Why doesn't FindClusters find any clusters in this case?

Question

I am trying to use FindClusters to segment data points into similar numbers but so far I couldn't get it work for this example:

l = {110, 111, 115, 117, 251, 254, 254, 259, 399, 400, 401, 
     402, 542, 546, 549, 554, 660, 660, 660, 660};
FindClusters[l]
(*
-> {{110, 111, 115, 117, 251, 254, 254, 259, 399, 400, 401, 402, 542, 
   546, 549, 554, 660, 660, 660, 660}}
*)

If I set the N parameter (to specify: Exactly N clusters), it works:

FindClusters[l, 5]
(*
-> {{110, 111, 115, 117}, {251, 254, 254, 259}, 
    {399, 400, 401, 402}, {542, 546, 549, 554}, {660, 660, 660, 660}}
*)

However, my intent was to use FindClusters to figure out N.

Answer 1

Use the Bray-Curtis distance Total[Abs[u-v]]/Total[Abs[u+v]]:

FindClusters[{110, 111, 115, 117, 251, 254, 254, 259, 399, 400, 401, 
              402, 542, 546, 549, 554, 660, 660, 660, 660}, 
              DistanceFunction -> BrayCurtisDistance]
(*
{{110, 111, 115, 117}, 
 {251, 254, 254, 259}, 
 {399, 400, 401, 402},
 {542, 546, 549, 554}, 
 {660, 660, 660, 660}}
*)

Edit:

Here you have an experimental setup to test the FindClusters[] options in problems like yours:

l1 = RandomInteger[{100, 1000}, 10];
l2 = Join @@ (IntegerPart /@ RandomVariate[NormalDistribution[#, 10], 10] & /@ l1);
l3 = FindClusters[l2, DistanceFunction -> CanberraDistance];
Framed@Show[MapIndexed[
            Graphics[{ColorData[3][#2[[1]]],
                     Line[{{#, 0}, {#, 1}}] & /@ #1}] &, l3], 
            PlotRange -> {0, 1}, AspectRatio -> 1/5]

Answer 2

I'm not really sure why the default option for FindClusters with EuclideanDistance and Method->"Optimize" fails to distinguish any clusters.

Here are some results which might add a little detail:

Here are the numeric distance functions:

dfs = {EuclideanDistance, SquaredEuclideanDistance, NormalizedSquaredEuclideanDistance, 
ManhattanDistance, ChessboardDistance, BrayCurtisDistance, CanberraDistance, 
CosineDistance, CorrelationDistance}

Applying the various distance functions and methods:

Length@FindClusters[l, DistanceFunction -> #, Method -> "Agglomerate"] & /@ dfs
Length@FindClusters[l, DistanceFunction -> #, Method -> "Optimize"] & /@ dfs

{5, 1, 1, 5, 5, 5, 5, 1, 1} {1, 1, 1, 1, 1, 5, 5, 1, 1}

And in tabular form:

So it is possible to use the EuclideanDistance function for this data, but only with agglomerative clustering.