IEEE Systems, Man and Cybernetics Magazine - April 2020 - 8

Clustering Delay Data
numbers of clusters becomes
In non-black-box machine learning,
increasingly important, for examIn non-black-box
experts often have substantial
ple, to identify large numbers of
machine learning,
prior knowledge about the memdifferent object types in images or
bership function shapes to be
large numbers of different customexperts often have
expected. Such knowledge can be
er needs and preferences in cussubstantial prior
easily taken into account in the
tomer relation management. The
ACE and SCE clustering schemes.
BIRCH data set [25] is an artificial
knowledge about
Consider, for example, the case of
data set that contains an array
the membership
an automated production line in a
of 10 × 10 = 100 clusters. Figure 3
factory that makes three different
shows the clusters found by FCM
function shapes to be
types of products with varying
and by SC clustering. In this examexpected.
manufacturing times. The producple, FCM (left) misses six of the
tion times are not deterministic but
100 clusters and assigns six duplivary probabilistically because the
cate cluster centers, while SC
necessary components are not
(right) correctly identifies all 100
immediately available or manufacturing cells or robots
clusters. These results vary slightly for different initializamay be momentarily busy. Such memoryless waiting times
tions (in 1,000 runs, FCM misses an average of 6.6 clusters
can be mathematically modeled by Poisson distributions.
with standard deviation 1.4), but on average, the SCE
Now consider the problem of identifying the waiting times
approach (here SC) yields a substantially higher cluster
for the three different manufacturing types from a data set
coverage than FCM, which indicates that SCE instances
of 500 production pieces (of several varieties). The blue
are better suited to find many clusters than ACE instances.
dots in Figure 4 illustrate this for the case of parts with target production times of 30, 40, and 50 s. For this data set, SP
clustering, an SCE instance with Poisson membership
functions, yields the three green membership functions
shown at the top of Figure 4, which precisely match the distributions used to generate this simulation data set (average deviation of the expected value is 2.3%). A similar
problem of clustering Poisson distributed data occurs in
transcriptome sequencing [26].

Figure 2. A cone is attracted by a data cluster (blue

points) until it matches the local cluster structure.

Conclusion
This article introduced a new generalized model for sequential clustering called SCE. The SP1M and MC methods are
instances of SCE with specific membership function shapes
(Cauchy and Gaussian). Moreover, two new SCE instances
(SC and SP clustering) have been introduced and experimentally validated. For finding large numbers of clusters,

0
(a)

(b)

Figure 3. The BIRCH data set clustering results for

(a) ACE/FCM and (b) SCE/SC. SCE/SC yields a much
better cluster coverage than ACE/FCM.

8	

IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE Apri l 2020

10

20

30

40

50

60

Figure 4. The simulation data from an automated

production line with three different product types. SP
is able to specifically look for Poisson clusters and
correctly identifies the Poisson distributions of the
three product specific delays.



IEEE Systems, Man and Cybernetics Magazine - April 2020

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