How does DTW penalize stretching ?

The DTW function from Signal Processing Toolbox seems to penalize stretching of a Time Series/Signal. Can i switch that off or define my own penalties/weights ?

ANSWER

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I believe you misunderstood the DTW principal. DTW does directly penalizes stretching in the signals, as intended. The best way to show this is by some examples.

Try following input were the stretching is done by repeating some samples (depending on the application: but might not be the stretching normally seen, e.g. speech signals):

s1 = [1 4 7]';
    s2 = [1 4 4 7]';

The error matrix will be (absolute error):

error = [0   3   6;...
          3   0   3;...
          3   0   3;...
          6   3   0]; % s1 horizontally, s2 vertically

So there is a path from upper left to bottom right (taking horizontally, vertically or diagonal steps) with only zero penalties and thus the DTW distance will be zero! And so, DTW does not penalize this kind of stretching.

But, now consider a second example:

s1 = [1 4 7];
 s2 = [1 3 5 7];

This is stretched as well. However, this is now done by interpolation (which might be a more normal kind of stretching, e.g. speech) the error matrix will now become:

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How does DTW penalize stretching ?

How does DTW penalize stretching ?