I'm trying to use DistanceMatrix to compute the matrix of pairwise distances between two generic vectors (a,b) and (c,d) in the following way:
I really can't understand the result. Comparing to EuclideanDistance as follows:
I would expect the following result or something like that:
where am I wrong?
Jason, the universe that you are looking for is the one that has the HammingDistance as a metric. Which in MMA is the default for symbolic vector input. Counting the number of inequalities between vector elements. From documentation:
HammingDistance
@Sander Huisman input/output does make sense, but it isn't the same example though. DistanceMatrix seems to return numeric output from purely symbolic input.
DistanceMatrix
Sorry my bad, I misread the question…
Sander: no, I was really interested in computing the matrix of distances between symbolic vectors but I wasn't able to.
Aah I see now, yes, by default it does HammingDistance for this type of input data rather than Euclidean distance. Set the DistanceFunction option.
I still don't understand in what universe the following makes sense:
In[1]:= DistanceMatrix[{{a, b}, {c, d}}] Out[1]= {{0, 2}, {2, 0}}
Perhaps something like:
{a, b, c, d} = List /@ Sqrt[{0, 2, 2, 0}] DistanceMatrix[{{a, b}, {c, d}}]
?
Massimiliano, this gives the result that you expect:
DistanceMatrix[{{a, b}, {c, d}}, DistanceFunction -> EuclideanDistance]
Thanks Hans, it works!
However it seems strange to me that DistanceMatrix adds an unnecessary (I think) Abs function under the square root:
As you can see, this happens using both EuclideanDistance and SquaredEuclideanDistance as distance function. I don't know if there is any mathematical or implementative reason for that.
Yes, I agree it is a bit strange. But Simplify cleans it up:
Simplify
DistanceMatrix[{{a, b}, {c, d}}, DistanceFunction -> EuclideanDistance] // Simplify
Perhaps more efficient is to use a substitution (though for 2x2 it doesn't matter):
DistanceMatrix[{{a, b}, {c, d}}, DistanceFunction -> EuclideanDistance] % //. Abs[x_] :> x
if you know for sure that a,b,c, and d are real numbers.
This is to take care of complex numbers…
Euclidean distance computes the distance between two points. While distance matrix computes all the distances between two sets of points.
