A Latent Variable Pairwise Classification Model of a Clustering Ensemble


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1 A atent Variabe Pairwise Cassification Mode of a Custering Ensembe Vadimir Berikov Soboev Institute of mathematics, Novosibirsk State University, Russia Abstract. This paper addresses some theoretica properties of custering ensembes. We consider the probem of custer anaysis from pattern recognition point of view. A atent variabe pairwise cassification mode is proposed for studying the efficiency (in terms of error probabiity ) of the ensembe. The notions of stabiity, homogeneity and correation between ensembe eements are introduced. An upper bound for miscassification probabiity is obtained. Numerica experiment confirms potentia usefuness of the suggested ensembe characteristics. Keywords: custering ensembe, atent variabe mode, miscassification probabiity, error bound, ensembe s homogeneity and correation. 1 Introduction Coective decisionmaking based on a combination of simpe agorithms is activey used in modern pattern recognition and machine earning. In ast decade, there is growing interest in custering ensembe agorithms [1,2]. In the ensembe design process, the resuts obtained by different agorithms, or by one agorithm with variousparameters settings are used. After construction of partia custering soutions, a fina coective decision is buit. Modern iterature on custering ensembes can be roughy divided into severa main categories. There are a great dea of works in which the ensembe methodoogy is adapted to new appication areas such as magnetic resonance imaging, sateite images anaysis, anaysis of genetic sequences etc (see, for exampe, [3,4,5]). Another direction aims to deveop custering ensembes methods of genera usage and eaborate efficient agorithms using various optimization techniques (e.g., [6]). Other categories of works are of more theoretica nature; their purpose is to study the properties of custering ensembes, improve measures of ensembe quaity, suggest the ways to achieve the best quaity (e.g., [7,8,9]). There is a arge number of experimenta evidences confirming a significant raise in stabiity of custering decisions for ensembe agorithms (see, for exampe, [2,10]). At the same time, theoretica grounds of custering ensembes agorithms, as opposed to the pattern cassifier ensembes theory (e.g., [11]), are sti in C. Sansone, J. Kitter, and F. Roi (Eds.): MCS 2011, NCS 6713, pp , c SpringerVerag Berin Heideberg 2011
2 280 V. Berikov the eary deveopment stage. Existing works consider mainy the asymptotic properties of custering ensembes (e.g., [7]). Custer anaysis probems are characterized by the compexity of formaization caused by substantiay subjective nature of grouping process. For the definition of custering quaity it is necessary to appy additiona a priori information in terms of natura cassification, to use data generation modes etc. In the given work we attempt to corroborate custering ensembe methodoogy by utiizing of a pattern recognition mode with atent cass abes. To avoid probems with cass renumbering, a pairwise cassification approach is used. The rest of the paper is organized as foows. In the next section we give basic definitions and introduce the mode of ensembe custer anaysis. In the third section we receive an upper bound for error probabiity (in cassifying a pair of arbitrary objects according to atent variabe abes) and give some quaitative consequences of the resut. In the forth section we introduce the estimates of ensembe characteristics. The next section describes numerica experiment that demonstrates the usage of these notions. The concusion summaries the work and gives possibe future directions. 2 Ensembe Mode et us consider a sampe s = {o (1),...,o (N) } of objects independenty and randomy seected from a genera popuation. The purpose of the anaysis is to group the objects into K 2 casses in accordance with some custering criterion; the number of casses may be either given beforehand or not (in the atter case an optima number of casses shoud be determined automaticay). et each of the objects be characterized by variabes X 1,...,X n.denoteby x = (x 1,...,x n ) the vector of these variabes for an object o, x j = X j (o), j =1,...,n. In many custering tasks it is aowabe to consider that there exists a ground truth (atent, directy unobserved) variabe Y {1,...,K} that determines to which cass an object beongs. Suppose that the observations of kth cass are distributed according to the conditiona dencity function p k (x) = p(x Y = k), k =1,...,K. Consider the foowing mode of data generation. et each object be assigned to cass k in accordance with a priori probabiities P k = P(Y = k), k =1,...,K, where K P k = 1. After the assignment, an observabe vaue of x is determined k=1 with use of p k (x). This procedure is repeated independenty for each object. For an arbitrary pair of different objects a, b s, their correspondent observations are denoted by x(a) and x(b). et Z = I(Y (a) Y (b)),
3 A atent Variabe Pairwise Cassification Mode of a Custering Ensembe 281 where I( ) is indicator function. Denote by P Z = P[Z =1 x(a),x(b)] the probabiity of the event a and b beong to different casses, given x(a) and x(b) : P Z =1 P[Y (a) =1 x(a)] P[Y (b) =1 x(b)]... P[Y (a) =K x(a)] P[Y (b) =K x(b)] = 1 where p(x(o)) = K p k (x(o))p k, o = a, b. k=1 K k=1 p k (x(a))p k (x(b))p 2 k p(x(a))p(x(b)) et a custering agorithm μ be run to partition s into K subsets (custers). Because the numberings of custers do not matter, it is convenient to consider the equivaence reation, i.e. to indicate whether the agorithm μ assigns each pair of objects to the same cass or to different casses. et h μ (a, b) =I[μ(a) μ(b)]. et us consider the foowing mode of ensembe custering. Suppose that agorithm μ is randomized, i.e. it depends from a random vaue Ω from a given set of aowabe vaues (parameters or more generay earning settings such as bootstrap sampes, order of input objects etc). In addition to Ω, the agorithm s decisions are dependent from the true status of the pair a, b (i.e., from Z): h μ (a, b) =h μ(ω) (Z, a, b). Hereinafter we wi denote h μ(ω) (Z, a, b) =h(ω,z). Suppose that P[h(Ω,Z) =1 Z =1]=P[h(Ω,Z) =0 Z =0]=q, i.e. the conditiona probabiities of correct decision (either partition or union of objects a,b) coincide. One can say that q refects the stabiity of agorithm under various earning settings. We sha suppose that q>1/2; it means that agorithm μ provides better custering quaity than just random guessing. In machine earning theory, such a condition is known as the condition of weak earnabiity. Denote P h = P[h(Ω,Z) = 1]. This quantity shows the homogeneity of agorithm s decisions: P h cose to 0 or 1; or homogeneity index I h =1 P h (1 P h ) cose to 1 means high agreement between the soutions. Note that P h = P[h(Ω,Z) =1 Z =1]P Z + P[h(Ω,Z) =1 Z =0](1 P Z )= qp Z +(1 q)(1 P Z ). Suppose that agorithm μ is running times under randomy and independenty chosen settings. As a resut, we get random decisions h(ω 1,Z),...,,
4 282 V. Berikov h(ω,z). By Ω 1,...,Ω we denote independent statistica copies of a random vector Ω. For every Ω, agorithm μ is running independenty (it does not use the resuts obtained with other Ω, ). Suppose that the decisions are conditionay independent: P[h(Ω i1,z)=h i1,...,h(ω ij,z)=h ij Z = z] = P[(h(Ω i1,z)=h i1 Z = z] P[h(Ω ij,z) =h ij Z = z], where Ω i1,...,ω ij are arbitrary earning settings, h i1,...,h ij,z {0, 1} (we sha assume that is odd). et P h,h = P[h(Ω,Z)=1,h(Ω,Z)=1],whereΩ,Ω havethesamedistribution as Ω, andω Ω. It foows from the assumptions of independence and stabiity that P h,h = P[h(Ω,Z)=1,h(Ω )=1 Z =1]P Z + P[h(Ω,Z)=1,h(Ω )=1 Z =0](1 P Z )= q 2 P Z +(1 q) 2 (1 P Z ). (1) Denote H = 1 h(ω,z). The function =1 c(h(ω 1,Z),...,h(Ω,Z)) = I[ H > 1 2 ] sha be caed the ensembe soution for a and b, based on the majority voting. For constructing a fina ensembe custering decision, various approaches can be utiized [2]. For exampe, it is possibe to use a methodoogy based on the pairwise dissimiarity matrix H = ( H(o (i 1),o (i2)),whereo (i1),o (i2) s, o (i1) o (i2). This matrix can be considered as a matrix of pairwise distances between objects and used as input information for a dendrogram construction agorithm to form a sampe partition on a desired number of custers. 3 An Upper Bound for Miscassification Probabiity et us consider the margin [11] of the ensembe: mg = 1 { number of votes for Z number of votes against Z}, where Z =0, 1. It is easy to show that the margin equas: mg = mg( H,Z)=(2Z 1)(2 H 1). Using the notion of margin, one can represent the probabiity of wrong prediction of the true vaue of Z: P err = P Ω1,...,Ω,Z[mg( H,Z) < 0].
5 A atent Variabe Pairwise Cassification Mode of a Custering Ensembe 283 It foows from the Tchebychev s inequaity that P[U <0] < VarU (EU) 2, where EU is popuation mean, VarU is variance of random vaue U (it is required that EU >0). Thus, P Ω1,...,Ω,Z[mg( H,Z) < 0] < Var mg( H,Z) (E mg( H,Z)) 2, provided that E mg( H,Z) > 0. Theorem. The expected vaue and variance of the margin are: E mg( H,Z)=2q 1, Var mg( H,Z)= 4 (P h P h,h ). Proof. We have: E mg( H,Z)=E(2Z 1)( 2 h(ω,z) 1) = 4 E Zh(Ω,Z) 2EZ 2 E h(ω,z)+1. Because a h(ω,z) are distributed in the same way as h(ω,z), we get: E mg( H,Z)=4EZh(Ω,Z) 2P Z 2E h(ω,z)+1= 4P[Z =1,h(Ω,Z) =1] 2P Z 2P[h(Ω,Z) =1]+1. As P[h(Ω,Z) =1]=P[Z =1,h(Ω,Z) =1]+P[Z =0,h(Ω,Z) =1]= qp Z +(1 q)(1 P Z )=2qP Z +1 q P Z, we obtain: E mg( H,Z)=4qP Z 2P Z 2(2qP Z +1 q P Z )+1=2q 1. Consider the variance of margin: Var mg( H,Z)=Var(4Z H 2 H 2Z) =E(4Z H 2 H 2Z) 2 (E (4Z H 2 H 2Z)) 2 =E(16Z 2 H2 +4 h 2 +4Z 2 16Z H 2 16Z 2 H +8Z H) (E mg( H,Z) 1) 2 =E(4 H 2 +4Z 8Z H) 4(1 q) 2
6 284 V. Berikov (we appy Z 2 = Z). Next, we have E H 2 = 1 2 E ( 1 E h(ω,z)+ 1 From this, we obtain: ) 2 ( ) h(ω,z) = 1 2 E h 2 (Ω,Z) E(h(Ω,Z)h(Ω,Z)) =, :,Z)h(Ω,Z)= P h + 1 E(h(Ω Ω,Ω : Ω Ω P h,h. Var mg( H,Z) = 4 P h P h,h + 4P Z 8qP Z 4(1 q) 2. Using (1) finay we get: Var mg( H,Z) = 4 P h P h,h 4P h,h = 4 (P h P h,h ). The theorem is proved. Evidenty, the requirement E mg( H,Z) > 0 is fufied if q>1/2. et us consider the correation coefficient ρ between h = h(ω,z)andh = h(ω,z), where Ω Ω.Wehave ρ = ρ h,h = P h,h P 2 h P h (1 P h ). Because P h P h,h = P h P 2 h + P 2 h P h,h, weobtain Var (mg( H,Z)) = 4 (1 ρ) P h(1 P h ). Note that P h P h,h = q(1 q), and after necessary transformations we get the foowing upper bound for error probabiity: P err < 1 ( ) 1 1 4(1 ρ)p h (1 P h ) 1. The obtained expression aows to make some quaitative concusions. Namey, if the mode assumptions are fufied and q>1/2, then under other conditions being equa the foowing statements are vaid:  the probabiity of error decreases with an increase in number of ensembe eements;  an increase in homogeneity of the ensembe and raise of correation between its outputs reduce the probabiity of error (note that a signed vaue of the correation coefficient is meant).
7 A atent Variabe Pairwise Cassification Mode of a Custering Ensembe Estimating Characteristics of a Custering Ensembe To evauate the quaity of a custering ensembe, it is necessary to estimate ensembe s characteristics (in our mode homogeneity and correation) from a finite number of ensembe eements. For an arbitrary pair of different objects a and b, the estimate of the ensembe s homogeneity can be found as foows: where Î h (a, b) =1 ˆP h (a, b)(1 ˆP h (a, b)), ˆP h (a, b) = 1 h (a, b). Unfortunatey, the straightforward estimation of the correation coefficient ρ(a, b) is impossibe: under a fixed sampe, every pair of custering agorithms give conditionay independent decisions. et us introduce a simiar notion: the averaged correation coefficient, where the averaging is done over a pairs of different objects: ρ = cov(h,h ) σ 2 ; (h) where the covariance cov(h,h )=h h h 2, and the variance =1 h h 2 2 = N(N 1) ( 1) a,b: a b 2 1 h = h (a, b) = N(N 1) a,b: a b σ 2 (h) =h 2 h 2 = h h 2. h (a, b) h (a, b),, : 2 ˆP h (a, b), N(N 1) a,b: a b Simiary, it is possibe to introduce the averaged homogeneity index: Ī h = 2 N(N 1) Î h (a, b). a,b: a b 5 Numerica Experiment To verify the appicabiity of the suggested methodoogy for the anaysis of custering ensembe behavior, the statistica modeing approach was used. In the modeing, artificia data sets are repeatedy generated according to certain distribution cass (a type of custering tasks ). Each data set is cassified by the ensembe agorithm. The correct cassification rate, averaged over a given number of trias, determines agorithm s performance for the given type of tasks.
8 286 V. Berikov The foowing experiment was performed. In each tria, two casses are independenty samped according to the Gauss mutivariate distributions where m 1,m 2 are vectors of means, N (m 1,Σ), N (m 2,Σ), Σ = σi is a diagona ndimensiona covariance matrix, σ =(σ 1,...,σ n ) is a vector of variances. Variabe X i sha be caed noisy, if for some i {1,..., n}, σ i = σ noise >> 1. In our experiment, the set of noisy variabes {X i1,..., X innoize } is chosen at random. For those variabes that are not noisy, we set σ i = σ 0 = const. Both casses have the same sampe size. The mixture of sampes is cassified by the ensembe of kmeans custering agorithms. Each agorithm performs custering in the randomy chosen variabe subspace of dimensionaity n ens. The ensembe decision for each pair of objects (i.e., either unite them or assign to different casses) is made by the majority voting procedure. The true overa performance P cor of the ensembe is determined as the proportion of correcty cassified pairs P cor 0.3 P ind 0.2 averaged ρ averaged I h n noise Fig. 1. Exampe of experiment resuts (averaged over 100 trias). Experiment settings: N = 60, n = 10, m 1 = 0, m 2 = 1, σ noise = 10, σ 0 =0.2, n ens = 2, ensembe size = 15.
9 A atent Variabe Pairwise Cassification Mode of a Custering Ensembe 287 An exampe of experiment resuts is shown in Figure 1. The graphs dispay the dependence of averaged vaues of P cor, ρ and Īh from the number of noisy variabes n noise. To demonstrate the effectiveness of the ensembe soution in comparison with individua custering, the averaged performance rate P ind of a singe kmeans custering agorithm is given (this agorithm performs custering in the whoe feature space of dimensionaity n). From this exampe, one can concude that a) in average, the ensembe agorithm has better performance than a singe custering agorithm (when a noise presents), and b) the dynamics of estimated ensembe characteristics (averaged homogeneity and correation) reproduces we the behavior of correct cassification rate (note that this rate is directy unobserved in rea custering probems). When averaged correation and homogeneity index are sufficienty arge, one can expect good cassification quaity. Concusion A atent variabe pairwise cassification mode is proposed for studying nonasymptotic properties of custering ensembes. In this mode, the notions of stabiity, homogeneity and correation between ensembe eements are utiized. An upper bound for probabiity of error is obtained. Theoretica anaysis of the suggested mode aows to make a concusion that the probabiity of correct decision increases with an increase in number of ensembe eements. It is aso found that a arge degree of agreement between partia custering soutions (expressed in our mode in terms of homogeneity and correation between ensembe eements), under condition of independence of base custering agorithms, indicates good cassification performance. Numerica experiment aso confirms this concusion. The foowing possibe future directions can be indicated. It is interesting to study intensiona connections between the notions used in the suggested mode (conditiona independence, stabiity, homogeneity and correation) and other known concepts such as mutua information [2] and diversity in custering ensembes (e.g., [8,9]). Another direction coud aim to improve the tightness of the obtained error bound. Acknowedgements This work was partiay supported by the Russian Foundation for Basic Research, projects a, a. References 1. Jain, A.K.: Data Custering: 50 Years Beyond KMeans. Pattern Recognition etters 31(8), (2010) 2. Streh, A., Ghosh, J.: Custering ensembes  a knowedge reuse framework for combining mutipe partitions. The Journa of Machine earning Research 3, (2002)
10 288 V. Berikov 3. Kuncheva,.I., Rodriguez, J.J., Pumpton, C.O., inden, D.E.J., Johnston, S.J.: Random Subspace Ensembes for fmri Cassification. IEEE Transactions on Medica Imaging 29(2), (2010) 4. Pestunov, I.A., Berikov, V.B., Kuikova, E.A.: Gridbased ensembe custering agorithm using sequence of fixed grids. In: Proc. of the 3rd IASTED Intern. Conf. on Automation, Contro, and Information Technoogy, pp ACTA Press, Cagary (2010) 5. Iamon, N., Boongoen, T., Garrett, S.: CE: a inkbased custer ensembe method for improved gene expression data anaysis. Bioinformatics 26(12), (2010) 6. Hong, Y., Kwong, S.: To combine steadystate genetic agorithm and ensembe earning for data custering. Pattern Recognition etters 29(9), (2008) 7. Topchy, A., aw, M., Jain, A., Fred, A.: Anaysis of Consensus Partition in Custer Ensembe. In: Fourth IEEE Internationa Conference on Data Mining, pp IEEE Press, New York (2004) 8. Hadjitodorov, S.T., Kuncheva,.I., Todorova,.P.: Moderate diversity for better custer ensembes. Information Fusion 7(3), (2006) 9. Azimi, J., Fern, X.: Adaptive Custer Ensembe Seection. In: Proceedings of Internationa Joint Conference on Artificia Inteigence, pp (2009) 10. Kuncheva,.: Combining Pattern Cassifiers. Methods and Agorithms. John Wiey & Sons, Hoboken (2004) 11. Breiman,.: Random Forests. Machine earning 45(1), 5 32 (2001)
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