Clustering Interval-valued Data Using an Overlapped Interval Divergence

Transcription

1 Poc. of the 8th Austalasian Data Mining Confeence (AusDM'9) Clusteing Inteval-valued Data Using an Ovelapped Inteval Divegence Yongli Ren Yu-Hsn Liu Jia Rong Robet Dew School of Infomation Engineeing, Zhengzhou Univesity, Zhengzhou 455, China School of Infomation Technology, Deakin Univesity, Buwood Highway, Vic 5, Austalia {yuhsnliu, jong, Abstact As a common poblem in data clusteing applications, how to identify a suitable poximity measue between data instances is still an open poblem. Especially when inteval-valued data is becoming moe and moe popula, it is expected to have a suitable distance fo intevals. Existing distance measues only conside the lowe and uppe bounds of intevals, but ovelook the ovelapped aea between intevals. In this pape, we intoduce a novel poximity measue fo intevals, called Ovelapped Inteval Divegence (OLID), which extends the existing distances by consideing the elationship between intevals and thei ovelapped aea. Futhemoe, the poposed OLID measue is also incopoated into diffeent adaptive clusteing famewoks. The expeiment esults show that the poposed OLID is moe suitable fo inteval data than the Hausdoff distance and the city-block distance. Keywods: Clusteing, Distance, Similaity, Inteval Valued Data. Intoduction The impotance of distance measues in machine leaning and data mining is clea: a lage numbe of leaning poblems, such as clusteing and lazy leaning, heavily ely on the similaity measuement ove the data instance space. Accodingly, one of the main issues in these poblems is the selection of a suitable metic fo the concened application domain. Most of existing distances have been designed fo a elatively simple way: the data instance is descibed by a vecto of andom vaiables, each of which esults in just one single value. Howeve, in eal life thee ae many situations whee the use of inteval-valued data is moe suitable. In geneal, inteval-valued data come fom two majo souces:. many phenomena cannot be explained by using single-valued vaiables, and fom thei outset some data sets will include inteval attibutes. Many of natual language ae expessed with intevals instead of single cisp values, e.g. I dink 4-6 cups of wate a day. Similaly, in medical and engineeing data, intevals also appea Copyight c 9, Austalian Compute Society, Inc. This pape appeaed at the Eighth Austalasian Data Mining Confeence (AusDM 9), Melboune, Austalia. Confeences in Reseach and Pactice in Infomation Technology (CRPIT), Vol., Paul J. Kennedy, Kok-Leong Ong and Pete Chisten, Ed. Repoduction fo academic, not-fo pofit puposes pemitted povided this text is included. Table : Sample Inteval Data Set No. Age Weight Blood Pessue [, 7] [45, 5] [9, ] [5, ] [7, 8] [8, 8] [, ] [65, 7] [, 5] [, ] [45, 7] [9, 7] [, 4] [7, 75] [7, ] fequently, because of some toleance in measuing eal paametes. Fo example, age could be ecoded as being in an inteval, such as [, ], [, 4] etc. In addition, it may not be possible to measue some chaacteistics accuately by a single value, e.g. the pulse ate at 7, but athe measues the vaiable as an (x ± δ) value, namely (7 ± ). The blood pessue may be ecoded by its [low, high] values, e.g. [8, 8]. These ae all inteval-valued attibutes. A typical data set with inteval-valued attibutes may follow the lines of Table.. As data sets inceasingly suffe fom the poblem of scale, in tems of eithe the numbe of attibutes o the numbe of instances. Reseaches and pactitiones fom moe divese disciplines than eve befoe ae attempting to use automated methods to analyze thei data. It is often desiable to educe the size of the data while maintaining thei essential infomation as much as possible. One appoach is to summaize lage data sets in such a way that the esulting data set is of a manageable size. In this situation, inteval data stoe vaiability bette than standad single value data when eal values descibing the individual obsevations esult in intevals in the desciption of the summaized data. Accodingly the summaized data could no longe be single values as in classical fomat, but instead be epesented as intevals (Billad 6). The statistical teatment of inteval-valued attibutes has been consideed in the context of Symbolic Data Analysis (SDA) (Diday 988), which is a domain elated to exploatoy data analysis, multivaiate analysis and patten ecognition. SDA aims to povide suitable methods fo analyzing data set descibed though multi-valued attibutes, including intevals, sets categoies, o weight distibutions. SDA has povided suitable tools fo clusteing intevalvalued data: in 4, Souza et al. poposed a clusteing algoithm fo inteval data based on the cityblock distance (de Souza & de A.T. de Cavalho 4), and they applied the dynamic adaptive clus- Page 5

2 CRPIT Vol AusDM'9 teing famewok which incopoate the city-block distance to measue the distance between intevals. In 6, De Cavalho et al. adopted a simila dynamic clusteing famewok but with the Hausdoff distance instead fo intevals (de A.T. de Cavalho et al. 6). Recently De Cavalho et al. futhe popose the single adaptive clusteing famewok, in which both the city-block and Hausdoff distances can be adopted (de A.T. De Cavalho & Lechevallie 9). The single adaptive distances in (de A.T. De Cavalho & Lechevallie 9) use the same adaptive paametes fo all clustes, while it is diffeent to the cluste adaptive distances in the ealy wok (de A.T. de Cavalho et al. 6), which use diffeent adaptive paametes fom cluste to cluste. Most distances used fo clusteing inteval data pesented thus fa have been designed fo a elatively simple way: given two intevals, only the cisp values of thei lowe and uppe bounds wee consideed, and the infomation about thei ovelapped aea has been lagely ovelooked. Howeve, in eal life thee ae many situations whee the ignoance of these ovelapped aea causes sevee loss infomation, especially when both the distance between inteval centes and the elative size of the ovelapped aea ae concened. In this pape, we aim to fill the void by poposing a new distance fo inteval-valued data. By consideing the intevals as a hypecube in a high dimensional space and take the ovelapped aea into consideation, the poposed Ovelapped Inteval Divegence is diffeent fom othe inteval distances which only conside thei lowe and uppe bounds as single high dimensional points. As we will see fom the late sections that by incopoating the poposed distances into both the single and the cluste adaptive clusteing famewoks, we can get moe accuate clusteing esults than existing distances. The est of the pape is oganized as follows. In section, the elated wok ae pesented. In section, we popose the Ovelapped Inteval Divegence with detailed analysis of its popeties and the adaptive clusteing algoithms employed in the wok. In section 4, we pesent the expeiment esults that evaluate the poposed algoithms compaed to single(cluste) adaptive Hausdoff distance and single(cluste) adaptive city-block distance unde the synthetic data sets. Finally conclusions and futue wok ae pesented in section 5. Dynamic Clusteing fo Inteval-valued Data Clusteing, patitioning data into sensible goupings accoding to measued o peceived intinsic chaacteistics o similaity, is one of the most fundamental unsupevised data mining tasks. It is useful fo helping use to undestand and intepet the geneal pattens in data when pio knowledge of the undelying distibution is missing. As the epesentation of data by means of intevals is becoming moe and moe fequent, eseaches and pactitiones fom moe divese disciplines than eve befoe ae attempting to extend existing methods fo the compaison of inteval data (Diday 988).. Inteval-Valued Data Accoding to symbolic data analysis (Diday 988), an inteval vaiable is a vaiable which takes the inteval values such as [a, b], whee a b and a, b R. When a = b, this inteval vaiable is becoming a nomal single valued vaiable. Let D be a data set descibed by p inteval vaiables. Each data instance x i D is epesented as a vecto of intevals: x i = (x i,, xp i ), whee x j i = [aj i, bj i ]. A distance o poximity measue d is a nonnegative function defined on each pai of intevalvalued data instances, such that the close the instances, the lowe the value assumed by d. Two popula distance measues which have been widely used ae the city-block distance (de Souza & de A.T. de Cavalho 4) and the Hausdoff distance (de A.T. de Cavalho et al. 6), which will be descibed late togethe with the dynamic clusteing algoithms.. Adaptive Distances fo Dynamic Clusteing Symbolic data analysis has povided clusteing methods in which inteval-valued data ae consideed. As the most influential symbolic data analysis method, the Dynamic Clusteing Algoithm epesents a goup of unsupevised patition-based clusteing algoithms. It can be poven that this goup of algoithms genealizes seveal clusteing algoithms including K- means and K-median algoithm. The geneal Dynamic Clusteing Algoithm looks fo the patition of data set D into K clustes, and each cluste is epesented by a single pototype vecto of intevals, such that the sum of distance measues between each instance belonging to a cluste and the cluste s pototype is minimized. Let y k = ([αk, β k ],..., [αp k, βp k ]) be the pototype fo the k-th cluste P k (k =,..., K). The Dynamic Clusteing Algoithm is then tying to minimize the following citeion: O = K k= x i P k d k (x i, y k ). () Popula distance measues fo inteval-valued data include the Hausdoff distance (de A.T. de Cavalho et al. 6) and the city-block distance (de Souza & de A.T. de Cavalho 4). When they ae used with Dynamic Clusteing Algoithms, they usually appea in one of two adaptive foms: the single adaptive distance uses the same paamete fo all clustes; the cluste adaptive distance uses diffeent paametes fom cluste to cluste (de A.T. De Cavalho & Lechevallie 9)... The Single Adaptive Distances Liteatue (de A.T. De Cavalho & Lechevallie 9) poposes the patitional clusteing algoithm fo inteval-valued data by using a single adaptive Hausdoff distance: j= λ j (max[ a j i αj k, bj i βj k ]), () in which λ j = (λ,..., λ p ) is a weight vecto fo p inteval vaiables. Similaly, if using the city-block distance we can get: j= λ j [ a j i αj k + bj i βj k ], () whee the weight vecto λ j = (λ,..., λ p ) is also fixed fo p inteval vaiables. Page 6

3 Poc. of the 8th Austalasian Data Mining Confeence (AusDM'9) De Cavalho et al. also popose an extended single adaptive city-block distance fo inteval-valued data clusteing (de A.T. De Cavalho & Lechevallie 9): 7 6 a 6 (λ j L aj i αj k + λj U bj i βj k ), (4) j= 5 4 a a 5 a 4 in which thee ae two vectos of weight, one fo the lowe boundayλ L = (λ L,..., λp L ), and the othe fo the uppe bounday λ U = (λ U,..., λp U ). These weight vectos ae the same fo each cluste... The Cluste-Adaptive Distances De Cavalho et al. intoduce the dynamic clusteing algoithm fo inteval data by adopting diffeent adaptive Hausdoff distances fo diffeent clustes (de A.T. de Cavalho et al. 6): j= λ j k (max[ aj i αj k, bj i βj k ]), (5) which is paameteized by K vectos λ k = (λk,..., λp k ) (k =,..., K), one fo each cluste. Similaly, we can have the cluste adaptive cityblock distance (de Souza & de A.T. de Cavalho 4): j= λ j k [ aj i αj k + bj i βj k ], (6) which is also paameteized by K vectos λ k = (λk,..., λp k ) (k =,..., K). Souza and De Cavalho also extended the cluste adaptive city-block distance by sepaately consideing the lowe and the uppe bounds (de Souza & de A.T. de Cavalho 4): (λ j kl aj i αj k + λj ku bj i βj k ), (7) j= whee each cluste P k is paameteized by two weight vectos: one fo lowe bounday λ kl = (λkl,..., λp kl ), the othe fo uppe bounday λ ku = (λku,..., λp ku ). Hee, the weight vectos ae also diffeent fom cluste to cluste... The Geneal Adaptive Clusteing Algoithm Once the stategy fo adaptive distances is detemined, the algoithm fo clusteing inteval-valued data can be geneated into a geneal pocess: it will andomly choose a patition of X into clustes P = (P,..., P k ), then iteate ove the following steps. In the fist step, detemine K cluste pototypes y = (y,..., y K ) to epesent each cluste. In the second step, fix the pototypes y = (y,..., y K ) and the patitions P = (P,..., P k ), and update the adaptive distances d k so that the adequacy citeion O is minimized. In the thid step, fix the pototypes and the adaptive distances, and detemine the best patition P = (P,..., P k ) which minimizes the adequacy citeion O. b a a Figue : The inteval a, a, a, a 4, a 5, a 6 and b. Clusteing with the Ovelapped Inteval Divegence Most distances defined fo inteval-valued data have been designed fo a elatively simple way: only the lowe and uppe bounds of the intevals ae consideed. Howeve, in eal life thee ae many situations whee thei ovelapped aea should also be consideed (Li & Tong, Li & Dai 4, Jiang et al. 5, Dai et al. 4). Fo example, consideing seven intevals as shown in Fig., the city-block and the Hausdoff distances fom any inteval {a, a, a, a 4, a 5, a 6 } to b ae pesented in Table. As we can see, intevals a, a, a and a 5 ae all ovelapping with inteval b, while intevals a 4 and a 6 have no ovelapped aea with b. Accoding to the city-block distance, we have d c (a, b) = d c (a, b) = and d c (a 4, b) = d c (a 5, b) = 8. It is evident that the city-block distance can not distinguish a fom a, o distinguish a 4 fom a 5. Similaly, if following the Hausdoff distance, we will have d H (a, b) = d H (a, b) = d H (a, b) = and d H (a 4, b) = d H (a 6, b) = 6, which means the Hausdoff can not distinguish among a, a and a, o between a 4 and a 6. This is contadict to the intuition that a, a, a ae diffeent fom each othe, especially when the elative size of the ovelapped aea is a concen. In this section, we ae addessing this poblem by poposing a new poximity measue fo intevals.. The Ovelapped Inteval Divegence (OLID) Any inteval-valued data genealizes a single-valued data because it epesents a ange of values, and have aea in natue. In addition, thee will be an ovelapped aea between any two intevals, even though the ovelapped aea might be empty. Fo intevals, two factos ae elated to the poximity between two intevals: one is the distance between thei centes; anothe one is the elative size of thei ovelapped aea. By consideing the above two factos togethe, we popose an Ovelapped Inteval Divegence (OLID) fo intevals. Definition Given two intevals a = [a, a ] and b = [b, b ], let c a = a+a, a = a a and c b = b +b, b = b b. Then the Ovelapped Inteval Divegence (OLID) fom inteval a to b is defined as: div(a, b) = l(a, b) ( OA(a, b) ). (8) a + Page 7

4 CRPIT Vol AusDM'9 Table : Distances to b = [, 5] Distances a = [, 6] a = [, 7] a = [, 7] a 4 = [6, 7] a 5 = [4, 9] a 6 = [6, 8] city-block Hausdoff OLID whee OA(a, b) is the Ovelapped Aea between a and b, and l(a, b) is a distance oiginated fom Hausdoff distance by consideing all points inside the intevals: l(a, b) = max a [a,a ] {min b [b,b ] {u(a, b )}}, (9) in which u(a, b ) is the Euclidean distance between a and b. Fo an inteval a = [a, a ], the elationship between a and any othe inteval b = [b, b ] could be divided into the following six types as shown in Fig. : Falling Inside This kind of elationship occus when inteval a is completely falling inside of inteval b, as shown in Fig. (a). In this situation we have the OLID div(a, b) =. Coveing This elationship happens when inteval b is completely falling into inteval a, as shown in Fig. (b). In this situation we have the OLID fom a to b as ( c a c b + a b )( b a+ ). Left Ovelapping This coesponds to the situation whee inteval b ovelaps with a on the left side of a, as shown in Fig. (c). We have a [b, b ] and a / [b, b ], then div(a, b) = ( c a c b + a b )( a+ b c a c b a+ ). Right Ovelapping This coesponds to the situation whee inteval b ovelaps with a on the ight side of a, as shown in Fig. (d). We have a / [b, b ] and a [b, b ], then div(a, b) = ( c a c b + a b )( a+ b c a c b a+ ). Left Neighboing This happens when inteval b is not ovelapping with a, and b is on the left side of a, as shown in Fig. (e). Right Neighboing This happens when inteval b is not ovelapping with a, and b is on the ight side of a, as shown in Fig. (f). By consideing all the types of situations, we can get the Ovelapped Inteval Divegence function as follows: div(a, b) = l(a, b) ( OA(a,b) a+ ) i ( c a c b + a b )( b a+ ) ii = c a c b iii ( c a c b + a b )( a+ b ca c b a+ ) iv ( c a c b + a b )( + c a c b (a+ b ) a+ ) v () in which, i denotes when c a c b b a ; ii denotes when c a c b a b ; iii denotes when a = b = ; iv denotes when a b < c a c b < a + b ; v denotes when c a c b a + b. It is inteesting to note that when both intevals degade into single values, the OLID divegence becomes the egula L distance.. The Dynamic Clusteing Algoithms based on Adaptive OLID Based on the geneal famewok of the adaptive clusteing algoithm intoduced in Section., the clusteing algoithms based on single o cluste adaptive OLID ae developed to discove the best patition of the oiginal data sets into K clustes, which holds the minimum adequacy citeion O min. in which O single = K k= i P k d single (x i, y k ), () d single (x i, y k ) = p j [λj (max{div(a j i, αj k ), div(αj k, aj i )})] () is the single adaptive OLID measuing the dissimilaity between an object x i (i =,, n) and a cluste pototype y k (k =,, K), which is the median of x P k and multiplied by a weight vecto λ j (j =,, p). Hee, since OLID is asymmetic, we use the max function to make it symmetic. In each iteation, the weight vecto λ j (j =,, p) is calculated accoding to the following expession: λ j = { p l= ( K k= [ i P k (max{div(a l i,αl k ),div(αl k,al i )})])} p Kk= [ i P k (max{div(a j i,αj k ),div(αj k,aj i )})], () which satisfies λ j > and p j= λj =. The Pseudo-code of the algoithms ae pesented in Alg., which is both fo single and cluste adaptive OLID algoithms. Fo the dynamic clusteing algoithm based on cluste adaptive distances shae the same algoithm schema with the one based on single adaptive distances, but using a specific adequacy citeion O cluste since the adaptive distances ae diffeent fom cluste to cluste: O cluste = K k= i P k d cluste (x i, y k ), (4) in which the dissimilaity between the object x i and the coesponding cluste pototype y k can be calculated by the cluste adaptive OLID equation as follows: d cluste (x i, y k ) = p j [λj k (max{div(aj i, αj k ), div(αj k, aj i )})]. (5) whee the weight vecto is λ j k = { p l= i P (max{div(a l k i,αl k ),div(αl k,al i )})} p i P k (max{div(a j i,αj k ),div(αj k,aj i )}), (6) which satisfies λ j k > and p j= λj k =. 4 Expeiment and Discussion To evaluate the pefomance of ou OLID measuement, we investigate it within the famewok of dynamic clusteing algoithms on synthetic data sets. Page 8

5 Poc. of the 8th Austalasian Data Mining Confeence (AusDM'9) b C b b a C a a O(a,b) b C b b a C a a O(a,b) b C b b a C a a O(a,b) (a) Inside (b) Coveing (c) Left Ovelapping b C b b a C a a O(a,b) b C b b a C a a a C a a b C b b (d) Right Ovelapping (e) Left Neighboing (f) Right Neighboing Figue : Diffeent Relationships Between Two Intevals Algoithm The adaptive OLID algoithm Input: Data Set X. The numbe of clustes K. Output: A patition P of X into K clustes. Algoithm Pocess: : Initialization: : P andom patition of input data X into K clustes; : Iteative Reseach: 4: F lag F alse; 5: while not F lag 6: F lag T RUE, change ; 7: Calculate the pototypes y k (k =,, K); 8: Calculate the weight vecto λ j (j =,..., p): λ j = { p o λ j k = { p l= ( K k= [ i P (max{div(a l k i,αl k ),div(αl k,al i )})])} p K, k= [ i P (max{div(a j k i,αj k ),div(αj k,aj i )})] l= i P (max{div(a l k i,αl k ),div(αl k,al i )})} p, i P (max{div(a j k i,αj k ),div(αj k,aj i )}) 9: fo each element x i X : k the label of the cluste which x i belongs to; : k new = agmin k (d single (x i, y k )) o k new = agmin k (d cluste (x i, y k )); : if k k new : Assign x i to P knew ; 4: change = change + ; 5: end if 6: end fo 7: if change =, then F lag T RUE; 8: end while The esults geneated based on the Hausdoff and the city-block distances ae also included fo compaison pupose. This section stats with an intoduction of the expeimental data sets, then we descibe the Coected Rand Index (CR Index), which is widely used in the simila studies to evaluate the pefomance of the clusteings; finally, the expeiments esults of ou OLID measuement ae shown togethe with a discussion based on the pefomance compaison with the othe popula measuements. We compae ou measuement with seveal popula distances, in the context of one/two weight single/cluste adaptive clusteing algoithms. As a andom initialization step in the dynamic clusteing famewok, we un each algoithm on each data set fo times, and use the aveage CR ove these unning fo compaison. 4. Data Sets In the expeiments, thee synthetic inteval data sets ae employed, which ae designed to be wellsepaated, not-so-well-sepaated and ove-lapping espectively. 4.. Synthetic Data Sets Following a simila stategy in (de Souza & de A.T. de Cavalho 4, de A.T. de Cavalho et al. 6, de A.T. De Cavalho & Lechevallie 9), thee types of the synthetic data sets ae geneated accoding to a bivaiate nomal distibution in a two-dimensional eal numbe space, R. The fist one epesents a well-sepaated data set. The second one epesents a not-so-well-sepaated data set, in which the class covaiance matices of the bivaiate distibution ae unequal; while fo the thid data set, the class covaiance matices ae nealy the same. The paametes listed in Table ae set up to geneate these thee data sets espectively. Each data set contains 45 data instances. A pioi classification is done fo evaluation convenience, by which fou labels ae set up to goup the data instances into fou classes with diffeent sizes: Class and Class have the same size of 5, Class contains 5 data instances, and Class 4 takes ones. As shown in Fig. (a), Fig. (c) and Fig. (e), each data instance (a i, b i ) in well-sepaated, not-so-wellsepaated and ove-lapping data sets is a seed of a vecto of intevals: ([a i γ, a i + γ ], [b i γ, b i + γ ]), whee γ and γ ae andomly picked up fom intevals of [, 5], [, ], [, 5] and [, ]. The data sets can be also epesented by inteval values as shown in Fig. (b), Fig. (d) and Fig. (f). 4. Clusteing Validation The Coected Rand (CR) index, which was intoduced in (Hubet & Aabie 985), is one of the most popula clusteing validation indexes (de Souza & de A.T. de Cavalho 4, de A.T. de Cavalho et al. 6, de A.T. De Cavalho & Lechevallie 9). Given two patitions of the same data set, U = {u, u,..., u R } and V = {v, v,..., v C }, which have Page 9

6 CRPIT Vol AusDM'9 Table : Paametes in the thee synthetic seed data sets Well-sepaated Not so Well-sepaated Ove-lapping µ µ σ σ µ µ σ σ µ µ σ σ Class Class Class Class Class Class Class Class 4 4 y y (a) Well-Sepaated Seeds (b) Well-Sepaated Intevals Class Class Class Class y y (c) Not so Well-Sepaated Seeds (d) Not so Well-Sepaated Intevals Class Class Class Class 4 5 y y (e) Ove-lapping Seeds (f) Ove-lapping Intevals Figue : Thee Synthetic Data Sets (γ, γ [, ]) R and C clustes espectively, the CR index can be estimated by the following equation: CR = ( ) [ R ni. i= ) ( n Ri= ( Cj= nij + ( C n.j j= ) Ri= ( ) ni. Cj= ( n.j ) ) ( ) n Ri= ( ) ni. Cj= ( n.j ] ) (7) in which ( ) n = n(n ), n ij epesents the numbe of instances that ae in clustes u i and v i ; n i. indicates the numbe of instances in cluste u i ; n.j indicates the numbe of instances in cluste v j ; and n is the Page 4

7 Poc. of the 8th Austalasian Data Mining Confeence (AusDM'9) total numbe of instances in the data set. The value of CR index fo a cetain clusteing algoithm falls into the ange of [, ]. A CR index value of indicates two clusteing esults ae exactly the same, wheeas the value o below indicates that the cluste ageement found by chance (Milligan 996). When compaing the clusteing esult with the tue clusteing patition, the highe the CR index value is, the bette the esult is. 4. Results and Discussion Each clusteing distance is incopoated with the single and cluste adaptive clusteing famewok, then un times befoe the aveaged CR is calculated. 4.. Results fo Synthetic Data Sets Table 4 pesents the values of the aveage and standad deviation (in paenthesis) of the CR index fo the well-sepaated data set. It is evident that the poposed OLID distance in the cluste famewok can get the best esults on all the testing data sets. This is consistent with De Cavalho s findings (de A.T. De Cavalho & Lechevallie 9), which discoveed that the cluste adaptive clusteing famewok pefoms well on the well-sepaated data sets. Table 5 pesents the values of the aveage and standad deviation (in paenthesis) of the CR index fo the not-so-well-sepaated data set. This is also consistent with De Cavalho s findings (de A.T. De Cavalho & Lechevallie 9), which discoveed that the cluste adaptive clusteing famewok pefoms well on the not-so-well-sepaated data sets. It is inteesting to note that the poposed OLID distance in this famewok can always lead to the best esults on all testing data sets. Table 6 pesents the values of the aveage and standad deviation(in paenthesis) of the CR index fo the ove-lapping data set. This is also consistent with De Cavalho s findings (de A.T. De Cavalho & Lechevallie 9) that the single adaptive clusteing famewok pefoms well on the ove-lapping data sets. It is as expected that the poposed OLID distance outpefoms all the othe distances in the single adaptive clusteing famewok on all data sets. 4.. Paied t-test Results The two-tailed, paied t-test with 95% confidence level has been used to evaluate OLID with cityblock and Hausdoff distance unde single and cluste famewoks. The esults ae pesented in Table 7. Fom the table, we can see that in the cluste adaptive clusteing famewok, the poposed OLID measue significantly impoves the existing city-block and Hausdoff distances. In the single adaptive clusteing famewok, the OLID measue pefoms significantly bette than the Hausdoff distance, and it is also significantly bette than the two weight city-block distance, though the diffeence between it and the one weight city-block distance is not significant. 5 Conclusion The choice of a distance measuement is essential fo the success of many machine leaning and data mining tasks, such as clusteing and lazy leaning. The tend of data epesentation as the intevalvalued data calls fo moe sophisticated methods to evaluate the distance o similaity between intevalvalued data instances. The wok is motivated by the fact that, most existing distance measues fo inteval-valued data only consideed the lowe and uppe bounds, and ovelooked the elative size of thei ovelapped aea. In this pape, we intoduce a new distance measuement based on the Hausdoff distance and the elative size of the ovelapped aea. We show its popeties, and use it into diffeent dynamic clusteing famewoks: the single adaptive OLID algoithm and the cluste adaptive OLID algoithm. Ou expeiment esults indicate the significant impovement of the poposed OLID measue ove existing distances. In addition, ou esults futhe confim that the single adaptive clusteing famewok is suitable fo the ovelapping data sets, while the cluste adaptive clusteing famewok is suitable fo the well-sepaated data sets (de A.T. De Cavalho & Lechevallie 9). Refeences Billad, L. (6), Symbolic data analysis: What is it?, in Poceedings of 7th Symposium on Computational Statistics (COMPSTAT 6), Physica- Velag HD, Rome, Italy, pp Dai, H., Li, G. & Zhou, Z.-H. (4), Ensembling mml causal discovey, in Poceedings of The Eighth Pacific-Asia Confeence on Knowledge Discovey and Data Mining (PAKDD 4), pp de A.T. de Cavalho, F., de Souza, R. M., Chavent, M. & Lechevallie, Y. (6), Adaptive hausdoff distances and dynamic clusteing of symbolic inteval data, Patten Recognitionn Lettes 7(), de A.T. De Cavalho, F. & Lechevallie, Y. (9), Patitional clusteing algoithms fo symbolic inteval data based on single adaptive distances, Patten Recognition 4(7), 6. de Souza, R. M. & de A.T. de Cavalho, F. (4), Clusteing of inteval data based on city-block distances, Patten Recognitionn Lettes 5(), Diday, E. (988), Classification methods of data analysis, Elesvie, Noth Holland, Amstedam, chapte The symbolic appoach in clusteing, elated methods of data analysis, pp Hubet, L. & Aabie, P. (985), Compaing patitions, Jounal of Clasification (), 9 8. Jiang, Y., Ling, J., Li, G., Dai, H. & Zhou, Z.-H. (5), Dependency bagging, in Poceedings of The Tenth Intenational Confeence on Rough Sets, Fuzzy Sets, Data Mining, and Ganula Computing (RSFDGC 5), pp Li, G. & Dai, H. (4), What will affect softwae euse: A causal model analysis, Intenational Jounal of Softwae Engineeing and Knowledge Engineeing 4(), Li, G. & Tong, F. (), Unsupevised discetization algoithm based on mixtue pobabilistic model, Jisuanji Xuebao/Chinese Jounal of Computes 5(), Milligan, G. (996), Clusteing and Classification, Wold Scientific, Singapoe, chapte Clusteing validation: esults and implications fo applied analysis, pp Page 4

8 CRPIT Vol AusDM'9 Table 4: Well-Sepaated Data Set: compaison of the distances OLID city-block Hausdoff Range of γ i One weight One weight Two weight One weight (i =, ) Single Cluste Single Cluste Single Cluste Single Cluste γ i [, 5] CR Index (.95) (.8) (.879) (.954) (.96) (.967) (.76) (.7) γ i [, ] CR Index (.8) (.895) (.97) (.8) (.96) (.768) (.58) (.998) γ i [, 5] CR Index (.98) (.7) (.88) (.86) (.57) (.99) (.49) (.96) γ i [, ] CR Index (.7) (.99) (.74) (.699) (.74) (.87) (.99) (.8) Table 5: Not So Well-Sepaated Data Set: compaison of the distances OLID city-block Hausdoff Range of γ i One weight One weight Two weight One weight (i =, ) Single Cluste Single Cluste Single Cluste Single Cluste γ i [, 5] CR Index (.47) (.589) (.8) (.88) (.54) (.768) (.5) (.84) γ i [, ] CR Index (.64) (.6) (.86) (.9) (.64) (.45) (.89) (.97) γ i [, 5] CR Index (.9) (.47) (.7) (.8) (.7) (.7) (.8) (.48) γ i [, ] CR Index (.4) (.487) (.5) (.84) (.) (.75) (.7) (.444) Table 6: Ove-Lapping Data Set: compaison of the distances OLID city-block Hausdoff Range of γ i One weight One weight Two weight One weight (i =, ) Single Cluste Single Cluste Single Cluste Single Cluste γ i [, 5] CR Index (.4) (.4) (.999) (.977) (.99) (.4) (.7) (.5) γ i [, ] CR Index (.56) (.765) (.67) (.8) (.66) (.89) (.9) (.548) γ i [, 5] CR Index (.) (.7) (.94) (.759) (.479) (.847) (.445) (.566) γ i [, ] CR Index (.845) (.947) (.654) (.799) (.654) (.9) (.64) (.684) Table 7: Paied t-test Results Algoithms Distance p-value t Single OLID vs. Single city-block One weight Two weight.49. Single OLID vs. Single Hausdoff one weight Cluste OLID vs. Cluste city-block One weight Two weight Cluste OLID vs. Cluste Hausdoff one weight Page 4