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- 1. Data Mining: Concepts and Techniques Lectures 17 - 20 — Cluster and Outlier Analysis — (Adapted from) Jiawei Han Department of Computer Science University of Illinois at Urbana-Champaign www.cs.uiuc.edu/~hanj ©2006 Jiawei Han and Micheline Kamber, All rights reserved November 10, 2013 Data Mining: Concepts and Techniques 1
- 2. Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. Types of Data in Cluster Analysis 3. A Categorization of Major Clustering Methods 4. Partitioning Methods 5. Hierarchical Methods 6. Outlier Analysis 7. Summary November 10, 2013 Data Mining: Concepts and Techniques 2
- 3. 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 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 November 10, 2013 Data Mining: Concepts and Techniques 3
- 4. Clustering: Rich Applications and Multidisciplinary Efforts Pattern Recognition Spatial Data Analysis Create thematic maps in GIS by clustering feature spaces Detect spatial clusters or for other spatial mining tasks Image Processing Economic Science (especially market research) WWW Document classification Cluster Weblog data to discover groups of similar access patterns November 10, 2013 Data Mining: Concepts and Techniques 4
- 5. 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 November 10, 2013 Data Mining: Concepts and Techniques 5
- 6. Quality: 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 November 10, 2013 Data Mining: Concepts and Techniques 6
- 7. Measure the Quality of Clustering Dissimilarity/Similarity metric: Similarity is expressed in terms of a distance function, 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, binary, categorical, ordinal ratio, and vector 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. November 10, 2013 Data Mining: Concepts and Techniques 7
- 8. 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 November 10, 2013 Data Mining: Concepts and Techniques 8
- 9. Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. Types of Data in Cluster Analysis 3. A Categorization of Major Clustering Methods 4. Partitioning Methods 5. Hierarchical Methods 6. Outlier Analysis 7. Summary November 10, 2013 Data Mining: Concepts and Techniques 9
- 10. Data Structures Data matrix (two modes) x11 ... x i1 ... x n1 Dissimilarity matrix (one mode) November 10, 2013 ... x1f ... ... ... ... xif ... ... ... ... ... xnf ... ... 0 d(2,1) 0 d(3,1) d ( 3,2) 0 : : : d ( n,1) d ( n,2) ... Data Mining: Concepts and Techniques x1p ... xip ... xnp ... 0 10
- 11. Type of data in clustering analysis Interval-scaled variables Binary variables Nominal, ordinal, and ratio variables Variables of mixed types November 10, 2013 Data Mining: Concepts and Techniques 11
- 12. Interval-valued variables Standardize data Calculate the mean absolute deviation: s f = 1 (| x1 f − m f | + | x2 f − m f | +...+ | xnf − m f |) n where m f = 1 x + x +...+ x n 1f 2f nf . Calculate the standardized measurement (z-score) xif − m f zif = sf Using mean absolute deviation is more robust than using standard deviation November 10, 2013 Data Mining: Concepts and Techniques 12
- 13. 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 j 2 ip jp where i = (xi1, xi2, …, xip) and j = (xj1, xj2, …, xjp) are two pdimensional data objects, and q is a positive integer If q = 1, d is Manhattan (or city block) distance d (i, j) =| x − x | + | x − x | + ...+ | x − x | i1 j1 i2 j 2 ip jp November 10, 2013 Data Mining: Concepts and Techniques 13
- 14. Fig 7.1
- 15. 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,i) = 0 d(i,j) = d(j,i) d(i,j) ≥ 0 d(i,j) ≤ d(i,k) + d(k,j) (triangle inequality) Also, one can use weighted distance, parametric Pearson product moment correlation, or other disimilarity measures November 10, 2013 Data Mining: Concepts and Techniques 15
- 16. Binary Variables A contingency table for binary Object i data Object j 1 0 1 q s 0 r t sum q + s r + t Distance measure for symmetric binary variables: Distance measure for asymmetric binary variables: Jaccard coefficient (similarity measure for asymmetric d (i, j) = November 10, 2013 r +s q + +s r simJaccard (i, j) = Data Mining: Concepts and Techniques p r +s q +r +s +t d (i, j) = binary variables): sum q+r s +t q q+r +s 16
- 17. 1 Dissimilarity between Binary 0 sum q + s r + t Example Name Jack Mary Jim 1 0 sum q r q+ Variablesr s t s +t 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 p 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 0 +1 =0.33 2 +0 +1 1 +1 d ( jack , jim ) = =0.67 1 +1 +1 1 +2 d ( jim , mary ) = =0.75 1 +1 +2 d ( jack , mary ) = November 10, 2013 Data Mining: Concepts and Techniques 17
- 18. 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 d (i, j) = p −m p Method 2: use a large number of binary variables creating a new binary variable for each of the M nominal states November 10, 2013 Data Mining: Concepts and Techniques 18
- 19. 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 rif ∈ 1,..., M f } { map the range of each variable onto [0, 1] by replacing i-th object in the f-th variable by r − 1 zif = if M − 1 f compute the dissimilarity using methods for intervalscaled variables November 10, 2013 Data Mining: Concepts and Techniques 19
- 20. 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 = loge(xif) treat them as continuous ordinal data or treat their ranks as interval-scaled November 10, 2013 Data Mining: Concepts and Techniques 20
- 21. Variables of Mixed Types A database may contain all the six types of variables symmetric binary, asymmetric binary, nominal, ordinal, interval-valued and ratio-scaled One may use a weighted formula to combine their p ( ( effects Σ =1δ f ) d ij f ) d (i, j ) = f p ij ( f ) Σ =1δ f ij f is binary or nominal: dij(f) = 0 if xif = xjf , or dij(f) = 1 otherwise f is interval-based: use the normalized distance f is ordinal or ratio-scaled compute ranks r and if zif = r −1 and treat z as interval-scaled M −1 if if f November 10, 2013 Data Mining: Concepts and Techniques 21
- 22. Vector Objects Vector objects: keywords in documents, gene features in micro-arrays, etc. Broad applications: information retrieval, biological taxonomy, etc. Cosine measure A variant: Tanimoto coefficient November 10, 2013 Data Mining: Concepts and Techniques 22
- 23. Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. Types of Data in Cluster Analysis 3. A Categorization of Major Clustering Methods 4. Partitioning Methods 5. Hierarchical Methods 6. Outlier Analysis 7. Summary November 10, 2013 Data Mining: Concepts and Techniques 23
- 24. Major Clustering Approaches (I) Partitioning approach: Construct various partitions and then evaluate them by some criterion, e.g., minimizing the sum of square errors Typical methods: k-means, k-medoids, CLARANS Hierarchical approach: Create a hierarchical decomposition of the set of data (or objects) using some criterion Typical methods: Diana, Agnes, BIRCH, ROCK, CAMELEON Density-based approach: Based on connectivity and density functions Typical methods: DBSACN, OPTICS, DenClue November 10, 2013 Data Mining: Concepts and Techniques 24
- 25. Major Clustering Approaches (II) Grid-based approach: based on a multiple-level granularity structure Typical methods: STING, WaveCluster, CLIQUE Model-based: A model is hypothesized for each of the clusters and tries to find the best fit of that model to each other Typical methods: EM, SOM, COBWEB Frequent pattern-based: Based on the analysis of frequent patterns Typical methods: pCluster User-guided or constraint-based: Clustering by considering user-specified or application-specific constraints Typical methods: COD (obstacles), constrained clustering November 10, 2013 Data Mining: Concepts and Techniques 25
- 26. Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. Types of Data in Cluster Analysis 3. A Categorization of Major Clustering Methods 4. Partitioning Methods 5. Hierarchical Methods 6. Outlier Analysis 7. Summary November 10, 2013 Data Mining: Concepts and Techniques 26
- 27. Partitioning Algorithms: Basic Concept Partitioning method: Construct a partition of a database D of n objects into a set of k clusters, s.t., minimize sum of squared distance Σ ik=1Σ p∈C p − mi 2 i 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 centre (or mean) 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 November 10, 2013 Data Mining: Concepts and Techniques 27
- 28. The K-Means Clustering Method Given k, the k-means algorithm is implemented in four steps: Partition objects into k non-empty subsets Compute seed points as the centroids of the clusters of the current partition (the centroid is the centre, i.e., mean point, of the cluster) Assign each object to the cluster with the nearest seed point based on distance Go back to Step 2, stop when no more movement between clusters November 10, 2013 Data Mining: Concepts and Techniques 28
- 29. Fig 7.3
- 30. 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 November 10, 2013 Data Mining: Concepts and Techniques 30
- 31. Variations of the K-Means Method A few variants of the k-means which differ in Dissimilarity calculations Selection of the initial k means 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 November 10, 2013 Data Mining: Concepts and Techniques 31
- 32. What Is the Problem of the 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 November 10, 2013 1 2 3 4 5 6 7 8 9 10 0 1 2 3 Data Mining: Concepts and Techniques 4 5 6 7 8 9 10 32
- 33. 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) November 10, 2013 Data Mining: Concepts and Techniques 33
- 34. A 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 1 0 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 0 K=2 Do loop Until no change 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 10 Compute total cost of swapping 8 4 5 6 7 8 9 10 7 6 5 9 8 7 6 5 4 4 3 3 2 2 1 1 0 0 0 November 10, 2013 3 10 9 If quality is improved. 2 Randomly select a nonmedoid object,Oramdom Total Cost = 26 Swapping O and Oramdom 1 1 2 3 4 5 6 7 8 9 10 Data Mining: Concepts and Techniques 0 1 2 3 4 5 6 7 8 9 10 34
- 35. PAM (Partitioning Around Medoids)
- 36. Four Cases of Potential Swapping
- 37. 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) November 10, 2013 Data Mining: Concepts and Techniques 37
- 38. Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. Types of Data in Cluster Analysis 3. A Categorization of Major Clustering Methods 4. Partitioning Methods 5. Hierarchical Methods 6. Outlier Analysis 7. Summary November 10, 2013 Data Mining: Concepts and Techniques 38
- 39. Hierarchical Methods Agglomerative Hierarchical Clustering Starts by placing each object in its own cluster and then merges these atomic clusters into larger and larger clusters Divisive Hierarchical Clustering Bottom-up strategy Top-down strategy Starts with all objects in one cluster and then subdivide the cluster into smaller and smaller pieces Stopping (or termination) conditions November 10, 2013 Data Mining: Concepts and Techniques 39
- 40. Stopping Conditions Agglomerative Hierarchical Clustering Merging continues until all objects are in a single cluster or until certain termination criteria are satisfied Divisive Hierarchical Clustering Subdividing continues until each object forms a cluster on its own or until it satisfies certain termination criteria: A desired number of clusters is obtained The diameter of each cluster is within a certain threshold November 10, 2013 Data Mining: Concepts and Techniques 40
- 41. 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 b Step 1 Step 2 Step 3 Step 4 ab abcde c cde d de e Step 4 November 10, 2013 agglomerative (AGNES) Step 3 Step 2 Step 1 Step 0 Data Mining: Concepts and Techniques divisive (DIANA) 41
- 42. AGNES (AGglomerative NESting) Introduced in Kaufmann and Rousseeuw (1990) Implemented in statistical analysis packages, e.g., Splus Use the Single-Link method and the Distance matrix. Merge nodes that have the least distance Go on in a non-decreasing (in terms of size) fashion Eventually all nodes belong to the same cluster 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 November 10, 2013 6 7 8 9 10 Data Mining: Concepts and Techniques 42
- 43. AGNES (AGglomerative NESting) Introduced in Kaufmann and Rousseeuw (1990) Implemented in statistical analysis packages, e.g., Splus Use the Single-Link method and the Distance matrix. Merge nodes that have the least distance Go on in a non-decreasing (in terms of size) 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 November 10, 2013 6 7 8 9 10 0 0 1 2 3 4 5 6 7 8 9 10 Data Mining: Concepts and Techniques 0 1 2 3 4 5 6 7 8 9 10 43
- 44. Dendrogram: Shows How the Clusters are Merged November 10, 2013 Data Mining: Concepts and Techniques 44
- 45. 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 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 November 10, 2013 5 6 7 8 9 10 Data Mining: Concepts and Techniques 45
- 46. 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 1 2 3 4 November 10, 2013 5 6 7 8 9 10 0 0 0 1 2 3 4 5 6 7 8 9 10 Data Mining: Concepts and Techniques 0 1 2 3 4 5 6 7 8 9 10 46
- 47. Recent Hierarchical Clustering Methods Major weakness of agglomerative clustering methods do not scale well: time complexity of at least O(n2), 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 ROCK (1999): clustering categorical data by neighbor and link analysis CHAMELEON (1999): hierarchical clustering using dynamic modeling November 10, 2013 Data Mining: Concepts and Techniques 47
- 48. Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. Types of Data in Cluster Analysis 3. A Categorization of Major Clustering Methods 4. Partitioning Methods 5. Hierarchical Methods 6. Outlier Analysis 7. Summary November 10, 2013 Data Mining: Concepts and Techniques 48
- 49. What Is Outlier Detection? What are outliers? The set of objects are considerably dissimilar from the remainder of the data Example: Sports: Shane Warne, Diego Maradona, ... Problem: Define and find outliers in large data sets Applications: Credit card fraud detection Telecom fraud detection Customer segmentation Medical analysis However, one person’s noise could be another person’s signal November 10, 2013 Data Mining: Concepts and Techniques 49
- 50. Outlier Detection: 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 November 10, 2013 Data Mining: Concepts and Techniques 50
- 51. Outlier Detection: Distance-Based Approach Introduced to counter the main limitations imposed by statistical methods Find those objects that do not have “enough” neighbours, where neighbours are defined based on distance from the given object 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 November 10, 2013 Data Mining: Concepts and Techniques 51
- 52. Density-Based Local Outlier Detection Distance-based outlier detection is based on global distance distribution It encounters difficulties to identify outliers if data is not uniformly distributed Ex. C1 contains 400 loosely distributed points, C2 has 100 tightly condensed points, 2 outlier points o1, o2 Distance-based method cannot identify o2 as an outlier Need the concept of local outlier November 10, 2013 Local outlier factor (LOF) Assume outlier is not crisp Each point has a LOF Data Mining: Concepts and Techniques 52
- 53. Outlier Detection: 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 November 10, 2013 Data Mining: Concepts and Techniques 53

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