Assessing the Number of Clusters in a Mixture Model with

Side-information

Edith Grall-Maes and Duc Tung Dao

ICD - LM2S - UMR 6281 CNRS - Troyes University of Technology, Troyes, France

Keywords:

Clustering, Model Selection, Mixture Model, Side-information, Criteria.

Abstract:

This paper deals with the selection of cluster number in a clustering problem taking into account the side-

information that some points of a chunklet arise from a same cluster. An Expectation-Maximization algorithm

is used to estimate the parameters of a mixture model and determine the data partition. To select the number of

clusters, usual criteria are not suitable because they do not consider the side-information in the data. Thus we

propose suitable criteria which are modiﬁed version of three usual criteria, the bayesian information criterion

(BIC), the Akaike information criterion (AIC), and the entropy criterion (NEC). The proposed criteria are

used to select the number of clusters in the case of two simulated problems and one real problem. Their

performances are compared and the inﬂuence of the chunklet size is discussed.

1 INTRODUCTION

Clustering is used in many ﬁelds with an increasing

interest. It aims to determine a partition rule of data

such that observations in the same cluster are similar

to each other. The estimation of mixture models has

been proposed for quite some time as an approach for

clustering. It assumes that data are from a mixture

of clusters in some proportions and that the probabil-

ity density function is a weighted sum of parameter-

ized probability density functions. When the number

of clusters is known, the problem consists in deter-

mining the parameters of the density functions and

the proportions of each cluster. However the number

of clusters is generally unknown and it has to be as-

sessed.

In this paper, we consider the problem of data

with side-information, which gives the constraint that

some data originate from the same source. In partic-

ular, when different measures are realizations of the

same random variable, these points belong a same

chunklet. It is the case when some spatiotemporal

measures are available and it is known that for exam-

ple the random variable does not depend on the time;

this means that all the measures originating from the

same position in space belong a same cluster. An ex-

ample is the temperature in a given month in different

towns and in different years. The temperature is a ran-

dom variable and the values for a same town and for

all the years make a chunklet. The clustering prob-

lem consists in grouping similar towns, considering

the values of the different years as a chunklet. It has

to be noticed that the number of samples in a chun-

klet is not ﬁxed. Another application is provided by

time series. In a series, all points arise from the same

system then they have to belong to the same cluster.

The mixture models using side-information have

already been studied. For a given number of clus-

ters, an algorithm for determining the parameters of

the probability density functions and the proportions

has been introduced in (Shental et al., 2003). A modi-

ﬁed version has been proposed in (Grall-Ma

es, 2014)

when partitioning the data is the main concern. How-

ever as in classical clustering problems, the number

of clusters is generally unknown.

This paper addresses the problem of assessing the

number of clusters in a mixture model for data with

the constraint that some points arise from the same

source. To select the number of clusters, usual crite-

ria are not suitable because they do not consider the

side-information in the data. Thus we propose suit-

able criteria which are based on usual criteria.

This paper is organized as follows. Section 2 de-

scribes the method for determining jointly the param-

eters of the mixture model and the cluster labels, with

the constraint that some points arise from the same

source, in the case of a known number of clusters.

At the same time it introduces notations. In section

3, three criteria based on usual criteria are proposed.

The Bayesian information criterion (BIC), the entropy

criterion (NEC), and the Akaike information criterion

Grall-Maes, E. and Dao, D.

Assessing the Number of Clusters in a Mixture Model with Side-information.

DOI: 10.5220/0005682000410047

In Proceedings of the 5th International Conference on Pattern Recognition Applications and Methods (ICPRAM 2016), pages 41-47

ISBN: 978-989-758-173-1

(AIC) are modiﬁed in order to be adapted to the prob-

lem of clustering with constraint. The results using

two examples on simulated data and one example on

real data are reported in section 4. The criteria are

compared and the inﬂuence of the chunklet size is dis-

cussed. We conclude the paper in section 5.

2 CLUSTERING WITH

SIDE-INFORMATION

The data we consider is a set of N observations X =

}

n=1..N

. Each observation s

is assumed to be a

chunklet, which is a set of |s

| independent points that

originate from the same source: s

= {x

}

i=1..|s

The observation set is assumed to be a sample

composed of K sub-populations which are all mod-

els of the same family. Each model corresponds to a

statistical law parameterized by θ

. The latent clus-

ter labels directly related to the parameters θ

are de-

scribed by Z = {z

}

n=1..N

where z

= k means that

the n

realization originates from the k

cluster. This

means that all points x

are within the k

cluster due

to the side-information.

Then in the case of data with side-information, the

observation set and the cluster label set are respec-

tively given by:

X = {s

}

n=1..N

with s

= {x

}

i=1..|s

(1)

and

Z = {z

}

n=1..N

. (2)

In order to compare this problem to an equivalent

case without side-information, we deﬁne the obser-

vation set X

which is composed of N

points with

∑

n=1

| and the cluster label set Z

respectively

by:

= {x

}

i=1..|s

|,n=1..N

(3)

and

= {z

}

i=1..|s

|,n=1..N

. (4)

The mixture model approach to clustering

(McLachlan and Basford, 1988) assumes that data

are from a mixture of a number K of clusters in

some proportions. The model is parameterized by

= {θ

, α

}

k=1..K

where α

is the probability that

a sample belongs to class k, α

= P(Z = k), and θ

the model parameter value for the class k. Then the

density function of a sample s given θ

writes as :

f (s|θ

) =

∑

k=1

(s|θ

)

where f

(s|θ

) is the density function of the compo-

nent k.

The maximum likelihood approach to the mixture

problem for a data set X and a given value K consists

of determining θ

that maximizes the log-likelihood.

The log-likelihood is given by:

(θ

) =

∑

n=1

log f (s

|θ

Due to the constraint given by the side-

information, and the independence of points within

a chunklet, we get

f (s

|θ

) =

∑

k=1

|θ

) =

∑

k=1

∏

i=1

|θ

) (5)

Then

(θ

) =

∑

n=1

log

∑

k=1

∏

i=1

|θ

)

(6)

The log-likelihood in the equivalent case without

side-information, for a data set X

and a parameter set

= {θ

, α

}

k=1..K

is:

(θ

) =

∑

n=1

∑

i=1

log f (x

|θ

)

∑

n=1

∑

i=1

log



∑

k=1

|θ

)



(7)

A common approach for optimizing the pa-

rameters of mixture models is the expectation-

maximization (EM) algorithm (Celeux and Govaert,

1992). This is an iterative method that produces

a set of parameters that locally maximizes the log-

likelihood of a given sample, starting from an arbi-

trary set of parameters.

A modiﬁed EM algorithm has been proposed in

(Grall-Ma

es, 2014) for taking into account the side-

information, as in (Shental et al., 2003), and with the

aim of getting a hard partition, as in (Celeux and Go-

vaert, 1995). It repeats an estimation step (E step), a

classiﬁcation step, and a maximization step (M step).

The E step at iteration m requires to compute the pos-

teriori probability c

(m)

that the n

chunklet originates

from the k

cluster. It is given by:

(m)

= p(Z

= k|s

, θ

(m−1)

)

(m−1)

∏

i=1

|θ

(m−1)

)

∑

r=1

(m−1)

∏

i=1

|θ

(m−1)

)

(8)

The M step consists in estimating the parameters

that maximize the expected value of log-likelihood

determined on the E step.

Let denote L

∗

(K) the maximized log-likelihood

for a given number K

∗

(K) = max

(θ

) = L

(θ

∗

) (9)

ICPRAM 2016 - International Conference on Pattern Recognition Applications and Methods

where θ

∗

is the optimal parameter set. One can com-

pare L

∗

(K) with the usual log-likelihood without side-

information L

0∗

(K) which is deﬁned similarly for the

set X

The model complexity increases with K and con-

sequently the maximized log-likelihood L

0∗

(K) is

generally an increasing function of K. Then in the

classical case (without side-information) the maxi-

mized log-likelihood cannot be used as a selection

criterion for choosing the number K. In the case with

side-information, this is the same then L

∗

(K) cannot

be used as a selection criterion for choosing the num-

ber K.

3 CRITERIA

In order to choose the number of clusters K, a crite-

rion for measuring the model’s suitability which bal-

ances the model ﬁt and the model complexity has to

be used. Various criteria have been previously pro-

posed for data without side-information. In this paper

we modify three criteria to adapt them to data with

side-information.

Generally a criterion allows to select a model in

a set. The complexity of a model depends on the

parameter dimension r of the model. For instance

for d-dimensional K components Gaussian mixture,

r = K − 1 + dK +

Kd(d+1)

Let {M

}

m=1,...,M

denote the set of candidate

models, where M

corresponds to a model of dimen-

sion r

parameterized by φ

in the space Φ

3.1 Criterion BIC

One of the most used information criterion is BIC

(Schwarz, 1978) which is a likelihood criterion penal-

ized by the number of parameters in the model. The

idea of BIC is to select a model from a set of candidate

models by maximizing the posterior probability:

P(M

|X ) =

P(X |M

)P(M

)

P(X )

With hypothesis that P(M

) = P(M

), ∀m =

(1, ..., M), the maximization of P(M

|X ) is equiva-

lent to the maximization of P(X |M

). It can be ob-

tained from the integration of the joint distribution :

P(X |M

) =

P(X , φ

)dφ

P(X |φ

, M

)P(φ

)dφ

The exact calculation of this integral can be ap-

proached by using the Laplace approximation (Lebar-

bier and Mary-Huard, 2006). The maximization

of P(X |M

) is equivalent to the maximization of

logP(X |M

). Neglecting error terms, it is shown that

logP(X |M

) ≈ logP(X |

, M

) −

log(N)

When the model depends directly to the number

of clusters, the criterion BIC for the set X

deﬁned by

relation 3 is given by:

BIC(K) = −2 L

0∗

(K) + r log(N

)

This is the value of the criterion in the case of no side-

information, for a model with r free parameters.

In order to take into account the side-information,

the criterion has to be adapted. For the set X given

by relation 1 the number of observations is N and the

maximum log-likelihood L

∗

(K), which takes into ac-

count the positive constraints is computed differently.

Then the BIC criterion is given by:

BIC(K) = −2 L

∗

(K) + r log(N) (10)

Then the criterion does not depend directly on the

total number of points

∑

n=1

|. It depends on the

number of chunklets N.

3.2 Criterion AIC

The criterion AIC proposed in (Akaike, 1974) is an-

other largely used information criterion to select a

model from a set. The chosen model is the one that

minimizes the Kullback-Leibler distance between the

model and the truth M

, M

) =

+∞

−∞

P(X |M

)log(X |M

)

−

+∞

−∞

P(X |M

)logP(X |M

)

It is equivalent to select the model giving the max-

imized value of

+∞

−∞

P(X |M

)P(X |M

)

The criterion AIC without side-information takes

the form:

AIC(K) = −2 L

0∗

(K) + 2 r

For taking into account that some points arise

from the same source, we propose to replace L

0∗

(K)

by L

∗

(K). Then the modiﬁed AIC is given by:

AIC(K) = −2 L

∗

(K) + 2 r (11)

= −2

∑

n=1

log

∑

k=1

∏

i=1

|θ

)

+ 2r

Assessing the Number of Clusters in a Mixture Model with Side-information

3.3 Criterion NEC

The normalized entropy criterion (NEC) proposed in

(Celeux and Soromenho, 1996) is derived from a re-

lation underscoring the differences between the max-

imum likelihood approach and the classiﬁcation max-

imum likelihood approach to the mixture problem.

In the case of a data set without side-information

, the criterion is deﬁned as :

NEC(K) =

0∗

(K)

0∗

(K) − L

0∗

(1)

(12)

where E

0∗

(K) denotes the entropy term which mea-

sures the overlap of the mixture components.

This criterion is expected to be minimized in or-

der to assess the number of clusters of the mixture

components. Because NEC(1) leads to an indeter-

minate form a new procedure has been proposed in

(Biernacki et al., 1999) to retain the number K. This

procedure is equivalent to setting NEC(1)=1 and to

retain the value K leading to the minimal NEC value.

Considering a data set with side-information X ,

we need to modify the computation of the terms of

entropy and of log-likelihood. Since

∑

k=1

= 1, we

can rewrite L

(θ

) given by relation (6) as :

(θ

) =

∑

n=1

∑

k=1

log

∑

r=1

∏

i=1

|θ

Using the value of c

adapted to data with side-

information:

∏

i=1

|θ

)

∑

r=1

∏

i=1

|θ

)

we obtain

(θ

) =

∑

k=1

∑

n=1

log

∏

i=1

|θ

)

which can be rewritten as

(θ

) = C

(θ

) + E

(θ

)

with

(θ

) =

∑

k=1

∑

n=1

log

∏

i=1

|θ

)

and

(θ

) = −

∑

k=1

∑

n=1

logc

Thus we propose to use the criterion given by:

NEC(K) =

∗

(K)

∗

(K) − L

∗

(1)

(13)

in which E

∗

(K) = E

(θ

∗

) and L

∗

(K) is given by (9).

−4 −2 0 2 4 6

−5

−4

−3

−2

−1

Figure 1: An example of a mixture of three Gaussian com-

ponents.

4 RESULTS

The performances of the three criteria have been as-

sessed using two simulated problems, a Gaussian

mixture and a Gamma process, and one real problem

using climatic data.

4.1 Gaussian Mixture

We considered a mixture of three two dimensional

Gaussian components. The observation data have

been generated with the following parameter values

= [0, 0], m

= [2, 2], m

= [2, −2],

= Σ

= I,

N = 150 (50 chunklets per cluster) and |s

| = 5 ∀n.

Consequently the total number of points is equal to

750. An example of data is given on ﬁgure 1. Let

note that on this ﬁgure we are not able to show the

chunklets and then we can only see the points. The

number of clusters was determined using each of the

three criteria. This experiment has been repeated 200

times in order to estimate the percent frequency of

choosing a K-component mixture for each criterion.

Three other cases have been tested changing the

number of chunklets and the parameter values of the

Gaussian components. The value of N and |s

| have

been changed to have the same total number of points

but a larger number of points in each chunklet. We

have used N = 15 (5 chunklets per cluster) and |s

| =

50. Thus the ”side-information” is more important

than in the reference case. Then the value of m

and

have been modiﬁed to have less separated clusters.

We have used m

= [1, 1] and m

= [1, −1].

The results for the four cases and for each of the

criteria BIC, AIC and NEC are given in table 1.

ICPRAM 2016 - International Conference on Pattern Recognition Applications and Methods

Table 1: Percent frequencies of choosing K clusters.

K BIC AIC NEC

1 0 0 0

= [0, 0] N = 150 2 0 0 40

= [2, 2] |s

| = 5 3 97 79 59

= [2, −2] 4 3 17 1

5 0 4 0

1 0 0 0

= [0, 0] N = 15 2 0 0 32

= [2, 2] |s

| = 50 3 99 95 68

= [2, −2] 4 1 5 0

5 0 0 0

1 0 0 0

= [0, 0] N = 150 2 0 0 78

= [1, 1] |s

| = 5 3 96 78 17

= [1, −1] 4 4 15 0

5 0 7 5

1 0 0 0

= [0, 0] N = 15 2 0 0 33

= [1, 1] |s

| = 50 3 97 94 67

= [1, −1] 4 3 6 0

5 0 0 0

The best criterion to evaluate the number of clus-

ters for data with side-information is the BIC; over

95% success is obtained in all cases. AIC has a

slight tendency to overestimate the number of clus-

ters, while NEC has a tendency to underestimate. This

conclusion is in accordance with the results given in

(Fonseca and Cardoso, 2007) obtained in case of the

classical criteria for mixture-model clustering without

side-information. It is also mentioned that for nor-

mal distributions mixtures, BIC performs very well.

The NEC criterion shows the worst behavior with a

rate success under 80%. In (Celeux and Soromenho,

1996), it is mentioned that this criterion is efﬁcient

when some cluster structure exists. Since the cluster

structure is not obvious in this experiment, this crite-

rion has low performances.

Comparing the four cases, the results are not sur-

prising. When the side-information increases, or

when the cluster overlapping decreases, it is easier to

determine the right number of clusters.

4.2 Gamma Process

The Gamma process is widely used for the

modeling of monotonic and gradual deterioration

(Van Noortwijk, 2009). This process is deﬁned by

parameters that describe the deterioration evolution

in time. The parameters depend on the properties or

operational conditions of the observed system. Their

values are usually estimated using data obtained on

the observed systems. In the case of data coming from

different systems, within an unknown ﬁnite number

of operational conditions, it is required to group sim-

0 5 10 15

Time

Realization value

Figure 2: An example of simulated data with three Gamma

processes, 10 realizations for each process, and 10 points

for each realization.

ilar systems to estimate the parameters of the Gamma

process model. Then it is necessary to determine the

number of components (i.e. process models), the pa-

rameters of each component, and the component used

to model each system.

An homogeneous Gamma process is parameter-

ized by a shape and a scale parameters respectively

noted a and b. Each observed increment of dete-

rioration ∆x observed for a time increment ∆t, is a

random variable which follows a Gamma distribution

Γ(a∆t, b). Instead of deﬁning the Gamma process by

the shape and the scale parameters, it is possible to

deﬁne it using the mean m and variance v of the dis-

tribution Γ(a, b). It is more convenient since m (resp.

v) corresponds to the mean value (resp. the variance

value) of the degradation per unit of time. m and v are

given by:

m =

v =

We used Monte-Carlo simulation to generate data

with mean m

= 2 k, variance σ

= 2 for k = 1. . . K,

where K is the number of clusters, i.e. the number

of processes. An example of simulated data is given

on ﬁgure 2, in the case of three Gamma processes

(clusters), a number of systems (chunklets) equal to

10 for each process, and a number of points per chun-

klet equal to 10, i.e. there are 10 measures for each

system. It means that K = 3, |s

| = 10 ∀n and N = 30

and the total sample size is equal to 300.

Two experiments have been done: one with con-

stant values for N and |s

| and varying the value of

K, and one with constant value for K and varying the

value of N and |s

In the ﬁrst experiment, we have used 20 realiza-

tions for each process and |s

| = 10. We considered

4 cases for the value of the cluster number: K = 1, 2,

3, and 4. Then the sample size was equal to 200K

points. A Gamma mixture model was used in the

clustering algorithm. The number of clusters was se-

lected using each of the three proposed criteria (mod-

Assessing the Number of Clusters in a Mixture Model with Side-information

Table 2: Percent frequencies of choosing K clusters in the

case of N = 20K and |s

| = 10 and different cluster num-

bers.

theoretical K chosen K BIC AIC NEC

1 96 96 44

2 4 4 42

3 0 0 8

4 0 0 6

1 0 0 0

2 100 98 78

3 0 2 22

4 0 0 0

1 0 0 0

2 0 0 46

3 96 94 52

4 4 6 2

1 0 0 0

2 0 0 68

3 0 0 8

4 98 94 24

5 2 4 0

6 0 2 0

iﬁed BIC, AIC, and NEC). The experiment has been

repeated 200 times for each value of K for estimat-

ing the percent frequency of choosing K clusters. The

results are reported in table 2.

As in the case of the Gaussian mixture experiment,

the best results are obtained with the criterion BIC

whatever the value of K. The results with the criterion

AIC are close to that with BIC. The results obtained

with the criterion NEC are not good. This is due to the

fact that the cluster overlapping is quite important.

In the second experiment, we have used K = 3.

We considered 6 cases for the couple (N, |s

|). The

experiment has been repeated 200 times for each case

for estimating the percent frequency of choosing 3

clusters. The results are given in table 3.

As expected, for a given value of N, the efﬁciency

of the clustering algorithm and the efﬁciency of the

criteria are increasing with the value of |s

|. For a

given value of |s

|, they are increasing with the value

of N. However the inﬂuence of |s

| is more important

than the value of N. This is due to the fact that |s

is related to the amount of side-information, while N

modiﬁes only the size of the learning data base.

4.3 Climatic Data

The public website donneespubliques.meteofrance.fr

provides climatic data in France. We have used the

average temperature and the average rainfall for the

months of January and July in 109 available towns.

Then the dimension of a point is equal to 4. The mea-

Table 3: Percent frequencies of choosing 3 clusters in the

case of 3 clusters and different couples (N, |s

|).

N |s

| BIC AIC NEC

20K 3 68 62 20

20K 5 92 78 20

20K 20 98 92 70

6K 10 96 94 24

10K 10 98 96 52

40K 10 98 96 78

sure for each town is a random variable in dimension

4. We have used the data for the years 2012 to 2014. It

is assumed that the random variable for a given town

has the same distribution within all years. Then we

can consider that 3 realizations are available for each

random variable. It also means that the number of

points for each town (chunklet) is equal to 3. And

we assume that we can group random variables which

follow the same distribution. We have used a Gaus-

sian mixture model for the clustering algorithm.

The selected number of clusters with the crite-

rion BIC is 5. The results are reported on ﬁgure 3.

We can see different towns with climates which are

rather mediterranean, maritime, mountainous, conti-

nental, and semi-continental. In a future work these

results will be compared with results obtained by in-

creasing the data dimension (adding data from other

months and from other parameters such as the wind),

and by increasing the numbers of points for each town

(adding data from other years).

The proposed approach for the clustering with

side-information allows to deal with a variable num-

ber of points within each chunklet (i.e. town). Thus

it can deal with missing data for some years in some

towns. It is very convenient since it occurs that sen-

sors are off.

−5 0 5 10

longitude

latitude

Figure 3: Classiﬁcation of climates in France.

ICPRAM 2016 - International Conference on Pattern Recognition Applications and Methods

5 CONCLUSION

In this paper, criteria are proposed for assessing the

number of clusters in a mixture-model clustering ap-

proach with side-information. The side-information

deﬁnes constraints, grouping some points in the same

cluster.

Three criteria used for assessing the number of

clusters in a mixture-model clustering approach with-

out side-information have been modiﬁed. These cri-

teria are the Bayesian information criterion (BIC), the

Akaike information criterion (AIC) and the entropy

criterion (NEC). For adapting the criteria, the com-

putation of the log-likelihood has been modiﬁed. It

takes into account that some points arise from the

same source. In addition the criteria depend on the

number of chunklets but not on the total number of

points.

Experiments have been done with simulated prob-

lems: Gaussian mixtures and Gamma processes, and

with a real problem. The simulations allowed to com-

pare the efﬁciency of the criteria to determine the right

number of clusters. The climatic data problem has

given an application example.

The side-information helps to determine the clus-

ters mainly when the clusters overlap. Thus the cri-

teria ﬁtted to such situations are the most efﬁcient.

The experiments have shown the best behavior of BIC

compared with the two other criteria. AIC presents

a slight tendency to overestimate the correct number

of clusters while NEC has an underestimating ten-

dency. Because NEC is strongly efﬁcient when the

mixture components are well separated, its perfor-

mance is quite poor for the considered experimental

cases.

The inﬂuence of point number per chunklet on the

performance of the proposed criteria has also been

studied. The larger the chunklet size is, the better the

clustering algorithm performances are, and the better

the estimated number of clusters is.

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Assessing the Number of Clusters in a Mixture Model with Side-information