1、第5课 数据聚类技术,徐从富，副教授 浙江大学人工智能研究所,浙江大学本科生数据挖掘导论课件,课程提纲,What is Cluster Analysis? Types of Data in Cluster Analysis A Categorization of Major Clustering Methods Partitioning Methods Hierarchical Methods Summary Reference,What is Cluster Analysis?,Cluster: a collection of data objects Similar to one anot
2、her 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 As a stand-alone tool to get insi
3、ght into data distribution As a preprocessing step for other algorithms,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 Proce
4、ssing Economic Science (especially market research) WWW Document classification Cluster Weblog data to discover groups of similar access patterns,Examples of Clustering Applications,Marketing: Help marketers discover distinct groups in their customer bases, and then use this knowledge to develop tar
5、geted 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 geograp
6、hical location Earth-quake studies: Observed earth quake epicenters should be clustered along continent faults,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
7、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,Measure the Quality of Clustering,Dissimilarity/Similarity metric: Similarity is expressed in terms of a
8、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, boolean, categorical, ordinal ratio, and vector variables. Weights should be associated
9、 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.,Requirements of Clustering in Data Mining,Scalability Ability to deal with different types of attributes Ability to handle dynamic dat
10、a 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,Data Structur
11、es,Data matrix (two modes)Dissimilarity matrix (one mode),Types of Data in Cluster Analysis,Type of data in clustering analysis,Interval-scaled variables（区间标度变量） Binary variables（二元变量） Nominal, ordinal, and ratio variables（标称型、序数型、比例标度型） Variables of mixed types,Interval-valued variables,区间标度变量是一个粗略
12、线性标度的连续度量 Standardize data Calculate the mean absolute deviation:where Calculate the standardized measurement (z-score)Using mean absolute deviation is more robust than using standard deviation,Similarity and Dissimilarity Between Objects,Distances are normally used to measure the similarity or diss
13、imilarity between two data objects Some popular ones include: Minkowski distance:where i = (xi1, xi2, , xip) and j = (xj1, xj2, , xjp) are two p-dimensional data objects, and q is a positive integer If q = 1, d is Manhattan distance,Similarity and Dissimilarity Between Objects (Cont.),If q = 2, d is
14、 Euclidean distance:Properties d(i,j) 0 d(i,i) = 0 d(i,j) = d(j,i) d(i,j) d(i,k) + d(k,j),Dissimilarity Between Binary Variables,A contingency table for binary dataDistance measure for symmetric binary variables: Distance measure for asymmetric binary variables: Jaccard coefficient (similarity measu
15、re for asymmetric binary variables):,Dissimilarity between Binary Variables,Examplegender 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,Nominal Variables（标称型）,A generalization of the binary variable in that it
16、can take more than 2 states, e.g., red, yellow, blue, green Method 1: Simple matching m: # of matches,p: total # of variablesMethod 2: use a large number of binary variables creating a new binary variable for each of the M nominal states,Ordinal Variables（序数型）,An ordinal variable can be discrete or
17、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 bycompute the dissimilarity using methods for interval-scaled variables,Ratio-Scaled Variables（比例标度型）,Ratio-s
18、caled variable: a positive measurement on a nonlinear scale, approximately at exponential scale, such as AeBt or Ae-Bt Methods: treat them like interval-scaled variablesnot a good choice! (why?the scale can be distorted) apply logarithmic transformation yif = log(xif) treat them as continuous ordina
19、l data treat their rank as interval-scaled,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 effectsf is binary or nominal: dij(f) = 0 if xif = xjf ,
20、 or dij(f) = 1 o.w. f is interval-based: use the normalized distance f is ordinal or ratio-scaled compute ranks rif and and treat zif as interval-scaled,Major Clustering Approaches,Partitioning approach: Construct various partitions and then evaluate them by some criterion, e.g., minimizing the sum
21、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 Typic
22、al methods: DBSACN, OPTICS, DenClue,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 oth
23、er 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,Typi
24、cal 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)
25、 = 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) =
26、dis(Mi, Mj) Medoid: one chosen, centrally located object in the cluster,Centroid, Radius and Diameter of a Cluster (for numerical data sets),Centroid: the “middle” of a clusterRadius: square root of average distance from any point of the cluster to its centroidDiameter: square root of average mean s
27、quared distance between all pairs of points in the cluster,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 distanceGiven a k, find a partition of k clusters that optimizes the chosen par
28、titioning criterion Global optimal: exhaustively enumerate all partitions Heuristic methods: k-means and k-medoids algorithms k-means (MacQueen67): Each cluster is represented by the center of the cluster k-medoids or PAM (Partition around medoids) (Kaufman & Rousseeuw87): Each cluster is represente
29、d by one of the objects in the cluster,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
30、the cluster) Assign each object to the cluster with the nearest seed point Go back to Step 2, stop when no more new assignment,The K-Means Clustering Method,Example,0,1,2,3,4,5,6,7,8,9,10,0,1,2,3,4,5,6,7,8,9,10,K=2 Arbitrarily choose K object as initial cluster center,Assign each objects to most sim
31、ilar center,Update the cluster means,Update the cluster means,reassign,reassign,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 ter
32、minates 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
33、outliers Not suitable to discover clusters with non-convex shapes,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 (Huang98) Replacing me
34、ans 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,What Is the Problem of the K-Means Method?,The k-means algorithm is sensitive to out
35、liers! 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.,The K-Medoids Cluste
36、ring 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
37、 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),A Typical K-Medoids Algorithm (PAM),Total Cost = 20,0,1,2,3,4,5,6,7,8,9,10,0,1,2,3,4,5,6,7,8,9,10,
38、K=2,Arbitrary choose k object as initial medoids,Assign each remaining object to nearest medoids,Randomly select a nonmedoid object,Oramdom,Compute total cost of swapping,Total Cost = 26,Swapping O and Oramdom If quality is improved.,Do loop Until no change,PAM (Partitioning Around Medoids) (1987),P
39、AM (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 h and selected object i, calculate the total swapping cost TCih For each pair of i and h, If TCih 0, i is replaced by h Then assi
40、gn each non-selected object to the most similar representative object repeat steps 2-3 until there is no change,PAM Clustering: Total swapping cost TCih=jCjih,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 ou
41、tliers 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),CLARA (Clustering Large Applications) (
42、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
43、 size A good clustering based on samples will not necessarily represent a good clustering of the whole data set if the sample is biased,CLARANS (“Randomized” CLARA) (1994),CLARANS (A Clustering Algorithm based on Randomized Search) (Ng and Han94) CLARANS draws sample of neighbors dynamically The clu
44、stering 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 Focusi
45、ng techniques and spatial access structures may further improve its performance (Ester et al.95),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,AGNES (Agglomerative Nesting),Intr
46、oduced 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,Dendrogram: Shows
47、 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.,DIANA (Divisive Analys
48、is),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,Summary,Cluster analysis groups objects based on their similarity and has wide applications Measure of similarity can be co
49、mputed 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,Problems and Challenges,
