Data Mining Concepts and Techniques Clustering 1272022 Data

Data Mining: Concepts and Techniques Clustering 1/27/2022 Data Mining: Concepts and Techniques 1

Chapter 8. Cluster Analysis n What is Cluster Analysis? n Types of Data in Cluster Analysis n A Categorization of Major Clustering Methods n Partitioning Methods n Hierarchical Methods n Density-Based Methods n Grid-Based Methods n Outlier Analysis n Summary 1/27/2022 Data Mining: Concepts and Techniques 2

What is Cluster Analysis? n n Cluster: a collection of data objects n Similar to one another within the same cluster n Dissimilar to the objects in other clusters Cluster analysis n Grouping a set of data objects into clusters Clustering is unsupervised classification: no predefined classes Typical applications n As a stand-alone tool to get insight into data distribution n As a preprocessing step for other algorithms

General Applications of Clustering n n n Pattern Recognition Spatial Data Analysis n create thematic maps in GIS by clustering feature spaces n detect spatial clusters and explain them in spatial data mining Image Processing Economic Science (especially market research) WWW n Document classification n Cluster Weblog data to discover groups of similar access patterns 1/27/2022 Data Mining: Concepts and Techniques 4

Examples of Clustering Applications n n n 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 1/27/2022 Data Mining: Concepts and Techniques 5

What Is Good Clustering? n n n A good clustering method will produce high quality clusters with n high intra-class similarity n 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. 1/27/2022 Data Mining: Concepts and Techniques 6

Requirements of Clustering in Data Mining n Scalability n Ability to deal with different types of attributes 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 1/27/2022 Data Mining: Concepts and Techniques 7

Chapter 8. Cluster Analysis n What is Cluster Analysis? n Types of Data in Cluster Analysis n A Categorization of Major Clustering Methods n Partitioning Methods n Hierarchical Methods n Density-Based Methods n Grid-Based Methods n Outlier Analysis n Summary 1/27/2022 Data Mining: Concepts and Techniques 8

Data Structures n n 1/27/2022 Data matrix n (two modes) Dissimilarity matrix n (one mode) Data Mining: Concepts and Techniques 9

Measure the Quality of Clustering n n n Dissimilarity/Similarity metric: Similarity is expressed in terms of a distance function, which is typically metric: d(i, j) There is a separate “quality” function that measures the “goodness” of a cluster. The definitions of distance functions are usually very different for interval-scaled, boolean, categorical, ordinal and ratio variables. Weights should be associated with different variables based on applications and data semantics. It is hard to define “similar enough” or “good enough” n the answer is typically highly subjective. 1/27/2022 Data Mining: Concepts and Techniques 10

Type of data in clustering analysis n Interval-scaled variables n Binary variables n Nominal, ordinal, and ratio variables n Variables of mixed types 1/27/2022 Data Mining: Concepts and Techniques 11

Interval-valued variables n Standardize data n Calculate the mean absolute deviation: where n n Calculate the standardized measurement (z-score) Using mean absolute deviation is more robust than using standard deviation 1/27/2022 Data Mining: Concepts and Techniques 12

Similarity and Dissimilarity Between Objects n n Distances are normally used to measure the similarity or dissimilarity between two data objects Some popular ones include: Minkowski distance: where i = (xi 1, xi 2, …, xip) and j = (xj 1, xj 2, …, xjp) are two p-dimensional data objects, and q is a positive integer n If q = 1, d is Manhattan distance 1/27/2022 Data Mining: Concepts and Techniques 13

Similarity and Dissimilarity Between Objects (Cont. ) n If q = 2, d is Euclidean distance: n Properties n n n d(i, j) 0 d(i, i) = 0 d(i, j) = d(j, i) d(i, j) d(i, k) + d(k, j) Also one can use weighted distance, parametric Pearson product moment correlation, or other dissimilarity measures. 1/27/2022 Data Mining: Concepts and Techniques 14

Binary Variables n A contingency table for binary data Object j Object i n Simple matching coefficient (invariant, if the binary variable is symmetric): n Jaccard coefficient (noninvariant if the binary variable is asymmetric): 1/27/2022 Data Mining: Concepts and Techniques 15

Dissimilarity Between Binary Variables: Example n n n 1/27/2022 gender is a symmetric attribute, the remaining attributes are asymmetric let the values Y and P be set to 1, and the value N be set to 0 consider only asymmetric attributes Data Mining: Concepts and Techniques 16

Nominal Variables n n 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 n n m: # of matches, p: total # of variables Method 2: use a large number of binary variables n 1/27/2022 creating a new binary variable for each of the M nominal states Data Mining: Concepts and Techniques 17

Ordinal Variables n An ordinal variable can be discrete or continuous n Order is important, e. g. , rank n Can be treated like interval-scaled n n n 1/27/2022 replacing xif by their rank map the range of each variable onto [0, 1] by replacing i-th object in the f-th variable by compute the dissimilarity using methods for intervalscaled variables Data Mining: Concepts and Techniques 18

Ratio-Scaled Variables n n Ratio-scaled variable: a positive measurement on a nonlinear scale, approximately at exponential scale, such as Ae. Bt or Ae-Bt Methods: n treat them like interval-scaled variables n apply logarithmic transformation yif = log(xif) n 1/27/2022 treat them as continuous ordinal data treat their rank as interval-scaled. Data Mining: Concepts and Techniques 19

Variables of Mixed Types n n A database may contain all the six types of variables n symmetric binary, asymmetric binary, nominal, ordinal, interval and ratio. One may use a weighted formula to combine their effects. n n n 1/27/2022 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 n compute ranks rif and treat zif as interval-scaled Data Mining: Concepts and Techniques 20

Chapter 8. Cluster Analysis n What is Cluster Analysis? n Types of Data in Cluster Analysis n A Categorization of Major Clustering Methods n Partitioning Methods n Hierarchical Methods n Density-Based Methods n Grid-Based Methods n Outlier Analysis n Summary 1/27/2022 Data Mining: Concepts and Techniques 21

Major Clustering Approaches n Partitioning algorithms: Construct various partitions and then evaluate them by some criterion n Hierarchy algorithms: Create a hierarchical decomposition of the set of data (or objects) using some criterion n Density-based: based on connectivity and density functions n Grid-based: based on a multiple-level granularity structure n Model-based: A model is hypothesized for each of the clusters and the idea is to find the best fit of that model to each other 1/27/2022 Data Mining: Concepts and Techniques 22

Chapter 8. Cluster Analysis n What is Cluster Analysis? n Types of Data in Cluster Analysis n A Categorization of Major Clustering Methods n Partitioning Methods n Hierarchical Methods n Density-Based Methods n Grid-Based Methods n Outlier Analysis n Summary 1/27/2022 Data Mining: Concepts and Techniques 23

Partitioning Algorithms: Basic Concept n n Partitioning method: Construct a partition of a database D of n objects into a set of k clusters Given k, find a partition of k clusters that optimizes the chosen partitioning criterion n Global optimality: exhaustively enumerate all partitions n Heuristic methods: k-means and k-medoids algorithms n k-means (Mac. Queen’ 67): Each cluster is represented by the center of the cluster n k-medoids or PAM (Partition around medoids) (Kaufman & Rousseeuw’ 87): Each cluster is represented by one of the objects in the cluster 1/27/2022 Data Mining: Concepts and Techniques 24

The K-Means Clustering Method n 1/27/2022 Given k, the k-means algorithm is implemented in 4 steps: n Partition objects into k nonempty subsets n Compute seed points as the centroids of the clusters of the current partition. The centroid is the center (mean point) of the cluster. n Assign each object to the cluster with the nearest seed point. n Go back to Step 2, stop when no more new assignment. Data Mining: Concepts and Techniques 25

The K-Means Clustering Method n Example 1/27/2022 Data Mining: Concepts and Techniques 26

Comments on the K-Means Method n Strength n n n Relatively efficient: O(tkn), where n is # objects, k is # clusters, and t is # iterations. Normally, k, t << n. 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 1/27/2022 Data Mining: Concepts and Techniques 27

Variations of the K-Means Method n n A few variants of the k-means which differ in n Selection of the initial k means n Dissimilarity calculations n Strategies to calculate cluster means Handling categorical data: k-modes 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 1/27/2022 Data Mining: Concepts and Techniques 28

The K-Medoids Clustering Method n Find representative objects, called medoids, in clusters n PAM (Partitioning Around Medoids) n n starts from an initial set of medoids and iteratively replaces one of the medoids by one of the nonmedoids 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 n n CLARANS 1/27/2022 Data Mining: Concepts and Techniques 29

PAM (Partitioning Around Medoids) 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 1/27/2022 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 Data Mining: Concepts and Techniques 30

PAM Clustering: Total swapping cost TCih= j. Cjih j t t j i h h i h j i i t h j t 1/27/2022 Data Mining: Concepts and Techniques 31

CLARA (Clustering Large Applications) (1990) n CLARA n n 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 n Strength: deals with larger data sets than PAM n Weakness: n n 1/27/2022 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 Data Mining: Concepts and Techniques 32

CLARANS (“Randomized” CLARA) n CLARANS (A Clustering Algorithm based on Randomized Search) n n n CLARANS draws sample of neighbors dynamically The clustering process can be presented as searching a graph where every node is a potential solution, that is, a set of k medoids If the local optimum is found, CLARANS starts with new randomly selected node in search for a new local optimum It is more efficient and scalable than both PAM and CLARA Focusing techniques and spatial access structures may further improve its performance 1/27/2022 Data Mining: Concepts and Techniques 33

Chapter 8. Cluster Analysis n What is Cluster Analysis? n Types of Data in Cluster Analysis n A Categorization of Major Clustering Methods n Partitioning Methods n Hierarchical Methods n Density-Based Methods n Grid-Based Methods n Outlier Analysis n Summary 1/27/2022 Data Mining: Concepts and Techniques 34

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 1/27/2022 agglomerative (AGNES) Step 3 Step 2 Step 1 Step 0 Data Mining: Concepts and Techniques divisive (DIANA) 35

More on Hierarchical Clustering Methods n n Major weakness of agglomerative clustering methods 2 n do not scale well: time complexity of at least O(n ), where n is the number of total objects n can never undo what was done previously Integration of hierarchical with distance-based clustering n BIRCH (1996): uses CF-tree and incrementally adjusts the quality of sub-clusters n CURE (1998): selects well-scattered points from the cluster and then shrinks them towards the center of the cluster by a specified fraction n CHAMELEON (1999): hierarchical clustering using dynamic modeling 1/27/2022 Data Mining: Concepts and Techniques 36

BIRCH (1996) 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 n Scales linearly: finds a good clustering with a single scan n Weakness: handles only numeric data, and sensitive to the and improves the quality with a few additional scans order of the data record. Data Mining: Concepts and Techniques 1/27/2022 37

Clustering Feature Vector Clustering Feature: CF = (N, LS, SS) N: Number of data points LS: Ni=1=Xi SS: Ni=1=Xi 2 CF = (5, (16, 30), (54, 190)) (3, 4) (2, 6) (4, 5) (4, 7) (3, 8) 1/27/2022 Data Mining: Concepts and Techniques 38

CF Tree 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 1/27/2022 CF 6 next Leaf node prev CF 1 CF 2 Data Mining: Concepts and Techniques CF 4 next 39

Insertion into CF-tree n Identify the appropriate leaf n n Starting from the root, descend the tree by choosing the closest child node Modify the leaf n n If the leaf can “absorb” (the radius has to be less than T), update the CF vector Else, add new entry to leaf n n n If there is room for the entry, DONE Else, split the leaf node by choosing the farthest pair of entries as seeds Modify the path to the leaf n n 1/27/2022 Update each nonleaf entry on the path to the leaf In case of split, do as the B+-trees do! Data Mining: Concepts and Techniques 40

Chapter 8. Cluster Analysis n What is Cluster Analysis? n Types of Data in Cluster Analysis n A Categorization of Major Clustering Methods n Partitioning Methods n Hierarchical Methods n Density-Based Methods n Grid-Based Methods n Outlier Analysis n Summary 1/27/2022 Data Mining: Concepts and Techniques 41

Density-Based Clustering Methods n n n 1/27/2022 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) Data Mining: Concepts and Techniques 42

Density Concepts n Core object (CO) – object with at least ‘M’ objects within a radius ‘E-neighborhood’ n Directly density reachable (DDR) – x is CO, y is in x’s ‘Eneighborhood’ n Density reachable – there exists a chain of DDR objects from x to y n Density based cluster – set of density connected objects that is maximal w. r. t. density-reachability 1/27/2022 Data Mining: Concepts and Techniques 43

Density-Based Clustering: Background n n n Two parameters: n Eps: Maximum radius of the neighbourhood n 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 wrt. Eps, Min. Pts if n 1) p belongs to NEps(q) n 2) core point condition: |NEps (q)| >= Min. Pts 1/27/2022 Data Mining: Concepts and Techniques p q Min. Pts = 5 Eps = 1 cm 44

Density-Based Clustering: Background (II) n Density-reachable: n n p A point p is density-reachable from a point q wrt. 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 1 q Density-connected n A point p is density-connected to a point q wrt. Eps, Min. Pts if there is a point o such that both, p and q are density-reachable from o wrt. Eps and Min. Pts. 1/27/2022 p Data Mining: Concepts and Techniques q o 45

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 1/27/2022 Min. Pts = 5 Data Mining: Concepts and Techniques 46

DBSCAN: The Algorithm n n n 1/27/2022 Select an arbitrary point p Retrieve all points density-reachable from p wrt 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. Data Mining: Concepts and Techniques 47

Chapter 8. Cluster Analysis n What is Cluster Analysis? n Types of Data in Cluster Analysis n A Categorization of Major Clustering Methods n Partitioning Methods n Hierarchical Methods n Density-Based Methods n Grid-Based Methods n Outlier Analysis n Summary 1/27/2022 Data Mining: Concepts and Techniques 48

Grid-Based Clustering Method n Using multi-resolution grid data structure n Several interesting methods n n STING (a STatistical INformation Grid approach) by Wang, Yang and Muntz (1997) Wave. Cluster by Sheikholeslami, Chatterjee, and Zhang (VLDB’ 98) n n 1/27/2022 A multi-resolution clustering approach using wavelet method CLIQUE: Agrawal, et al. (SIGMOD’ 98) Data Mining: Concepts and Techniques 49

CLIQUE (Clustering In QUEst) n Automatically identifying subspaces of a high dimensional data space that allow better clustering than original space n CLIQUE can be considered as both density-based and gridbased n It partitions each dimension into the same number of equal length interval n It partitions an m-dimensional data space into nonoverlapping rectangular units n A unit is dense if the fraction of total data points contained in the unit exceeds the input model parameter n A cluster is a maximal set of connected dense units within a subspace 1/27/2022 Data Mining: Concepts and Techniques 50

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 1/27/2022 Data Mining: Concepts and Techniques 51

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Strength and Weakness of CLIQUE n n Strength n It automatically finds subspaces of the highest dimensionality such that high density clusters exist in those subspaces n It is insensitive to the order of records in input and does not presume some canonical data distribution n It 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 1/27/2022 Data Mining: Concepts and Techniques 53

Chapter 8. Cluster Analysis n What is Cluster Analysis? n Types of Data in Cluster Analysis n A Categorization of Major Clustering Methods n Partitioning Methods n Hierarchical Methods n Density-Based Methods n Grid-Based Methods n Outlier Analysis n Summary 1/27/2022 Data Mining: Concepts and Techniques 54

What Is Outlier Discovery? n n n 1/27/2022 What are outliers? n The set of objects are considerably dissimilar from the remainder of the data n Example: Sports: Michael Jordon, Wayne Gretzky, . . . Problem n Find top n outlier points Applications: n Credit card fraud detection n Telecom fraud detection n Customer segmentation n Medical analysis Data Mining: Concepts and Techniques 55

Outlier Discovery: Statistical Approaches § § § Assume a model underlying distribution that generates data set (e. g. normal distribution) Use discordancy tests depending on § data distribution § distribution parameter (e. g. , mean, variance) § number of expected outliers Drawbacks § most tests are for single attribute § in many cases, data distribution may not be known 1/27/2022 Data Mining: Concepts and Techniques 56

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 n Index-based algorithm n Nested-loop algorithm n Cell-based algorithm

Chapter 8. Cluster Analysis n What is Cluster Analysis? n Types of Data in Cluster Analysis n A Categorization of Major Clustering Methods n Partitioning Methods n Hierarchical Methods n Density-Based Methods n Grid-Based Methods n Outlier Analysis n Summary 1/27/2022 Data Mining: Concepts and Techniques 59

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, such as constraint-based clustering 1/27/2022 Data Mining: Concepts and Techniques 60