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DisTools examples: Pseudo-Euclidean Embedding

Instead of constructing a dissimilarity space, the dissimilarity matrix may be used for finding an embedding. Non-Euclidean dissimilarities may be embedded in a pseudo-Euclidean space if they are symmetric. Some examples of such an embedding will be shown. It is assumed that readers are familiar with PRTools and will consult the following pages where needed:

The main DisTools routine for P-embedding is pe_em:

d  = chickenpieces(15,60)*makesym;
w = d*pe_em;  % compute PE mapping
getsig(w)     % get signature of the mapping
x = d*w;      % map dismat in PE space
getsig(x)     % get signature of the data

Note that many dissimilarity datasets are not symmetric. For embedding symmetry is necessary. pe_em symmetrizes by default, but further results are confusing if we map non-symmetric data n a PE space. The routine makesym takes care of this (by default averaging of the matrix and its transpose). The signature (dimensionalities of the positive and negative space) are stored in the mapping as well as in the projected data. The accuracy of a PE mapping and the reconstruction of the dissimilarities can be inspected in the following way:

[p,q] = getsig(x); % get the signature
+d(1:5,1:5).^2     % show part of original data  squared
% recompute squared dissimilarities
r = distm(x(:,1:p)) - distm(x(:,p+1:p+q));
+r(1:5,1:5)        % show and compare
mean(mean(abs(+d.^2-(+r)))) % total average difference is small!

The routine pe_distm does exactly the same as the above construct based on distm and automatically extracts the signature.

Let us now compute a PE space defined by all data except the first 5 objects and then project these objects in the PE space, followed by a reconstruction of their distances.

w = d(4:end,4:end)*pe_em; % mapping defined by all but 3 objects
x = d(:,4:end)*w;         % map all objects
+d(1:5,1:5).^2            % show part of original data  squared
r = pe_distm(x);          % recompute squared dissimilarities
+r(1:5,1:5)               % show and compare

Note that the reconstructed distance between the objects 4 and 5 is correct but that all distances with and between the object 1, 2 and 3 are entirely wrong: objects not participating in the construction of the space cannot be mapped correctly. This behavior can be significantly improved by the construction of approximate mappings to the PE space and using just a subset of the eigenvectors. One way to do this is by setting a parameter in pe_em which determines the retained eigen-fraction:

alf = 0.99;
w = d(4:end,4:end)*pe_em([],alf);
x = d(:,4:end)*w;         % map all objects
+d(1:5,1:5).^2            % show part of original data, squared
r = pe_distm(x);          % recompute squared dissimilarities
+r(1:5,1:5)               % show and compare

Exercise_1

Try to find an acceptable value for alf.

Exercise_2

  1. Split the dataset 50-50 in both directions obtaining four matrices D(S,S), D(S,T), D(T,S) and D(T,T). genddat may be used to find proper, randomized sets of indices T and S.
  2. Use D(T,T) to find a PE mapping.
  3. Project D(S,T) as well as D(T,T) in the PE space
  4. Reconstruct D, obtaining a distance matrix R. Compare the four submatrices of D with the corresponding four submatrices of R. Define an accuracy measure. Plot the four accuracies as a function of alf.

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