Description

Generic reimplementation of the Framework for Optimal Selection of Clusters (FOSC; Campello et al, 2013) to extract clusterings from hierarchical clustering (i.e., hclust objects). Can be parameterized to perform unsupervised cluster extraction through a stability-based measure, or semisupervised cluster extraction through either a constraint-based extraction (with a stability-based tiebreaker) or a mixed (weighted) constraint and stability-based objective extraction.

Usage

Arguments

Details

Campello et al (2013) suggested a Framework for Optimal Selection of Clusters (FOSC) as a framework to make local (non-horizontal) cuts to any cluster tree hierarchy. This function implements the original extraction algorithms as described by the framework for hclust objects. Traditional cluster extraction methods from hierarchical representations (such as hclust objects) generally rely on global parameters or cutting values which are used to partition a cluster hierarchy into a set of disjoint, flat clusters. This is implemented in R in function cutree(). Although such methods are widespread, using global parameter settings are inherently limited in that they cannot capture patterns within the cluster hierarchy at varying local levels of granularity.

Rather than partitioning a hierarchy based on the number of the cluster one expects to find (k) or based on some linkage distance threshold (H), the FOSC proposes that the optimal clusters may exist at varying distance thresholds in the hierarchy. To enable this idea, FOSC requires one parameter (minPts) that represents the minimum number of points that constitute a valid cluster. The first step of the FOSC algorithm is to traverse the given cluster hierarchy divisively, recording new clusters at each split if both branches represent more than or equal to minPts. Branches that contain less than minPts points at one or both branches inherit the parent clusters identity. Note that using FOSC, due to the constraint that minPts must be greater than or equal to 2, it is possible that the optimal cluster solution chosen makes local cuts that render parent branches of sizes less than minPts as noise, which are denoted as 0 in the final solution.

Traversing the original cluster tree using minPts creates a new, simplified cluster tree that is then post-processed recursively to extract clusters that maximize for each cluster Ci the cost function

J = ∑ δ S(Ci) for all i clusters,

where S(Ci) is the stability-based measure as

S(Ci) = ∑ (1/Hmin(Xj, Ci) - 1/Hmax(Ci)) for all Xj in Ci.

δ represents an indicator function, which constrains the solution space such that clusters must be disjoint (cannot assign more than 1 label to each cluster). The measure S(Ci) used by FOSC is an unsupervised validation measure based on the assumption that, if you vary the linkage/distance threshold across all possible values, more prominent clusters that survive over many threshold variations should be considered as stronger candidates of the optimal solution. For this reason, using this measure to detect clusters is referred to as an unsupervised, stability-based extraction approach. In some cases it may be useful to enact instance-level constraints that ensure the solution space conforms to linkage expectations known a priori. This general idea of using preliminary expectations to augment the clustering solution will be referred to as semisupervised clustering. If constraints are given in the call to extractFOSC(), the following alternative objective function is maximized:

J = 1/(2 * nc) ∑ γ(Xj)

nc is the total number of constraints given and γ(Xj) represents the number of constraints involving object Xj that are satisfied. In the case of ties (such as solutions where no constraints were given), the unsupervised solution is used as a tiebreaker. See Campello et al (2013) for more details.

As a third option, if one wishes to prioritize the degree at which the unsupervised and semisupervised solutions contribute to the overall optimal solution, the parameter α can be set to enable the extraction of clusters that maximize the mixed objective function

J = α S(Ci) + (1 - α) γ(Ci).

FOSC expects the pairwise constraints to be passed as either 1) an n(n-1)/2 vector of integers representing the constraints, where 1 represents should-link, -1 represents should-not-link, and 0 represents no preference using the unsupervised solution (see below for examples). Alternatively, if only a few constraints are needed, a named list representing the (symmetric) adjacency list can be used, where the names correspond to indices of the points in the original data, and the values correspond to integer vectors of constraints (positive indices for should-link, negative indices for should-not-link). Again, see the examples section for a demonstration of this.

The parameters to the input function correspond to the concepts discussed above. The minPts parameter to represent the minimum cluster size to extract. The optional constraints parameter contains the pairwise, instance-level constraints of the data. The optional alpha parameters controls whether the mixed objective function is used (if alpha is greater than 0). If the validate_constraints parameter is set to true, the constraints are checked (and fixed) for symmetry (if point A has a should-link constraint with point B, point B should also have the same constraint). Asymmetric constraints are not supported.

Unstable branch pruning was not discussed by Campello et al (2013), however in some data sets it may be the case that specific subbranches scores are significantly greater than sibling and parent branches, and thus sibling branches should be considered as noise if their scores are cumulatively lower than the parents. This can happen in extremely nonhomogeneous data sets, where there exists locally very stable branches surrounded by unstable branches that contain more than minPts points. prune_unstable = TRUE will remove the unstable branches.

Value

A list with the elements:

Author(s)

Matt Piekenbrock

References

Campello, Ricardo JGB, Davoud Moulavi, Arthur Zimek, and Joerg Sander (2013). A framework for semi-supervised and unsupervised optimal extraction of clusters from hierarchies. Data Mining and Knowledge Discovery 27(3): 344-371. doi: 10.1007/s10618-013-0311-4

See Also

hclust(), hdbscan(), stats::cutree()

Other clustering functions: dbscan(), hdbscan(), jpclust(), optics(), sNNclust()

Examples