How to implement density-based clustering?

I’m looking to implement density-based clustering with R or Mathematica on a giant file (600,000 points on a 3 billion x 3 billion plane). Is DBSCAN the right method for data that is this sparse? I am also anticipating a huge amount of noise. Which parameters would I tweak to account for this?

• Why the close votes? OK, so the user wants to ultimately use R or Mathematica, this is not a code request question.There are some clear statistical aspects in the OP's question (ie. the use of DBSCAN with a large sparse noisy dataset). Maybe one can suggest the question here as being a possible duplicate (I think it is not) but that's a totally different criticism. – usεr11852 Jul 20 '16 at 22:37

Given you have a large dataset data as well as noisy data I strongly recommend that a dimensional reduction step is done prior to clustering. This should allow potentially irrelevant variation to be filtered out and the clustering algorithm to work in a lower dimensional space. Standard dimensional reduction techniques like Principal Component Analysis (PCA) and Locality-sensitive hashing (LSH) are two standard approaches.

Detecting density-based clusters in high-dimensional spaces, even when having noiseless data can be very demanding. High-dimensional density estimation is a typical scenario where the curse of dimensionality manifests. DBSCAN ultimately relies on finding fixed-radius nearest neighbours for each point. As the dimensionality of the data increases this nearest neighbour (NN) finding task becomes more and more attenuated. In addition (and most importantly) a standard distance metric as Euclidean distance gets potentially increasingly irrelevant. Therefore even if we have a distance and neighbourhood to work with that information is not very useful. This association between curse of dimensionality and NN-related tasks has been touched upon many time in CV, eg. see 1, 2, 3, 4.

By the way, something "simple" like the following script where $N$ is quite larger than just 600$k$ as in your case, runs on my laptop (Intel i5 U-series) in under 5 minutes using ~10 GB of RAM. This is because the fixed-radius nearest neighbour problem mentioned above is solved within the library dbscan using $k$-d trees; most NN-finding routines use some approximation approach; otherwise even cases with just a few more than tenths of thousands of points would get prohibitively large to work with when involving $O(n^2)$ requirements. So while not "instant" a use-case with 600$k$ points is definitely doable in R given a standard workstation and some appropriate dimensional reduction.

N = 10^7; p = 50;
Q = matrix(nrow = N, rt(N*p, df = 2))
library(dbscan)
W = dbscan(Q, eps= 0.01, minPts = 100) # About 4.5'


DBSCAN will work, but you need a good index structure. Without a good index, it will be $O(n^2)$ which is too expensive.

If you have a lot of noise, increase minPts.

• Can you kind of expand a bit? I always enjoy reading you insightful answers/comments on clustering issue but this post needs some extra context. How does minPts associate with what the user might experience noise-wise? How an index would help the time complexity? How much? Why would it be $O(n^2)$ to begin with? – usεr11852 Jul 20 '16 at 22:32
• See the Wikipedia article for DBSCAN complexity. If you have a lot of noise, a larger sample provides a more reliable density estimate. – Has QUIT--Anony-Mousse Jul 21 '16 at 7:41
• That section of the Wikipedia article, while relating to your answer on the use of index structures, has nothing to do with the noise against minPts relation. You might want(ed) to comment that with larger minPts parameters is more probable that outliers/noisy data are not incorporated erroneously in a cluster/neighbourhood but that is not explained in your post or your comment... – usεr11852 Apr 1 '17 at 10:14