measure similarity between 1 dimensional vectors - matlab

Question

EDITED QUESTION
I have n signals of equal length.
X_signal
Y_signal
...
Z_signal
I calculate minima of these signals and I store their location (in time) in the vectors
X = [x1 x2 x3 x4 ... x100]
Y = [y1 y2 y3 y4 ... y150]
...
Z = [z1 z2 z3 z4 ... z110]
You can think about X,Y,..Z as time series that can have different lenght.
I assume that the original signals are similar if they have their minima almost at the same locations.
I would like to know what would be a smart approach to measure this kind of similarity keeping in mind that some minima in X,Y,Z can be just noise.
For example if X = [1 5 8 12 15 20] and Y = [1.5 5.5 7.5 10 12 15.5 20.2] they should be similar since almost all the points have the same value exept for Y(4) = [10].
If you have time code or pseudo code in Matlab is appreciated, otherwise also a suggestion, link etc. is super fine.
Thanks
ORIGINAL QUESTION
I have n vectors of different length.
X = [x1 x2 x3 x4 ... x100]
Y = [y1 y2 y3 y4 ... y150]
...
Z = [z1 z2 z3 z4 ... z110]
Vectors (X Y ... Z) represent minima values of the energy of the corresponding signals (X_energy, Y_energy, etc).
To recap starting from the signals X_signal, Y_signal ... Z_signal I compute the energy in windows of 20 samples and I calculate the minima of the resulting energy signals.
Assuming that 2 or more vector are similar if they have almost equal values (i.e. X and Y are similar if x1 ~= y1, x2 ~=y2, etc.) In other words I assume that the original signals are similar if they have minimum energy at the same (or almost at the same) time instant. I would like to know what would be a smart approach to measure this kind of similarity.
PS.
It is almost impossible that two vectors are equal so I would like to have just an idea of how close their "points" are.
X and Y could be similar also if they are shifted (i.e. x1~=y3, x2=~y4, etc)
It is always the case that the values are in ascending order (x1<x2<...<x100)
If you have time code or pseudo code in Matlab is appreciated, otherwise also a suggestion, link etc. is super fine.
Thanks

Answer 1

One possible approach (particularly if you do not have the Statistics and/or Signal Processing toolbox) is to generate a correlation matrix for all of your vectors with the Matlab function corrcoef
Since your vectors are different sizes, you would have to either
zero pad the smaller vectors so they are the same size as the largest
Or take an aligned sample of values less than or equal to the number
of values in the smallest vector, out of each of them before
computing correlation.
It depends on your application which procedure is more suitable. Since your vectors are ordered in ascending order, likely zero padding would be inappropriate.
Then you would need to create a matrix M with the rows corresponding to the elements, and the columns corresponding to each (zero padded or sampled) vector.
You could do that with the Matlab function horzcat:
M=horzcat(V1,V2,...Vn)
where V1, V2, ..Vn are each column vectors of the same size.
Finally you could get a correlation matrix for all of your vectors with corrcoef:
Cmat=corrcoef(M)
Matlab docs for corrcoef at this link will help you understand how to interpret the results statistically.
Note that this approach would not take into account any correlation between lagged versions of your vectors.

Answer 2

Edited answer
Now that it is clear that X vector is the time positions of all minima of signal 'X', Y vector is the time positions of all minima of signal 'Y', etc... Here is some updated code.
In fact the idea is still the same ... we build a linearly sampled time vector from all time positions of the minima in all signals (+ from some time sampling precision)... then we build new signals being 1.0 everywhere expect at minima time locations (set to 0.0) ... finally we use the same correlation code as before.
NB Speed and memory optimized version is now available here
function [RMax] = MinimaCorrelation(c, ts)
%[
% Some default resolution and time-location of minima positions
if (nargin < 2), ts = 0.1; end
if (nargin < 1), c = { [1 3 8 7 3 4 12]; [3 8 7 3]; [4 12]; [5 3 8 -3 12]; [1 3 8 7 3 4 12]; }; end
% Number of channels
n = length(c);
% Build linearly sample time vector for all time locations
minTime = min(cellfun(#min, c));
maxTime = max(cellfun(#max, c));
timeVector = minTime:ts:maxTime;
timeVector(end+1) = timeVector(end) + ts; % just to really include min and max if step is not ok
% Build new signals being '1' everywhere except at minima locations (set to '0')
s = ones(n, length(timeVector));
for ni = 1:n
for mv = c{ni}
[~, ind] = min(abs(timeVector - mv));
s(ni, ind) = 0;
end
end
% Correlation (copied 3 times to avoid biased effect on sides ==> circular shifting is ok this way)
s = [s, s, s].';
RMax = max(xcorr(s, 'coeff'), [], 1);
% Put in R(i,j) format
RMax = reshape(RMax, [n n]);
%]
end
With default data, one obtains:
1.0000 0.9899 0.9866 0.9829 1.0000
0.9899 1.0000 0.9833 0.9865 0.9899
0.9866 0.9833 1.0000 0.9832 0.9866
0.9829 0.9865 0.9832 1.0000 0.9829
1.0000 0.9899 0.9866 0.9829 1.0000
Careful, it is brute force solution (time & memory consumption increases quickly with the number of signal and time resolution to have). Now that question is more clear, maybe someone will find smarter answer.
Original answer
Here is coarse-code for an approach using the maximum of cross-correlation and xcorr routine (in signal processing toolbox):
function [RMax] = xcorrmax(c)
%[
% Default signals for test
if (nargin < 1),
c = cell(0,0);
c{end+1} = [1 3 8 7 3 4 12];
c{end+1} = [3 8 7 3];
c{end+1} = [4 12];
c{end+1} = [5 3 8 -3 12];
c{end+1} = [1 3 8 7 3 4 12];
end
% Number of channels
n = length(c);
% Padding to have vectors all of the same length
% See also `padarray` to do circular/symmetric padding (i don't have image toolbox)
maxlength = max(cellfun(#length, c));
c = cellfun(#(x)myquickpad(x, maxlength), c, 'UniformOutput', false);
c = cell2mat(c.').';
% Compute cross correlation (multichannel case) and keep max value
% NB1: May also use xcov if signal mean is not important
% NB2: Normalization at lag = 0
RMax = max(xcorr(c, 'coeff'), [], 1);
% Put in R(i,j) format
RMax = reshape(RMax, [n n]);
%]
end
function [a] = myquickpad(a, maxlength)
%[
if (length(a) < maxlength)
a(maxlength) = 0;
end
%]
end
For the following signals:
(1) [1 3 8 7 3 4 12]
(2) [3 8 7 3]
(3) [4 12]
(4) [5 3 8 -3 12]
(5) [1 3 8 7 3 4 12]
It returns the following correlation matrix R(i,j) between ith and jth signals:
1.0000 0.6698 0.7402 0.8016 1.0000
0.6698 1.0000 0.8012 0.4853 0.6698
0.7402 0.8012 1.0000 0.6587 0.7402
0.8016 0.4853 0.6587 1.0000 0.8016
1.0000 0.6698 0.7402 0.8016 1.0000
Some remarks:
It looks coherent, for instance signal (1) and (5) are identical and correlation is 1.0.
Because of normalization used it considers (1) closer to (3) than (2) ... so should be reviewed upon your needs (see normalization as in xcorrcoef for instance as shown by #paisanco).
You can use xcov instead of xcorr if signal shifts in amplitude are not important.
Again, this is a coarse approach, not speed/memory optimized at all, nor accounting for the fact that values are sorted, and may be not fully inline with what you'd really like to have.