Determine how different are some vectors

I want to differentiate data vectors to find those that are similar. For example:

A=[4,5,6,7,8];
B=[4,5,6,6,8];
C=[4,5,6,7,7];

D=[1,2,3,9,9];
E=[1,2,3,9,8];

In the previous example I want to distinguish that A,B,C vectors are similar (not the same) to each other and D,E are similiar to each other. The result should be something like: A,B,C are similar and D,E are similar, but the group A,B,C is not similar to the group of D,E. Matlab c开发者_如何学JAVAan do this? I was thinking using some classification algorithm or Kmeans,ROC,etc.. but I'm not sure which one will be the best one.

Any suggestion? Thanks in advance

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One of my new favourite methods for this sort of thing is agglomerate clustering.

First, concatenate all your vectors into a matrix, where each row is a separate vector. This makes such methods much easier to use:

F = [A; B; C; D; E];

Then the linkages can be found:

Z = linkage(F, 'ward', 'euclidean');

This can be plotted using:

dendrogram(Z);

This shows a tree, where each leaf at the bottom is one of the original vectors. Lengths of the branches show similarities and dissimilarities.

As you can see, 1, 2 and 3 are shown to be very close, as are 4 and 5. This even gives a measure of closeness, and shows that vectors 1 and 3 are deemed to be closer than vectors 2 and 3 (in the sense that, percentagewise, 7 is closer to 8 than 6 is to 7).

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If all the vectors you are comparing are of the same length, a suitable norm on pairwise differences may well be enough. The norm to choose will depend on your particular criteria of closeness, of course, but with the examples you show, simply summing the absolute values of the components of the pairwise differences gives:

  A B C  D  E
A 0 1 1 12 11
B   0 2 13 12
C     0 13 12
D       0  1
E          0

which doesn't need a particularly well-tuned threshold to work.

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You can use pdist(), this function gives you the pairwise distances.

Various distance (opposite of similarity) metrics are already implemented, 'euclidean' seems appropriate for your situation, although you may want to try out the effect of different metrics.

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Here it goes the solution I propose based on your results:

Z = [A;B;C;D;E];
Y = pdist(Z);
matrix = SQUAREFORM(Y);
matrix_round = round(matrix);

Now that we have the vector we can set the threshold based on the maximun value and decide with which theshold is the most appropriate.

It would be nice to create some cluster plot showing the differences between them.

Best regards