Formalization of Data Stream Clustering Properties and Analysis of

Algorithms

Marcelo Keese Albertini e Rodrigo Fernandes de Mello

Department of Computer Sciences

Institute of Mathematics and Computer Sciences

University of Sao Paulo

Av.Trabalhador S˜aocarlense 400,S˜ao Carlos - SP,Brazil

{albertini,mello}@icmc.usp.br

Abstract—The understanding of several phenom-

ena requires unbounded data collections,called data

streams.These phenomena often present unstable

behavior and are studied by means of unsupervised

induction processes based on data clustering.Cur-

rently,clustering processes have shown serious lim-

itations in their applications to data streams due

to the demands imposed by behavioral changes and

unlimited data collection.However,despite the key

distinctions in between traditional data sets,which

are ﬁnite and unordered,and data streams,which are

essentially inﬁnite sequences,studies have overlooked

the dynamic and transient nature of streams,lim-

iting the appropriate understanding of phenomena.

The lack of a theoretical analysis for the problem

of data streams clustering led us to propose,in this

paper,a formalization based on Set Theory.This

formalization made it possible to identify and propose

basic properties for the design and comparison of

data stream clustering algorithms.It is expected to

be a starting point to understand the foundations

of unsupervised induction based on clustering and,

mainly,the modeling of phenomena.

Keywords:Artiﬁcial Intelligence;Machine Learning;

Data Mining;Unsupervised Learning;Data Clustering;

Data Streams;Set Theory

I.Introduction

The literature from several research ﬁelds has de-

scribed two main types of phenomena that produce

endless sequences of data also referred to as data streams

[11],[12].The ﬁrst type is characterized by the need for

data storage space and fast computation.In this situa-

tion,data is stored in secondary memory,which presents

low transfer rates,and usually accessed in a contiguous

manner.The second type is even more computationally

demanding:data is collected at high rates and shortly

afterwards is disposed.In this type,clustering models

must be continuously obtained throughout the endless

data-gathering process,whose dynamical properties,i.e.,

behavior,are expected to evolve over time [7],[12].

Data streams are frequently found in computationally

intensive environments,such as climate and weather

analysis [2],text mining [10],[9],genomic analysis,and

advanced scientiﬁc experiments [18],[19].

The typical clustering process has been designed to ap-

proach ﬁnite and unordered data sets and,consequently,

does not meet data streams requirements [20].There

exist great distinctions between clustering requirements

for data sets and data streams.Firstly,a data stream

must be accessed and processed sequentially as it cannot

be completely stored in memory.Secondly,the open-

endedness nature of data streams demands continuous

and automatic analysis.Thirdly,while stable phenomena

can be accurately represented by bounded data sets

and are suitable to traditional data clustering,data

streams,on the other hand,usually represent unstable

phenomena [7],whose characteristics tend to change

over the collection.This tendency indicates the transient

nature of data streams and demands a continuous re-

evaluation of clustering models.Finally,the role of do-

main experts when clustering data streams is diﬀerent

from clustering data sets.In the latter,specialists are

often required to empirically extract,select and analyze

data features in order to deﬁne the clustering algorithm

and the validation criterion.Conversely,in data streams,

the fast and continuous production of large amounts of

data restricts human intervention,due to the limited

capability of specialists to make well-founded decisions

under such constraints.

In order to better understand data set clustering,

Kleinberg [8] formalized three properties,which allow

the analysis of algorithm capabilities independently of

the target application.The formalization of the data-set

clustering problem is an important step towards its deep

understanding.Recently,several studies concerning the

usage and proposition of properties to related problems

have been developed [5],[4],[22].These studies have

broadened the possibilities to analyze clustering algo-

rithms,although very few of them have been conducted

in the context of data streams.

The lack of formal studies on the problem motived

our formalization of data stream clustering as an ex-

tension of Kleinberg’s approach to data sets [8].In

such a deﬁnition,data stream clustering is described in

terms of an inﬁnite sequence of partitions,which are

modiﬁed along time.The endless and changing nature

of data streams requires properties diﬀerent than those

proposed by Kleinberg for ﬁnite and unordered data sets.

Data streams,as inﬁnite and ordered sequences,need

clustering properties representative of the time evolution

and behavior changing.

We have extended Kleinberg’s properties to repre-

sent clustering partitions evolving according to the data

stream behavior.In addition,we introduce the Coher-

ence property,which states that partitions must conti-

nously evolve over time in order to preserve the meaning

of the clustering process.However,we observed that this

property is incompatible with the Kleinberg’s Richness,

which states that a data-set clustering function must

be capable of generating any partition.This conﬂict

indicates that trade-oﬀ analyses are needed in order

to demand properties from algorithms.Additionally,we

noticed that it is diﬃcult to ﬁnd an algorithm to comply

with Richness in a data stream context.

The remainder of this paper is organized as follows:

Section II introduces studies on data set clustering;

Section III describes our formalization approach to data

streams;Section IV draws conclusions and present ideas

for future work;and,ﬁnally,references are listed.

II.Related Work

Kleinberg [8] considered concepts of Set Theory [14]

to describe clustering algorithms in general terms,i.e.,

without relying on a speciﬁc algorithm,objective func-

tion for optimization,or statistical model.This analysis

is important to understand the functioning of algorithms

beforehand,i.e.,without the need for experimental trials,

and also to provide design guidelines for new approaches.

The concepts involved in Kleinberg’s formalization are

illustrated in Figure 1.This formalization assumes data

elements are organized in set E,whose measurements

of dissimilarity in between elements are provided in

a square matrix.This matrix,d,is described in the

form of a function d:E × E → R

+

,and is applied

to two elements {i,j} ∈ E to obtain a dissimilarity

value d(i,j) in [0,∞).The construction of d determines

the possibility of a clustering algorithm to organize the

elements in E.A cluster is essentially an organization

of elements of E into subsets,so that each element is

contained in one,and only one,subset.This structure is

known as the partition of a set.Partitions are denoted

by uppercase Greek letters,such as Γ (Gamma) and Υ

(Upsilon).Aclustering algorithmis,therefore,a function

f(d),which considers a distance matrix d for mapping

elements in E into a partition belonging to the universe

of partitions U.

E d:E ×E →R

+

d(♠,♣) = 9

f:d →U

f

f(d) = Γ

i

j

Figure 1.Illustration of concepts of data set clustering

By using this formalization,Kleinberg proposed prop-

erties to data set clustering introduced in the form of

statements of principles or self-evident conditions to a

well-succeeded clustering algorithm.The author initially

proposed three properties:Scale-Invariance,Richness

and Consistency.

Scale-Invariance refers to the ability of algorithms to

abstract the measurement scale of elements,i.e.,for

any distance function d and constant α > 0,we have

f(d) = f(α ∙ d).This property reﬂects the expectation

that the magnitude of the scale of elements should not

change the partition;for example,adapting elements

from centimeters to inches should not modify the parti-

tions obtained.

Richness refers to the ability of an algorithm to in-

duce all possible partitions for a set of elements.When

Richness holds,for any partition Γ of E,there exists a

distance matrix d for which f(d) results in Γ.It applies

the concept of surjection of Set Theory [14],which

deﬁnes the image of the surjective function as being

equal to its counter-domain.This property assumes it is

possible to arrange elements by modifying their relative

distances in order to obtain all partitions.A clustering

algorithm complies with Richness if and only if it

permits obtaining all partitions for the elements in E,

otherwise some partitions are impossible to be found.

Ensuring the possibility of ﬁnding all partitions for a

data set,as required by Richness,appears,at ﬁrst,de-

sirable for an algorithm whose data nature is unknown.

However,this property may be diﬃcult to comply with

[8].

Consistency refers to the ability algorithms have to

maintain the same partition when distances among ele-

ments within the same group are reduced while distances

among elements of diﬀerent groups are increased.For

example,consider a partition Γ generated by a clustering

function f,according to a distance function deﬁned by d.

Also consider changing d into d

′

in a way that it increases

the distances among elements of distinct subsets of Γ

and reduces the distances among elements of the same

subset.A consistent clustering function f(d

′

) always

generates a partition Γ

′

equivalent to Γ,in the sense

that f(d

′

) = f(d).

The previously described Kleinberg’s properties have

proven too demanding for a clustering function to com-

ply with,resulting in the incompatibility among them.

Kleinberg [8] proved,in an impossibility theorem,that

only two out of these three properties could be satisﬁed

by any algorithm.Although the practical consequences

of the impossibility theorem are limited,mainly because

small adaptations to the properties avoid the impossibil-

ity,Kleinberg’s approach has motived several studies to

further understand data clustering [5],[4],[22].Among

those studies,very few are relevant to data streams.

To the best of our knowledge,the only exception is

the paper by Ryabko [15].It deals with the problem

of clustering stationary stochastic processes,in which

the author claims that two elements must be associated

with the same cluster if and only if they are generated

by the same probability distribution.Nonetheless,the

usefulness of this result is limited in the context of data

streams,in which the behavior of sequences changes over

time.

III.Clustering data streams

Data streams diﬀer from other types of data tra-

ditionally considered in Machine Learning.They are

inﬁnite sequences with unknown and unstable behavior

[6].These characteristics contrast with traditional data

sets,which are ﬁnite,have no particular order,and are

characterized by a stable behavior.These distinctions

have motivated the adaptation of Kleinberg’s data set

properties to data streams.

In this context,a data stream is deﬁned as an inﬁnite

and ordered set of elements,that is,a sequence S =

(s

t

−∞

,...,s

t

−k

,...,s

t

0

),in which elements are indexed

by t ∈ R.The data collection is performed at time

instants t whose order is given by integers k ∈ (0,∞).

The complete sequence is represented by S,while a

sub-sequence of data collected up to a time instant t

is S

t

.Every element s

t

in this sequence consists of a

vector of values v,i.e.,the data features obtained.The

elements in S

t

are comparable by a dissimilarity function

d:S ×S →R

+

.

The main goal of a data-stream clustering algorithm

f(d,S) is to generate a sequence of partitions

¯

Γ

t

0

=

{Γ

t

−∞

,...,Γ

t

0

} for elements in S

t

0

,in which Γ

t

0

=

f(d,S

t

0

).The sequence of partitions is created from the

ﬁrst data-stream element S

t

−∞

until the most recent

element S

t

0

.Although S is inﬁnite,partitions in

¯

Γ have

a ﬁnite number of non-intersecting subsets.As a result

of practical requirements of the data streams analysis,

although S is inﬁnite,every partition in

¯

Γ may use only a

ﬁnite subset of S.Therefore,the design of f can consider

the option of removing a convenient subset of elements R

if they are represented by other elements or even expired.

In summary,the main diﬀerence of the problem of

data stream clustering,when compared to data set clus-

tering,originates from the inﬁnite and ordered nature of

such scenarios,which demands a sequence of partitions

over time,instead of a single one.

A.Data-stream clustering properties

The diﬀerences between the data-set clustering prob-

lem and the data stream one motivated the adaptation

of Kleinberg’s properties.We observed these properties

do not assure that clustering partitions smoothly evolve

according to the data stream behavior.Therefore,we

introduce the property of Coherence,which states that

a coherent algorithm for data-stream clustering creates

partition sequences in which elements do not drastically

change from one cluster to another.This property en-

sures that clustering algorithms maintain continuity in

between consecutive partitions.

Firstly,the Time-space Scale Invariance property,

equivalent to Kleinberg’s property of Scale Invariance,

suggests that algorithms should produce the same par-

tition if the measurement scale of space or times is

transformed by a multiplicative constant.

Property 1:Time-space Scale Invariance – Consider

the multiplication of distances d by a positive constant

α,α ∙ d,and the time indexes of each element s

t

∈ S

by another positive constant β,s

βt

.The sequence of

partitions f(d,s

t

) is equal to f(α ∙ d,s

βt

).

Similarly,the properties of Richness in Data Streams

and Time-space Consistency are adaptations of Richness

and Consistency for data sets.Both properties consider a

notion of temporal proximity,and,therefore,diﬀer from

properties to data set clustering in terms of a reference

point in time to conduct comparisons among partitions.

Property 2:Richness in Data Streams – Consider

a reference point in time t

0

,a sequence S

t

0

=

(s

−∞

,...,s

t

0

) observed until t

0

,and a matrix of dissim-

ilarity d among elements in S

t

.The clustering function

f complies with Richness in Data Streams at instant

t

0

through an arbitrary transformation of d and in the

order of elements in S

t

if it is capable of obtaining all

possible partitions for (s

−∞

,...,s

t

0

).

The Time-space Consistency property extends Consis-

tency by including the notion of temporal proximity.It

states that if elements in the same cluster are temporal

and spatially closer to each other and,at the same

time,elements of diﬀerent clusters are farther,then the

partition sequence is maintained.

Property 3:Time-space Consistency – Consider d

′

and S

′

t

are transformations of matrix dissimilarity d

and observation sequence S

t

such that the intervals

of the occurrence of elements within a cluster become

shorter and those of diﬀerent clusters become longer.

A clustering function f complies with space-temporal

consistency if and only if f(d,S

t

) = f(d

′

,S

′

t

).

The proposed properties are directly related to those

deﬁned for data-set clustering.Nonetheless,the data

stream properties based on Kleinberg’s study do not

oblige a clustering algorithmto obtain a logical sequence

of partitions for data streams.For practical purposes,the

guarantee that an algorithm will not randomly assign

samples to distinct clusters in consecutive partitions is

desirable.For example,in the context of concept drift

[13],the evaluation of clusters over time is required.

However,if consecutive partitions do not share a sense

of continuity,the evaluation becomes meaningless.

The continuity of partitions (Deﬁnition 2) is for-

malized by a relation named reach,represented by ⊲,

according to Deﬁnition 1.It is inspired in the concept

of reﬁnement by Kleinberg [8],which states that a

partition Γ is a reﬁnement of Υ if and only if each subset

in Γ either belongs to Υ or is contained in one of its

subsets.

Deﬁnition 1:A partition Γ reaches another partition

Υ if,for every subset A ∈ Γ,there is another subset B ∈

Υ,such as (A\R

Γ

) ⊆ (B\R

Υ

) or (B\R

Υ

) ⊆ (A\R

Γ

),

given that operation\is deﬁned by B\A = {s ∈ B|s/∈

A} and R

Γ

is the subset of elements in Γ that do not

belong to Υ,and R

Υ

is the subset of elements in Υ that

are not in Γ.The relation reach is denoted by Γ ⊲ Υ.

Continuity (see Deﬁnition 2) is relevant to the analysis

of capability of clustering algorithms to capture the

behavior evolution of data streams.

Deﬁnition 2:A partition sequence

¯

Γ

t

is continuous

if and only if for all consecutive partitions,i.e.,Γ

t−i−1

and Γ

t−i

,in which i ≥ 0,the relation Γ

t−1

⊲ Γ

t

is true.

The guarantee of generating continuous partitions is

stated by the Coherence property.The Coherence of a

clustering algorithmallows,for example,employing mea-

surements to evaluate partition sequences and support

the exploratory analysis of phenomena.

Property 4:Coherence – For any d,S

t

,and Γ

t

=

f(d,S

t

),the partition sequence

¯

Γ

t

is always continuous.

The time dimension is included in the ﬁrst three

properties proposed to formalize the data stream clus-

tering.However,these properties do not approach the

generation of an inﬁnite sequence of partitions,which

contrasts with the data set clustering deﬁned only by

one partition.In this sense,the coherence of data-stream

clustering algorithms is probably the most important

property.The main relevance of the Coherence property

relies on the fact it provides a parallel between clustering

continuity and function continuity,allowing to evaluate

diﬀerences in clustering models over time.By measuring

such diﬀerences,it is possible to better the understand

phenomena represented by data streams.However,as

shown in the next subsections,Coeherence can be in-

compatible with other properties,and,sometimes,is

not respected by some current important algorithms.

For example,the usage of the k-means algorithm for

clustering data streams,whose data are pre-organized in

the form of micro-clusters,as performed by Birch [23]

and Clustream [1] will not guarantee the continuity of

partitions and,eventually,may not represent phenomena

behavior.

B.Analysis of clustering algorithm properties

The formalization we have proposed is aimed at

evaluating and supporting the design of data-stream

clustering algorithms.In a similar approach,Kleinberg

[8] considered the three previously mentioned data-set

clustering properties to prove no algorithm respects

them simultaneously.

Based on Kleinberg’s study,we have observed a similar

impossibility theorem to obtain a data-stream cluster-

ing algorithm for the ﬁrst three properties proposed.

Furthermore,we have also observed that Properties 2,

Richness in data streams,and 4,Coherence,are mutually

exclusive,because the latter limits the possibilities to

produce partitions.In order to prove it,we show that if

a data-stream clustering function f complies with Rich-

ness,then it necessarily generates sequences of partitions

¯

Γ in which at least a pair of consecutive partitions does

not respect the relation reach;and,also,if f generates

only continuous sequences of partitions,then it does not

complies with Richness.Theorem1 shows the impossibil-

ity of designing a data-stream clustering algorithm with

the properties of Coherence and Richness in data

streams.

Theorem 1:There is no data-stream clustering func-

tion f that complies with Properties 2 and 4.

Proof:

First part:suppose f complies with Richness in data

streams,then f generates a partition sequence

¯

Γ

t

con-

taining consecutive partitions that do not reach each

other.

We will show that there is a sequence

¯

Γ

t

with parti-

tions that do not reach each other,regardless of changes

in both distances d and the sequence of elements S

t

.A

partition Γ

t

is unreachable when there is no partition

Γ

t−1

preceding it,so that the relation Γ

t−1

⊲ Γ

t

is not

respected,that is,the set of all possible partitions at

instant t −1 is empty.

It suﬃces to provide an example in order to prove

that there is an unreachable Γ

t

.Take a partition at time

instant t,Γ

t

= {{a

1

,b

2

},{a

2

,b

1

}},where elements a

i

,∀i

and b

j

,∀j in Γ

t−1

= {{a

1

,a

2

},{b

1

,b

2

}} are in diﬀerent

clusters.

It is known that {{a

1

,a

2

},{b

1

,b

2

}} ⊲

{{a

1

,b

2

},{a

2

,b

1

}} is not valid,because,according

to the deﬁnition of the relation reach,no subset of the

ﬁrst partition is contained in a subset of the second one,

and reciprocally.It is also known that,by deﬁnition,

any f that complies with Richness in data streams is

supposed to generate such a sequence of partitions.

Second part:if f generates only continuous parti-

tion sequences,then f does not comply with Rich-

ness in data streams.Consider the following partition

Γ

t−1

= {{a

1

,a

2

},{b

1

,b

2

}},then f cannot generate

Γ

t

= {{a

1

,b

2

},{b

1

,a

2

}} in the next time instant t,and,

similarly,if we take Γ

t−1

= {{a

1

,b

2

},{b

1

,a

2

}},then f

will not generate Γ

t

= {{a

1

,a

2

},{b

1

,b

2

}}.

Therefore,no f that generates continuous partition

sequences may comply with Property 2.

C.Analysis of data stream clustering algorithms

The use of properties for the analysis of clustering

algorithms is incipient.However,such properties allow

the understanding of theoretical principles for the design

and selection of algorithms,taking into account utility

and economic factors inherent to the application domain

[3].

The properties we have introduced are the ﬁrst to

represent the inherent characteristics of data streams.

In summary,these properties are Time-space Scale In-

variance,Richness in Data Streams,Time-space Consis-

tency and Coherence.We present a comparison among

the most relevant data stream clustering algorithms,

which is summarized in

1

Table I based on the proper-

ties proposed.Among the algorithms are Birch [23],

WaveCluster [16],CluStream [1],Olindda [17]

and Starvation Wta [21].Observe that none of the

algorithms complies with the property of Richness.

Table I

Evaluation of properties for data stream clustering

algorithms

Algorithm T-S Scale T-S Coherence

Invariance Consistency

Birch N N N

WaveCluster N — N

CluStream N N N

Olindda Y N N

Wta Y Y Y

(1) T-S means Time-space.

(2) Value ’Y’ means yes and ’N’ means no.

Usually,to verify that algorithms comply with such

properties,one can prove a theorem or present a coun-

terexample.However,in some cases none of the two

options is possible.On the other hand,there are other

options to check whether an algorithm respects a given

property.For example,when an algorithm limits the

1

The symbol ‘—’ represents that it was not possible to achieve

an evaluation.We omit further details on verifying algorithm

properties due to lack of space.

number of groups,it does not comply with the property

of Richness.An algorithm does not comply with Prop-

erty 1,i.e.,Time-space Invariance,if it considers any

threshold for accepting elements in clusters as there is

always a scalar constant that,multiplied by the distances

among elements,will modify the partition produced.

Still,an algorithm does not comply with Property 1 and

Property 4 if it does not consider the order of data during

clustering.

Another analytic option considers the evidence pre-

viously established for algorithms used to cluster tra-

ditional data sets,such as k-means and hierarchical

algorithms (e.g.,Single-linkage) [1],[23].For example,

algorithms that use k-means in the clustering process

do not comply with Property 4 because k-means does

not guarantee the continuity in sequences of partitions.

Among such algorithms,Birch and CluStream are

some of the most commonly considered data stream

clustering algorithms.

IV.Conclusions and Future Work

Despite the fundamental diﬀerences between data set

and data stream clustering,many studies have over-

looked the inﬁnite,dynamic and transient nature of the

latter.In this paper,we formally tackled the problem of

data-stream clustering as an inﬁnite sequence of parti-

tions.This approach is an extension of Kleinberg’s prop-

erties.Besides adapting Kleinberg’s properties to data

streams,we also proposed a new property referred to as

Coherence,which deals with the inﬁnite and sequen-

tial properties of data streams.This new property was

proven to be incompatible with Richness,evidencing

the trade-oﬀ in between both properties when designing

data-stream clustering algorithms.The existence of few

related studies in this theoretical branch indicates that

this is a seminal study and also that there are plenty of

possibilities towards developing a clustering theory.

Acknowledments

This paper is based upon work supported by FAPESP

– S˜ao Paulo Research Foundation,Brazil,under grants

no.2006/05939-0 and 2011/19459-8,”CAPES – Brazil-

ian Federal Agency for Support and Evaluation of Grad-

uate Education”research funding agency under grant no.

PDEE-4443-08-0,CNPq – National Council for Scientiﬁc

and Technological Development research funding agency

under grant no.304338/2008-7.Any opinions,ﬁndings,

and conclusions or recommendations expressed in this

material are those of the authors and do not necessarily

reﬂect the views of FAPESP,CAPES and CNPq.

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