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MIS 542 Data Mining Concepts and Techniques — Chapter 5 — Clustering 2014/2015 Fall 1 Chapter 5. Cluster Analysis What is Cluster Analysis? Types of Data in Cluster Analysis A Categorization of Major Clustering Methods Partitioning Methods Hierarchical Methods Density-Based Methods Grid-Based Methods Model-Based Clustering Methods Outlier Analysis Summary 2 What is Cluster Analysis? Cluster: a collection of data objects Similar to one another within the same cluster Dissimilar to the objects in other clusters Cluster analysis Finding similarities between data according to the characteristics found in the data and grouping similar data objects into clusters Clustering is unsupervised learning: no predefined classes Typical applications As a stand-alone tool to get insight into data distribution As a preprocessing step for other algorithms 3 General Applications of Clustering Pattern Recognition Spatial Data Analysis create thematic maps in GIS by clustering feature spaces detect spatial clusters and explain them in spatial data mining Image Processing Economic Science (especially market research) WWW Document classification Cluster Weblog data to discover groups of similar access patterns 4 Examples of Clustering Applications Marketing: Help marketers discover distinct groups in their customer bases, and then use this knowledge to develop targeted marketing programs Land use: Identification of areas of similar land use in an earth observation database Insurance: Identifying groups of motor insurance policy holders with a high average claim cost City-planning: Identifying groups of houses according to their house type, value, and geographical location Earth-quake studies: Observed earth quake epicenters should be clustered along continent faults 5 What Is Good Clustering? A good clustering method will produce high quality clusters with high intra-class similarity low inter-class similarity The quality of a clustering result depends on both the similarity measure used by the method and its implementation. The quality of a clustering method is also measured by its ability to discover some or all of the hidden patterns. 6 Requirements of Clustering in Data Mining Scalability Ability to deal with different types of attributes Ability to handle dynamic data Discovery of clusters with arbitrary shape Minimal requirements for domain knowledge to determine input parameters Able to deal with noise and outliers Insensitive to order of input records High dimensionality Incorporation of user-specified constraints Interpretability and usability 7 Chapter 5. Cluster Analysis What is Cluster Analysis? Types of Data in Cluster Analysis A Categorization of Major Clustering Methods Partitioning Methods Hierarchical Methods Density-Based Methods Grid-Based Methods Model-Based Clustering Methods Outlier Analysis Summary 8 Data Structures Data matrix (two modes) Dissimilarity matrix (one mode) x11 ... x i1 ... x n1 ... x1f ... ... ... ... xif ... ... ... ... ... xnf ... ... 0 d(2,1) 0 d(3,1) d ( 3,2) 0 : : : d ( n,1) d ( n,2) ... x1p ... xip ... xnp ... 0 9 Properties of Dissimilarity Measures Properties d(i,j) 0 for i j d(i,i) = 0 d(i,j) = d(j,i) symmetry d(i,j) d(i,k) + d(k,j) triangular inequality Exercise: Can you find examples where distance between objects are not obeying symmetry property 10 Measure the Quality of Clustering Dissimilarity/Similarity metric: Similarity is expressed in terms of a distance function, which is typically metric: d(i, j) There is a separate “quality” function that measures the “goodness” of a cluster. The definitions of distance functions are usually very different for interval-scaled, boolean, categorical, ordinal and ratio variables. Weights should be associated with different variables based on applications and data semantics. It is hard to define “similar enough” or “good enough” the answer is typically highly subjective. 11 Type of data in clustering analysis Interval-scaled variables: Binary variables: Nominal, ordinal, and ratio variables: Variables of mixed types: 12 Classification by Scale Nominal scale:merely distinguish classes: with respect to A and B XA=XB or XAXB e.g.: color {red, blue, green, …} gender { male, female} occupation {engineering, management. .. } Ordinal scale: indicates ordering of objects in addition to distinguishing XA=XB or XAXB XA>XB or XA<XB e.g.: education {no school< primary sch. < high sch. < undergrad < grad} age {young < middle < old} income {low < medium < high } 13 Interval scale: assign a meaningful measure of difference between two objects Not only XA>XB but XAis XA – XB units different from XB e.g.: specific gravity temperature in oC or oF Boiling point of water is 100 oC different then its melting point or 180 oF different Ratio scale: an interval scale with a meaningful zero point XA > XB but XA is XA/XB times greater then XB e.g.: height, weight, age (as an integer) temperature in oK or oR Water boils at 373 oK and melts at 273 oK Boiling point of water is 1.37 times hotter then melting poing 14 Comparison of Scales Strongest scale is ratio weakest scale is ordinal Ahmet`s height is 2.00 meters HA Mehmet`s height is 1.50 meter HM HA HM nominal: their heights are different HA > HM ordinal Ahmet is taller then Mehmet HA - HM =0.50 meters interval Ahmet is 50 cm taller then Mehmet HA / HM =1.333 ratio scale, no mater height is measured in meter or inch … 15 Interval-valued variables Standardize data Calculate the mean absolute deviation: sf 1 n (| x1 f m f | | x2 f m f | ... | xnf m f |) where m f 1n (x1 f x2 f ... xnf ) . Calculate the standardized measurement (z-score) xif m f zif sf Using mean absolute deviation is more robust than using standard deviation 16 Other Standardizations Min Max scale between 0 and 1 or -1 and 1 Decimal scale For Ratio Scaled variales Mean transformation zi,f = xi,f/mean_f Measure in terms of means of variable f Log transformation zi,f = logxi,f 17 X2 * * * * * X2 * * * * * * * X1 * *** * * * * ** ** Both has zero mean and standardized by Z scores X1 and X2 are unity in both cases I and II X1.X2=0 in case I whereas X1.X2 1 in case II Shell we use the same distance measure in both cases After obtaining the z scores X1 18 Exercise X2 * * A* * * * * * * * X2 * * X1 *** * * * * A* ** ** X1 Suppose d(A,O) = 0.5 in case I and II Does it reflect the distance between A and origin? Suggest a transformation so as to handle correlation Between variables 19 Similarity and Dissimilarity Between Objects Distances are normally used to measure the similarity or dissimilarity between two data objects Some popular ones include: Minkowski distance: d (i, j) q (| x x |q | x x |q ... | x x |q ) i1 j1 i2 j2 ip jp where i = (xi1, xi2, …, xip) and j = (xj1, xj2, …, xjp) are two p-dimensional data objects, and q >=1 If q = 1, d is Manhattan distance d (i, j) | x x | | x x | ... | x x | i1 j1 i2 j 2 i p jp 20 Similarity and Dissimilarity Between Objects (Cont.) If q = 2, d is Euclidean distance: d (i, j) (| x x |2 | x x |2 ... | x x |2 ) i1 j1 i2 j2 ip jp Properties d(i,j) 0 d(i,i) = 0 d(i,j) = d(j,i) d(i,j) d(i,k) + d(k,j) Also, one can use weighted distance, parametric Pearson product moment correlation, or other disimilarity measures 21 Similarity and Dissimilarity Between Objects (Cont.) Weights can be assigned to variables d (i, j) (w | x x |2 w | x x |2 ... w | x x |2 ) i1 j1 i2 j 2 ip jp 1 2 p Where wi i = 1…P weights showing the importance of each variable 22 XA XA XB Manhatan distance between XA and XB XB Euclidean distance between XA and XB 23 When q = Mincovsly distance becomes Chebychev distance or L metric limq=(|Xi1-Xj1|q+|Xi2-Xj2|q+…+|Xip-Xjp|q)1/q =maxp |Xip-Xjp| 24 Exercise Take one of these points as origin and draw the locus of points that are 1,2 ,3 units away from the oirgin with two dimensions according to Menhattan distance Euchlidean distance Chebychev distance 25 Binary Variables Symmetric asymmetric Symmetric: both of its states are equally valuable and carry the same weight gender: male female 0 male 1 female arbitarly coded as 0 or 1 Asymmetric variables Outcomes are not equally important Encoded by 0 and 1 E.g. patient smoker or not 1 for smoker 0 for nonsmoker asymmetric Positive and negative outcomes of a disease test HIV positive by 1 HIV negative 0 26 Binary Variables A contingency table for binary data Object j Object i 1 0 1 a b 0 c d sum a c b d sum a b cd p Simple matching coefficient (invariant, if the binary bc variable is symmetric): d (i, j) a bc d Jaccard coefficient (noninvariant if the binary variable is asymmetric): d (i, j) bc a bc 27 Dissimilarity between Binary Variables Example Name Jack Mary Jim Gender M F M Fever Y Y Y Cough N N P Test-1 P P N Test-2 N N N Test-3 N P N Test-4 N N N gender is a symmetric attribute the remaining attributes are asymmetric binary let the values Y and P be set to 1, and the value N be set to 0 01 0.33 2 01 11 d ( jack , jim ) 0.67 111 1 2 d ( jim , mary ) 0.75 11 2 d ( jack , mary ) 28 Nominal Variables A generalization of the binary variable in that it can take more than 2 states, e.g., red, yellow, blue, green Method 1: Simple matching m: # of matches, p: total # of variables m d (i, j) p p Higher weights can be assigned to variables with large number of states Method 2: use a large number of binary variables creating a new binary variable for each of the M nominal states 29 Example 2 nominal variables Faculty and country for students Faculty {eng, applied Sc., Pure Sc., Admin., } 5 distinct values Country {Turkey, USA} 10 distinct values P = 2 just two varibales Weight of country may be increased Student A (eng, Turkey) B(Applied Sc, Turkey) m =1 in one variable A and B are similar D(A,B) = (2-1)/2 =1/2 30 Example cont. Different binary variables for each faculty Eng 1 if student is in engineering 0 otherwise AppSc 1 if student in MIS, 0 otherwise Different binary variables for each country Turkey 1 if sturent Turkish, 0 otherwise USA 1 if student USA ,0 otherwise 31 Ordinal Variables An ordinal variable can be discrete or continuous Order is important, e.g., rank Can be treated like interval-scaled replace xif by their rank map the range of each variable onto [0, 1] by replacing i-th object in the f-th variable by zif rif {1,...,M f } rif 1 M f 1 compute the dissimilarity using methods for intervalscaled variables 32 Example Credit Card type: gold > silver > bronze > normal, 4 states Education: grad > undergrad > highschool > primary school > no school, 5 states Two customers A(gold,highschool) B(normal,no school) rA,card = 1 , rB,card = 4 rA,edu = 3 , rA,card = 5 zA,card = (1-1)/(4-1)=0 zB,card = (4-1)/(4-1)=1 zA,edu = (3-1)/(5-1)=0.5 zB,edu = (5-1)/(5-1)=1 Use any interval scale distance measure on z values 33 Exercise Find an attribute having both ordinal and nominal charecterisitics define a similarity or dissimilarity measure for to objects A and B 34 Ratio-Scaled Variables Ratio-scaled variable: a positive measurement on a nonlinear scale, approximately at exponential scale, such as AeBt or Ae-Bt Methods: treat them like interval-scaled variables—not a good choice! (why?—the scale can be distorted) apply logarithmic transformation yif = log(xif) treat them as continuous ordinal data treat their rank as interval-scaled 35 Example Cluster individuals based on age weights and heights All are ratio scale variables Mean transformation Zp,i = xp,i/meanp As absolute zero makes sense measure distance by units of mean for each variable Then you may apply z`= logz Use any distance measure for interval scales then 36 Example cont. A weight difference of 0.5 kg is much more important for babies then for adults d(3kg,3.5kg) = 0.5 (3.5-3)/3 percentage difference d(71.5kg,70.0kg) =0.5 d`(3kg,3.5kg) = (3.5-3)/3 percentage difference very significant approximately log(3.5)-log3 d(71.5kg,71.0kg) = (71.5-70.0)/70.0 Not important log71.5 – log71 almost zero 37 Examples from Sports Boxing wrestling 48 48 51 52 54 56 57 62 60 68 63.5 74 67 82 71 90 75 100 81 130 38 Variables of Mixed Types A database may contain all the six types of variables symmetric binary, asymmetric binary, nominal, ordinal, interval and ratio One may use a weighted formula to combine their effects pf 1 ij( f ) d ij( f ) d (i, j) pf 1 ij( f ) f is binary or nominal: dij(f) = 0 if xif = xjf , or dij(f) = 1 o.w. f is interval-based: use the normalized distance f is ordinal or ratio-scaled compute ranks rif and r 1 z if and treat zif as interval-scaled M 1 if f 39 fij is count variable fij = 0 if f is binary and asymmetric variable and Xif = Xjf = 0 fij = 1 o.w. 40 Exercise Construct an example data containiing all types of variables Define variables in that data set and compute distance between two sample objects 41 Chapter 6. Cluster Analysis What is Cluster Analysis? Types of Data in Cluster Analysis A Categorization of Major Clustering Methods Partitioning Methods Hierarchical Methods Density-Based Methods Grid-Based Methods Model-Based Clustering Methods Outlier Analysis Summary 42 Basic Measures for Clustering Clustering: Given a database D = {t1, t2, .., tn}, a distance measure dis(ti, tj) defined between any two objects ti and tj, and an integer value k, the clustering problem is to define a mapping f: D {1, …, k} where each ti is assigned to one cluster Kf, 1 ≤ f ≤ k, such that tfp,tfq ∈ Kf and ts ∉ Kf, dis(tfp,tfq) ≤ dis(tfp,ts) Centroid, radius, diameter Typical alternatives to calculate the distance between clusters Single link, complete link, average, centroid, medoid 43 Centroid, Radius and Diameter of a Cluster (for numerical data sets) Centroid: the “middle” of a cluster ip ) N Radius: square root of average distance from any point of the cluster to its centroid Cm iN 1(t N (t cm ) 2 Rm i 1 ip N Diameter: square root of average mean squared distance between all pairs of points in the cluster N N (t t ) 2 Dm i 1 i 1 ip iq N ( N 1) 44 Typical Alternatives to Calculate the Distance between Clusters Single link: smallest distance between an element in one cluster and an element in the other, i.e., dis(Ki, Kj) = min(tip, tjq) Complete link: largest distance between an element in one cluster and an element in the other, i.e., dis(Ki, Kj) = max(tip, tjq) Average: avg distance between an element in one cluster and an element in the other, i.e., dis(Ki, Kj) = avg(tip, tjq) Centroid: distance between the centroids of two clusters, i.e., dis(Ki, Kj) = dis(Ci, Cj) Medoid: distance between the medoids of two clusters, i.e., dis(Ki, Kj) = dis(Mi, Mj) Medoid: one chosen, centrally located object in the cluster 45 Major Clustering Approaches Partitioning algorithms: Construct various partitions and then evaluate them by some criterion Hierarchy algorithms: Create a hierarchical decomposition of the set of data (or objects) using some criterion Density-based: based on connectivity and density functions Grid-based: based on a multiple-level granularity structure Model-based: A model is hypothesized for each of the clusters and the idea is to find the best fit of that model to each other 46 Chapter 6. Cluster Analysis What is Cluster Analysis? Types of Data in Cluster Analysis A Categorization of Major Clustering Methods Partitioning Methods Hierarchical Methods Density-Based Methods Grid-Based Methods Model-Based Clustering Methods Outlier Analysis Summary 47 Partitioning Algorithms: Basic Concept Partitioning method: Construct a partition of a database D of n objects into a set of k clusters, s.t., min sum of squared distance k m1 tmiKm (Cm tmi ) 2 Given a k, find a partition of k clusters that optimizes the chosen partitioning criterion Global optimal: exhaustively enumerate all partitions Heuristic methods: k-means and k-medoids algorithms k-means (MacQueen’67): Each cluster is represented by the center of the cluster k-medoids or PAM (Partition around medoids) (Kaufman & Rousseeuw’87): Each cluster is represented by one of the objects in the cluster 48 The K-Means Clustering Method Given k, the k-means algorithm is implemented in four steps: Partition objects into k nonempty subsets Compute seed points as the centroids of the clusters of the current partition (the centroid is the center, i.e., mean point, of the cluster) Assign each object to the cluster with the nearest seed point Go back to Step 2, stop when no more new assignment 49 The K-Means Clustering Method Example 10 10 9 9 8 8 7 7 6 6 5 5 10 9 8 7 6 5 4 4 3 2 1 0 0 1 2 3 4 5 6 7 8 K=2 Arbitrarily choose K object as initial cluster center 9 10 Assign each objects to most similar center 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 Update the cluster means 4 3 2 1 0 0 1 2 3 4 5 6 reassign 10 9 9 8 8 7 7 6 6 5 5 4 3 2 1 0 1 2 3 4 5 6 7 8 8 9 10 reassign 10 0 7 9 10 Update the cluster means 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 50 Comments on the K-Means Method Strength: Relatively efficient: O(tkn), where n is # objects, k is # clusters, and t is # iterations. Normally, k, t << n. Comparing: PAM: O(k(n-k)2 ), CLARA: O(ks2 + k(n-k)) Comment: Often terminates at a local optimum. The global optimum may be found using techniques such as: deterministic annealing and genetic algorithms Weakness Applicable only when mean is defined, then what about categorical data? Need to specify k, the number of clusters, in advance Unable to handle noisy data and outliers Not suitable to discover clusters with non-convex shapes 51 Variations of the K-Means Method A few variants of the k-means which differ in Selection of the initial k means Dissimilarity calculations Strategies to calculate cluster means Handling categorical data: k-modes (Huang’98) Replacing means of clusters with modes Using new dissimilarity measures to deal with categorical objects Using a frequency-based method to update modes of clusters A mixture of categorical and numerical data: k-prototype method 52 What is the problem of k-Means Method? The k-means algorithm is sensitive to outliers ! Since an object with an extremely large value may substantially distort the distribution of the data. K-Medoids: Instead of taking the mean value of the object in a cluster as a reference point, medoids can be used, which is the most centrally located object in a cluster. 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 53 Exercise Show by designing simple examples: The k-means algorithm may converge to different local optima starting from different initial assignments of objects into different clusters Suitable only clusters are spherical and equal sized 54 The K-Medoids Clustering Method Find representative objects, called medoids, in clusters PAM (Partitioning Around Medoids, 1987) starts from an initial set of medoids and iteratively replaces one of the medoids by one of the non-medoids if it improves the total distance of the resulting clustering PAM works effectively for small data sets, but does not scale well for large data sets CLARA (Kaufmann & Rousseeuw, 1990) CLARANS (Ng & Han, 1994): Randomized sampling Focusing + spatial data structure (Ester et al., 1995) 55 Typical k-medoids algorithm (PAM) Total Cost = 20 10 10 10 9 9 9 8 8 8 Arbitrary choose k object as initial medoids 7 6 5 4 3 2 7 6 5 4 3 2 1 1 0 0 0 1 2 3 4 5 6 7 8 9 0 10 1 2 3 4 5 6 7 8 9 10 Assign each remainin g object to nearest medoids 7 6 5 4 3 2 1 0 0 K=2 Until no change 2 3 4 5 6 7 8 9 10 Randomly select a nonmedoid object,Oramdom Total Cost = 26 Do loop 1 10 10 Compute total cost of swapping 9 9 Swapping O and Oramdom 8 If quality is improved. 5 5 4 4 3 3 2 2 1 1 7 6 0 8 7 6 0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 56 PAM (Partitioning Around Medoids) (1987) PAM (Kaufman and Rousseeuw, 1987), built in Splus Use real object to represent the cluster Select k representative objects arbitrarily For each pair of non-selected object random and selected object j, calculate the total swapping cost TCj,radom For each pair of j and random, If TCj,rand < 0, j is replaced by random J such that min TCj,random Then assign each non-selected object to the most similar representative object repeat steps 2-3 until there is no change 57 four possibilities (1 and 2) p any object, j current medoid random candidate modoid (1): p belongs to j but p close to iij i any other medoid so p is reassigned to i d(p,i)<d(p, random) TCp,j.random d(p,i)-d(p,j)= < 0 (2): p belongs to j but p close to random so p is reassigned to random d(p,i)>d(p, random) TCp,j.random=d(p,random)d(p,j)<0 58 Case 1 P random i j random is candidate to replace medoid j but any object p close to i another medoid i such that d(p,i)<d(p, random) P i random j Seperate p from j assign to i as distance between p and i is smaller then between P and random 59 Case 2 P i random j random is candidate to replace medoid j but any object p is closer to random then any other Medoid i d(p, random)<d(p,i) i P i j random Seperate p from j assign to random as distance between p and i İs greater than between P and medoid i 60 four possibilities (3 and 4) (3): p belongs to i ij but p close to i so p is still belongs to I d(p,i)<d(p, random) TCp,j.random =0 (4): p belongs to i ij but p close to random so p is reassigned to random d(p,i)>d(p, random) TCp,j.random =d(p,random)d(p,i)<0 61 Case 3 P random random is candidate to replace medoid j but any object p is closer to another medoid i i j d(p, random)>d(p,i) P random i reassing p to its previous mediod i j 62 Case 4 P random random is candidate to replace medoid j but any object p is closer to another medoid i i j d(p, random)<d(p,i) P i random assing p to random as it is closer to random than mediod i j 63 Example A B C D E A 0 B 1 0 C 2 2 0 D 2 4 1 0 E 3 3 5 3 0 A and B are initial medoids Initial clusters C1{A C D} C2{B E} Note randomly assign C to C1 and E to C2 Three non medoids C D E are examined whether they replace A or B 64 Example cont. Six cost to determine TCAC, TCAD, TCAE,TCBC, TCBD, TCBE, TCAC= TCAAC + TCBAC + TCCAC + TCDAC + TCEAC TCAAC: A was a medoid it is assigned to either B or C d(A,B)=1 d(A,C)=2 assign to B So TCAAC is 1-0 =1 TCBAC: B is still a medoid So TCBAC =0 TCCAC: C is the new medoid it is assinged to nowhere but it is seperated from A releasing 2 So TCCAC is 0-2=-2 65 Example cont. TCDAC: D was a nonmedoid it was assigned to A First it is released from A -2 It will be assigned to either B or C d(D,B)=1 d(D,C)=4 So assign to B with a cost of 1 So TCDAC is 1-2 =-1 TCEAC: E was a nonmedoid it was assigned to B It will be assigned to either B or C d(E,B)=3 d(E,C)=5 So reassign to B or no change So TCEAC 2 -2 = 0 66 Example cont. So TCAC= TCAAC + TCBAC + TCCAC + TCDAC + TCEAC = 1 + 0 + -2 + -1 + 0 =-2 67 Replacing A with C A 2 C D 2 C A 1 D 1 B 3 E A and B are medoids TC 7 C is a candidate medoid to Replace A B 3 E After replacing A with C B and C are medoids TC = 5 TCAC = 5-7=-2 68 Replacing A with D A 2 C D 2 C A 1 D 1 B 3 E A and B are medoids TC 7 D is a candidate medoid to Replace A B 3 E After replacing A with D B and D are medoids TC = 5 TCAD = 5-7=-2 69 Replacing A with E A 2 C D 2 B 3 E A and B are medoids TC 7 E is a candidate medoid to Replace A A 1 B C 2 D 3 E After replacing A with C B and C are medoids TC = 6 TCAE = 5-6=-1 70 Replacing B with C A 2 C D 2 C A 1 D 1 B 3 E A and B are medoids TC 7 C is a candidate medoid to Replace B B 3 E After replacing B with C A and C are medoids TC = 5 TCBC = 5-7=-2 71 Replacing B with D A 2 C D 2 C A 1 D 1 B 3 E A and B are medoids TC 7 D is a candidate medoid to Replace B B 3 E After replacing B with D A and D are medoids TC = 5 TCBD = 5-7=-2 72 Replacing B with E A 2 C D 2 2 A 1 B 3 E A and B are medoids TC 7 E is a candidate medoid to Replace B B C D 2 E After replacing B with E A and E are medoids TC = 5 TCBE = 5-7=-2 73 TCAC=-2 TCAD=-2 TCAE=-1 TCBC,=-2 TCBD =-2 TCBE=-2 Fife of six possibilities has a reduction of 2 Arbitrarily choose one say AC Replace A with C so B and C are new medoids and C1 {B A E} C2 {C D} Continue the PAM until no reduction in costs 74 What is the problem with PAM? PAM is more robust than k-means in the presence of noise and outliers because a medoid is less influenced by outliers or other extreme values than a mean PAM works efficiently for small data sets but does not scale well for large data sets. O(k(n-k)2 ) for each iteration where n is # of data,k is # of clusters Sampling based method, CLARA(Clustering LARge Applications) 75 CLARA (Clustering Large Applications) (1990) CLARA (Kaufmann and Rousseeuw in 1990) Built in statistical analysis packages, such as S+ It draws multiple samples of the data set, applies PAM on each sample, and gives the best clustering as the output Strength: deals with larger data sets than PAM Weakness: Efficiency depends on the sample size A good clustering based on samples will not necessarily represent a good clustering of the whole data set if the sample is biased 76 CLARANS (“Randomized” CLARA) (1994) CLARANS (A Clustering Algorithm based on Randomized Search) (Ng and Han’94) CLARANS draws sample of neighbors dynamically The clustering process can be presented as searching a graph where every node is a potential solution, that is, a set of k medoids If the local optimum is found, CLARANS starts with new randomly selected node in search for a new local optimum It is more efficient and scalable than both PAM and CLARA Focusing techniques and spatial access structures may further improve its performance (Ester et al.’95) 77 Chapter 6. Cluster Analysis What is Cluster Analysis? Types of Data in Cluster Analysis A Categorization of Major Clustering Methods Partitioning Methods Hierarchical Methods Density-Based Methods Grid-Based Methods Model-Based Clustering Methods Outlier Analysis Summary 78 Hierarchical Clustering Use distance matrix as clustering criteria. This method does not require the number of clusters k as an input, but needs a termination condition Step 0 a Step 1 Step 2 Step 3 Step 4 ab b abcde c cde d de e Step 4 agglomerative (AGNES) Step 3 Step 2 Step 1 Step 0 divisive (DIANA) 79 AGNES (Agglomerative Nesting) Introduced in Kaufmann and Rousseeuw (1990) Implemented in statistical analysis packages, e.g., Splus Use the Single-Link method and the dissimilarity matrix. Merge nodes that have the least dissimilarity Go on in a non-descending fashion Eventually all nodes belong to the same cluster 10 10 10 9 9 9 8 8 8 7 7 7 6 6 6 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1 0 0 0 1 2 3 4 5 6 7 8 9 10 0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 80 A Dendrogram Shows How the Clusters are Merged Hierarchically Decompose data objects into a several levels of nested partitioning (tree of clusters), called a dendrogram. A clustering of the data objects is obtained by cutting the dendrogram at the desired level, then each connected component forms a cluster. 81 Example A A 0 B 1 C 2 D 2 E 3 B C D E 0 2 4 3 0 1 5 0 3 0 82 Single Link Distance Measure E A 2 3 3 1 B 2 A C 4 5 2 1 3 D E {A}{B}{C}{D}{E} A 2 E 1 2 {ABCD}{E} B 2 C D 1 1 B C 1 D {AB}{CD}{E} 1 B A 2 2 3 3 C E 2 1 3 D 3 2 1 {ABCDE} A B C D E 83 Complete Link Distance Measure E A 2 3 3 1 B 2 A C 4 5 2 1 3 D E {A}{B}{C}{D}{E} 3 A 1 3 C E D {ABE}{CD} B 1 1 B C 1 D {AB}{CD}{E} 1 B A 2 2 3 3 5 C E 2 4 1 3 D 5 3 1 {ABCDE} C D A B E 84 Average Link distance measure E A 2 3 3 1 B 2 A C 4 5 2 1 3 D E {A}{B}{C}{D}{E} A 2 E 1 2 B 4 D 2 C 1 {ABCD}{E} Average link Between {AB} and{CD} Is 2.5 1 B C 1 D {AB}{CD}{E} average link {AB} and {CD} is 1 1 B A 2 2 3 3.5 3 5 C E 2 4 1 2.5 3 D {ABCDE} Average link between {ABCD} and {E} A Is 3.5 1 B C D E 85 DIANA (Divisive Analysis) Introduced in Kaufmann and Rousseeuw (1990) Implemented in statistical analysis packages, e.g., Splus Inverse order of AGNES Eventually each node forms a cluster on its own 10 10 10 9 9 9 8 8 8 7 7 7 6 6 6 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1 0 0 0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 86 More on Hierarchical Clustering Methods Major weakness of agglomerative clustering methods 2 do not scale well: time complexity of at least O(n ), where n is the number of total objects can never undo what was done previously Integration of hierarchical with distance-based clustering BIRCH (1996): uses CF-tree and incrementally adjusts the quality of sub-clusters CURE (1998): selects well-scattered points from the cluster and then shrinks them towards the center of the cluster by a specified fraction CHAMELEON (1999): hierarchical clustering using dynamic modeling 87 BIRCH (1996) Birch: Balanced Iterative Reducing and Clustering using Hierarchies, by Zhang, Ramakrishnan, Livny (SIGMOD’96) Incrementally construct a CF (Clustering Feature) tree, a hierarchical data structure for multiphase clustering Phase 1: scan DB to build an initial in-memory CF tree (a multi-level compression of the data that tries to preserve the inherent clustering structure of the data) Phase 2: use an arbitrary clustering algorithm to cluster the leaf nodes of the CF-tree Scales linearly: finds a good clustering with a single scan Weakness: handles only numeric data, and sensitive to the and improves the quality with a few additional scans order of the data record. 88 Clustering Feature Vector Clustering Feature: CF = (N, LS, SS) N: Number of data points LS: Ni=1=Xi SS: Ni=1=Xi2 CF = (5, (16,30),(54,190)) 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 (3,4) (2,6) (4,5) (4,7) (3,8) 89 CF-Tree in BIRCH Clustering feature: summary of the statistics for a given subcluster: the 0-th, 1st and 2nd moments of the subcluster from the statistical point of view. registers crucial measurements for computing cluster and utilizes storage efficiently A CF tree is a height-balanced tree that stores the clustering features for a hierarchical clustering A nonleaf node in a tree has descendants or “children” The nonleaf nodes store sums of the CFs of their children A CF tree has two parameters Branching factor: specify the maximum number of children. threshold: max diameter of sub-clusters stored at the leaf nodes 90 CF Tree Root B=7 CF1 CF2 CF3 CF6 L=6 child1 child2 child3 child6 Non-leaf node CF1 CF2 CF3 CF5 child1 child2 child3 child5 Leaf node prev CF1 CF2 CF6 next Leaf node prev CF1 CF2 CF4 next 91 Contributions of BIRCH BIRCH is local in that each clustering decision is made without scanning all data points. BIRCH exploits the observation that the data space is usually not uniformly occupied, and hence not every data point is equally important for clustering purposes. BIRCH makes full use of available memory to derive the finest possible subclusters ( to ensure accuracy) while minimizing I/O costs ( to ensure efficiency). 92 Background knowledge { x Given N d-dimensional data points in a cluster: i }, where i = 1,2,…,N, the centroid, radius and diameter is defined as: x0 N x i 1 i N 2 i 1 ( xi x0 ) N R( N 2 ( x i 1 j 1 i x j ) N D( We treat x0 ) 1 2 N N ( N 1) ) 1 2 , R and D as properties of a single cluster. 93 Background knowledge (Cont.) Next between two clusters, we define 5 alternative distances for measuring their closeness. Given the centroids of two clusters: , the x0 and x0 two alternative distances D0 and D1 are defined as: 1 2 2 12 D0 (( x01 x02 ) ) d (i ) (i ) D1 | x01 x02 | | x01 x02 | i 1 94 Background knowledge (Cont.) – Given N1 d-dimensional data points in a cluster:{xi } where i = 1,2,…N1, and N2 data points in another cluster: { x j } where j = N1+1,N1+2,…,N1+N2, the average inter-cluster distance D2, average intra-cluster distance D3 and variance increase distance D4 of the two clusters are defined as: 2 ( x i 1 j N 1 i x j ) N1 D2 ( N1 N 2 1 N1 N 2 D3 ( N1 N 2 i 1 N1 N 2 j 1 ) 1 2 ( xi x j ) 2 1 2 ) ( N1 N 2 )( N1 N 2 1) N1 N 2 N1 N 2 N1 N1 N 2 N N N x 1 1 2 l 1 xl 2 l 1 xl 2 l N1 1 l 2 D 4 ( xk ) ( xi ) (x j ) N N N N k 1 i 1 j N1 1 1 2 1 2 Actually, D3 is D of the merged cluster. D0,D1,D2,D3,D4 is some measurement of two clusters. We can use them to determine if two clusters are close. 95 Clustering Feature A Clustering Feature is a tirple summarizing the information that we maintain about a cluster. Definition. Given N d-dimensional data points in a { x cluster: i } where i = 1,2,…N, the Clustering Feature (CF) vector of the cluster is defined as a triple: , where N is the number of data points in the cluster, CF ( sum N , LS , SS is the linear of) the N data points, i.e., and x SS isthe square sum of LSthe N data points, i.e., N i 1 i 2 i1 xi N 96 Clustering Feature (Cont.) CF Additive Theorem: Assume that CF1=(N1L , S ,SS1), and CF2 =(N2, LS ,SS2) are the CF vectors of two disjoint clusters. Then the CF vector of the cluster that is formed by merging the two disjoint clusters, is: CF1 CF2 ( N1 N 2 , LS1 LS 2 , SS1 SS 2 ) From the CF definition and additive theorem, we know that the corresponding X0, R,D,D0,D1,D2,D3 and D4,can all be calculated easily. 97 CF Tree A CF tree is a height-balanced tree with two parameters: branching factor B. Each nonleaf node contains at most B entries of the form[cfi,childi], where i = 1,2,…,B, childi is a pointer to its i-th child node, and CFi is the CF of the sub-cluster represented by this child. A leaf node contains at most L entries, each of the form [CFi], where i = 1,2,…L. Initial threshold T. Which is an initial threshold for the radius ( or diameter ) of any cluster. The value of T is an upper bound on the radius of any cluster and controls the number of clusters that the algorithm discovers. 98 CF Tree (Cont.) CF Tree will be built dynamically as new data objects are inserted. It is used to guide a new insertion into the correct subcluster for clustering purpose. Actually, all clusters are formed as each data point is inserted into the CF Tree. CF Tree is a very compact representation of the dataset because each entry in a leaf node is not a single data point but a subcluster. 99 Insertion into a CF Tree 1. 2. we now present the algorithm for inserting an entry into a CF tree.Given entry “Ent”, it proceeds as below: Identifying the appropriate leaf: Starting from the root, according to a chosen distance metric D0 to D4 as defined before, it recursively descends the CF tree by choosing the closest child node . Modifying the leaf: When it reaches a leaf node, it finds the closest leaf entry, and tests whether the node can absorb it without violating the threshold condition. If so, the CF vector for the node is updated to reflect this. If not, a new entry for it is added to the leaf. If there is space on the leaf for this new entry, we are done. Otherwise we must split the leaf node. Node splitting is done by choosing the farthest pair of entries based on the closest criteria. 100 Insertion into a CF Tree 3. (Cont.) Modifying the path to the leaf: After inserting “Ent” into a leaf, we must update the CF information for each nonleaf entry on the path to the leaf. In the absence of a split, this simply involves adding CF vectors to reflect the additions of “Ent”. A leaf split requires us to insert a new nonleaf entry into the parent node, to describe the newly created leaf. If the parent has space for this entry, at all higher levels, we only need to update the CF vectors to reflect the addition of “Ent”. Otherwise, we may have to split the parent as well, and so on up to the root. 101 Insertion into a CF Tree 4. (Cont.) Merging Refinement: In the presence of skewed data input order, split can affect the clustering quality, and also reduce space utilization. A simple additional merging often helps ameliorate these problems: suppose the propagation of one split stops at some nonleaf node Nj, i.e., Nj can accommodate the additional entry resulting from the split. Scan Nj to find the two closest entries. 2. If they are not the pair corresponding to the split, merge them. 3. If there are more entries than one page can hold, split it again. 4. During the resplitting, one of the seeds attracts enough merged entries, the other receives the rest entries. In summary, it improve the distribution of entries in the closest two children, even free a node space for later use. 1. 102 Insertion into a CF Tree (Cont.) Notes: Some problems which will be resolved in the later phases of the entire algorithm. Depending upon the order of data input and the degree of skew, two subclusters that should not be in one cluster are kept in the same node. It will be remedied with a global algorithm that arranges leaf entries across nodes.( Phase 3) If the same data point is inserted twice, but at two different times, the two copies might be entered into distinct leaf entries. It will be addressed with further refinement passes over the data.(Phase 4) 103 Figure: Effect of Split, Merge and Resplit CF1 CF3 CF4 CF2 Split CF1 CF3 CF5 CF1 CF3 CF5 CF6 CF7 CF8 CF4 CF2 CF4 CF2 Split CF1 CF3 CF5 CF6 CF7 CF8 CF4 CF2 CF9 Merge CF1 CF3 CF5 CF6 CF7 CF8 CF4 CF2 CF9 Resplit CF1 CF3 CF5 CF6 CF7 CF8 CF4 CF2 CF9 104 BIRCH Clustering Algorithm - Overview Data Phase1: Load into memory by building a CF tree Initial CF tree Phase2(Optional): Condense into desirable range by building a smaller CF tree Smaller CF tree Phase3: Global clustering Good cluster Phase4(optional and off line): Clustering refinement Better Cluster Click Here 105 BIRCH Clustering Algorithm – Phase 2,3 & 4 Phase 2: Condense into desirable range by building a smaller CF tree Phase 3: Global clustering To meet the need of the input size range of Phase 3 To solve the problem 1 Approach: re-cluster all subclusters by using the existing global or semi-global clustering algorithms Phase 4: Global clustering To solve the problem 2 Approach: use the centroids of the cluster produced by phase 3 as seeds, and redistributes the data points to its closest seed to obtain a set of new clusters. This can use existing algorithm. 106 BIRCH Clustering Algorithm – Control flow of Phase1 Start CF tree t1 of Initial T Continue scanning data and insert to t1 Out of memory (1) (2) (3) Result? Finish scanning data Increase T. Rebuild CF tree t2 of new T from CF tree t1: if a leaf entry of t1 is potential outlier and disk space available, write to disk; otherwise use it to rebuild t2 T1< - t2 Otherwise Result? Out of disk space Re-absorb potential outliers in to t1 Re-absorb potential outliers in to t1 107 CURE (Clustering Using REpresentatives ) CURE: proposed by Guha, Rastogi & Shim, 1998 Stops the creation of a cluster hierarchy if a level consists of k clusters Uses multiple representative points to evaluate the distance between clusters, adjusts well to arbitrary shaped clusters and avoids single-link effect 108 Drawbacks of Distance-Based Method Drawbacks of square-error based clustering method Consider only one point as representative of a cluster Good only for convex shaped, similar size and density, and if k can be reasonably estimated 109 Cure: The Algorithm Draw random sample s. Partition sample to p partitions with size s/p Partially cluster partitions into s/pq clusters Eliminate outliers By random sampling If a cluster grows too slow, eliminate it. Cluster partial clusters. Label data in disk 110 Data Partitioning and Clustering s = 50 p=2 s/p = 25 s/pq = 5 y y y x y y x x x x 111 Cure: Shrinking Representative Points y y x x Shrink the multiple representative points towards the gravity center by a fraction of . Multiple representatives capture the shape of the cluster 112 Clustering Categorical Data: The ROCK Algorithm ROCK: RObust Clustering using linKs S. Guha, R. Rastogi & K. Shim, ICDE’99 Major ideas Use links to measure similarity/proximity Not distance-based 2 O ( n nmmma Computational complexity: Algorithm: sampling-based clustering Draw random sample Cluster with links Label data in disk Experiments Congressional voting, mushroom data n2 log n) 113 Similarity Measure in ROCK Traditional measures for categorical data may not work well, e.g., Jaccard coefficient Example: Two groups (clusters) of transactions C1. <a, b, c, d, e>: {a, b, c}, {a, b, d}, {a, b, e}, {a, c, d}, {a, c, e}, {a, d, e}, {b, c, d}, {b, c, e}, {b, d, e}, {c, d, e} C2. <a, b, f, g>: {a, b, f}, {a, b, g}, {a, f, g}, {b, f, g} Jaccard co-efficient may lead to wrong clustering result C1: 0.2 ({a, b, c}, {b, d, e}} to 0.5 ({a, b, c}, {a, b, d}) C1 & C2: could be as high as 0.5 ({a, b, c}, {a, b, f}) New Similarity function: Sim(T , T ) T1 T2 1 2 T1 T2 Ex. Let T1 = {a, b, c}, T2 = {c, d, e} Sim (T 1, T 2) {c} {a, b, c, d , e} 1 0.2 5 114 Link Measure in ROCK Links: # of common neighbors given a treshold value C1 <a, b, c, d, e>: {a, b, c}, {a, b, d}, {a, b, e}, {a, c, d}, {a, c, e}, {a, d, e}, {b, c, d}, {b, c, e}, {b, d, e}, {c, d, e} C2 <a, b, f, g>: {a, b, f}, {a, b, g}, {a, f, g}, {b, f, g} Let T1 = {a, b, c}, T2 = {c, d, e}, T3 = {a, b, f} link(T1, T2) = 4, since they have 4 common neighbors link(T1, T3) = 3, since they have 3 common neighbors {a, c, d}, {a, c, e}, {b, c, d}, {b, c, e} {a, b, d}, {a, b, e}, {a, b, g} Thus link is a better measure than Jaccard coefficient 115 Goodness Measure to Join Clusters Goodness measure between two clusters i and j g(Ci,Cj) = _______links(Ci,Cj)_______ (ni+nj)1+2f()-ni1+2f()-nj1+2f() Links(Ci,Cj): number of links betweeen two clusters ni nj number of object in each cluster ni1+2f() is number of expected links in cluster i f () function of shape of the cluster 116 CHAMELEON (Hierarchical clustering using dynamic modeling) CHAMELEON: by G. Karypis, E.H. Han, and V. Kumar’99 Measures the similarity based on a dynamic model Two clusters are merged only if the interconnectivity and closeness (proximity) between two clusters are high relative to the internal interconnectivity of the clusters and closeness of items within the clusters Cure ignores information about interconnectivity of the objects, Rock ignores information about the closeness of two clusters A two-phase algorithm 1. Use a graph partitioning algorithm: cluster objects into a large number of relatively small sub-clusters 2. Use an agglomerative hierarchical clustering algorithm: find the genuine clusters by repeatedly combining these sub-clusters 117 Overall Framework of CHAMELEON Construct Partition the Graph Sparse Graph Data Set Merge Partition Final Clusters 118 Chapter 6. Cluster Analysis What is Cluster Analysis? Types of Data in Cluster Analysis A Categorization of Major Clustering Methods Partitioning Methods Hierarchical Methods Density-Based Methods Grid-Based Methods Model-Based Clustering Methods Outlier Analysis Summary 119 Density-Based Clustering Methods Clustering based on density (local cluster criterion), such as density-connected points Major features: Discover clusters of arbitrary shape Handle noise One scan Need density parameters as termination condition Several interesting studies: DBSCAN: Ester, et al. (KDD’96) OPTICS: Ankerst, et al (SIGMOD’99). DENCLUE: Hinneburg & D. Keim (KDD’98) CLIQUE: Agrawal, et al. (SIGMOD’98) 120 Density Concepts Core object (CO)–object with at least ‘M’ objects within a radius ‘E-neighborhood’ Directly density reachable (DDR)–x is CO, y is in x’s ‘Eneighborhood’ Density reachable–there exists a chain of DDR objects from x to y Density based cluster–density connected objects maximum w.r.t. reachability 121 Density-Based Clustering: Background Two parameters: Eps: Maximum radius of the neighbourhood MinPts: Minimum number of points in an Epsneighbourhood of that point NEps(p): {q belongs to D | dist(p,q) <= Eps} Directly density-reachable: A point p is directly densityreachable from a point q wrt. Eps, MinPts if 1) p belongs to NEps(q) 2) core point condition: |NEps (q)| >= MinPts p q MinPts = 5 Eps = 1 cm 122 Density-Based Clustering: Background (II) Density-reachable: p A point p is density-reachable from a point q wrt. Eps, MinPts if there is a chain of points p1, …, pn, p1 = q, pn = p such that pi+1 is directly density-reachable from pi p1 q Density-connected A point p is density-connected to a point q wrt. Eps, MinPts if there is a point o such that both, p and q are density-reachable from o wrt. Eps and MinPts. p q o 123 Let MinPts =3 * * Q *M * * P ** ** * * *M,P are core objects since each is in an neighborhood containing at least 3 points *Q is directly density-reachable from M M is directly density-reachable form P and vice versa *Q is indirectly density reachable form P since Q is directly density-reachable from M and M is directly density reacable From P but p is not density reacable from Q sicne Q is not a core object 124 DBSCAN: Density Based Spatial Clustering of Applications with Noise Relies on a density-based notion of cluster: A cluster is defined as a maximal set of density-connected points Discovers clusters of arbitrary shape in spatial databases with noise Outlier Border Eps = 1cm Core MinPts = 5 125 DBSCAN: The Algorithm Arbitrary select a point p Retrieve all points density-reachable from p wrt Eps and MinPts. If p is a core point, a cluster is formed. If p is a border point, no points are density-reachable from p and DBSCAN visits the next point of the database. Continue the process until all of the points have been processed. 126 OPTICS: A Cluster-Ordering Method (1999) OPTICS: Ordering Points To Identify the Clustering Structure Ankerst, Breunig, Kriegel, and Sander (SIGMOD’99) Produces a special order of the database wrt its density-based clustering structure This cluster-ordering contains info equiv to the density-based clusterings corresponding to a broad range of parameter settings Good for both automatic and interactive cluster analysis, including finding intrinsic clustering structure Can be represented graphically or using visualization techniques 127 OPTICS: Some Extension from DBSCAN Index-based: k = number of dimensions N = 20 p = 75% M = N(1-p) = 5 D Complexity: O(kN2) Core Distance p1 Reachability Distance o p2 Max (core-distance (o), d (o, p)) r(p1, o) = 2.8cm. r(p2,o) = 4cm o MinPts = 5 = 3 cm 128 Reachability -distance undefined ‘ Cluster-order of the objects 129 Density-Based Cluster analysis: OPTICS & Its Applications 130 DENCLUE: Using density functions DENsity-based CLUstEring by Hinneburg & Keim (KDD’98) Major features Solid mathematical foundation Good for data sets with large amounts of noise Allows a compact mathematical description of arbitrarily shaped clusters in high-dimensional data sets Significant faster than existing algorithm (faster than DBSCAN by a factor of up to 45) But needs a large number of parameters 131 Denclue: Technical Essence Uses grid cells but only keeps information about grid cells that do actually contain data points and manages these cells in a tree-based access structure. Influence function: describes the impact of a data point within its neighborhood. Overall density of the data space can be calculated as the sum of the influence function of all data points. Clusters can be determined mathematically by identifying density attractors. Density attractors are local maximal of the overall density function. 132 Gradient: The steepness of a slope Example f Gaussian ( x , y ) e f D Gaussian f d ( x , y )2 2 2 ( x ) i 1 e N d ( x , xi ) 2 2 2 ( x, xi ) i 1 ( xi x) e D Gaussian N d ( x , xi ) 2 2 2 133 Density Attractor 134 Center-Defined and Arbitrary 135 Chapter 6. Cluster Analysis What is Cluster Analysis? Types of Data in Cluster Analysis A Categorization of Major Clustering Methods Partitioning Methods Hierarchical Methods Density-Based Methods Grid-Based Methods Model-Based Clustering Methods Outlier Analysis Summary 136 Grid-Based Clustering Method Using multi-resolution grid data structure Several interesting methods STING (a STatistical INformation Grid approach) by Wang, Yang and Muntz (1997) WaveCluster by Sheikholeslami, Chatterjee, and Zhang (VLDB’98) A multi-resolution clustering approach using wavelet method CLIQUE: Agrawal, et al. (SIGMOD’98) 137 STING: A Statistical Information Grid Approach Wang, Yang and Muntz (VLDB’97) The spatial area area is divided into rectangular cells There are several levels of cells corresponding to different levels of resolution 138 STING: A Statistical Information Grid Approach (2) Each cell at a high level is partitioned into a number of smaller cells in the next lower level Statistical info of each cell is calculated and stored beforehand and is used to answer queries Parameters of higher level cells can be easily calculated from parameters of lower level cell count, mean, s, min, max type of distribution—normal, uniform, etc. Use a top-down approach to answer spatial data queries Start from a pre-selected layer—typically with a small number of cells For each cell in the current level compute the confidence interval 139 STING: A Statistical Information Grid Approach (3) Remove the irrelevant cells from further consideration When finish examining the current layer, proceed to the next lower level Repeat this process until the bottom layer is reached Advantages: Query-independent, easy to parallelize, incremental update O(K), where K is the number of grid cells at the lowest level Disadvantages: All the cluster boundaries are either horizontal or vertical, and no diagonal boundary is detected 140 WaveCluster (1998) Sheikholeslami, Chatterjee, and Zhang (VLDB’98) A multi-resolution clustering approach which applies wavelet transform to the feature space A wavelet transform is a signal processing technique that decomposes a signal into different frequency subband. Both grid-based and density-based Input parameters: # of grid cells for each dimension the wavelet, and the # of applications of wavelet transform. 141 WaveCluster (1998) How to apply wavelet transform to find clusters Summaries the data by imposing a multidimensional grid structure onto data space These multidimensional spatial data objects are represented in a n-dimensional feature space Apply wavelet transform on feature space to find the dense regions in the feature space Apply wavelet transform multiple times which result in clusters at different scales from fine to coarse 143 Wavelet Transform Decomposes a signal into different frequency subbands (can be applied to n-dimensional signals) Data are transformed to preserve relative distance between objects at different levels of resolution Allows natural clusters to become more distinguishable 144 What Is Wavelet (2)? 145 Quantization 146 Transformation 147 WaveCluster (1998) Why is wavelet transformation useful for clustering Unsupervised clustering It uses hat-shape filters to emphasize region where points cluster, but simultaneously to suppress weaker information in their boundary Effective removal of outliers Multi-resolution Cost efficiency Major features: Complexity O(N) Detect arbitrary shaped clusters at different scales Not sensitive to noise, not sensitive to input order Only applicable to low dimensional data 148 CLIQUE (Clustering In QUEst) Agrawal, Gehrke, Gunopulos, Raghavan (SIGMOD’98). Automatically identifying subspaces of a high dimensional data space that allow better clustering than original space CLIQUE can be considered as both density-based and gridbased It partitions each dimension into the same number of equal length interval It partitions an m-dimensional data space into nonoverlapping rectangular units A unit is dense if the fraction of total data points contained in the unit exceeds the input model parameter A cluster is a maximal set of connected dense units within a subspace 149 CLIQUE: The Major Steps Partition the data space and find the number of points that lie inside each cell of the partition. Identify the subspaces that contain clusters using the Apriori principle Identify clusters: Determine dense units in all subspaces of interests Determine connected dense units in all subspaces of interests. Generate minimal description for the clusters Determine maximal regions that cover a cluster of connected dense units for each cluster Determination of minimal cover for each cluster 150 =3 30 40 Vacation 20 50 Salary (10,000) 0 1 2 3 4 5 6 7 30 Vacation (week) 0 1 2 3 4 5 6 7 age 60 20 30 40 50 age 60 50 age 151 Strength and Weakness of CLIQUE Strength It automatically finds subspaces of the highest dimensionality such that high density clusters exist in those subspaces It is insensitive to the order of records in input and does not presume some canonical data distribution It scales linearly with the size of input and has good scalability as the number of dimensions in the data increases Weakness The accuracy of the clustering result may be degraded at the expense of simplicity of the method 152 Chapter 6. Cluster Analysis What is Cluster Analysis? Types of Data in Cluster Analysis A Categorization of Major Clustering Methods Partitioning Methods Hierarchical Methods Density-Based Methods Grid-Based Methods Model-Based Clustering Methods Outlier Analysis Summary 153 Model-Based Clustering Methods Attempt to optimize the fit between the data and some mathematical model Statistical and AI approach Conceptual clustering A form of clustering in machine learning Produces a classification scheme for a set of unlabeled objects Finds characteristic description for each concept (class) COBWEB (Fisher’87) A popular a simple method of incremental conceptual learning Creates a hierarchical clustering in the form of a classification tree Each node refers to a concept and contains a probabilistic description of that concept 154 COBWEB Incremental learning to form hierarchy of concepts Tree structure Root node represents highest level of concept generalization Summary information for all instances Each object is presented to the existing concept hierarchy in a sequential manner In the beginning the tree consists of the root along Objects are added one by one The tree is updated at each stage 155 Create a cluster with the first instance as its only member For each new instance Place the new instance into an existing cluster Create a new concept cluster having the new instance as its only member At each level of the hierarchy an evaluation function is used (category utility) To include the object in an existing cluster To create a new cluster the object being its only member 156 Category Utility Attribute-value format categorical values Ai i =1..n attributes Vij j th value of attribute i Three probabilities: Conditional probability of attribute Ai having value Vij given in class Ck P(Ai=Vij|Ck) Note that if P(Ai = Vij|Ck) = 1 Each instance of class Ck will always have Vij as the value for attribute Ai 157 Second probability: P(Ck|Ai = Vij) conditional probability that An object is in class Ck given that attribute Ai has value Vij If P(Ck|Ai = Vij) = 1 if Ai has vaue Vij, The class must be Ck Attribute Ai having vaue vij is sufficient condition for defining class Ck 158 Partition quality KijP(Ai=Vij) P(Ck|Ai = Vij) P(Ai = Vij|Ck) Category utility is defined as Kk=1 P(Ck) ij P(Ai = Vij|Ck)2- ij P(Ai = Vij)2 K First numerator term is partition-quality expression by applying Bayes rule Second term is probability of correctly guessing attribute values without any category knowledge Division by K total number of classes: per cluster utility 159 Category utility is to be expressed as Kk=1 P(Ck)[ij P(Ai = Vij|Ck)2- ij P(Ai = Vij)2] K ij P(Ai = Vij|Ck)2- ij P(Ai = Vij)2 The cluster give some advantage in predicting the values of attributes of objects in that cluster In double summation: Improvement in information in predicting the values of attributes by knowing their clusters 160 Example Object 1 2 3 4 5 6 7 Tails one two two one one one one color light light dark dark light light light nuclei one two two three two two three 161 P(N)=7/7 N1 P(N)=3/7 P(V/C) P(C/V) Tails Öne Two .71 .29 1.0 1.0 Color Light Dark .71 .29 1.0 1.0 Nuclei Öne Two Three .14 .57 .29 1.0 1.0 1.0 N4 N2 P(V/C) P(C/V) P(N)=2/7 P(V/C) P(C/V) P(N)=2/7 P(V/C) P(C/V) Tails Öne Two 1.0 0.0 0.6 0.0 Tails Öne Two 0.0 1.0 0.0 1.0 Tails Öne Two 1.0 0.0 0.4 0.0 Color Light Dark 1.0 0.0 0.6 0.0 Color Light Dark 0.5 0.5 0.2 0.5 Color Light Dark 0.5 0.5 0.2 0.5 Nuclei Öne Two Three 0.33 0.67 0.0 1.0 0.5 0.0 Nuclei Öne Two Three 0.0 1.0 0.0 0.0 0.5 0.0 Nuclei Öne Two Three 0.0 0.0 1.0 0.0 0.0 1.0 P(V/C) P(C/V) N5 N3 P(N)=1/3 P(V/C) P(C/V) P(N)=2/3 Tails Öne Two 1.0 0.0 0.33 0.0 Tails Öne Two 1.0 0.0 0.67 0.0 Color Light Dark 1.0 0.0 0.33 0.0 Color Light Dark 1.0 0.0 0.67 0.0 Nuclei Öne Two Three 1.0 0.0 0.0 1.0 0.0 0.0 Nuclei Öne Two Three 0.0 1.0 0.0 0.0 1.0 0.0 162 A new object to be placed in the hierarchy Enters at the root node N N` statistics are updated As the object enters to the second level of the hierarchy one of four actions are chosen (1)If the new object is similar enough to one of N1 N2 N4 Incorporated with the preferred node Proceeds to the second level of the hierarchy (2) the new object is unique enough to merit the creation of a new first level concept node The object becomes a first level concept node Classification process terminates 163 (3) merging the two-best scoring nodes into a single nodes (4) removes the best-scoring node from the hierarchy If (3) or (4) is processed The new object is once again presented for classification to the modified hierarchy This continues at each level of the concept tree until the new object becomes a terminal node 164 Example 2 Weather data without the play attribute At the beginning when new objects are presented in the the structure they each form their own subcluster under the root 165 humıdıty windy play hot high false no sunny hot high true no c overcast hot high false yes d rainy mild high false yes e rainy cool normal false yes f rainy cool normal true no g overcast cool normal true yes h sunny mild high false no i sunny cool normal false yes j rainy mild normal false yes k sunny mild normal true yes l overcast mild high true yes m overcast hot normal false yes n rainy mild high true no ID outlook a sunny b temperature 166 abcde a:no a:no b:no c:yes d:yes e:yes For each of the first five objects It is better in terms of category utility to form A new leaf for each object 167 abcdef a:no b:no c:yes d:yes ef e:yes f:no It finally becomes beneficial to form a cluster, joining the new Object f with the old host e e and f are very similar differing only in windy attribute 168 abcdefg a:no b:no c:yes d:yes e:yes efg f:no g:yes g is placed in the same cluster It differs from e only in outlook When g is introduced First which of the five children of the root makes the best host it turns out to be the rightmost 169 abcdefg a:no b:no c:yes d:yes e:yes efg f:no g:yes Then in the second level root is {ed} Its two children are evaluated to see which will be the better host It it best in terms of category utility that to add g as a subcluster of its own 170 abcdefgh adh a:no d:yes b:no h:no c:yes e:yes efg f:no g:yes When h is introduced 171 Nodes a and d are merged into the single cluster before the new object h is added One Method Consider all pairs of nodes for merging and evaluate the category utility of each pair Computationally expensive COBWEB: Best host for h that is a Second best host for h that is d Merge a and d to form a node Test the category utility for placing h into ad as well 172 Splitting Operation Converse of merging Whenever the best host is identified and merging has not proven beneficial, Consider splitting the host node Take a node and replace it with its children 173 abcdefg a:no adh b:no d:yes h:no c:yes e:yes f:no g:yes Splitting example 174 COBWEB Clustering Method A classification tree 175 More on Statistical-Based Clustering Limitations of COBWEB The assumption that the attributes are independent of each other is often too strong because correlation may exist Not suitable for clustering large database data – skewed tree and expensive probability distributions CLASSIT an extension of COBWEB for incremental clustering of continuous data suffers similar problems as COBWEB AutoClass (Cheeseman and Stutz, 1996) Uses Bayesian statistical analysis to estimate the number of clusters Popular in industry 176 Shortcomings of Conceptual Clustering Systems Object ordering can have a marked impacted on the results of the clustering merge and split operations to overcome this problem Not useful in all cases 177 Probability Based Clustering A mixture is a set of n probability distributions where each distribution represents a cluster The mixture model assigns each object a probability of being a member of cluster k Rather than assigning the object or not assigning the object into cluster k as in K-mean, k-medoids… 178 Simplest Case K = 2 two clusters There is one real valued attribute X Each having Normally distributed Five parameters to determine: mean and standard deviation for cluster A A A mean and standard deviation for cluster B B B Sampling probability for cluster A pA As P(A) + P(B) = 1 179 There are two populations each normally distributed with mean and standard deviation An object comes from one of them but which one is unknown Assign a probability that an object comes from A or B That is equivalent to assign a cluster for each object 180 In parametrıc statıstıcs Assume that objects come from a distribution usually Normal distribution with unknown parameters and •Estimate the parameters from data •Test hypothesis based on the normality assumption •Test the normality assumption 181 Two normal dirtributions Bi modal Arbitary shapes can be captured by mixture of normals 182 Unbiased Estimators of parmeters If you know which distribution samples come from estimating the parameters mean and standard deviation is easy X_bar = x1+x2+...+xn n s2_bar = (x1- X-b)2+(x2- X-b)2+…+(xn- X-b)2 n-1 Unbiased estimator of variance X_bar and s2_bar are unbiased estimators of the unknown parameters and 2 E(x_bar) = and E s2_bar)= 2 183 Maximum Likelihood estimators Overall likelihood that the data came from dataset, given the values for the two parameters And assuming a probability density function Multiplying the probabilities of the individual objects i (assuming they are independent) L =iP(Xi|,) P(Xi| ,)=f(Xi;A,A) Likelihood is a measure of the likelihood of sample points x1,x2…xn 184 Maximum Likelihood estimators For continuous variables X f(X;,) is not a probability but probability density so likelihood function does not lay between 0 and 1 In practice its logarithm is calculated logL = iLog(P(Xi| ,)) 185 ML for discrete variables Tossing a coin PH probability of head 1- PH probability of tail if PH is unknown how do you estimate PH from sample observations of heads and tails Three tosses HHT Likelihood of that is PH PH PT= PH PH(1-PH) Estimate the probability of head by maximizing the likelihood of that particular sample In this case likelihood is interpreted as probability of observing that sample 186 Maximum Likelihood estimators Observing the sample Estimate the values of the unknown parameters Assuming that the observations are coming from a normal distribution Take derivatives of the likelihood function with respect to equate to zero dL/d = 0 dL/d = 0 or dlogL/d =0, dlogL/d = 0 187 Maximum Likelihood estimators X_ML = x1+x2+...+xn n s2_ML = (x1- X-b)2+(x2- X-b)2+…+(xn- X-b)2 n Notice that ML estimate of is the same as its point estimate ML estimate of is biased Then data comes from a normal distribution with Mean X_ML and variance s2_ML 188 Returning to Mixtures Given the five parameters Finding the probabilities that a given object comes from each distribution or cluster is easy Given an object X, The probability that it belongs to cluster A P(A|X) = P(A,X) = P(X|A)*P(A) = f(X;A,A)P(A) P(X) P(X) P(X) P(B|X) = P(B,X) = P(X|B)*P(B) = f(X;B,B)P(B) P(X) P(X) P(X) 189 f(X;A,A) is the normal distribution function for cluster A: 1 f ( X ; , ) e 2 ( X )2 2 The denominator P(x) will disappear; Calculate the numerators for both P(A|X) and P(B|X) And normalize them by dividing their sum 190 P(A|X) or P(B|X) is the conditional probability that the object X whose data values are known is in cluster A or B Assume a density for each cluster Say a normal distribution so P(A/x) is the normal distrubution for cluster A Whose parameters its mean and standard deviation is unknown to be estimated Similarly for P(B/X) 191 P(A) or P(B) is the unconditional probability that ta randomly selected object is in cluster A or B without knowing its data point its value Here P(A) + P(B) = 1 So only one parameter is unknown 192 P(A|X) = _ P(X|A)*P(A) _ P(X|A)*P(A)+ P(X|B)*P(B) P(B|X) = _ P(X|B)*P(B) _ P(X|A)*P(A)+ P(X|B)*P(B) Note that the final outcome P(A|X) and P(B|X) gives the probabilities that given object X belongs to cluster A and B Note that these can be calculated only by knowing the priori probabilities P(A) and P(B) 193 EM Algorithm for the Simple Case We do not know neither of these things Distribution that each object came from Nor the five parameters of the mixture model P(X/A), P(X/B) A, B,A,B,P(A) Adapt the procedure used for k-means algorithm 194 EM Algorithm for the Simple Case Initially guess initial values for the five parameters A, B,A,B,P(A) Until a specified termination criterion Compute P(A/Xi) and P(B/Xi) for each Xi i =1..n Use these probabilities to reestimate the parameters A, B,A,B,P(A) Class probabilities for each object As these are normal distributions with The expectation step Maximization of the likelihood of the distributions Until likelihood converges or does not improve so much 195 Estimation of new means If wi is the probability that object i belongs to cluster A WiA = P(A|Xi) calculated in E step A = w1Ax1+ w2Ax2+ …+ wnAxn w1A+ w2A+…+ wnA WiB = P(B|Xi) calculated in E step B = w1Bx1+ w2Bx2+ …+ wnBxn w1B+ w2B+…+ wnB 196 Estimation of new standard deviations 2A=w1A(X1- A)2+w2A(X2- A)2+…+wnA(Xn- A)2 w1A+ w2A+…+ wnA 2B=w1B(X1- B)2+w2B(X2- B)2+…+wnB(Xn- B)2 w1B+ w2B+…+ wnB Xi are all the objects not just belonging to A or B These are maximum likelihood estimators for variance If the weights are equal denominator sum up to n rather then n-1 197 Estimation of new priori probabilities P(A) = P(A|X1)+ P(A|X2)+… +P(A|Xn) n pA = w1A+ w2A+…+ wnA n P(B) = P(B|X1)+ P(B|X2)+… +P(B|Xn) n pB = w1B+w2B+…+ wnB n Or P(B) = 1-P(A) 198 How to terminate the iteratin K-means stops when the classes of the objects do not change from one iteration to the next Overall likelihood that the data came from this dataset, given the values for the five parameters Multiplying the probabilities of the individual objects I L =i(pAP(Xi|A)+pBP(Xi|B)) The probabilities given the cluster A and B are determined from the normal distribution function P(Xi|A)=f(Xi;A,A) Likelihood is a measure of the goodness of clustering and increases at each iteration of the EM algorithm 199 For continuous variables X f(X;A,A) is not a probability but probability density so likelihood function does not lay between 0 and 1 In practice its logarithm is calculated Since Y =logX is a monotonically increasing function logL = ilog(pAP(Xi|A)+pBP(Xi|B)) Iterate until the increase in likelihood or log-likelihood becomes negligible Maximization of L is equivalent to maximization of log of L 200 General Considerations EM converges to a local optimum maximum likelihood score May not be global EM may be run with different initial values of the parameters and the clustering structure with the highest likelihood is chosen Or an initial k-means is conducted to get the mean and standard deviation parameters 201 Other Model-Based Clustering Methods Neural network approaches Represent each cluster as an exemplar, acting as a “prototype” of the cluster New objects are distributed to the cluster whose exemplar is the most similar according to some distance measure Competitive learning Involves a hierarchical architecture of several units (neurons) Neurons compete in a “winner-takes-all” fashion for the object currently being presented 202 Model-Based Clustering Methods 203 Self-Organizing feature Maps (SOMs) Clustering is also performed by having several units competing for the current object The unit whose weight vector is closest to the current object wins The winner and its neighbors learn by having their weights adjusted SOMs are believed to resemble processing that can occur in the brain Useful for visualizing high-dimensional data in 2or 3-D space 204 Chapter 6. Cluster Analysis What is Cluster Analysis? Types of Data in Cluster Analysis A Categorization of Major Clustering Methods Partitioning Methods Hierarchical Methods Density-Based Methods Grid-Based Methods Model-Based Clustering Methods Outlier Analysis Summary 205 What Is Outlier Discovery? What are outliers? The set of objects are considerably dissimilar from the remainder of the data Example: Sports: Michael Jordon, Wayne Gretzky, ... Problem Find top n outlier points Applications: Credit card fraud detection Telecom fraud detection Customer segmentation Medical analysis 206 Outlier Discovery: Statistical Approaches Assume a model underlying distribution that generates data set (e.g. normal distribution) Use discordancy tests depending on data distribution distribution parameter (e.g., mean, variance) number of expected outliers Drawbacks most tests are for single attribute In many cases, data distribution may not be known 207 Outlier Discovery: Distance-Based Approach Introduced to counter the main limitations imposed by statistical methods We need multi-dimensional analysis without knowing data distribution. Distance-based outlier: A DB(p, D)-outlier is an object O in a dataset T such that at least a fraction p of the objects in T lies at a distance greater than D from O Algorithms for mining distance-based outliers Index-based algorithm Nested-loop algorithm Cell-based algorithm 208 Outlier Discovery: Deviation-Based Approach Identifies outliers by examining the main characteristics of objects in a group Objects that “deviate” from this description are considered outliers sequential exception technique simulates the way in which humans can distinguish unusual objects from among a series of supposedly like objects OLAP data cube technique uses data cubes to identify regions of anomalies in large multidimensional data 209 Problems and Challenges Considerable progress has been made in scalable clustering methods Partitioning: k-means, k-medoids, CLARANS Hierarchical: BIRCH, CURE Density-based: DBSCAN, CLIQUE, OPTICS Grid-based: STING, WaveCluster Model-based: Autoclass, Denclue, Cobweb Current clustering techniques do not address all the requirements adequately Constraint-based clustering analysis: Constraints exist in data space (bridges and highways) or in user queries 210 Constraint-Based Clustering Analysis Clustering analysis: less parameters but more user-desired constraints, e.g., an ATM allocation problem 211 Clustering With Obstacle Objects Not Taking obstacles into account Taking obstacles into account 212 Chapter 6. Cluster Analysis What is Cluster Analysis? Types of Data in Cluster Analysis A Categorization of Major Clustering Methods Partitioning Methods Hierarchical Methods Density-Based Methods Grid-Based Methods Model-Based Clustering Methods Outlier Analysis Summary 213 Summary Cluster analysis groups objects based on their similarity and has wide applications Measure of similarity can be computed for various types of data Clustering algorithms can be categorized into partitioning methods, hierarchical methods, density-based methods, grid-based methods, and model-based methods Outlier detection and analysis are very useful for fraud detection, etc. and can be performed by statistical, distance-based or deviation-based approaches There are still lots of research issues on cluster analysis, such as constraint-based clustering 214 References (1) R. Agrawal, J. Gehrke, D. Gunopulos, and P. Raghavan. Automatic subspace clustering of high dimensional data for data mining applications. 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BIRCH : an efficient data clustering method for very large databases. SIGMOD'96. 216 www.cs.uiuc.edu/~hanj Thank you !!! 217