TWO-STAGE CLUSTERING OF A HUMAN BRAIN TUMOUR

DATASET USING MANIFOLD LEARNING MODELS

∗

Ra´ul Cruz-Barbosa and Alfredo Vellido

Dept. de Llenguatges i Sistemes Inform`atics, Universitat Polit`ecnica de Catalunya

Ediﬁci Omega, Campus Nord, C. Jordi Girona, 1-3, Barcelona, 08034, Spain

Keywords:

Brain tumours, MRS, Generative Topographic Mapping, two-stage clustering, outliers.

Abstract:

This paper analyzes, through clustering and visualization, Magnetic Resonance spectra of a complex multi-

center human brain tumour dataset. Clustering is performed as a two-stage process, in which the models used

in the ﬁrst stage are variants of Generative Topographic Mapping (GTM). Class information-enriched variants

of GTM are used to obtain a primary cluster description of the data. The number of clusters used by GTM

is usually large and does not necessarily correspond to the overall class structure. Consequently, in a second

stage, clusters are agglomerated using K-means with different initialization strategies, some of them deﬁned ad

hoc for the GTM models. We evaluate if the use of class information inﬂuence the brain tumour cluster-wise

class separability resulting from the process. We also resort to a robust variant of GTM that detects outliers

while effectively minimizing their negative impact in the clustering process.

1 INTRODUCTION

Medical decision making is usually riddled with un-

certainty, especially in sensitive settings such as non-

invasive brain tumour diagnosis. The brain tumour

data analysed in this study are obtained by Magnetic

Resonance Spectroscopy (MRS). Information derived

from the MR spectra can contribute to the evidence

base available for a particular patient, providing sup-

port to clinicians.

The ﬁelds of Machine Learning and Statistics co-

exist with data analysis as a common target. An

example can be found in Finite Mixture Models

(Figueiredo and Jain, 2002). In practical scenarios,

such as medical decision making, these models could

beneﬁt from data visualization. Finite Mixture Mod-

els can be endowed with visualization capabilities

provided certain constrains are enforced, such as forc-

ing the mixture components to be centred in a low-

dimensional manifold embedded in the observed data

∗

Alfredo Vellido is a Ram´on y Cajal researcher and

acknowledges funding from the MEC project TIN2006-

08114. Ra´ul Cruz-Barbosa acknowledges SEP-SESIC

(PROMEP program) of M´exico for his PhD grant. Authors

gratefully acknowledge the former INTERPRET project

centres (http://azizu.uab.es/INTERPRET) for making avail-

able and managing the data for this study.

space, as in Generative Topographic Mapping (GTM)

(Bishop et al., 1998), which can be seen as a prob-

abilistic alternative to Self-Organizing Maps (SOM)

(Kohonen, 1995) for data clustering and visualiza-

tion. When available class information can also be

integrated as part of the GTM training to enrich the

cluster structure deﬁnition (Cruz and Vellido, 2006).

The resulting models will be used in our experiments

to analyze a complex MRS dataset.

GTM-based models do not place any strong re-

striction on the number of mixture components (or

clusters), in order to achieve an appropriate visual-

ization of the data. This richly detailed cluster struc-

ture does not necessarily match the more global clus-

ter and class structures of the data. In this scenario, a

two-stage clustering procedure may be useful to un-

cover such global structure (Vesanto and Alhoniemi,

2000). GTM and its variants can be used in the ﬁrst

stage to generate a detailed cluster partition in the

form of a mixture of components. The centres of

these components can be further clustered in the sec-

ond stage. For that role, the well-known K-means al-

gorithm is used in this study.

The ﬁrst goal of the paper is assessing to what

extent the introduction of class information improves

the ﬁnal cluster-wise class separation. The issue re-

mains of how we should initialize K-means in the sec-

191

Cruz-Barbosa R. and Vellido A. (2008).

TWO-STAGE CLUSTERING OF A HUMAN BRAIN TUMOUR DATASET USING MANIFOLD LEARNING MODELS.

In Proceedings of the First Inter national Conference on Bio-inspired Systems and Signal Processing, pages 191-196

DOI: 10.5220/0001058501910196

Copyright

c

SciTePress

ond clustering stage. Random initialization (Vesanto

and Alhoniemi, 2000) does not make use of the prior

knowledge generated in the ﬁrst stage of the proce-

dure and requires a somehow exhaustive search of the

initialization space. Here, we propose two different

ways of introducing such prior knowledge as ﬁxed

initialization. These procedures, resulting from GTM

properties, allow signiﬁcant computational savings.

In section 2, we summarily introduce the GTM

and its t-GTM and class-enriched variants, as well as

the two-stage clustering procedure with its alternative

initialization strategies. Several experimental results

are provided and discussed in section 3, while a ﬁnal

section outlines some conclusions.

2 TWO-STAGE CLUSTERING

2.1 The GTM Models

The standard GTM is a non-linear latent variable

model deﬁned as a mapping from a low dimensional

latent space onto the multivariate data space. The

mapping is carried through by a set of basis functions

generating a constrained mixture density distribution.

It is deﬁned as a generalized linear regression model:

y = φ(u)W, (1)

where φ are M basis functions φ(u) =

(φ

1

(u),...,φ

M

(u)). For continuous data of di-

mension D, spherically symmetric Gaussians are an

obvious choice of basis function; W is a matrix of

adaptive weights w

md

that deﬁnes the mapping, and

u is a point in latent space. To avoid computational

intractability a regular grid of K points u

k

can be

sampled from the latent space. Each of them, which

can be considered as the representative of a data

cluster, has a ﬁxed prior probability p(u

k

) = 1/K and

is mapped, using (1), into a low dimensional manifold

non-linearly embedded in the data space. This latent

space grid is similar in design and purpose to that

of the visualization space of the SOM. A probability

distribution for the multivariate data X= {x

n

}

N

n=1

can

then be deﬁned, leading to the following expression

for the log-likelihood:

L =

N

∑

n=1

ln

1

K

K

∑

k=1

β

2π

D/2

exp

−βky

k

−x

n

k

2

2

(2)

where y

k

, usually known as reference or prototype

vectors, are obtained for each u

k

using (1); and β is

the inverse of the noise model variance. The EM al-

gorithm is an straightforward alternative to obtain the

Maximum Likelihood (ML) estimates of the adaptive

parameters of the model, namely W and β.

The class-GTM model is an extension of GTM

that makes use of the available class information. The

main goal of this extension is to improve class separa-

bility in the clustering results of GTM. For the Gaus-

sian version of the GTM model (Sun et al., 2002;

Cruz and Vellido, 2006), this entails the calculation

of the posterior probability of a cluster representative

u

k

given the data point x

n

and its class label c

n

, or

class-conditional responsibility ˆz

c

kn

= p(u

k

|x

n

,c

n

), as

part of the E step of the EM algorithm. It can be cal-

culated as:

ˆz

c

kn

=

p(x

n

,c

n

|u

k

)

∑

K

k

′

=1

p(x

n

,c

n

|u

k

′

)

=

p(x

n

|u

k

)p(c

n

|u

k

)

∑

K

k

′

=1

p(x

n

|u

k

′

)p(c

n

|u

k

′

)

=

p(x

n

|u

k

)p(u

k

|c

n

)

∑

K

k

′

=1

p(x

n

|u

k

′

)p(u

k

′

|c

n

)

,

(3)

and, being T

i

each class,

p(u

k

|T

i

)=

∑

n;c

n

=T

i

p(x

n

|u

k

)

∑

n

p(x

n

|u

k

)

∑

k

′

∑

n;c

n

=T

i

p(x

n

|u

k

′

)

∑

n

p(x

n

|u

k

′

)

(4)

The rest of the model’s parameters are estimated fol-

lowing the standard EM procedure.

For the Gaussian GTM, the presence of outliers is

likely to negatively bias the estimation of the adaptive

parameters, distorting the clustering results. In order

to overcome this limitation, the GTM was recently re-

deﬁned (Vellido, 2006; Vellido and Lisboa, 2006) as a

constrained mixture of Student’s t distributions: the t-

GTM, aiming to increase the robustness of the model

towards outliers. The mapping described by Equation

(1) remains, with the basis functions now being Stu-

dent’s t distributions and leading to the deﬁnition of

the following mixture density:

p(x|W,β,ν

k

)=

1

K

∑

K

k=1

Γ(

ν

k

+D

2

)β

D/2

Γ(

ν

k

2

)(ν

k

π)

D/2

(1+

β

ν

k

ky

k

−x

n

k

2

)

ν

k

+D

2

(5)

where Γ(·) is the gamma function and the parame-

ter ν = (ν

1

,... ,ν

K

) represents the degrees of free-

dom for each component k of the mixture, so that it

can be viewed as a tuner that adapts the level of ro-

bustness (divergence from normality) for each com-

ponent. This density leads to the redeﬁnition of the

model log-likelihood and, again, the estimation of the

correspondingadaptiveparameters using EM. The ex-

tension to class-t-GTM is straightforward and is omit-

ted here for the sake of brevity.

2.2 Two-Stage Clustering based on

GTM

In the ﬁrst stage of the proposed two-stage cluster-

ing procedure, the GTM models are trained to ob-

tain the representative prototypes (detailed clustering)

BIOSIGNALS 2008 - International Conference on Bio-inspired Systems and Signal Processing

192

of the observed dataset. In this study, the resulting

prototypes y

k

of the GTM models are further clus-

tered using the K-means algorithm. In a similar two-

stage procedure to the one described in (Vesanto and

Alhoniemi, 2000), based on SOM, the second stage

K-means initialization in this study is ﬁrst randomly

replicated 100 times, subsequently choosing the best

available result, which is the one that minimizes the

error function E =

∑

C

c=1

∑

x∈G

c

kx− µ

c

k

2

, where C is

the ﬁnal number of clusters in the second stage and

µ

c

is the centre of the K-means cluster G

c

. This ap-

proach seems somehow wasteful, though, as the use

of GTM instead of SOM can provide us with richer a

priori information to be used for ﬁxing the K-means

initialization in the second stage.

Two novel ﬁxed initialization strategies that use

the prior knowledge obtained by GTM in the ﬁrst

stage are proposed. They are based on the Magni-

ﬁcation Factors (MF) and the Cumulative Responsi-

bility (CR). The MF measure the level of stretching

that the mapping undergoes from the latent to the data

spaces. Areas of low data density correspond to high

distorsions of the mapping (high MF), whereas areas

of high data density correspond to low MF. The MF

is described in terms of the derivatives of the basis

functions φ

j

(u) in the form:

dA

′

dA

= det

1/2

ψ

T

W

T

Wψ

, (6)

where ψ has elements ψ

ji

= ∂φ

j

/∂u

i

(Bishop et al.,

1997) and dA

′

and dA are, in turn, inﬁnitesimal rect-

angles in the manifold and latent spaces. If we choose

C to be the ﬁnal number of clusters for K-means in

the second stage, the ﬁrst proposed ﬁxed initialization

strategy will consist on the selection of the class-GTM

prototypes corresponding to the C non-contiguous la-

tent points with lowest MF for K-means initialization.

That way, the second stage algorithm is meant to start

from the areas of highest data density.

The CR is the sum of responsibilities over all data

points in X for each cluster k:

CR

k

=

N

∑

n=1

ˆz

c

kn

(7)

The second proposed ﬁxed initialization strategy,

based on CR, is similar in spirit to that based on MF.

Again, if we choose C to be the ﬁnal number of clus-

ters for K-means in the second stage, the ﬁxed ini-

tialization strategy will now consist on the selection

of the GTM prototypes corresponding to the C non-

contiguous latent points with highest CR. That is, the

second stage is meant to start from those prototypes

that are found in the ﬁrst stage to be most responsible

for the generation of the observed data.

3 EXPERIMENTS

3.1 Human Brain Tumour Data

The multi-center data used in this study consists of

217 single voxel PROBE (PROton Brain Exam sys-

tem) MR spectra acquired in vivo for six brain tumour

types: meningiomas (58 cases), glioblastomas (86),

metastases (38), astrocytomas (22), oligoastrocy-

tomas (6), and oligodendrogliomas (7). For the analy-

ses, the spectra were grouped into three types (typol-

ogy that will be used in this study as class informa-

tion), as in (Tate et al., 2006): high grade malignant

(metastases and glioblastomas), low grade gliomas

(astrocytomas, oligodendrogliomas and oligoastrocy-

tomas) and meningiomas. The clinically relevant re-

gions of the spectra were sampled to obtain 200 fre-

quency intensity values. The high dimensionality of

the problem was compounded by the small number of

spectra available, which is commonplace in MRS data

analysis.

3.2 Experimental Design and Settings

The GTM, t-GTM and their class-enriched counter-

parts were implemented in MATLAB

R

. For the

experiments reported next, the adaptive matrix W

was initialized, following a PCA-based procedure de-

scribed in (Bishop et al., 1998). This ensures the

replicability of the results. The grid of latent points

u

k

was ﬁxed to a square 20x20 layout for the MRS

dataset. The corresponding grid of basis functions φ

was equally ﬁxed to a 5x5 square layout.

The goals of these experiments are fourfold. First,

we aim to assess whether the inclusion of class infor-

mation in the ﬁrst stage of the procedure results in any

improvement in terms of cluster-wise class separabil-

ity (and under what circumstances) compared to the

procedure using standard GTM. Second, we aim to

assess whether the two-stage procedure improves, in

the same terms, on the use of direct clustering of the

data using K-means. Third, we aim to test whether the

second stage initialization procedures based on MF

and the CR of the class-GTM, described in section

2.2, retain the cluster-wise class separability capabil-

ities of the two-stage clustering procedure in which

K-means is randomly initialized. In fourth place, we

aim to explore the properties of the structure of the

dataset concerning atypical data. For that, we use the

t-GTM (Vellido, 2006), as described in section 2.1.

The clustering results of all models will be ﬁrst

compared visually, which should help to illustrate the

visualization capabilities of the models. Beyond the

visual exploration, the second stage clustering results

TWO-STAGE CLUSTERING OF A HUMAN BRAIN TUMOUR DATASET USING MANIFOLD LEARNING

MODELS

193

Figure 1: Representation, on the 2-dimensional latent space of GTM and its variants, of a part of the tumour dataset. It is

based on the mean posterior distributions for the data points belonging to low grade gliomas (‘*’) and meningiomas (‘o’). The

axes of the plot convey no meaning by themselves and are kept unlabeled. (Top left): GTM without class information. (Top

right): class-GTM. (Bottom left): t-GTM without class information. (Bottom right): class-t-GTM.

should be explicitly quantiﬁed in terms of cluster-

wise class separability. For that purpose, the follow-

ing entropy-like measure is proposed:

E

G

c

({T

i

})= −

∑

{G

c

}

P(G

c

)

∑

{T

i

}

P(T

i

|G

c

)lnP(T

i

|G

c

)

= −

C

∑

c=1

K

G

c

K

|{T

i

}|

∑

i=1

p

ci

ln p

ci

(8)

Sums are performed over the set of classes (tumour

types) {T

i

} and the K-means clusters {G

c

}; K is the

total number of prototypes; K

G

c

is the number of pro-

totypes assigned to the c

th

cluster; p

ci

=

K

G

c

i

K

G

c

, where

K

G

c

i

is the number of prototypes from class i assigned

to cluster c; and, ﬁnally, |{T

i

}| is the cardinality of the

set of classes. An entropy of 0 corresponds to the case

of no clusters being assigned prototypes correspond-

ing to more than one class.

Given that the use of a second stage in the cluster-

ing procedure is intended to provide ﬁnal clusters that

best reﬂect the overall structure of the data, the prob-

lem remains of what is the most adequate number of

clusters. In this paper we do not use any cluster va-

lidity index and we just evaluate the entropy measure

for solutions from 2 up to 10 clusters.

3.3 Results and Discussion

In the ﬁrst stage of the two-stage clustering procedure,

GTM, t-GTM and their class-enriched variants class-

GTM and class-t-GTM were trained to model the hu-

man brain tumour dataset. The resulting prototypes y

k

were then clustered in the second stage using the K-

means algorithm. This last stage was performed with

three different initializations, as described in section

2.2. In all cases, K-means was forced to yield a given

number of ﬁnal clusters, from 2 up to 10. The entropy

was calculated for all settings.

Before considering the entropy results, visualiza-

tion maps (obtained using the mean of the poste-

rior distribution:

∑

K

k=1

u

k

ˆz

kn

or

∑

K

k=1

u

k

ˆz

c

kn

) of all the

trained models in the ﬁrst stage were generated. Three

hypotheses were made for the clustering results vi-

sualized here. First, the use of class information

in the clustering models should yield visualization

maps where classes are separated better than in those

models which do not use it. Second, the use of t-

BIOSIGNALS 2008 - International Conference on Bio-inspired Systems and Signal Processing

194

GTM should help to diminish the inﬂuence of out-

liers and the visualization maps generated with these

models should show the data more homogeneously

distributed throughout the visualization maps than in

Gaussian GTM, which do no use it. Thirdly, since

the tumour dataset is stronly class-unbalanced, we hy-

pothesized that the small classes would consist mainly

of atypical data. The second and third hypotheses will

be tested using the t-GTM variants.

For the sake of brevity, we only provide one of

these illustrative visualizations in Fig. 1.

Here, two tumour groups (low grade gliomas and

meningiomas) are shown. The right column of Fig.

1, where the models that include class information

are located, provides some preliminary support for

the ﬁrst hypothesis since the class separation between

both classes is better than that of the models that

do not use class information, located in the left col-

umn. This can be observed in the form of a more pro-

nounced overlapping of both classes in the left hand-

side models of Fig. 1. This is reinforced by the en-

tropy results provided later on in the paper.

The use of t distributions in the models repre-

sented in the bottom row yields a similar data spread

to that of the standard Gaussian GTM models of the

top row. This is an indication that there might be not

too many clear outliers in the two classes visually rep-

resented. Therefore, the second hypothesis cannot be

supported at this stage.

We now turn our attention to the third hypothe-

sis. In (Vellido and Lisboa, 2006) it was shown that a

given data instance could be characterized as an out-

lier if the value of

O

∗

n

=

∑

k

ˆz

kn

βky

k

− x

n

k

2

(9)

was sufﬁciently large. The histogram in Fig. 2 dis-

plays the values of O

∗

n

from (9) for the brain tumour

dataset. We did the same for the class-t-GTM model

and the corresponding values of O

∗

n

are displayed in

Fig. 3.

First of all, and supporting our previous impres-

sion, not too many data could be clearly characterized

as outliers according to these histograms. Somehow

surprisingly, given the complex tumour typology of

the dataset under study, these results do not support

the third hypothesis, as most of the spectra that might

be considered as outliers actually belong to the largest

and best represented tumour types, such as menin-

giomas and glioblastomas. Interestingly, few metas-

tases and astrocytomas are amongst the most extreme

outliers.

The entropy measurements quantifying the

cluster-wise class separation for the brain tumour

dataset are shown in Fig. 4. Two immediate conclu-

0 200 400 600 800 1000 1200 1400 1600

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

−−−−−meningioma

−−−−−−−−−−glioblastoma

−−−−−−−−−−−−−−−−−−−−−−astrocytoma

−−−glioblastoma, metastases

−−−−−meningioma

−−−−−−−−−−glioblastoma

−−−−−meningioma

−−−glioblastoma

−−−glioblastoma

Figure 2: Histogram of the statistic (9) for the t-GTM

model; outliers are characterized by its large values. As

an example, the ten largest values are labeled.

0 500 1000 1500 2000 2500 3000

0

5

10

15

20

25

−−−−glioblastoma

−−−−meningioma

−−−−astrocytoma

−−−−meningioma

Figure 3: Histogram of (9) for class-t-GTM. As an example,

the four largest values are labeled.

sions can be drawn: First, all the two-stage clustering

procedures based on GTM perform much better than

the direct clustering of the data through K-means in

terms of cluster-wise class separation. The two-stage

procedure based on class-GTM also performs much

better than its counterpart without class information

based on the standard GTM (right hand side of Fig.

4). On the contrary, it can also be observed that

the two-stage clustering based on class-t-GTM does

not perform better than the t-GTM model. This

is explained by the fact that the adjustment of the

model provided by t-GTM, which is blind to class

information by itself, alters the accordance between

class and cluster distributions, especially in a strongly

class-unbalanced dataset such as the one under

analysis. This result draws the limits out of which the

addition of class information is not necessarily useful

in terms of cluster-wise separation. The second

main conclusion to be drawn is that the random

initialization in the second stage of the clustering

procedure, with or without class information, does

not entail any signiﬁcant advantage over the proposed

ﬁxed initialization strategies across the whole range

of possible ﬁnal number of clusters, while being far

more costly in computational terms.

TWO-STAGE CLUSTERING OF A HUMAN BRAIN TUMOUR DATASET USING MANIFOLD LEARNING

MODELS

195

1 2 3 4 5 6 7 8 9 10

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

clusters

entropy

km alone

rand init−c

MF init−c

CR init−c

rand init−nc

MF init−nc

CR init−nc

rand init−ct

MF init−ct

CR init−ct

rand init−nct

MF init−nct

CR init−nct

1 2 3 4 5 6 7 8 9 10

0.17

0.175

0.18

0.185

0.19

0.195

0.2

0.205

clusters

entropy

rand init−c

MF init−c

CR init−c

rand init−nc

MF init−nc

CR init−nc

rand init−ct

MF init−ct

CR init−ct

rand init−nct

MF init−nct

CR init−nct

Figure 4: Entropy for the two-stage clustering of the tumour dataset, with different initializations (MF init, CR init and rand

init) and K-means alone. The ‘c’ and ‘nc’ symbols refer to models that, in turn, use and not use class information. The ‘t’ in

the legend means that t-GTM was used in the ﬁrst stage. (Left): all models are shown. (Right): only the GTM, t-GTM and

their class-enriched variants are shown.

The entropy measure in (8) quantiﬁes the level

of agreement between the clustering solutions and

the class distributions. In terms of the overall

cluster-wise class separation provided by the Gaus-

sian distributions-based GTM clustering models, it

has been shown that the addition of class information

consistently helps. As a result, these class-enriched

models would be useful in a semi-supervised setting

in which new undiagnosed tumour cases were added

to the database.

4 CONCLUSIONS

In this paper we have analyzed the inﬂuence exerted

by the inclusion of class information in the two-stage

clustering of a complex human brain tumour MRS

dataset. We have also introduced two economical and

principled ﬁxed initialization procedures for the sec-

ond stage of the procedure. The existence of atypi-

cal data or outliers in the human brain tumours MRS

dataset under study and its inﬂuence on the clustering

have also been explored.

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