# Talk:Chromosome structure via Euclidean Distance Matrices

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

-                                                                                frame = 4;                                          % 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 == - [[Image:E.coli-4.jpg|thumb|right|560px|Test image E.coli]] - I regard the autocorrelation data you provided as a Gram matrix. - - Then conversion to an EDM is straightforward - Chapter 5.4.2 of [http://meboo.convexoptimization.com/BOOK/EuclideanDistanceMatrix.pdf Convex Optimization & Distance Geometry]. - - The program calculates only the first 20 eigenvalues of the projection of the EDM on a positive semidefinite (PSD) cone. - - 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 project the EDM on the PSD cone, rank 3 subset; this means, precisely, we truncate eigenvalues. - - It is unlikely that this picture is an accurate representation unless the number of eigenvalues of the EDM projection approaches 3 to begin with. - - 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.