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That's a clustering problem.

How can we clusterize a union of clusters that are bounded boxes in N dimensions.

More explicitly, the data is generated the following way: $N$ is an arbitrary integer being the number of dimensions $k$ is the number of clusters For $i$ from $1$ to $k$, we define the $i$th cluster as :

For $d$ from $1$ to $N$, draw $m_{i, d}$ and $M_{i, d}$ the bounds of the cluster $i$ on dimension $d$. Then, draw a random number of points $x = (x_1, ..., x_N)$ in $\mathbb{R}^N$ such that, for all $d$, $m_{i, d} < x_d < M_{i, d}$

So the data is a union of clusters that are bounded boxes in $N$ dimensions.

The number of clusters is unknown and possibly large (up to 10.000).

So far, I tried density based clustering algorithm like DBSCAN and HDBSCAN but the problem is that it's hard to define distances on this dataset, I have some variable that are much larger than the others and the algorithm use the bigger variable more often. I tried to normalize the data but it didn't help that much.

As the data is very structure (bounded boxes), I think there might be better algorithm that would exploit that.

How can we clusterize datas of this type ?

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    $\begingroup$ Why are distances hard to define? $\endgroup$ – user20160 Nov 11 '19 at 13:53
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Define a distance. Its not that hard.

For example, you can use the distance of the box centers, or the minimum distance between the boxes, or the average distances of the points of each box.

Then use HAC, DBSCAN, OPTICS, ...

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