return, end; | ||
close all | close all | ||
| - | + | Xs = [zeros(3,1) sqrt(real(evals(1:3,1:3)))*real(evec(:,1:3))']; % Projection of -VDV on PSD cone rank 3 | |
| - | 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,:), '.') | plot3(Xs(1,:), Xs(2,:), Xs(3,:), '.') | ||
</pre> | </pre> | ||
Revision as of 13:56, 9 August 2008
%%% 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 -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 conversion to a Euclidean distance matrix (EDM) is straightforward -
Chapter 5.4.2 of 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 -
Chapter 7.0.4 - 7.1 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.