1st commit of rdpg. Steps 1 and 2 as outlined in email (see analysis.m)

Author: dpmcsuss

rdpg/analysis.m

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+% %% Pipeline as outlined below by Jovo (via e-mail) 
+%
+% 1) for each graph, compute the scaled sign positive graph laplacian of
+% each graph (call it L_i).  one obtains this by filling in the diagonal of
+% each adjacency matrix by the degree of that vertex, divided by n-1 (this
+% is the expected weight of the self loop).
+% 
+% 2) compute an svd of L_i for all i.  look at the scree plots.
+% presumably, a relatively small number of eigenvalues should suffice to
+% capture most of the variance. 
+% 
+% 3) build a classifier using only the first d singular vectors.  we
+% haven't really thought much about this, but svd implicitly assumes
+% gaussian noise, so, the optimal classifier under this assumption is a
+% discriminant classifier.  the code i have in the repo for discriminant
+% classifiers is still in progress, but if you want to fix it up, that'd be
+% fine.  report results for the discriminant classifiers (LDA and QDA, but
+% with diagonal covariance matrices and whitened covariance matrices).
+% 
+% 4) cluster the singular vectors.  do this by initializing with a
+% hierarchical clustering algorithm, and then use that to seed a k-means
+% algorithm.  you can do this for a variety of values of k and d.  choose
+% the one with the best BIC. marchette will send details for computing BIC
+% values here, and for implementing the "k-means" algorithm, which is, in
+% fact, more like a constrained finite gaussian mixture model algorithm.
+% 
+% 5) given a particular choice of k and d, build a classifier.  here, each
+% class is a mixture of gaussians, so a naive discriminant classifier is
+% insufficient.  instead, one can implement a bayes plugin classifier. so,
+% given the fit of the mixture distributions for the two classes, compute
+% whether a test graph is closer to class 0 or class 1 distribution.
+
+%% Setup - Change to fit your desires
+
+% this is just where i load stuff from my computer
+load /users/dsussman/documents/MATLAB/Brain_graphs/BLSA79_data.mat;
+
+% you can set up stuff however you want here
+
+%% Step 1 - Compute Laplacians
+Ls = graph_laplacian(As);
+
+%% Step 2 - Compute SVDs and make scree plots
+[Us, Ss, Vs] = graph_svd(Ls);
+sz =  size(Ss);
+singVals =  zeros(sz(1),sz(3)); % col vectors of singular vals
+for k=1:sz(3)
+    singVals(:,k) = diag(squeeze(Ss(:,:,k)));
+end
+
+
+% show scree plots of for all the graphs, red is targs is true blue is
+% targs is false
+figure(401);
+hold all;
+plot( singVals(:,targs==1), 'r');
+plot( singVals(:,targs==0), 'b');
+
+%% 

rdpg/graph_laplacian.m

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+function [ L ] = graph_laplacian( A )
+%graph_laplacian Comput scaled sign positive graph laplacian
+%   compute the scaled sign positive graph laplacian of each graph (call it
+%   L_i).  one obtains this by filling in the diagonal of each adjacency
+%   matrix by the degree of that vertex, divided by n-1 (this is the 
+%   expected weight of the self loop).
+d = size(A);
+L=A;
+if numel(d)==2
+for k=1:d(1)
+    L(k,k) = sum(A(k,:))/(d(1)-1);
+end
+end
+if numel(d) == 3
+   for g=1:d(3)
+       for k=1:d(1)
+           L(k,k,g) = sum(A(k,:,g))/(d(1)-1);
+       end
+   end   
+end
+end
+

rdpg/graph_svd.m

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+function [ U, S, V ] = graph_svd( L )
+%graph_svd Computes the SVD of the Laplacian of an adjacancy matrix or
+%matrices
+%   Returns the standard U,S,V matrix or matrices for each Laplacian matrix
+%   in L. Ie L can be either a 2d square matrix or a 3d array of square
+%   matrices.
+%   Q: Do we need to keep U or V or should we just return a vector of
+%   singular values?
+d =  size(L);
+if numel(d) == 2
+    [U,S,V] = svd(L);
+    return;
+end
+U = zeros(d);
+S = U;
+V = U;
+for k=1:d(3)
+    [U(:,:,k),S(:,:,k),V(:,:,k)] = svd(L(:,:,k)); 
+end
+
+end
+