Distributed Clustering using Semi-supervised Fusion and Feature

Reduction Preprocessing

Huaying Li and Aleksandar Jeremic

∗

Department of Electrical and Computer Engineering, McMaster University, Hamilton, Ontario, Canada

Keywords:

Clustering, Information Fusion, Cluster Ensemble, Semi-supervised Learning.

Abstract:

In the recent years there has been tremendous development of data acquisition system resulting in a whole

new set of so called big data problems. In addition to other techniques data analysis of these data sets involves

signiﬁcant amount of clustering and/or classiﬁcation. Due to a heterogeneous nature of the data sets the perfor-

mance of these algorithms can vary signiﬁcantly in different applications. In our previous work we proposed

semi-supervised information fusion system and demonstrated its performance in various applications. In this

paper we proposed to improve the performance of the proposed system by applying data preprocessing algo-

rithms using feature reduction as well as various base clustering techniques. We demonstrate the applicability

of the proposed techniques using real data sets.

1 INTRODUCTION

Thee major goal of data clustering is to ﬁnd the hid-

den structure of a given data set by dividing data

points into distinct clusters based on certain criteria.

Data points in the same cluster are expected to be sim-

ilar to each other than to a data point from another

cluster. Although many clustering algorithms exist

in the literature, in practice no single algorithm can

correctly identify the underlying structure of all data

sets (Jain and Dubes, 1988), (Xu and Wunsch, 2008).

Furthermore, it is usually difﬁcult to select a suit-

able clustering algorithm for a given data set when the

prior information about cluster shape and size is not

available. Therefore, in many applications one option

to improve the clustering results is to generate multi-

ple base clusterings and combine them into a consen-

sus clustering (Strehl and Ghosh, 2003),(Vega-Pons

and Ruiz-Shulcloper, 2011). This is often referred to

as clustering ensemble. Many existing clustering en-

semble methods consist of two major steps: genera-

tion and fusion of multiple base clusterings. Nowa-

days, there is a growing interest in utilizing additional

supervision information in the unsupervised learning

process (such as clustering) to improve the perfor-

mance. This is often referred to as semi-supervised

clustering (Chapelle et al., 2006).

Motivated by the success of both approaches, re-

searchers become interested in combining the ben-

eﬁts of both techniques to further improve cluster-

∗

This work was supported by Natural Sciences and Engi-

neering Research Council of Canada.

ing results. The supervision information of semi-

supervised learning can be provided and utilized in

either step of clustering ensemble methods. In (Iqbal

et al., 2012), the supervision information is utilized

in the base clustering generation step, i.e., applying

semi-supervised clustering algorithms to generate the

set of base clusterings and fuse the cluster labels with-

out supervision. In this paper, we propose to utilize

the supervision information in the fusion step, i.e.,

applying unsupervised clustering algorithms to gen-

erated the set of base clusterings and fuse the cluster

labels with supervision. The remainder of this paper

is organized as follows. In Section 2, we propose the

modiﬁed semi-supervised clustering ensemble algo-

rithm using data preprocessing based on variable base

clustering generation and normalization. In Section 3,

we demonstrate the performance of our proposed al-

gorithms and the effect of normalization in clustering

ensemble. In the last section, we give the summary

of current research work and also list some future re-

search direction we will continue to work on.

2 SEMI-SUPERVISED

CLUSTERING ENSEMBLE

Clustering ensemble methods usually consists of two

major steps: the generation and fusion of base cluster-

ings, as shown in Fig. 1. In this section, we propose

four different ways to generate a set of base cluster-

ings and two different ways to combine the set into a

consensus clustering.

Li, H. and Jeremic, A.

Distributed Clustering using Semi-supervised Fusion and Feature Reduction Preprocessing.

DOI: 10.5220/0006658002330239

In Proceedings of the 11th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2018) - Volume 4: BIOSIGNALS, pages 233-239

ISBN: 978-989-758-279-0

Copyright © 2018 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

233

Figure 1: Distributed clustering system.

2.1 Base Clustering Generation

In this paper, the term clusterer represent the process-

ing unit that produces cluster labels for the given data

input. The set of D local clusterers (Fig. 1) is viewed

as a black box, which takes data set X as the input

and produces a set of base clusterings as the output.

We name it as the base clustering generator Φ (BCG).

The internal structure of base clustering generator is

shown in Fig. 2.

Figure 2: Base clustering generator with D local clusterers.

As a preliminary approach we apply K-means al-

gorithm in each local clusterer. In practice, different

clustering algorithms can be implemented in the local

clusterers to generate base clusterings. In order to de-

scribe the setting of base clustering generator, we ﬁrst

deﬁne some necessary parameters as follow:

• φ

( j)

: j-th local clusterer

• D: total number of local clusterers in Φ

• I

( j)

: input of local clusterer φ

( j)

• λ

( j)

: output of local clusterer φ

( j)

• K

( j)

: number of clusters in λ

( j)

One possible way to design the base clustering

generator is to build D identical local clusterers and

apply the same clustering algorithm with different ini-

tializations in each local clusterer. We set D = 21

and denote this base clustering generator as Φ

1

. The

set of base clusterings generated by Φ

1

is named as

“BASE1”. The parameter settings of Φ

1

is listed

in Table 1. In this design, the clustering processes

are distributed over different local clusterers. The

advantage is that each local clusterer has the access

to the entire data matrix and generates base cluster-

ings based on all the information. In the literature,

Table 1: Base clustering generators: X represents the input

data matrix, F represents the number of features (columns)

of X, and x

( j)

represents the j-th feature (column) of X.

Base

Set Name

No. of Local Local Clusterer φ

( j)

Clustering Clusterers Input No. of Clusters

Generator (D) (I

( j)

) (K

( j)

)

Φ

1

BASE1 21 X K

( j)

= K

0

Φ

2

BASE2 F x

( j)

K

( j)

= K

0

Φ

3

BASE3 21 X K

( j)

∈ [K

0

,40]

Φ

4

BASE4 F x

( j)

K

( j)

∈ [K

0

,40]

many clustering ensemble methods are evaluated by

generating base clusterings in this way (Strehl and

Ghosh, 2003),(Fred and Jain, 2005),(Visalakshi and

Thangavel, 2009).

Another way to design the base clustering gener-

ator is to apply clustering algorithm to only one of

the data features in each local clusterer. For a data set

containing F features, there are D = F local clusterers

in the generator. We denote this base clustering gen-

erator as Φ

2

and the set of base clusterings generated

by this generator as “BASE2”. The parameter settings

of Φ

2

is listed in Table 1. In this design, data features

are distributed over different local clusterers. Each lo-

cal clusterer only has the access to one of the features

and partitions data points from a speciﬁc aspect of the

data. It is suitable for data sets whose features are

measured in diverse scales. It is also suitable for data

sets whose features are heterogeneous or categorical

when the dissimilarity measure based on all features

does not have a real meaning. Furthermore, the afore-

mentioned approach may be the only choice when the

features or attributes of the data set are not shareable

between organizations due to privacy, ownership or

other reasons.

Note that recently proposed MCLA algorithm

(Strehl and Ghosh, 2003) is also based on clustering

clusters. Similar clusters (from different clusterings)

are grouped together to form a meta-cluster, which is

ﬁnally collapsed into a consensus cluster. Intuitively,

it is easier to identify similar clusters with less num-

ber of data points. Therefore, we modify base cluster-

ing generator Φ

1

and Φ

2

by setting K

( j)

to relatively

larger integers. Due to the fact that optimal value of

K

( j)

is data dependent and to avoid the selection of

a suitable value for K

( j)

, we propose to randomly se-

lect an integer value for K

( j)

of each local clusterer.

The parameter settings of the modiﬁed base cluster-

ing generators Φ

3

and Φ

4

are also listed in Table 1.

The sets of base clusterings generated by the modiﬁed

generators are denoted as “BASE3” and “BASE4” re-

spectively.

Suppose the input data set X is the combination of

a training set X

r

and a testing set X

u

. The training set

X

r

contains data points {x

1

,. ..,x

N

r

}, for which labels

are provided in a label vector λ

r

. The testing data set

BIOSIGNALS 2018 - 11th International Conference on Bio-inspired Systems and Signal Processing

234

X

u

contains data points {x

N

r

+1

,. ..,x

N

}, the labels of

which are unknown. The consensus cluster label vec-

tor (output of SEA) of testing set X

u

is denoted by λ

u

.

The size of training set X

r

is measured by the number

of data points in the training set and denoted by N

r

,

i.e., |X

r

| = N

r

. Similarly, the size of testing set X

u

is measured by the number of data points in the test-

ing set and denoted by N

u

, i.e., |X

u

| = N

u

. Accord-

ing to the training and testing sets, the label matrix

F can be partitioned into two block matrices F

r

and

F

u

, each of which contains all the labels correspond-

ing to the data points in the training set X

r

and testing

set X

u

respectively. Suppose training data points be-

long to K

0

classes and all training points from the k-th

class form one cluster, denoted by C

k

r

(k = 1,...,K

0

).

Therefore, the training set X

r

consists of a set of K

0

clusters {C

1

r

,. ..,C

k

r

,. ..,C

K

0

r

}. If the size of cluster C

k

r

is denoted by N

k

r

, the total number of training points

equals to the sum of N

k

r

, i.e., N

k

r

=

∑

K

0

k=1

N

k

r

. We re-

arrange label matrix F

r

to form K

0

block matrices:

[F

1

r

,. ..,F

k

r

,. ..,F

K

0

r

]. Each block matrix F

k

r

con-

tains the base cluster labels of data points in the k-th

training cluster C

k

r

.

The SHSEA is deﬁned as follows: (1) for a par-

ticular data point count the number of agreements be-

tween its label and the labels of training points in each

training cluster, according to an individual base clus-

tering (2) calculate the association vector between this

data point and the corresponding base clustering, (3)

compute the average association vector by averaging

the association vectors between this data point and all

base clusterings and (4) repeat for all data points and

derive the soft consensus clustering for the testing set.

Since the overall consensus cluster labels are derived

from the fuzzy(soft) label matrix, we name this ap-

proach as the soft-to-hard semi-supervised clustering

ensemble algorithm (SHSEA).

According to the j-th clustering λ

( j)

, we compute

the association vector a

( j)

i

for the i-th unlabelled data

point x

i

, where i = 1,...,N

u

and j = 1,...,D. Since

there are K

0

training clusters, the association vector

a

( j)

i

has K

0

entries. Each entry describes the asso-

ciation between data point x

i

and the corresponding

training cluster. The k-th entry of the association vec-

tor a

( j)

i

is measured as the occurrence of cluster label

of data point x

i

among the labels of reference data

points in the k-th training cluster (according to base

clustering λ

( j)

), i.e.,

a

( j)

i

(k) =

occurrence of F

u

(i, j)in F

k

r

(:, j)

N

k

r

, (1)

where F

u

(i, j) represents the cluster label of data

point x

i

and F

k

r

(:, j) represents the labels of reference

points in the k-th training category generated accord-

ing to base clustering λ

( j)

. In order to fuse the set

of base clusterings, the weighted average association

vector a

i

of data point x

i

is computed by averaging D

association vectors a

( j)

i

, i.e.,

a

i

=

D

∑

j=1

ω

j

a

( j)

i

, (2)

where ω

j

is the corresponding weight of the j-th lo-

cal clusterer. When local clusterers are equally impor-

tant, ω

j

= 1/D. Each entry of a

i

describes the overall

association between data point x

i

and the correspond-

ing training cluster. As a consequence, the summation

of all the entries of a

i

could be used to describe the

association between data point x

i

and all the training

clusters quantitatively. We deﬁne it as the association

level of data point x

i

to all training clusters and denote

it as γ

i

, i.e.,

γ

i

=

K

0

∑

k=1

a

i

(k). (3)

By computing association level for all data observa-

tions, the association level vector γ

u

for the testing set

X

u

is made up by stacking association level γ

i

for all

i = 1,...,N

u

, i.e., γ

u

= [γ

1

,γ

2

,. ..,γ

N

u

]

T

. We have two

options to present the overall consensus clustering for

testing set X

u

. One option is to produce a soft consen-

sus label matrix Λ

u

. The i-th row of Λ

u

is computed

by normalizing the average association vector a

i

, i.e.,

Λ

u

(i,:) = a

T

i

/γ

i

. (4)

The other option is to produce a hard consensus la-

bel vector λ

u

. The consensus cluster label assigned

to each data point is its most associated category la-

bels in the corresponding average association vector.

Since the overall hard cluster labels are assigned ac-

cording to the soft label matrix, we name this algo-

rithm as the soft-to-hard semi-supervised clustering

ensemble algorithm (SHSEA). The normalized soft

consensus label matrix (Λ

u

) can be used as the out-

put of the algorithm.

Following the naming convention, the other

semi-supervised ensemble method is called hard-to-

hard semi-supervised clustering algorithm (HHSEA),

since the overall cluster labels are assigned based on

hard label matrix. The HHSEA is deﬁned as follows:

(1) for a particular data point count the number of

agreements between its label and the labels of train-

ing points in each training cluster, according to an in-

dividual base clustering, (2) calculate the association

vector between this data point and the corresponding

base clustering, (3) assign this data point to its most

associated cluster label (4) repeat for all data points

Distributed Clustering using Semi-supervised Fusion and Feature Reduction Preprocessing

235

and all base clusterings to relabel the labels in ma-

trix F

u

and (5) apply majority voting to derive hard

consensus clustering. The details of both SHSEA

and HHSEA are given in our previous work (Li and

Jeremi

´

c, 2017).

3 NUMERICAL EXAMPLES

In this section, we evaluate the performance of

the proposed distributed clustering system using the

breast cancer data cells. This data is used to study hu-

man breast cancer cells undergoing treatment of dif-

ferent drugs. The cancer cells are plated into clear-

bottom well plates and 10 types of treatments are

taken placed to the cells. Images of the untreated

and treated cells are captured using the high content

imaging system and processed by the CAFE (Clas-

siﬁcation and Feature Extraction of micro-graphs of

cells) software to extract useful information. In total

705 attributes/features per cell are recorded for further

analysis (Razeghi Jahromi, 2014).

Since the ground truth of class assignments for

each data set are available, we use micro-precision

(Modha and Spangler, 2003) as our metric to mea-

sure the accuracy of clustering result with respect to

the expected (true) labelling. Recall that data set X

contains N data points that belong to K

0

classes and

N

k

represents the number of data points in the k-th

cluster that are correctly assigned to the correspond-

ing class. Corresponding class here represents the

true class that has the largest overlap with the k-th

cluster. The micro-precision (MP) is calculated by

MP =

∑

K

0

k=1

N

k

/N. The data set that are used in this

paper are listed Table 2, including the number of data

points, features and classes. The available data points

are divided in testing and training sets (data points

with know reference labels).

Table 2: Data Information I: the number of data points, fea-

tures and classes.

Data Sets Data Points Features Classes

DataSet1 300 705 2

DataSet2 300 705 2

DataSet3 300 705 2

DataSet4 450 705 3

3.1 Data Pre-processing

Data pre-processing is a necessary step to improve the

results of cluster analysis (Liu and Motoda, 1998),

(Pyle, 1999). In practice, many data sets to be clus-

tered contain features that are measured in different

units and scales. Features measured in relatively large

scales may play a dominant role in the similarity mea-

sure and inﬂuence the accuracy of the clustering re-

sults. As a consequence, normalizing the features

is an important pre-processing procedure, especially

when the similarity measure is based on Euclidean

distances (de Souto et al., 2008). Min-max normaliza-

tion is a linear transformation of features into a spec-

iﬁed range, which equalize the magnitude of the fea-

tures and prevents over weighting features measured

in relatively large scale over features measured in rel-

atively small scale. Suppose x

( f )

represents the f -th

feature of data set X. Let x

( f )

max

and x

( f )

min

represent the

maximum and minimum value of the f -th feature re-

spectively. Min-max normalization maps the f -th fea-

ture into range [0, 1] by

x

( f )

Norm

=

x

( f )

− x

( f )

max

x

( f )

max

− x

( f )

min

. (5)

In this paper, we demonstrate the effect of normal-

ization in clustering ensemble methods by comparing

the clustering results using original data sets (without

any pre-processing) and normalized data sets.

3.2 Original Data Sets

To study the effect of base clusterings on clustering

ensemble problem, we generate four different sets of

base clusterings (BASE1 to BASE4) for each data

set. Note that base clustering generator F

1

is de-

signed based on the common way used in the litera-

ture to generate base clusterings (Strehl and Ghosh,

2003),(Wang et al., 2011),(Dudoit and Fridlyand,

2003),(Fred and Jain, 2005). To evaluate different

clustering ensemble methods, we apply the unsuper-

vised HGPA, CSPA, MCLA (Strehl and Ghosh, 2003)

and BCE (Wang et al., 2011) in the fusion center and

compare the performance to the proposed SHSEA and

HHSEA. Recall that the ratio of number of reference

data points (N

r

) to number of testing data points (N

u

)

is denoted by P. We set P = 25% in the experiments

and repeat each experiment 100 times to calculate the

average micro-precision.

The micro-precision of K-means clustering algo-

rithm using all original features is listed in Table 3.

The maximum and minimum micro-precision of K-

means using features individually are also listed in

Table 3. Among all 11 data sets maximum MP of

K-means using single feature is higher than MP of K-

means using all features. Recall that BASE1 set of

base clusterings is generated by repetitively applying

K-means to all features together, while BASE2 is gen-

erated by applying K-means to each feature individu-

ally. Therefore, we expect the micro-precision of en-

semble methods using BASE2 to be higher than that

BIOSIGNALS 2018 - 11th International Conference on Bio-inspired Systems and Signal Processing

236

Table 3: Micro-precision of K-means using all features and

single feature of original data.

Data Sets

Kmeans

All Features

Single Feature

Max Min

DWALabSet1 0.5033 0.7917 0.5000

DWALabSet2 0.5033 0.7233 0.5000

DWALabSet3 0.5367 0.7933 0.5000

DWALabSet4 0.3400 0.5642 0.3333

of BASE1 since BASE2 contains a certain number

of “better” base clusterings. In addition, the perfor-

mance of SHSEA using BASE2 is expected to be bet-

ter than HHSEA, since base clusterings with higher

MP are given larger weights in the consensus fusion

step. Furthermore, recall that BASE3 (BASE4) is

generated in the same way as BASE1 (BASE2) re-

spectively expect that K

( j)

(number of clusters in each

local clusterers) are set to be greater than K

0

(ex-

pected number of clusters). Therefore, we expect the

performance of SHSEA and HHSEA using BASE3

(BASE4) to be better than BASE1 (BASE2), since

the proposed semi-supervised methods are expected

to perform better when data points are divided into

smaller groups.

The micro-precision of our proposed system

(four unsupervised and two semi-supervised ensem-

ble methods) using four sets of base clusterings

(BASE1 to BASE4) is illustrated by sub-ﬁgure (a) of

Fig. 3 to Fig. 6. The performance of SHSEA and HH-

SEA is represented by series SH(P25) and HH(P25)

respectively and P25 means the ratio of reference and

testing points is P = 25%. Among four groups of

clustering results, the bar corresponding to the high-

est average MP of the unsupervised ensemble meth-

ods and the bars corresponding to the highest MP of

SHSEA and HHSEA are labelled in each chart. It

is clear that the performance of the proposed semi-

supervised methods conforms with our expectations.

Compared to the micro-precision of K-means al-

gorithm (Table 3), the clustering results has been im-

proved by both operational modes of the proposed

system. The performance of the semi-supervised

mode is better than the unsupervised mode (except

“DataSet1”). The winning set of base clusterings is

either BASE2 or BASE4. In all the example the best

performance is achieved by utilizing SHSEA.

To study the effect of quantity of reference points

on semi-supervised clustering ensemble methods, we

repeat the experiments in semi-supervised mode by

selecting different numbers of reference points, i.e.,

by varying the value of P in N

r

= P · N

u

. Compared

to the performance of K-means (Table 3), micro-

precision of SHSEA or HHSEA increases dramati-

cally when P is relatively small. It becomes steady

and sometimes starts to decrease as P increases.

Therefore, for the purpose of improving the perfor-

mance of semi-supervised ensemble algorithms may

not be beneﬁcial to label more data points. It is due

to the facts that more reference points do not guaran-

tee the improvement and obtaining additional labels

is time-consuming and expensive.

Recall that the number of clusters in the j-th base

clustering K

( j)

is randomly generated in the base clus-

tering generator Φ

3

and Φ

4

. To study the effect of

randomized K

( j)

on the clustering ensemble methods,

we repeat the experiments by setting the number of

clusters in each base clustering the same and varying

the value of K

( j)

. Among these data sets, the high-

est MP occurs at different K

( j)

. The performance of

the proposed system using randomized K

( j)

is either

the best of all tested values of K

( j)

or it is very closed

to best. Due to the fact that we lack the knowledge

on how to select the optimal K

( j)

, we use randomized

K

( j)

in the following experiments to avoid the selec-

tion of K

( j)

for each data set.

3.3 Normalized Data Sets

The micro-precision of K-means using all normal-

ized features and normalized features individually is

shown in Table 4. The performance of K-means using

all features has been improved signiﬁcantly by nor-

malization except the ﬁrst three data sets, as compared

to Table 3. As discussed earlier the performance of

distance-based clustering algorithms may be affected

when data sets to be clustered contains features mea-

sured in diverse scales. By investigating features of

each data set, we noticed that the data sets contain fea-

tures measured in quite different ranges. Moreover,

the performance of K-means using normalized fea-

tures individually is similar to the performance of K-

means using original features individually. This result

is expected since similarity measure for single feature

is based on 1-dimensional distance calculation and it

is invariant to the feature scales.

Table 4: Micro-precision of K-means using all features and

single feature or normalized data.

Data Sets

Kmeans

All Features

Single Feature

(Normalized) Max Min

DWALabSet1 0.6628 0.7920 0.5000

DWALabSet2 0.5609 0.7233 0.5000

DWALabSet3 0.6120 0.7933 0.5000

DWALabSet4 0.5058 0.5644 0.3333

To study the effect of normalization on clustering

ensemble methods, we repeat the experiments previ-

ously described in Section 3.2 using normalized data

sets. The micro-precision of the proposed system is

illustrated by sub-ﬁgures (b) of Fig. 3 to Fig. 6.

Distributed Clustering using Semi-supervised Fusion and Feature Reduction Preprocessing

237

Figure 3: Data Set: DWALabSet1.

Figure 4: Data Set: DWALabSet2.

Figure 5: Data Set: DWALabSet3.

Figure 6: Data Set: DWALabSet4.

Note that the system performance using BASE1 and

BASE3 has been improved by normalization, while

the system performance using BASE2 and BASE4

stays close to the system performance using the corre-

sponding sets of base clusterings obtained by cluster-

ing original data sets. It is also expected since normal-

ization does not affect the performance of K-means

using single feature. Overall it can be observed that

the performance of SHSEA is very close to the per-

formance of HHSEA using normalized data.

4 CONCLUSIONS

In this paper we have proposed semi-supervised clus-

tering ensemble algorithms based on utilizing labelled

training data to improve the clustering results. We

designed four different ways to generate base clus-

terings and two ways to fuse them in the fusion cen-

ter with supervision. We provided numerical exam-

ples to demonstrate the effect of base clusterings on

the clustering ensemble methods and the performance

BIOSIGNALS 2018 - 11th International Conference on Bio-inspired Systems and Signal Processing

238

of semi-supervised clustering algorithms. We also

demonstrated the effect of normalization in the clus-

tering ensemble. In the future, we will focus on uti-

lizing the supervision information in both steps of the

clustering ensemble methods.

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