IEEE Systems, Man and Cybernetics Magazine - April 2020 - 7

Cauchy Gaussian
ACE

FCM

PCM

SCE

SP1M

Cone

Poisson

DC

MC

SC

SP

Figure 1. The instances of ACE and SCE for different membership

function shapes. New terms introduced in this article are shown in red.
DC: dancing cones; SP1M: sequential possibilistic one-means; MC:
mountain clustering; SC: sequential cones; SP: sequential Poisson.

and PCM are defined by squared error functionals whose
minimization yields specific membership function shapes.
Instead, squared error functionals may be abandoned, and
fuzzy clustering may be defined directly by membership
function shapes (which may or may not correspond to
solutions of squared error functionals). This leads to the
general scheme of ACE [8], where memberships are updated using the chosen function shape and clusters are updated as the centroid of each cluster, as shown on the left in
Table  1. FCM, PCM, and infinitely many other clustering
methods are special cases of ACE with specific membership function shapes. Figure 1 shows a taxonomy of ACE
(and SCE, as will be discussed later) instances for different membership function shapes such as Cauchy, Gaussian, cone (triangular), Poisson, and many more. The white
boxes in the second row of Figure 1 show the ACE instances FCM, PCM (Cauchy membership functions), and dancing cones for conical (i.e., radially triangular) membership
functions. Clustering with a conical membership function
is illustrated in Figure 2. ACE instances for Gaussian, Poisson, or other membership functions exist but have not
obtained specific names (yet). ACE has been successfully
applied to a large variety of problems such as function
approximation [10], [11], relational clustering [12], data
compression [13], web mining [14], keyword extraction
[15], or news analysis [16], [17].
SCE
For each update of an FCM cluster estimate, all other clusters are taken into account, so FCM clusters are mutually
coupled. In PCM (and many other ACE instances), the clusters are completely independent of each other. This may
yield coinciding, almost identical, clusters [18], [19], but it
also enables PCM to find only one single cluster, termed
possibilistic one-mean (P1M) [20], where additional preand/or postprocessing is needed to find all desired clusters. One approach to do so is sequential possibilistic
one-means (SP1M) [21], where the initial cluster centers
are randomly chosen from the given data set-with
	

probabilities proportional to one minus the already
assigned memberships-and where the cluster parameters
are adapted during the clustering process [22], [23]. An
equivalent approach, with Gaussian instead of Cauchy
membership functions, is termed mountain clustering
(MC) [24]. SCE is a generalization of the SP1M method for
arbitrarily defined membership function shapes, as listed
on the right in Table 1. Notice the differences between the
ACE and SCE algorithms on the left and right sides of
Table 1, which contain the same commands, but ACE iterates the clusters simultaneously, while SCE finds clusters
sequentially, one at a time. The third row of Figure 1 shows
the SCE instances SP1M and MC and also two new
instances of SCE that will be introduced and experimentally validated in the next sections: sequential cones (SC)
and sequential Poisson (SP) clustering. The new contributions of this article (SCE, SC, and SP) are displayed in red.
Finding Many Clusters
With the ability of modern computer systems and cloud
services to process large amounts of data, finding large
Table 1. The ACE and SCE algorithms.
ACE

SCE

Initialize cluster centers

For each cluster

Repeat

  Initialize cluster center

  For each cluster

 Repeat

  Update memberships

  Update memberships

 End

   Update cluster center

  For each cluster

  Until termination

   Update cluster center

End

 End
Until termination

Ap ri l 2020

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IEEE Systems, Man and Cybernetics Magazine - April 2020

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