Patent Application: US-85854104-A

Abstract:
unlike traditional clustering methods that focus on grouping objects with similar values on a set of dimensions , clustering by pattern similarity finds objects that exhibit a coherent pattern of rise and fall in subspaces . pattern - based clustering extends the concept of traditional clustering and benefits a wide range of applications , including e - commerce target marketing , bioinformatics , and automatic computing , etc . however , state - of - the - art pattern - based clustering methods can only handle datasets of thousands of records , which makes them inappropriate for many real - life applications . furthermore , besides the huge data volume , many data sets are also characterized by their sequentiality , for instance , customer purchase records and network event logs are usually modeled as data sequences . hence , it becomes important to enable pattern - based clustering methods i ) to handle large datasets , and ii ) to discover pattern similarity embedded in data sequences . there is presented herein a novel method that offers this capability .

Description:
the choice of distance functions has great implications on the meaning of similarity , and this is particularly important in subspace clustering because of computational complexity . hence , there is broadly contemplated in accordance with at least one preferred embodiment of the present invention a distance function that makes measuring of the similarity between two objects in high dimensional space meaningful and intuitive , and at the same time yields to an efficient implementation . finding objects that exhibit coherent patterns of rise and fall in a tabular dataset ( e . g . table 1 ) is similar to finding subsequences in a sequential dataset ( e . g . table 2 ). this indicates that one should preferably unify the data representation of tabular and sequential datasets so that a single similarity model and algorithm can apply to both tabular and sequential datasets for clustering based on pattern similarity . preferably , sequences are used to represent objects in a tabular dataset . it is assumed that here is a total order among its attributes . for instance , let a ={ c 1 , . . . , c n } be the set of attributes . it is also assumed that c 1 . . . c n is the total order . thus , one can represent any object x by a sequence : where x c i is the value of x in column c i . one can then concatenate objects in d into a long sequence , which is a sequential representation of the tabular data . ( one may also use x c1 , . . . , x cn to represent x if no confusion arises .) after the conversion , pattern discovery on tabular datasets is no different from pattern discovery in a sequential dataset . for instance , in the yeast dna micro - array , one can use the following sequence to represent a pattern : to express this in words , for genes that exhibit this pattern , their expression levels under condition ch2b , ch2i , and ch1i must be 180 , 205 , 280 units higher than that under ch1d . there is broadly contemplated in accordance with at least one preferred embodiment of the present invention a new distance measure that is capable of capturing subspace pattern similarity and is conducive to an efficient implementation . here , only the shifting pattern of fig1 ( b ) is considered , as scaling patterns are equivalent to shifting patterns after a logarithmic transformation of the data . to tell whether two objects exhibit a shifting pattern in a given subspace , the simplest way is to normalize the two objects by subtracting x s from each of their coordinate value x i ( i ∈ ), where x s is the average coordinate value of x in subspace . this , however , requires one to compute and keep track of x s for each subspace . as there are as many as 2 | a | − 1 different ways of normalization , it makes the computation of such similarity model impractical for large datasets . to find a distance function that is conducive to an efficient implementation , one may choose an arbitrary dimension k ∈ for normalization . it can be shown that the choice of k has very limited impact on the similarity measure . more formally , given two objects x and y , a subspace , a dimension k ∈ , one defines the sequence - based distance between x and y as follows : fig2 demonstrates the intuition behind eq ( 1 ). let ={ k , a , b , c }. with respect to dimension k , the distance between x and y in is less than δ if the difference between x and y on any dimension of is within δ ± δ , where δ is the difference of x and y on dimension k . clearly , with a different choice of dimension k , one may find the distance between two objects different . however , such different is bounded by a factor of 2 , as is shown by the following property . property 1 . for any two objects x , y , and a subspace , if ∃ k ∈ such that dist k , s ( x , y )≦ δ , then ∀ j ∈ s , dist j , s ( x , y )≦ 2δ . since δ is but a user - defined threshold , property 1 shows that eq ( 1 )&# 39 ; s capability of capturing pattern similarity does not depend on the choice of k , which can be an arbitrary dimension in . as a matter of fact , as long as one uses a fixed dimension k for any given subspace , then , with a relaxed δ , one can always find those clusters discovered by eq ( 1 ) where a different dimension k is used . this gives one great flexibility in defining and mining clusters based on subspace pattern similarity . turning to a clustering algorithm , the concept of pattern is first defined herebelow and then the pattern space is divided into grids . a tree structure is then constructed which provides a compact summary of all of the frequent patterns in a data set . it is shown that the tree structure enables one to find efficiently the number of occurrences of any specified pattern , or equivalently , the density of any cell in the grid . a density and grid based clustering algorithm can then be applied to merge dense cells into clusters . finally , there is introduced an apriori - like method to find clusters in any subspace . let be a dataset in a multidimensional space a . a pattern p is a tuple ( t ., δ ), where δ is a distance threshold and t is an ordered sequence of ( column , value ) pairs , that is , t = ( t 1 , 0 ),( t 2 , v 2 ), . . . ,( t k , v k ) where t i ∈ a , and t 1 . . . t k . let ={ t 1 , . . . , t k }. an object x ∈ exhibits pattern p in subspace if v i − δ ≦ x t i − x t 1 ≦ v i + δ , 1 ≦ i ≦ k ( 2 ) apparently , if two objects x , y ∈ are both instances of pattern p =( t , δ ), then one has in order to find clusters , we start with high density patterns : a pattern p =( t , δ ) is of high density if given p , the number of objects that satisfy eq ( 2 ) reaches a user - defined density threshold . preferably , the dataset is discretized so that patterns fall into grids . for any given subspace , after one finds the dense cells in , there is preferably employed a grid and density based clustering algorithm to find the clusters ( fig3 ). the difficult part , however , lies in finding the dense cells efficiently for all subspaces . further discussion herebelow deals with this issue . a counting tree provides a compact summary of the dense patterns in a dataset . it is motivated by the suffix trie , which , given a string , indexes all of its substrings . here , each record in the dataset is represented by a sequence , but sequences are different from strings , as the interest is essentially in non - contiguous sub - sequence match , while suffix tries only handle contiguous substrings . before introducing the structure of the counting tree , by way of example , table 3 shows a dataset of 3 objects in a 4 dimensional space . preferably , one starts with the relevant subsequences of each object . the relevant subsequences of an object o in an n - dimensional space are : x i = x i + 1 − x i , . . . , x n − x i 1 ≦ i & lt ; n in relevant subsequence x i , column c i is used as the base for comparison . s , wherein i is the minimal dimension , we shall search for c in dataset { x i |∀ x ∈ }. in any such subspace , one preferably uses c i as the base for comparison ; in other words , c i serves as the dimension k in eq ( 1 ). as an example , the relevant subsequences of object z in table 3 are : to create a counting tree for a dataset , for each object z ∈ , insert its relevant subsequences into a tree structure . also , assuming the insertion of a sequence , say z 1 , ends up at node t in the tree ( fig4 ), increase the count associated with node t by 1 . more often than not , the interest is in patterns equal to or longer than a given size , say ξ ≧ 1 . a relevant subsequence whose length is shorter than ξ cannot contain patterns longer than ξ . thus , if ξ is known beforehand , one only needs to insert x i where 1 ≦ i & lt ; n − ξ + 1 for each object x . fig4 shows the counting tree for the dataset of table 3 where ξ = 2 . in the second step , label each tree node t with a triple : ( id ├ , id ┤ , count ). the first element of the triple , id ├ , uniquely identifies node t , and the second element , id ┤ , is the largest id ├ of t &# 39 ; s descendent nodes . the ids are assigned by a depth - first traversal of the tree structure , during which one preferably assigns sequential numbers ( starting from 0 , which is assigned to the root node ) to the nodes as they are encountered one by one . if t is a leaf node , then the 3rd element of the triple , count , is the number of objects in t &# 39 ; s object set , otherwise , it is the sum of the counts of its child nodes . apparently , one can label a tree with a single depth - first traversal . fig4 shows a labeled tree for the sample dataset . to count pattern occurrences using the tree structure , there are preferably introduced counting lists . for each column pair ( c i , c j ), i & lt ; j , and each possible value v = x j − x i ( after data discretization ), create a counting list ( c i , c j , v ). the counting lists are also constructed during the depth - first traversal . suppose during the traversal , one encounters node t , which represents sequence element x j − x i = v . assuming t is to be labeled ( id ├ , id ┤ , cnt ), and the last element of counting list ( c i , c j , v ) is (-,-, cnt ′), one preferably appends a new element ( id ├ , id ┤ , cnt + cnt ′) into the list . ( if list ( c i , c j , v ) is empty , then make ( id ├ , id ┤ , count ) the first element of the list .) above is a part of the counting lists for the tree structure in fig4 . for instance , link ( c 2 , c 4 ,− 1 ) contains two nodes , which are created during the insertion of x 2 and z 2 ( relevant subsequences of x and z in table 3 ). the two nodes represent element x 4 − x 2 =− 1 and z 4 − z 2 =− 1 in sequence x 2 and z 2 respectively . the process of building the counting tree is summarized in algorithm 1 ( all algorithms appear in the appendix hereto ). thus , the counting tree is composed of two structures , the tree and the counting lists . one can observe the following properties of the counting tree : 1 . for any two nodes x , y labeled ( id x ├ , id x ┤ , count x ) and ( id y ├ , id y ┤ , count y ) respectively , node y is a descendent of node x if id y ├ ∈[ id x ├ , id x ┤ ]. 2 . each node appears once and only once in the counting lists . 3 . nodes in any counting list are in ascending order of their id ├ . these properties are important in finding the dense patterns efficiently , as presented herewbelow . herebelow , there is described “ seqclus ”, an efficient algorithm for finding the occurrence number of a specified pattern using the counting tree structure introduced above . each node s in the counting tree represents a pattern p , which is embodied by the path leading from the root node to t . for instance , the node s in fig4 represents pattern ( c 1 , 0 ),( c 2 , 1 ) . how can one find the number of occurrence of pattern p ′ which is one element longer than p ? that is , the counting tree structure makes this operation very easy . first , one only needs to look for nodes in counting list ( c i , c k , v ), since all nodes of x k − x i = v are in that list . second , the interest is essentially in nodes that are under node s , because only those nodes satisfy pattern p , a prefix of p ′. assuming s is labeled ( id s ├ , id s ┤ , count ), we know s &# 39 ; s descendent nodes are in the range of [ id s ├ , id s ┤ ]. according to the counting properties , elements in any counting list are in ascending order of their id ├ values , which means one can binary - search the list . finally , assume list ( c i , c k , v ) contain the following nodes : then , it is known altogether that there are cnt w − cnt u objects ( or just cnt w objects if id v ├ is the first element of the list ) that satisfy pattern p ′. one may denote the above process by count ( r , c k , v ), where r is a range , and in this case r =[ id s ├ , id s ┤ ]. if , however , one is looking for patterns even longer than p ′, then instead of returning cnt w − cnt u , one preferably shall continue the search . let l denote the list of the sub - ranges represented by the nodes within range [ id s ├ , id s ┤ ] in list ( c i , c k , v ), that is , l ={[ id v ├ , id v ┤ ], . . . ,[ id w ├ , id w ┤ ]} then , repeat the above process for each range in l , and the final count comes to turning now to clustering , the counting algorithm hereinabove finds the number of occurrences of a specified pattern , or the density of the cells in the pattern grids of a given subspace ( fig3 ). one can then use a density and grid based clustering algorithm to group the dense cells together . start with patterns containing only two columns ( in a 2 - dimensional subspace ), and grow the patterns by adding new columns into them . during this process , patterns that correspond to no more than minrows objects are pruned , as introducing new columns into the pattern will only reduce the number of objects . fig5 shows a tree structure for growing the clusters . each node t in the tree is a triple ( item , count , range - list ). the items in the nodes along the path from the root node to node t constitutes the pattern represented by t . for instance , the node in the 3rd level in fig5 represents ( c 0 , 0 ),( c 1 , 0 ),( c 2 , 0 ) , a pattern in a 3 - dimensional space . the value count in the triple represents the number of occurrences of the pattern in the dataset , and range - list is the list of ranges of the ids of those objects . both count and range - list are computed by the count ( ) routine in algorithm 2 . first of all , count the occurrences of all patterns containing 2 columns , and insert them under the root node if they are frequent ( count ≧ minrows ). note there is no need to consider all the columns . as any c i − c j = v to be the first item in a pattern with at least mincols columns , c i must be less than c n − mincols + 1 and c j must be less than c n − mincols . in the second step , for each node p on the current level , join p with its eligible nodes to derive nodes on the next level . a node q is node p &# 39 ; s eligible nodes if it satisfies the following criteria : q is on the same level as p ; if p denotes item a − b = v and q denotes c − d = v ′, then a c , b = d . besides p &# 39 ; s eligible nodes , we also join p with item in the form of c n − mincols + k − b = v , since column c n − mincols + k does not appear in levels less than k . the join operation is easy to perform . assume p , represented by triple ( a − b = v , count , range - list ), is to be joined with item c − b = v ′, we simply compute count ( r , c , v ′) for each range r in range - list . if the sum of the returned counts is larger than minrows , then insert a new node ( c − b = v ′, count ′, range - list ′) under p , where count ′ is the sum of the returned counts , and range - list ′ is the union of all the ranges returned by count ( ). algorithm 3 summarizes the clustering process described above . in experimentation , the algorithms in c were implemented on a linux machine with a 700 mhz cpu and 256 mb main memory . it was tested on both synthetic and real life data sets . an overview of the experimentation is provided herebelow . synthetic datasets are generated in tabular and sequential forms . for real life datasets , there are preferably used use time - stamped event sequences generated by a production network ( sequential data ), and dna micro - arrays of yeast and mouse gene expressions under various conditions ( tabular data ). with regard to tabular forms , initially , the table is filled with random values ranging from 0 to 300 , and then there are embedded a fixed number of clusters in the raw data . the clusters embedded can also have varying quality . perfect clusters are embedded in the matrix , i . e ., the distance between any two objects in the embedded cluster is 0 ( i . e ., δ = 0 ). also embedded are clusters whose distance threshold among the objects is δ = 2 , 4 , 6 , . . . . also generated are synthetic sequential datasets in the form of . . . ( id , timestamp ) . . . , where instead of embedding clusters , there are simply modeled the sequences by probabilistic distributions . here , the ids are randomly generated ; however , the occurrence rate of different ids follows either a uniform or a zipf distribution . generated are ascending timestamps in such a way that the number of elements in a unit window follows either uniform or poisson distribution . gene expression data are presented as a matrix . the yeast microarray [ 15 ] can be converted to a weighted - sequence of 49 , 028 elements ( 2 , 884 genes under 17 conditions ). the expression levels of the yeast genes ( after transformation ) range from 0 - 600 , and they are discretized into 40 bins . the mouse cdna array [ 10 ] is 535 , 766 in size ( 10 , 934 genes under 49 conditions ) and it is pre - processed in the same way . the data sets are taken from a production computer network at a financial service company . netview [ 14 ] has six attributes : timestamp , eventtype , host , severity , interestingness , and dayofweek . of import are attribute timestamp and eventtype , which has 241 distinctive values . tec [ 14 ] has attributes timestamp , eventtype , source , severity , host , and dayofyear . in tec , there are 75 distinctive values of eventtype and 16 distinctive types of source . it is often interesting to differentiate same type of events from different sources , and this is realized by combining eventtype and source to produce 75 × 16 = 1200 symbols . by way of performance analysis , the scalability of the clustering algorithm on synthetic tabular datasets is evaluated , and compared with pcluster [ 16 ]. the number of objects in the dataset increases from 1 , 000 to 100 , 000 , and the number of columns from 20 to 120 . the results presented in fig6 ( a )- 6 ( c ) are average response times obtained from a set of 10 synthetic data . data sets used for fig6 ( a ) are generated with number of columns fixed at 30 . embedded are a total of 10 perfect clusters ( δ = 0 ) in the data . the minimal number of columns of the embedded cluster is 6 , and the minimal number of rows is set to 0 . 01n , where n is the number of rows of the synthetic data . the pcluster algorithm is invoked with mincols = 5 , minrows = 0 . 01n , and δ = 3 , and the seqclus algorithm is invoked with δ = 3 . fig6 ( a ) shows that there is almost a linear relationship between the time and the data size for the seqclus algorithm . the pcluster algorithm , on the other hand , is not scalable , and it can only handle datasets with size in the range of thousands . for fig6 ( b ), there is increased the dimensionality of the synthetic datasets from 20 to 120 . each embedded cluster is in subspace whose dimensionality is at least 0 . 02c , where c is the number of columns of the data set . the pcluster algorithm is invoked with δ = 3 , min cols = 0 . 02c , and minrows = 30 . the curve of seqclus exhibits quadratic behavior . however , it shows that , with increasing dimensionality , seqclus can almost handle datasets of size an order of magnitude larger than pcluster ( 30k vs . 3k ). next , there is studied the impact of the quality of the embedded clusters on the performance of the clustering algorithms . there are generated synthetic datasets containing 3k / 30k objects , 30 columns with 30 embedded clusters ( each on average contains 30 objects , and the clusters are in subspace whose dimensionality is 8 on average ). within each cluster , the maximum distance ( under the pcluster model ) between any two objects ranges from δ = 2 to δ = 6 . fig7 shows that , while the performance of the pcluster algorithm degrades with the increase of δ , the seqclus algorithm is more robust under this situation . the reason is because much of the computation of seqclus is performed on the counting tree , which provides a compact summary of the dense patterns in the dataset , while for pcluster , a higher δ value has a direct , negative impact on its pruning effect [ 16 ]. there is also studied clustering performance on timestamped sequential datasets . the dataset in use is in the form of . . . ( id , timestamp ) . . . , where every minute contains on average 10 ids ( uniform distribution ). there is placed a sliding window of size 1 minute on the sequence , and there is created a counting tree for the subsequences inside the windows . the scalability result is shown in fig8 . also attempted were different distributions of id and timestamp , but there were not observed significant differences in performance . with regard to cluster analysis , there are reported clusters found in real life datasets . table 4 shows the number of clusters found by the pcluster and seqclus algorithm in the raw yeast micro - array dataset . for mincols = 9 and minrows = 30 , the two algorithms found the same clusters . but in general , using the same parameters , seqclus produces more clusters . this is because the similarity measure used in the pcluster model is more restrictive . it is found that the objects ( genes ) in those clusters overlooked by the pcluster algorithm but discovered by the seqclus method exhibit easily perceptible coherent patterns . for instance , the genes in fig9 shows a coherent pattern in the specified subspace , and this subspace cluster is discovered by seqclus but not by pcluster . this indicates the relaxation of the similarity model not only improves the performance but also provides extra insight in understanding the data . the seqclus algorithm works directly on both tabular and sequential datasets . table 5 shows event sequence clusters found in the netview dataset [ 14 ]. the algorithm is applied on 10 days &# 39 ; worth of event logs ( around 41 m bytes ) of the production computer network . herebelow is a discussion of a comparison of seqclus with previous approaches , while highlighting its advantage in discovering pattern similarity . the pcluster algorithm [ 16 ] was among the first efforts to discover clusters based on pattern similarity . however , due to the limitation of the similarity model in use , neither the pciuster model , nor its predecessors , which include the bicluster [ 8 ], the δ - cluster [ 17 ], and their variations [ 13 , 12 ], provide a scalable solution to discovering clusters based on pattern similarity in large datasets . the distance function used in the pcluster model [ 16 ] for measuring the similarity between two objects x and y in a subspace s a can be expressed as follows : where x i is object x &# 39 ; s value on coordinate i . a set of objects form a δ - pcluster if the distance between any of its two objects is less than δ . the advantage of seqclus over pcluster is due to the following two important differences between eq ( 3 ) and eq ( 1 ): to compute the distance between two objects x and y , eq ( 3 ) compares x and y for every two dimensions in s , while eq ( 1 ) is linear in the size of s . for pcluster , the fact that both { x . y } and { x , z } are δ - pclusters in s does not necessarily mean { x , y , z } is a δ - pcluster in s . because of this , pcluster resorts to a pair - wise clustering method consisting of two steps . first , it finds , for every two objects x and y , the subspaces in which they form a δ - cluster . the complexity of this process is o ( n 2 ). second , it finds , for each subspace s , sets of objects in which each pair forms a δ - pcluster in s . this process is np - complete , as it is tantamount to finding cliques ( complete subgraphs ) in a graph of objects ( two objects are linked by an edge if they form a δ - pcluster in s ). clearly , pcluster has scalability problems . even if objects form only a sparsely connected graph , which makes the second step possible , the o ( n 2 ) complexity of the first step still prevents it from clustering large datasets . effectiveness given that pcluster is computationally more expensive than seqclus , does pcluster find more clusters or clusters of higher quality than seqclus does ? the answer is no . as a matter a fact , with a relaxed user - defined similarity threshold , seqclus can find any cluster found by pcluster . property 2 . the clusters found in a dataset by seqclus with distance threshold 2δ contain all δ - pclusters . it is easy to see that dist s ( x , y )≦ δ ∃ k such that dist k , s ( x , y )≦ 2δ property 1 establishes a semi - equivalence between eq ( 3 ) and eq ( 1 ). but the latter is conducive to a much more efficient way of implementation . furthermore , seqclus can find meaningful clusters that pcluster is unable to discover ( e . g ., the cluster in fig9 ) since pcluster &# 39 ; s pair - wise model is often times too rigid . in many applications , for instance , system management , where one monitors various system events , data are coming in continuously in the form of data streams . herebelow is a discussion of how to adapt our algorithm to the data stream environment . the main data structure for clustering by pattern similarity is the counting tree . in order support data insertion and data deletion dynamically , one needs to support dynamic labeling . recall that a node is labeled by a triple : ( id ├ , id ┤ , count ). the ids are assigned by a depthfirst traversal of the tree structure , during which we assign sequential numbers ( starting form 0 , which is assigned to the root node ) to the nodes as they are encountered one by one . it is clear to see that such a labeling schema will prevent dynamic data insertion and deletion . one must instead pre - allocate space for future insertions . in order to do this , one can rely on an estimated probability distribution of the data . more specifically , one needs to estimate the probability p ( c i = v i | c i − 1 = v i − 1 ), which can be derived from data sampling or domain knowledge . one then uses such probability to pre - allocate label spaces . in addition , one needs to keep dynamic counts of candidate clusters . this can be achieved by checking whether the incoming data are instances of any particular clusters we are keeping track of multiple index structures . in this vein , incremental index maintenance might be costly if the data arrival rate is high . in the data stream environment , one will be interested to find out the clusters in the most recent window of size t . one can still perform clustering in the batch mode on data chunks of a fixed size τ . the clustering however , will use a reduced threshold of δ × τ / t . we combine the clusters found in different chunks to form the final clusters of the entire window . the benefits of this approach is that at any point of time , one only need worry about the data chunks that are moving into or out of the window ; thus , one will not incur global changes on the index structure . by way of recapitulation and conclusion , clustering by pattern similarity is an interesting and challenging problem . the computational complexity problem of subspace clustering is further aggravated by the fact that one is generally concerned with patterns of rise and fall instead of value similarity . the task of clustering by pattern similarity can be converted into a traditional subspace clustering problem by ( i ) creating a new dimension ij for every two dimension i and j of any object x , and set x ij , the value of the new dimension , to x i − x j ; or ( ii ) creating | a | copies ( a is the entire dimension set ) of the original dataset , where x k , the value of x on the k th dimension in the i th copy is changed to x k − x i , for k ∈ a . for both cases , we need to find subspace clusters in the transformed dataset , which is | a | times larger . it is to be understood that the present invention , in accordance with at least one presently preferred embodiment , includes an arrangement for accepting input data , an arrangement for discerning pattern similarity in the input data , and an arrangement for clustering the data on the basis of discerned pattern similarity . together , these elements may be implemented on at least one general - purpose computer running suitable software programs . these may also be implemented on at least one integrated circuit or part of at least one integrated circuit . thus , it is to be understood that the invention may be implemented in hardware , software , or a combination of both . if not otherwise stated herein , it is to be assumed that all patents , patent applications , patent publications and other publications ( including web - based publications ) mentioned and cited herein are hereby fully incorporated by reference herein as if set forth in their entirety herein . although illustrative embodiments of the present invention have been described herein with reference to the accompanying drawings , it is to be understood that the invention is not limited to those precise embodiments , and that various other changes and modifications may be affected therein by one skilled in the art without departing from the scope or spirit of the invention . alison abbott . bioinformatics institute plans public database for gene expression data . nature , 398 : 646 , 1999 . c . c . aggarwal , c . procopiuc , j . wolf , p . s . yu , and j . s . park . fast algorithms for projected clustering . in sigmod , 1999 . c . c . aggarwal and p . s . yu . finding generalized projected clusters in high dimensional spaces . in sigmod , pages 70 - 81 , 2000 . r . agrawal , j . gehrke , d . gunopulos , and p . raghavan . authomatic subspace clustering of high dimensional data for data mining applications . in sigmod , 1998 . alvis brazma , alan robinson , graham cameron , and michael ashburner . one - stop shop for microarray data . nature , 403 : 699 - 700 , 2000 . p . o . brown and d . botstein . exploring the new world of the genome with dna microarrays . nature genetics , 21 : 33 - 37 , 1999 . c . h . cheng , a . w . fu , and y . zhang . entropy - based subspace clustering for mining numerical data . in sigkdd , pages 84 - 93 , 1999 . y . cheng and g . church . biclustering of expression data . in proc . of 8 th international conference on intelligent system for molecular biology , 2000 . p . d &# 39 ; 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