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Data Mining: Concepts and Techniques — Chapter 7 — Jiawei Han and Micheline Kamber Department of Computer Science University of Illinois at Urbana-Champaign www. cs. uiuc. edu/~hanj © 2006 Jiawei Han and Micheline Kamber. All rights reserved. 15 March 2018 Data Mining: Concepts and Techniques 1
15 March 2018 Data Mining: Concepts and Techniques 2
Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. A Categorization of Major Clustering Methods 3. Partitioning Methods 4. Hierarchical Methods 5. Density-Based Methods 6. Grid-Based Methods 7. Model-Based Methods 8. Clustering High-Dimensional Data 9. Constraint-Based Clustering 10. Link-based clustering 11. Outlier Analysis 12. Summary 3
What is Cluster Analysis? n Cluster: A collection of data objects n n n similar (or related) to one another within the same group dissimilar (or unrelated) to the objects in other groups Cluster analysis n Finding similarities between data according to the characteristics found in the data and grouping similar data objects into clusters n Unsupervised learning: no predefined classes n Typical applications n As a stand-alone tool to get insight into data distribution n As a preprocessing step for other algorithms 4
Clustering for Data Understanding and Applications n n n n Biology: taxonomy of living things: kindom, phylum, class, order, family, genus and species Information retrieval: document clustering Land use: Identification of areas of similar land use in an earth observation database Marketing: Help marketers discover distinct groups in their customer bases, and then use this knowledge to develop targeted marketing programs 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 Climate: understanding earth climate, find patterns of atmospheric and ocean Economic Science: market resarch 5
Clustering as Preprocessing Tools (Utility) n Summarization: n n Compression: n n Preprocessing for regression, PCA, classification, and association analysis Image processing: vector quantization Finding K-nearest Neighbors n Localizing search to one or a small number of clusters 6
Quality: What Is Good Clustering? n A good clustering method will produce high quality clusters n n n high intra-class similarity: cohesive within clusters low inter-class similarity: distinctive between clusters The quality of a clustering result depends on both the similarity measure used by the method and its implementation n The quality of a clustering method is also measured by its ability to discover some or all of the hidden patterns 15 March 2018 Data Mining: Concepts and Techniques 7
Measure the Quality of Clustering n n Dissimilarity/Similarity metric n Similarity is expressed in terms of a distance function, typically metric: d(i, j) n The definitions of distance functions are usually rather different for interval-scaled, boolean, categorical, ordinal ratio, and vector variables n Weights should be associated with different variables based on applications and data semantics Quality of clustering: n There is usually a separate “quality” function that measures the “goodness” of a cluster. n It is hard to define “similar enough” or “good enough” n 15 March 2018 The answer is typically highly subjective Data Mining: Concepts and Techniques 8
Distance Measures for Different Kinds of Data Discussed in Chapter 2: Data Preprocessing n n n n Numerical (interval)-based: n Minkowski Distance: n Special cases: Euclidean (L 2 -norm), Manhattan (L 1 norm) Binary variables: n symmetric vs. asymmetric (Jaccard coeff. ) Nominal variables: # of mismatches Ordinal variables: treated like interval-based Ratio-scaled variables: apply log-transformation first Vectors: cosine measure Mixed variables: weighted combinations 15 March 2018 Data Mining: Concepts and Techniques 9
Requirements of Clustering in Data Mining n Scalability n Ability to deal with different types of attributes n Ability to handle dynamic data n Discovery of clusters with arbitrary shape n Minimal requirements for domain knowledge to determine input parameters n Able to deal with noise and outliers n Insensitive to order of input records n High dimensionality n Incorporation of user-specified constraints n Interpretability and usability 15 March 2018 Data Mining: Concepts and Techniques 10
Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. A Categorization of Major Clustering Methods 3. Partitioning Methods 4. Hierarchical Methods 5. Density-Based Methods 6. Grid-Based Methods 7. Model-Based Methods 8. Clustering High-Dimensional Data 9. Constraint-Based Clustering 10. Link-based clustering 11. Outlier Analysis 12. Summary 15 March 2018 Data Mining: Concepts and Techniques 11
Major Clustering Approaches (I) n n Partitioning approach: n Construct various partitions and then evaluate them by some criterion, e. g. , minimizing the sum of square errors n Typical methods: k-means, k-medoids, CLARANS Hierarchical approach: n Create a hierarchical decomposition of the set of data (or objects) using some criterion n Typical methods: Diana, Agnes, BIRCH, ROCK, CAMELEON Density-based approach: n Based on connectivity and density functions n Typical methods: DBSACN, OPTICS, Den. Clue Grid-based approach: n based on a multiple-level granularity structure n Typical methods: STING, Wave. Cluster, CLIQUE 15 March 2018 Data Mining: Concepts and Techniques 12
Major Clustering Approaches (II) n n Model-based: n A model is hypothesized for each of the clusters and tries to find the best fit of that model to each other n Typical methods: EM, SOM, COBWEB Frequent pattern-based: n Based on the analysis of frequent patterns n Typical methods: p-Cluster User-guided or constraint-based: n Clustering by considering user-specified or application-specific constraints n Typical methods: COD (obstacles), constrained clustering Link-based clustering: n Objects are often linked together in various ways n Massive links can be used to cluster objects: Sim. Rank, Link. Clus 15 March 2018 Data Mining: Concepts and Techniques 13
Calculation of Distance between Clusters n Single link: smallest distance between an element in one cluster and an element in the other, i. e. , dist(Ki, Kj) = min(tip, tjq) n Complete link: largest distance between an element in one cluster and an element in the other, i. e. , dist(Ki, Kj) = max(tip, tjq) n Average: avg distance between an element in one cluster and an element in the other, i. e. , dist(Ki, Kj) = avg(tip, tjq) n Centroid: distance between the centroids of two clusters, i. e. , dist(Ki, Kj) = dist(Ci, Cj) n Medoid: distance between the medoids of two clusters, i. e. , dist(Ki, Kj) = dist(Mi, Mj) n Medoid: one chosen, centrally located object in the cluster 15 March 2018 Data Mining: Concepts and Techniques 14
Centroid, Radius and Diameter of a Cluster (for numerical data sets) n Centroid: the “middle” of a cluster n Radius: square root of average distance from any point of the cluster to its centroid n Diameter: square root of average mean squared distance between all pairs of points in the cluster 15 March 2018 Data Mining: Concepts and Techniques 15
Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. A Categorization of Major Clustering Methods 3. Partitioning Methods 4. Hierarchical Methods 5. Density-Based Methods 6. Grid-Based Methods 7. Model-Based Methods 8. Clustering High-Dimensional Data 9. Constraint-Based Clustering 10. Link-based clustering 11. Outlier Analysis 12. Summary 15 March 2018 Data Mining: Concepts and Techniques 16
Partitioning Algorithms: Basic Concept n n 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 Given a k, find a partition of k clusters that optimizes the chosen partitioning criterion n Global optimal: exhaustively enumerate all partitions n Heuristic methods: k-means and k-medoids algorithms n n k-means (Mac. Queen’ 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 15 March 2018 Data Mining: Concepts and Techniques 17
The K-Means Clustering Method n Given k, the k-means algorithm is implemented in four steps: n n 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 15 March 2018 Data Mining: Concepts and Techniques 18
The K-Means Clustering Method n Example 10 10 9 9 8 8 7 7 6 6 5 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 Assign each objects to most similar center Update the cluster means reassign 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 reassign K=2 Arbitrarily choose K object as initial cluster center 15 March 2018 Update the cluster means Data Mining: Concepts and Techniques 19
Comments on the K-Means Method n Strength: Relatively efficient: O(tkn), where n is # objects, k is # clusters, and t is # iterations. Normally, k, t << n. n n n Comparing: PAM: O(k(n-k)2 ), CLARA: O(ks 2 + 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 n Applicable only when mean is defined, then what about categorical data? n Need to specify k, the number of clusters, in advance n Unable to handle noisy data and outliers n Not suitable to discover clusters with non-convex shapes 15 March 2018 Data Mining: Concepts and Techniques 20
Variations of the K-Means Method n A few variants of the k-means which differ in n n Dissimilarity calculations n n Selection of the initial k means Strategies to calculate cluster means Handling categorical data: k-modes (Huang’ 98) n Replacing means of clusters with modes n Using new dissimilarity measures to deal with categorical objects n Using a frequency-based method to update modes of clusters n A mixture of categorical and numerical data: k-prototype method 15 March 2018 Data Mining: Concepts and Techniques 21
What Is the Problem of the K-Means Method? n The k-means algorithm is sensitive to outliers ! n Since an object with an extremely large value may substantially distort the distribution of the data. n 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 15 March 2018 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 22
The K-Medoids Clustering Method n Find representative objects, called medoids, in clusters n PAM (Partitioning Around Medoids, 1987) n 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 n PAM works effectively for small data sets, but does not scale well for large data sets n CLARA (Kaufmann & Rousseeuw, 1990) n CLARANS (Ng & Han, 1994): Randomized sampling n Focusing + spatial data structure (Ester et al. , 1995) 15 March 2018 Data Mining: Concepts and Techniques 23
A Typical K-Medoids Algorithm (PAM) Total Cost = 20 10 9 8 Arbitrary choose k object as initial medoids 7 6 5 4 3 2 Assign each remainin g object to nearest medoids 1 0 0 1 2 3 4 5 6 7 8 9 10 K=2 Randomly select a nonmedoid object, Oramdom Total Cost = 26 Do loop Until no change 10 10 9 Swapping O and Oramdom Compute total cost of swapping 8 7 6 9 8 7 6 5 4 4 3 3 2 2 1 If quality is improved. 5 1 0 0 0 15 March 2018 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 24
PAM (Partitioning Around Medoids) (1987) n PAM (Kaufman and Rousseeuw, 1987), built in Splus n Use real object to represent the cluster n n n Select k representative objects arbitrarily For each pair of non-selected object h and selected object i, calculate the total swapping cost TCih For each pair of i and h, n n n If TCih < 0, i is replaced by h Then assign each non-selected object to the most similar representative object repeat steps 2 -3 until there is no change 15 March 2018 Data Mining: Concepts and Techniques 25
PAM Clustering: Finding the Best Cluster Center n Case 1: p currently belongs to oj. If oj is replaced by orandom as a representative object and p is the closest to one of the other representative object oi, then p is reassigned to oi 15 March 2018 Data Mining: Concepts and Techniques 26
What Is the Problem with PAM? n n 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. n O(k(n-k)2 ) for each iteration where n is # of data, k is # of clusters è Sampling-based method CLARA(Clustering LARge Applications) 15 March 2018 Data Mining: Concepts and Techniques 27
CLARA (Clustering Large Applications) (1990) n CLARA (Kaufmann and Rousseeuw in 1990) n n Built in statistical analysis packages, such as SPlus It draws multiple samples of the data set, applies PAM on each sample, and gives the best clustering as the output n Strength: deals with larger data sets than PAM n Weakness: n n 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 15 March 2018 Data Mining: Concepts and Techniques 28
CLARANS (“Randomized” CLARA) (1994) n n n CLARANS (A Clustering Algorithm based on Randomized Search) (Ng and Han’ 94) n Draws sample of neighbors dynamically n The clustering process can be presented as searching a graph where every node is a potential solution, that is, a set of k medoids n If the local optimum is found, it starts with new randomly selected node in search for a new local optimum Advantages: More efficient and scalable than both PAM and CLARA Further improvement: Focusing techniques and spatial access structures (Ester et al. ’ 95) 15 March 2018 Data Mining: Concepts and Techniques 29
Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. A Categorization of Major Clustering Methods 3. Partitioning Methods 4. Hierarchical Methods 5. Density-Based Methods 6. Grid-Based Methods 7. Model-Based Methods 8. Clustering High-Dimensional Data 9. Constraint-Based Clustering 10. Link-based clustering 11. Outlier Analysis 12. Summary 15 March 2018 Data Mining: Concepts and Techniques 30
Hierarchical Clustering n 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 15 March 2018 agglomerative (AGNES) Step 3 Step 2 Step 1 Step 0 Data Mining: Concepts and Techniques divisive (DIANA) 31
AGNES (Agglomerative Nesting) n Introduced in Kaufmann and Rousseeuw (1990) n Implemented in statistical packages, e. g. , Splus n Use the Single-Link method and the dissimilarity matrix n Merge nodes that have the least dissimilarity n Go on in a non-descending fashion n Eventually all nodes belong to the same cluster 15 March 2018 Data Mining: Concepts and Techniques 32
Dendrogram: Shows How the Clusters are Merged 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. 15 March 2018 Data Mining: Concepts and Techniques 33
DIANA (Divisive Analysis) n Introduced in Kaufmann and Rousseeuw (1990) n Implemented in statistical analysis packages, e. g. , Splus n Inverse order of AGNES n Eventually each node forms a cluster on its own 15 March 2018 Data Mining: Concepts and Techniques 34
Extensions to Hierarchical Clustering n Major weakness of agglomerative clustering methods n n n Do not scale well: time complexity of at least O(n 2), where n is the number of total objects Can never undo what was done previously Integration of hierarchical & distance-based clustering n n n 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 15 March 2018 Data Mining: Concepts and Techniques 35
BIRCH (Zhang, Ramakrishnan & Livny, SIGMOD’ 96) n n Birch: Balanced Iterative Reducing and Clustering using Hierarchies Incrementally construct a CF (Clustering Feature) tree, a hierarchical data structure for multiphase clustering n n 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 and improves the quality with a few additional scans Weakness: handles only numeric data, and sensitive to the order of the data record 15 March 2018 Data Mining: Concepts and Techniques 36
Clustering Feature Vector in BIRCH Clustering Feature (CF): CF = (N, LS, SS) N: Number of data points LS: linear sum of N points: SS: square sum of N points CF = (5, (16, 30), (54, 190)) (3, 4) (2, 6) (4, 5) (4, 7) (3, 8) 15 March 2018 Data Mining: Concepts and Techniques 37
CF-Tree in BIRCH n Clustering feature: n n Summary of the statistics for a given subcluster: the 0 -th, 1 st and 2 nd 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 n n n 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 n Branching factor: specify the maximum number of children n Threshold: max diameter of sub-clusters stored at the leaf nodes 15 March 2018 Data Mining: Concepts and Techniques 38
The CF Tree Structure Root B=7 CF 1 CF 2 CF 3 CF 6 L=6 child 1 child 2 child 3 child 6 CF 1 Non-leaf node CF 2 CF 3 CF 5 child 1 child 2 child 3 child 5 Leaf node prev CF 1 CF 2 15 March 2018 CF 6 next Leaf node prev CF 1 CF 2 Data Mining: Concepts and Techniques CF 4 next 39
Birch Algorithm n n Cluster Diameter For each point in the input n Find closest leaf entry n Add point to leaf entry, Update CF n If entry diameter > max_diameter n n n split leaf, and possibly parents Algorithm is O(n) Problems n Sensitive to insertion order of data points n We fix size of leaf nodes, so clusters my not be natural n Clusters tend to be spherical given the radius and diameter measures 15 March 2018 Data Mining: Concepts and Techniques 40
ROCK: Clustering Categorical Data n n ROCK: RObust Clustering using lin. Ks n S. Guha, R. Rastogi & K. Shim, ICDE’ 99 Major ideas n Use links to measure similarity/proximity n Not distance-based Algorithm: sampling-based clustering n Draw random sample n Cluster with links n Label data in disk Experiments n Congressional voting, mushroom data 15 March 2018 Data Mining: Concepts and Techniques 41
Similarity Measure in ROCK n n Traditional measures for categorical data may not work well, e. g. , Jaccard coefficient Example: Two groups (clusters) of transactions n C 1.
Link Measure in ROCK n n Clusters n C 1:
CHAMELEON: Hierarchical Clustering Using Dynamic Modeling (1999) n CHAMELEON: by G. Karypis, E. H. Han, and V. Kumar, 1999 n Measures the similarity based on a dynamic model n n n 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 (Hierarchical clustering with multiple representative objects) ignores information about interconnectivity of the objects, Rock ignores information about the closeness of two clusters A two-phase algorithm 1. 2. Use a graph partitioning algorithm: cluster objects into a large number of relatively small sub-clusters Use an agglomerative hierarchical clustering algorithm: find the genuine clusters by repeatedly combining these sub-clusters 15 March 2018 Data Mining: Concepts and Techniques 45
Overall Framework of CHAMELEON Construct (K-NN) Partition the Graph Sparse Graph Data Set K-NN Graph p, q connected if q among the top k closest neighbors of p Merge Partition Final Clusters • Relative interconnectivity: connectivity of c 1, c 2 over internal connectivity • Relative closeness: closeness of c 1, c 2 over internal closeness 15 March 2018 Data Mining: Concepts and Techniques 46
CHAMELEON (Clustering Complex Objects) 15 March 2018 Data Mining: Concepts and Techniques 47
Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. A Categorization of Major Clustering Methods 3. Partitioning Methods 4. Hierarchical Methods 5. Density-Based Methods 6. Grid-Based Methods 7. Model-Based Methods 8. Clustering High-Dimensional Data 9. Constraint-Based Clustering 10. Link-based clustering 11. Outlier Analysis 12. Summary 15 March 2018 Data Mining: Concepts and Techniques 48
Density-Based Clustering Methods n n n Clustering based on density (local cluster criterion), such as density-connected points Major features: n Discover clusters of arbitrary shape n Handle noise n One scan n Need density parameters as termination condition Several interesting studies: n DBSCAN: Ester, et al. (KDD’ 96) n OPTICS: Ankerst, et al (SIGMOD’ 99). n DENCLUE: Hinneburg & D. Keim (KDD’ 98) n CLIQUE: Agrawal, et al. (SIGMOD’ 98) (more gridbased) 15 March 2018 Data Mining: Concepts and Techniques 49
Density-Based Clustering: Basic Concepts n Two parameters: n n Eps: Maximum radius of the neighbourhood Min. Pts: 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 w. r. t. Eps, Min. Pts if n p belongs to NEps(q) n core point condition: |NEps (q)| >= Min. Pts 15 March 2018 Data Mining: Concepts and Techniques p q Min. Pts = 5 Eps = 1 cm 50
Density-Reachable and Density-Connected n Density-reachable: n n A point p is density-reachable from a point q w. r. t. Eps, Min. Pts if there is a chain of points p 1, …, pn, p 1 = q, pn = p such that pi+1 is directly density-reachable from pi p p 1 q Density-connected n A point p is density-connected to a p point q w. r. t. Eps, Min. Pts if there is a point o such that both, p and q are density-reachable from o w. r. t. Eps and Min. Pts 15 March 2018 Data Mining: Concepts and Techniques q o 51
DBSCAN: Density Based Spatial Clustering of Applications with Noise n n 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 = 1 cm Core 15 March 2018 Min. Pts = 5 Data Mining: Concepts and Techniques 52
DBSCAN: The Algorithm n n n Arbitrary select a point p Retrieve all points density-reachable from p w. r. t. Eps and Min. Pts. 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. 15 March 2018 Data Mining: Concepts and Techniques 53
DBSCAN: Sensitive to Parameters 15 March 2018 Data Mining: Concepts and Techniques 54
CHAMELEON (Clustering Complex Objects) 15 March 2018 Data Mining: Concepts and Techniques 55
OPTICS: A Cluster-Ordering Method (1999) n OPTICS: Ordering Points To Identify the Clustering Structure n Ankerst, Breunig, Kriegel, and Sander (SIGMOD’ 99) n Produces a special order of the database wrt its density -based clustering structure n This cluster-ordering contains info equiv to the densitybased clusterings corresponding to a broad range of parameter settings n Good for both automatic and interactive cluster analysis, including finding intrinsic clustering structure n Can be represented graphically or using visualization techniques 15 March 2018 Data Mining: Concepts and Techniques 56
OPTICS: Some Extension from DBSCAN n Index-based: n k = number of dimensions n N = 20 n p = 75% n M = N(1 -p) = 5 n n Complexity: O(Nlog. N) Core Distance: n D p 1 min eps s. t. point is core o n Reachability Distance p 2 Max (core-distance (o), d (o, p)) r(p 1, o) = 2. 8 cm. r(p 2, o) = 4 cm 15 March 2018 o Min. Pts = 5 e = 3 cm Data Mining: Concepts and Techniques 57
Reachability -distance undefined ‘ 15 March 2018 Data Mining: Concepts and Techniques Cluster-order of the objects 58
Density-Based Clustering: OPTICS & Its Applications 15 March 2018 Data Mining: Concepts and Techniques 59
DENCLUE: Using Statistical Density Functions n n n DENsity-based CLUst. Ering by Hinneburg & Keim (KDD’ 98) total influence on x Using statistical density functions: Major features influence of y on x n Solid mathematical foundation n Good for data sets with large amounts of noise n gradient of x in the direction of xi Allows a compact mathematical description of arbitrarily shaped clusters in high-dimensional data sets n Significant faster than existing algorithm (e. g. , DBSCAN) n But needs a large number of parameters 15 March 2018 Data Mining: Concepts and Techniques 60
Denclue: Technical Essence n n n 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 n Clusters can be determined mathematically by identifying density attractors n Density attractors are local maximal of the overall density function n n Center defined clusters: assign to each density attractor the points density attracted to it Arbitrary shaped cluster: merge density attractors that are connected through paths of high density (> threshold) 15 March 2018 Data Mining: Concepts and Techniques 61
Density Attractor 15 March 2018 Data Mining: Concepts and Techniques 62
Center-Defined and Arbitrary 15 March 2018 Data Mining: Concepts and Techniques 63
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. Density-Based Methods 7. Grid-Based Methods 8. Model-Based Methods 9. Clustering High-Dimensional Data 10. Constraint-Based Clustering 11. Outlier Analysis 12. Summary 15 March 2018 Data Mining: Concepts and Techniques 64
Grid-Based Clustering Method n n Using multi-resolution grid data structure Several interesting methods n STING (a STatistical INformation Grid approach) by Wang, Yang and Muntz (1997) n Wave. Cluster by Sheikholeslami, Chatterjee, and Zhang (VLDB’ 98) n n A multi-resolution clustering approach using wavelet method CLIQUE: Agrawal, et al. (SIGMOD’ 98) n 15 March 2018 On high-dimensional data (thus put in the section of clustering high-dimensional data Data Mining: Concepts and Techniques 65
STING: A Statistical Information Grid Approach n n n Wang, Yang and Muntz (VLDB’ 97) The spatial area is divided into rectangular cells There are several levels of cells corresponding to different levels of resolution 15 March 2018 Data Mining: Concepts and Techniques 66
The STING Clustering Method n n n 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 is used to answer queries Parameters of higher level cells can be easily calculated from parameters of lower level cell n count, mean, s, min, max n 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 15 March 2018 Data Mining: Concepts and Techniques 67
STING Algorithm and Its Analysis n n n 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: n Query-independent, easy to parallelize, incremental update n O(K), where K is the number of grid cells at the lowest level Disadvantages: n All the cluster boundaries are either horizontal or vertical, and no diagonal boundary is detected 15 March 2018 Data Mining: Concepts and Techniques 68
Wave. Cluster: Clustering by Wavelet Analysis (1998) n n n Sheikholeslami, Chatterjee, and Zhang (VLDB’ 98) A multi-resolution clustering approach which applies wavelet transform to the feature space How to apply wavelet transform to find clusters n Summarizes the data by imposing a multidimensional grid structure onto data space n These multidimensional spatial data objects are represented in a n-dimensional feature space n Apply wavelet transform on feature space to find the dense regions in the feature space n Apply wavelet transform multiple times which result in clusters at different scales from fine to coarse 15 March 2018 Data Mining: Concepts and Techniques 69
Wavelet Transform n n n Wavelet transform: A signal processing technique that decomposes a signal into different frequency sub -band (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 15 March 2018 Data Mining: Concepts and Techniques 70
The Wave. Cluster Algorithm n n Input parameters n # of grid cells for each dimension n the wavelet, and the # of applications of wavelet transform Why is wavelet transformation useful for clustering? n Use hat-shape filters to emphasize region where points cluster, but simultaneously suppress weaker information in their boundary n Effective removal of outliers, multi-resolution, cost effective Major features: n Complexity O(N) n Detect arbitrary shaped clusters at different scales n Not sensitive to noise, not sensitive to input order n Only applicable to low dimensional data Both grid-based and density-based 15 March 2018 Data Mining: Concepts and Techniques 71
Quantization & Transformation n First, quantize data into m-D grid structure, then wavelet transform n a) scale 1: high resolution n b) scale 2: medium resolution n c) scale 3: low resolution 15 March 2018 Data Mining: Concepts and Techniques 72
Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. A Categorization of Major Clustering Methods 3. Partitioning Methods 4. Hierarchical Methods 5. Density-Based Methods 6. Grid-Based Methods 7. Model-Based Methods 8. Clustering High-Dimensional Data 9. Constraint-Based Clustering 10. Link-based clustering 11. Outlier Analysis 12. Summary 15 March 2018 Data Mining: Concepts and Techniques 73
Model-Based Clustering n n What is model-based clustering? n Attempt to optimize the fit between the given data and some mathematical model n Based on the assumption: Data are generated by a mixture of underlying probability distribution Typical methods n Statistical approach n EM (Expectation maximization), Auto. Class n Machine learning approach n COBWEB, CLASSIT n Neural network approach n SOM (Self-Organizing Feature Map) 15 March 2018 Data Mining: Concepts and Techniques 74
EM — Expectation Maximization n EM — A popular iterative refinement algorithm n An extension to k-means n n n New means are computed based on weighted measures General idea n n n Assign each object to a cluster according to a weight (prob. distribution) Starts with an initial estimate of the parameter vector Iteratively rescores the patterns against the mixture density produced by the parameter vector The rescored patterns are used to update the parameter updates Patterns belonging to the same cluster, if they are placed by their scores in a particular component Algorithm converges fast but may not be in global optima 15 March 2018 Data Mining: Concepts and Techniques 75
The EM (Expectation Maximization) Algorithm n n Initially, randomly assign k cluster centers Iteratively refine the clusters based on two steps n Expectation step: assign each data point Xi to cluster Ci with the following probability n Maximization step: n Estimation of model parameters 15 March 2018 Data Mining: Concepts and Techniques 76
Conceptual Clustering n n Conceptual clustering n A form of clustering in machine learning n Produces a classification scheme for a set of unlabeled objects n Finds characteristic description for each concept (class) COBWEB (Fisher’ 87) n A popular a simple method of incremental conceptual learning n Creates a hierarchical clustering in the form of a classification tree n Each node refers to a concept and contains a probabilistic description of that concept 15 March 2018 Data Mining: Concepts and Techniques 77
COBWEB Clustering Method A classification tree 15 March 2018 Data Mining: Concepts and Techniques 78
More on Conceptual Clustering n Limitations of COBWEB n n n Not suitable for clustering large database data – skewed tree and expensive probability distributions CLASSIT n n n The assumption that the attributes are independent of each other is often too strong because correlation may exist an extension of COBWEB for incremental clustering of continuous data suffers similar problems as COBWEB Auto. Class (Cheeseman and Stutz, 1996) n Uses Bayesian statistical analysis to estimate the number of clusters n Popular in industry 15 March 2018 Data Mining: Concepts and Techniques 79
Neural Network Approach n n Neural network approaches n Represent each cluster as an exemplar, acting as a “prototype” of the cluster n New objects are distributed to the cluster whose exemplar is the most similar according to some distance measure Typical methods n SOM (Soft-Organizing feature Map) n Competitive learning n Involves a hierarchical architecture of several units (neurons) n Neurons compete in a “winner-takes-all” fashion for the object currently being presented 15 March 2018 Data Mining: Concepts and Techniques 80
Self-Organizing Feature Map (SOM) n n SOMs, also called topological ordered maps, or Kohonen Self. Organizing Feature Map (KSOMs) It maps all the points in a high-dimensional source space into a 2 to 3 -d target space, s. t. , the distance and proximity relationship (i. e. , topology) are preserved as much as possible Similar to k-means: cluster centers tend to lie in a low-dimensional manifold in the feature space Clustering is performed by having several units competing for the current object n The unit whose weight vector is closest to the current object wins n The winner and its neighbors learn by having their weights adjusted n SOMs are believed to resemble processing that can occur in the brain n Useful for visualizing high-dimensional data in 2 - or 3 -D space 15 March 2018 Data Mining: Concepts and Techniques 81
Web Document Clustering Using SOM n The result of SOM clustering of 12088 Web articles n The picture on the right: drilling down on the keyword “mining” n Based on websom. hut. fi Web page 15 March 2018 Data Mining: Concepts and Techniques 82
Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. A Categorization of Major Clustering Methods 3. Partitioning Methods 4. Hierarchical Methods 5. Density-Based Methods 6. Grid-Based Methods 7. Model-Based Methods 8. Clustering High-Dimensional Data 9. Constraint-Based Clustering 10. Link-based Clustering 11. Outlier Analysis 12. Summary 15 March 2018 Data Mining: Concepts and Techniques 83
Clustering High-Dimensional Data n n Clustering high-dimensional data n Many applications: text documents, DNA micro-array data n Major challenges: n Many irrelevant dimensions may mask clusters n Distance measure becomes meaningless—due to equi-distance n Clusters may exist only in some subspaces Methods n Feature transformation: only effective if most dimensions are relevant n PCA & SVD useful only when features are highly correlated/redundant n Feature selection: wrapper or filter approaches n useful to find a subspace where the data have nice clusters n Subspace-clustering: find clusters in all the possible subspaces n CLIQUE, Pro. Clus, and frequent pattern-based clustering 15 March 2018 Data Mining: Concepts and Techniques 84
The Curse of Dimensionality (graphs adapted from Parsons et al. KDD Explorations 2004) n n Data in only one dimension is relatively packed Adding a dimension “stretch” the points across that dimension, making them further apart Adding more dimensions will make the points further apart—high dimensional data is extremely sparse Distance measure becomes meaningless—due to equi-distance 15 March 2018 Data Mining: Concepts and Techniques 85
Why Subspace Clustering? (adapted from Parsons et al. SIGKDD Explorations 2004) n n 15 March 2018 Clusters may exist only in some subspaces Subspace-clustering: find clusters in all the subspaces Data Mining: Concepts and Techniques 86
CLIQUE (Clustering In QUEst) n n n 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 grid-based n n It partitions each dimension into the same number of equal length interval It partitions an m-dimensional data space into non-overlapping 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 15 March 2018 Data Mining: Concepts and Techniques 87
CLIQUE: The Major Steps n n n 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 n n n Determine dense units in all subspaces of interests Determine connected dense units in all subspaces of interests. Generate minimal description for the clusters n Determine maximal regions that cover a cluster of connected dense units for each cluster n Determination of minimal cover for each cluster 15 March 2018 Data Mining: Concepts and Techniques 88
30 40 50 20 a al ry 30 50 S 15 March 2018 30 40 50 age 60 Vacation =3 Vacation (week) 0 1 2 3 4 5 6 7 Salary (10, 000) 0 1 2 3 4 5 6 7 20 age 60 Data Mining: Concepts and Techniques age 89
Strength and Weakness of CLIQUE n n Strength n automatically finds subspaces of the highest dimensionality such that high density clusters exist in those subspaces n insensitive to the order of records in input and does not presume some canonical data distribution n scales linearly with the size of input and has good scalability as the number of dimensions in the data increases Weakness n The accuracy of the clustering result may be degraded at the expense of simplicity of the method 15 March 2018 Data Mining: Concepts and Techniques 90
Frequent Pattern-Based Approach n Clustering high-dimensional space (e. g. , clustering text documents, microarray data) n Projected subspace-clustering: which dimensions to be projected on? n CLIQUE, Pro. Clus n n n Feature extraction: costly and may not be effective? Using frequent patterns as “features” Clustering by pattern similarity in micro-array data (p. Clustering) [H. Wang, W. Wang, J. Yang, and P. S. Yu. Clustering by pattern similarity in large data sets, SIGMOD’ 02] 15 March 2018 Data Mining: Concepts and Techniques 91
Clustering by Pattern Similarity (p-Clustering) n Right: The micro-array “raw” data shows 3 genes and their values in a multi-dimensional space n n Difficult to find their patterns Bottom: Some subsets of dimensions form nice shift and scaling patterns 15 March 2018 Data Mining: Concepts and Techniques 92
Why p-Clustering? n Microarray data analysis may need to n n n Clustering on thousands of dimensions (attributes) Discovery of both shift and scaling patterns Clustering with Euclidean distance measure? — cannot find shift patterns Clustering on derived attribute Aij = ai – aj? — introduces N(N-1) dimensions Bi-cluster (Y. Cheng and G. Church. Biclustering of expression data. ISMB’ 00) using transformed mean-squared residue score matrix (I, J) n n n Where A submatrix is a δ-cluster if H(I, J) ≤ δ for some δ > 0 Problems with bi-cluster n No downward closure property n Due to averaging, it may contain outliers but still within δ-threshold 15 March 2018 Data Mining: Concepts and Techniques 93
p-Clustering: Clustering by Pattern Similarity n n n Given object x, y in O and features a, b in T, p. Cluster is a 2 by 2 matrix A pair (O, T) is in δ-p. Cluster if for any 2 by 2 matrix X in (O, T), p. Score(X) ≤ δ for some δ > 0 Properties of δ-p. Cluster n n Downward closure Clusters are more homogeneous than bi-cluster (thus the name: pair-wise Cluster) Pattern-growth algorithm has been developed for efficient mining For scaling patterns, one can observe, taking logarithmic on will lead to the p. Score form 15 March 2018 Data Mining: Concepts and Techniques 94
Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. A Categorization of Major Clustering Methods 3. Partitioning Methods 4. Hierarchical Methods 5. Density-Based Methods 6. Grid-Based Methods 7. Model-Based Methods 8. Clustering High-Dimensional Data 9. Constraint-Based Clustering 10. Link-based clustering 11. Outlier Analysis 12. Summary 15 March 2018 Data Mining: Concepts and Techniques 95
Why Constraint-Based Cluster Analysis? Need user feedback: Users know their applications the best n Less parameters but more user-desired constraints, e. g. , an ATM allocation problem: obstacle & desired clusters n 15 March 2018 Data Mining: Concepts and Techniques 96
A Classification of Constraints in Cluster Analysis n n Clustering in applications: desirable to have user-guided (i. e. , constrained) cluster analysis Different constraints in cluster analysis: n Constraints on individual objects (do selection first) n n Constraints on distance or similarity functions n n # of clusters, Min. Pts, etc. User-specified constraints n n Weighted functions, obstacles (e. g. , rivers, lakes) Constraints on the selection of clustering parameters n n Cluster on houses worth over $300 K Contain at least 500 valued customers and 5000 ordinary ones Semi-supervised: giving small training sets as “constraints” or hints 15 March 2018 Data Mining: Concepts and Techniques 97
Clustering With Obstacle Objects n n n Tung, Hou, and Han. Spatial Clustering in the Presence of Obstacles, ICDE'01 K-medoids is more preferable since kmeans may locate the ATM center in the middle of a lake Visibility graph and shortest path Triangulation and micro-clustering Two kinds of join indices (shortestpaths) worth pre-computation n VV index: indices for any pair of obstacle vertices n MV index: indices for any pair of micro-cluster and obstacle indices 15 March 2018 Data Mining: Concepts and Techniques 98
An Example: Clustering With Obstacle Objects Not Taking obstacles into account 15 March 2018 Taking obstacles into account Data Mining: Concepts and Techniques 99
User-Guided Clustering Open-course Course Work-In Professor person name course-id group office semester name position instructor area Advise Group professor name student area degree User hint Publish author title year conf Register student Student Target of clustering n n n Publication course name office semester position unit grade X. Yin, J. Han, P. S. Yu, “Cross-Relational Clustering with User's Guidance”, KDD'05 User usually has a goal of clustering, e. g. , clustering students by research area User specifies his clustering goal to Cross. Clus 15 March 2018 Data Mining: Concepts and Techniques 100
Comparing with Classification User hint n User-specified feature (in the form of attribute) is used as a hint, not class labels n The attribute may contain too many or too few distinct values, e. g. , a user may want to cluster students into 20 clusters instead of 3 n All tuples for clustering 15 March 2018 Additional features need to be included in cluster analysis Data Mining: Concepts and Techniques 101
Comparing with Semi-Supervised Clustering n n Semi-supervised clustering: User provides a training set consisting of “similar” (“must-link) and “dissimilar” (“cannot link”) pairs of objects User-guided clustering: User specifies an attribute as a hint, and more relevant features are found for clustering User-guided clustering All tuples for clustering Semi-supervised clustering 15 March 2018 x All tuples for clustering Data Mining: Concepts and Techniques 102
Why Not Semi-Supervised Clustering? n n n Much information (in multiple relations) is needed to judge whether two tuples are similar A user may not be able to provide a good training set It is much easier for a user to specify an attribute as a hint, such as a student’s research area Tom Smith Jane Chang SC 1211 BI 205 TA RA Tuples to be compared User hint 15 March 2018 Data Mining: Concepts and Techniques 103
Cross. Clus: An Overview n n Measure similarity between features by how they group objects into clusters Use a heuristic method to search for pertinent features n n Use tuple ID propagation to create feature values n n Start from user-specified feature and gradually expand search range Features can be easily created during the expansion of search range, by propagating IDs Explore three clustering algorithms: k-means, k-medoids, and hierarchical clustering 15 March 2018 Data Mining: Concepts and Techniques 104
Multi-Relational Features n n A multi-relational feature is defined by: n A join path, e. g. , Student → Register → Open. Course → Course n An attribute, e. g. , Course. area n (For numerical feature) an aggregation operator, e. g. , sum or average Categorical feature f = [Student → Register → Open. Course → Course, Course. area, null] areas of courses of each student Tuple Areas of courses DB AI 5 5 0 t 2 0 3 t 3 1 t 4 t 5 Tuple TH t 1 15 March 2018 f(t 1) Values of feature f Feature f DB AI TH t 1 0. 5 0 7 t 2 0 0. 3 0. 7 5 4 t 3 0. 1 0. 5 0. 4 5 0 5 t 4 0. 5 0 0. 5 3 3 4 t 5 0. 3 0. 4 Data Mining: Concepts and Techniques f(t 2) f(t 3) f(t 4) f(t 5) 105
Representing Features n Similarity between tuples t 1 and t 2 w. r. t. categorical feature f n Cosine similarity between vectors f(t 1) and f(t 2) Similarity vector Vf n n 15 March 2018 Most important information of a feature f is how f groups tuples into clusters f is represented by similarities between every pair of tuples indicated by f The horizontal axes are the tuple indices, and the vertical axis is the similarity This can be considered as a vector of N x N dimensions Data Mining: Concepts and Techniques 106
Similarity Between Features Vf Values of Feature f and g Feature f (course) Feature g (group) DB AI TH Info sys Cog sci Theory t 1 0. 5 0 1 0 0 t 2 0 0. 3 0. 7 0 0 1 t 3 0. 1 0. 5 0. 4 0 0. 5 t 4 0. 5 0 0. 5 t 5 0. 3 0. 4 0. 5 0 Vg Similarity between two features – cosine similarity of two vectors 15 March 2018 Data Mining: Concepts and Techniques 107
Computing Feature Similarity Feature f Tuples Feature g DB Info sys AI Cog sci TH Similarity between feature values w. r. t. the tuples sim(fk, gq)=Σi=1 to N f(ti). pk∙g(ti). pq Theory Tuple similarities, hard to compute DB Info sys AI Cog sci TH Theory 15 March 2018 Info sys DB Feature value similarities, easy to compute Compute similarity between each pair of feature values by one scan on data Data Mining: Concepts and Techniques 108
Searching for Pertinent Features n Different features convey different aspects of information Academic Performances Research area Research group area Conferences of papers Advisor n n Demographic info GPA Permanent address GRE score Nationality Number of papers Features conveying same aspect of information usually cluster tuples in more similar ways n Research group areas vs. conferences of publications Given user specified feature n Find pertinent features by computing feature similarity 15 March 2018 Data Mining: Concepts and Techniques 109
Heuristic Search for Pertinent Features Open-course Course Work-In person Overall procedure Professor name course-id group office semester name position instructor area 2 1. Start from the user- Group specified feature name 2. Search in neighborhood area of existing pertinent features User hint 3. Expand search range gradually Target of clustering n Advise Publish professor student author 1 degree title Publication title year conf Register student Student name office position course semester unit grade Tuple ID propagation is used to create multi-relational features n IDs of target tuples can be propagated along any join path, from which we can find tuples joinable with each target tuple 15 March 2018 Data Mining: Concepts and Techniques 110
Clustering with Multi-Relational Features n Given a set of L pertinent features f 1, …, f. L, similarity between two tuples n n Weight of a feature is determined in feature search by its similarity with other pertinent features Clustering methods n CLARANS [Ng & Han 94], a scalable clustering algorithm for non-Euclidean space n K-means n Agglomerative hierarchical clustering 15 March 2018 Data Mining: Concepts and Techniques 111
Experiments: Compare Cross. Clus with n n n Baseline: Only use the user specified feature PROCLUS [Aggarwal, et al. 99]: a state-of-the-art subspace clustering algorithm n Use a subset of features for each cluster n We convert relational database to a table by propositionalization n User-specified feature is forced to be used in every cluster RDBC [Kirsten and Wrobel’ 00] n A representative ILP clustering algorithm n Use neighbor information of objects for clustering n User-specified feature is forced to be used 15 March 2018 Data Mining: Concepts and Techniques 112
Measure of Clustering Accuracy n Measured by manually labeled data n n We manually assign tuples into clusters according to their properties (e. g. , professors in different research areas) Accuracy of clustering: Percentage of pairs of tuples in the same cluster that share common label n n 15 March 2018 This measure favors many small clusters We let each approach generate the same number of clusters Data Mining: Concepts and Techniques 113
DBLP Dataset 15 March 2018 Data Mining: Concepts and Techniques 114
Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. A Categorization of Major Clustering Methods 3. Partitioning Methods 4. Hierarchical Methods 5. Density-Based Methods 6. Grid-Based Methods 7. Model-Based Methods 8. Clustering High-Dimensional Data 9. Constraint-Based Clustering 10. Link-based clustering 11. Outlier Analysis 12. Summary 15 March 2018 Data Mining: Concepts and Techniques 115
Link-Based Clustering: Calculate Similarities Based On Links Authors Tom Mike Cathy John Mary Proceedings Conferences sigmod 03 sigmod 04 sigmod 05 vldb 03 vldb 04 vldb 05 aaai 04 aaai 05 The similarity between two objects x and y is defined as the average similarity between objects linked with x and those with y: vldb aaai n Jeh & Widom, KDD’ 2002: Sim. Rank Two objects are similar if they are linked with the same or similar objects 15 March 2018 n Disadv: Expensive to compute: n For a dataset of N objects and M links, it takes O(N 2) space and O(M 2) time to compute all similarities. Data Mining: Concepts and Techniques 116
Observation 1: Hierarchical Structures n Hierarchical structures often exist naturally among objects (e. g. , taxonomy of animals) Relationships between articles and words (Chakrabarti, Papadimitriou, Modha, Faloutsos, 2004) A hierarchical structure of products in Walmart grocery electronics TV DVD 15 March 2018 apparel Articles All camera Data Mining: Concepts and Techniques Words 117
Observation 2: Distribution of Similarity Distribution of Sim. Rank similarities among DBLP authors n Power law distribution exists in similarities n 56% of similarity entries are in [0. 005, 0. 015] n 1. 4% of similarity entries are larger than 0. 1 n Can we design a data structure that stores the significant similarities and compresses insignificant ones? 15 March 2018 Data Mining: Concepts and Techniques 118
A Novel Data Structure: Sim. Tree Each non-leaf node represents a group of similar lower-level nodes Each leaf node represents an object Similarities between siblings are stored Canon A 40 digital camera Digital Sony V 3 digital Cameras Consumer camera electronics Apparels TVs 15 March 2018 Data Mining: Concepts and Techniques 119
Similarity Defined by Sim. Tree Similarity between two sibling nodes n 1 and n 2 n 1 Adjustment ratio for node n 7 0. 8 0. 9 n 7 n Path-based node similarity n n 4 n 2 0. 9 0. 3 0. 8 0. 9 n 5 n 3 n 6 n 8 1. 0 n 9 simp(n 7, n 8) = s(n 7, n 4) x s(n 4, n 5) x s(n 5, n 8) Similarity between two nodes is the average similarity between objects linked with them in other Sim. Trees Average similarity between x and all other nodes Adjustment ratio for x = 15 March 2018 Average similarity between x’s parent and all other nodes Data Mining: Concepts and Techniques 120
Link. Clus: Efficient Clustering via Heterogeneous Semantic Links X. Yin, J. Han, and P. S. Yu, “Link. Clus: Efficient Clustering via Heterogeneous Semantic Links”, VLDB'06 Method n Initialize a Sim. Tree for objects of each type n Repeat n For each Sim. Tree, update the similarities between its nodes using similarities in other Sim. Trees n Similarity between two nodes x and y is the average similarity between objects linked with them n Adjust the structure of each Sim. Tree n Assign each node to the parent node that it is most similar to 15 March 2018 Data Mining: Concepts and Techniques 121
Initialization of Sim. Trees n n Initializing a Sim. Tree n Repeatedly find groups of tightly related nodes, which are merged into a higher-level node Tightness of a group of nodes n For a group of nodes {n 1, …, nk}, its tightness is defined as the number of leaf nodes in other Sim. Trees that are connected to all of {n 1, …, nk} Leaf nodes in Nodes another Sim. Tree n 1 n 2 15 March 2018 1 2 3 4 5 The tightness of {n 1, n 2} is 3 Data Mining: Concepts and Techniques 122
Finding Tight Groups by Freq. Pattern Mining n Finding tight groups Frequent pattern mining Reduced to The tightness of a g 1 group of nodes is the support of a frequent pattern g 2 n n 1 n 2 n 3 n 4 Transactions 1 2 3 4 5 6 7 8 9 {n 1} {n 1, n 2} {n 2, n 3, n 4} {n 3, n 4} Procedure of initializing a tree n Start from leaf nodes (level-0) n At each level l, find non-overlapping groups of similar nodes with frequent pattern mining 15 March 2018 Data Mining: Concepts and Techniques 123
Updating Similarities Between Nodes n n The initial similarities can seldom capture the relationships between objects Iteratively update similarities n Similarity between two nodes is the average similarity between objects linked with them 0 2 1 4 5 ST 2 3 6 7 8 sim(na, nb) = average similarity between 9 a b o p q r 15 March 2018 14 e f l m n and 13 takes O(3 x 2) time ST 1 d c 11 12 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 z 10 g s h t u v w k x y Data Mining: Concepts and Techniques 124
Aggregation-Based Similarity Computation 0. 2 4 0. 9 10 1. 0 0. 8 11 a 12 5 0. 9 1. 0 13 ST 2 14 b ST 1 For each node nk ∈ {n 10, n 11, n 12} and nl ∈ {n 13, n 14}, their pathbased similarity simp(nk, nl) = s(nk, n 4)·s(n 4, n 5)·s(n 5, nl). takes O(3+2) time After aggregation, we reduce quadratic time computation to linear time computation. 15 March 2018 Data Mining: Concepts and Techniques 125
Computing Similarity with Aggregation Average similarity and total weight sim(na, nb) can be computed from aggregated similarities a: (0. 9, 3 ) 4 10 11 0. 2 12 a b: (0. 95, 2) 5 13 14 b sim(na, nb) = avg_sim(na, n 4) x s(n 4, n 5) x avg_sim(nb, n 5) = 0. 9 x 0. 2 x 0. 95 = 0. 171 To compute sim(na, nb): n n n Find all pairs of sibling nodes ni and nj, so that na linked with ni and nb with nj. Calculate similarity (and weight) between na and nb w. r. t. ni and nj. Calculate weighted average similarity between na and nb w. r. t. all such pairs. 15 March 2018 Data Mining: Concepts and Techniques 126
Adjusting Sim. Tree Structures n 1 n 4 0. 8 n 7 n 0. 9 n 2 n 5 n 7 n 8 n 3 n 6 n 9 After similarity changes, the tree structure also needs to be changed n If a node is more similar to its parent’s sibling, then move it to be a child of that sibling n Try to move each node to its parent’s sibling that it is most similar to, under the constraint that each parent node can have at most c children 15 March 2018 Data Mining: Concepts and Techniques 127
Complexity For two types of objects, N in each, and M linkages between them. Time Space Updating similarities O(M(log. N)2) O(M+N) Adjusting tree structures O(N) Link. Clus O(M(log. N)2) O(M+N) Sim. Rank O(M 2) O(N 2) 15 March 2018 Data Mining: Concepts and Techniques 128
Experiment: Email Dataset n n n F. Nielsen. Email dataset. Approach www. imm. dtu. dk/~rem/data/Email-1431. zip Link. Clus 370 emails on conferences, 272 on jobs, and 789 spam emails Sim. Rank Accuracy: measured by manually labeled Re. Com data F-Sim. Rank Accuracy of clustering: % of pairs of objects in the same cluster that share common label CLARANS Accuracy time (s) 0. 8026 1579. 6 0. 7965 39160 0. 5711 74. 6 0. 3688 479. 7 0. 4768 8. 55 Approaches compared: n Sim. Rank (Jeh & Widom, KDD 2002): Computing pair-wise similarities n Sim. Rank with Finger. Prints (F-Sim. Rank): Fogaras & R´acz, WWW 2005 n n pre-computes a large sample of random paths from each object and uses samples of two objects to estimate Sim. Rank similarity Re. Com (Wang et al. SIGIR 2003) n 15 March 2018 Iteratively clustering objects using cluster labels of linked objects Data Mining: Concepts and Techniques 129
Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. A Categorization of Major Clustering Methods 3. Partitioning Methods 4. Hierarchical Methods 5. Density-Based Methods 6. Grid-Based Methods 7. Model-Based Methods 8. Clustering High-Dimensional Data 9. Constraint-Based Clustering 10. Link-based clustering 11. Outlier Analysis 12. Summary 15 March 2018 Data Mining: Concepts and Techniques 130
What Is Outlier Discovery? n What are outliers? n n The set of objects are considerably dissimilar from the remainder of the data Example: Sports: Michael Jordon, Wayne Gretzky, . . . n Problem: Define and find outliers in large data sets n Applications: n Credit card fraud detection n Telecom fraud detection n Customer segmentation n Medical analysis 15 March 2018 Data Mining: Concepts and Techniques 131
Outlier Discovery: Statistical Approaches Assume a model underlying distribution that generates data set (e. g. normal distribution) n Use discordancy tests depending on n data distribution n distribution parameter (e. g. , mean, variance) n number of expected outliers n Drawbacks n most tests are for single attribute n In many cases, data distribution may not be known 15 March 2018 Data Mining: Concepts and Techniques 132
Outlier Discovery: Distance-Based Approach n n n Introduced to counter the main limitations imposed by statistical methods n 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 [Knorr & Ng, VLDB’ 98] n Index-based algorithm n Nested-loop algorithm n Cell-based algorithm 15 March 2018 Data Mining: Concepts and Techniques 133
Density-Based Local Outlier Detection n M. M. Breunig, H. -P. Kriegel, R. Ng, J. Sander. LOF: Identifying Density-Based Local Outliers. SIGMOD 2000. n Distance-based outlier detection is based on global distance distribution n It encounters difficulties to identify outliers if data is not uniformly distributed n n Ex. C 1 contains 400 loosely distributed points, C 2 has 100 tightly condensed n Need the concept of local outlier Local outlier factor (LOF) n Assume outlier is not crisp n Each point has a LOF points, 2 outlier points o 1, o 2 n Distance-based method cannot identify o 2 as an outlier 15 March 2018 Data Mining: Concepts and Techniques 134
Outlier Discovery: Deviation-Based Approach n n n Identifies outliers by examining the main characteristics of objects in a group Objects that “deviate” from this description are considered outliers Sequential exception technique n n simulates the way in which humans can distinguish unusual objects from among a series of supposedly like objects OLAP data cube technique n uses data cubes to identify regions of anomalies in large multidimensional data 15 March 2018 Data Mining: Concepts and Techniques 135
Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. A Categorization of Major Clustering Methods 3. Partitioning Methods 4. Hierarchical Methods 5. Density-Based Methods 6. Grid-Based Methods 7. Model-Based Methods 8. Clustering High-Dimensional Data 9. Constraint-Based Clustering 10. Link-based clustering 11. Outlier Analysis 12. Summary 15 March 2018 Data Mining: Concepts and Techniques 136
Summary n n n 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 15 March 2018 Data Mining: Concepts and Techniques 137
Problems and Challenges n Considerable progress has been made in scalable clustering methods n n Hierarchical: BIRCH, ROCK, CHAMELEON n Density-based: DBSCAN, OPTICS, Den. Clue n Grid-based: STING, Wave. Cluster, CLIQUE n Model-based: EM, Cobweb, SOM n Frequent pattern-based: p. Cluster n n Partitioning: k-means, k-medoids, CLARANS Constraint-based: COD, constrained-clustering Current clustering techniques do not address all the requirements adequately, still an active area of research 15 March 2018 Data Mining: Concepts and Techniques 138
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References (3) n n n G. J. Mc. Lachlan and K. E. Bkasford. Mixture Models: Inference and Applications to Clustering. John Wiley and Sons, 1988. P. Michaud. Clustering Techniques. Future Generation Computer Systems, 13, 1997. R. Ng and J. Han. Efficient and effective clustering method for spatial data mining. VLDB'94. L. Parsons, E. Haque and H. Liu, Subspace Clustering for High Dimensional Data: A Review, SIGKDD Explorations, 6(1), June 2004 E. Schikuta. Grid clustering: An efficient hierarchical clustering method for very large data sets. Proc. 1996 Int. Conf. on Pattern Recognition, . G. Sheikholeslami, S. Chatterjee, and A. Zhang. Wave. Cluster: A multi-resolution clustering approach for very large spatial databases. VLDB’ 98. A. K. H. Tung, J. Han, L. V. S. Lakshmanan, and R. T. Ng. Constraint-Based Clustering in Large Databases, ICDT'01. A. K. H. Tung, J. Hou, and J. Han. Spatial Clustering in the Presence of Obstacles, ICDE'01 H. Wang, W. Wang, J. Yang, and P. S. Yu. Clustering by pattern similarity in large data sets, SIGMOD’ 02. W. Wang, Yang, R. Muntz, STING: A Statistical Information grid Approach to Spatial Data Mining, VLDB’ 97. T. Zhang, R. Ramakrishnan, and M. Livny. BIRCH : An efficient data clustering method for very large databases. SIGMOD'96. 15 March 2018 Data Mining: Concepts and Techniques 141
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Clustering with User-Specified Constraints n n n Tung, Han, Lakshmanan, and Ng. Constraint-Based Clustering in Large Databases, ICDT'01 Example: Locating k delivery centers, each serving at least m valued customers and n ordinary ones Proposed approach n n n Find an initial “solution” by partitioning the data set into k groups and satisfying user-constraints Iteratively refine the solution by micro-clustering relocation (e. g. , moving δ micro-clusters from cluster Ci to Cj) and “deadlock” handling (breaking the microclusters when necessary) Efficiency is improved by micro-clustering 15 March 2018 Data Mining: Concepts and Techniques 143