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8 Cluster Analysis
8.1 What is cluster analysis? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Types of data in clustering analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Interval-scaled variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Binary variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Nominal, ordinal, and ratio-scaled variables . . . . . . . . . . . . . . . . . . . . . 8.2.4 Variables of mixed types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 A categorization of major clustering methods . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Partitioning methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Classical partitioning methods: k-means and k-medoids . . . . . . . . . . . . . . 8.4.2 Partitioning methods in large databases: from k-medoids to CLARANS . . . . . 8.5 Hierarchical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Agglomerative and divisive hierarchical clustering . . . . . . . . . . . . . . . . . . 8.5.2 BIRCH: Balanced Iterative Reducing and Clustering using Hierarchies . . . . . . 8.5.3 CURE: Clustering Using REpresentatives . . . . . . . . . . . . . . . . . . . . . . 8.5.4 CHAMELEON: A hierarchical clustering algorithm using dynamic modeling . . 8.6 Density-based methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 DBSCAN: A density-based clustering method based on connected regions with su density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 OPTICS: Ordering Points To Identify the Clustering Structure . . . . . . . . . . 8.6.3 DENCLUE: Clustering based on density distribution functions . . . . . . . . . . 8.7 Grid-based methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.1 STING: A STatistical INformation Grid approach . . . . . . . . . . . . . . . . . 8.7.2 WaveCluster: Clustering using wavelet transformation . . . . . . . . . . . . . . . 8.7.3 CLIQUE: Clustering high-dimensional space . . . . . . . . . . . . . . . . . . . . . 8.8 Model-based clustering methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.1 Statistical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.2 Neural network approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 Outlier analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9.1 Statistical-based outlier detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9.2 Distance-based outlier detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9.3 Deviation-based outlier detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ciently high ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ 7 9 9 11 12 14 14 16 16 18 19 20 21 22 24 25 25 26 27 29 29 31 32 33 33 34 35 36 37 38 39
List of Figures
8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15 8.16 8.17 8.18 8.19 The k-means algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Clustering of a set of objects based on the k-means method. . . . . . . . . . . . . . . . . . . . Four cases of the cost function for k-medoids clustering. . . . . . . . . . . . . . . . . . . . . . The k-medoids algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Agglomerative and divisive hierarchical clustering. . . . . . . . . . . . . . . . . . . . . . . . . A CF-tree structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Clustering of a set of points or objects by CURE. . . . . . . . . . . . . . . . . . . . . . . . . CHAMELEON: Hierarchical clustering based on k-nearest neighbors and dynamic modeling. Density reachability and density connectivity in density-based clustering. . . . . . . . . . . . OPTICS terminology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cluster ordering in OPTICS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Possible density functions for a 2-D data set. . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of center-de ned clusters and arbitrary-shaped clusters. . . . . . . . . . . . . . . . A hierarchical structure for STING clustering. . . . . . . . . . . . . . . . . . . . . . . . . . . . A sample of 2-dimensional feature space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiresolution of the feature space in Figure 8.15. . . . . . . . . . . . . . . . . . . . . . . . . CLIQUE: determining dense units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A classi cation tree. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An architecture for competitive learning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 17 18 19 20 21 23 24 26 27 27 28 29 30 31 32 43 44 44
LIST OF FIGURES
List of Tables
8.1 A contingency table for binary variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 8.2 A relational table containing mostly binary attributes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 8.3 A relational table containing binary attributes for a penpal matching service. . . . . . . . . . . . . . . 41
LIST OF TABLES
c J. Han and M. Kamber, 2000, DRAFT!! DO NOT COPY!! DO NOT DISTRIBUTE!!
January 16, 2000
Imagine that you are given a set of data objects for analysis where, unlike in classi cation, the class label of each object is not known. Clustering is the process of grouping the data into classes or clusters so that objects within a cluster have high similarity in comparison to one another, but are very dissimilar to objects in other clusters. Dissimilarities are assessed based on the attribute values describing the objects. Often, distance measures are used. Clustering has its roots in many areas, including data mining, statistics, biology, and machine learning. In this chapter, you will learn the requirements of clustering methods for operating on large amounts of data. You will also study how to compute dissimilarities between objects represented by various attribute or variable types. You will study several clustering techniques, organized into the following categories: partitioning methods, hierarchical methods, density-based methods, grid-based methods, and model-based methods. Clustering can also be used for outlier detection, which forms the nal topic of this chapter.
8.1 What is cluster analysis?
The process of grouping a set of physical or abstract objects into classes of similar objects is called clustering. A cluster is a collection of data objects that are similar to one another within the same cluster and are dissimilar to the objects in other clusters. A cluster of data objects can be treated collectively as one group in many applications. Cluster analysis is an important human activity. Early in childhood, one learns how to distinguish between cats and dogs, or between animals and plants, by continuously improving subconscious classi cation schemes. Cluster analysis has been widely used in numerous applications, including pattern recognition, data analysis, image processing, and market research. By clustering, one can identify crowded and sparse regions, and therefore, discover overall distribution patterns and interesting correlations among data attributes. In business, clustering may help marketers discover distinct groups in their customer bases and characterize customer groups based on purchasing patterns. In biology, it can be used to derive plant and animal taxonomies, categorize genes with similar functionality, and gain insight into structures inherent in populations. Clustering may also help in the identi cation of areas of similar land use in an earth observation database, and in the identi cation of groups of motor insurance policy holders with a high average claim cost, as well as the identi cation of groups of houses in a city according to house type, value, and geographical location. It may also help classify documents on the WWW for information discovery. As a data mining function, cluster analysis can be used as a stand-alone tool to gain insight into the distribution of data, to observe the characteristics of each cluster, and to focus on a particular set of clusters for further analysis. Alternatively, it may serve as a preprocessing step for other algorithms, such as classi cation and characterization, operating on the detected clusters. Data clustering is under vigorous development. Contributing areas of research include data mining, statistics, machine learning, spatial database technology, biology, and marketing. Owing to the huge amounts of data collected in databases, cluster analysis has recently become a highly active topic in data mining research. As a branch of statistics, cluster analysis has been studied extensively for many years, focusing mainly on distancebased cluster analysis. Cluster analysis tools based on k-means, k-medoids, and several other methods have also 7
What are some typical applications of clustering?"
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been built into many statistical analysis software packages or systems, such as S-Plus, SPSS, and SAS. In machine learning, clustering is an example of unsupervised learning. Unlike classi cation, clustering and unsupervised learning do not rely on prede ned classes and class-labeled training examples. For this reason, clustering is a form of learning by observation, rather than learning by examples. In conceptual clustering, a group of objects forms a class only if it is describable by a concept. This di ers from conventional clustering which measures similarity based on geometric distance. Conceptual clustering consists of two components: 1 it discovers the appropriate classes, and 2 it forms descriptions for each class, as in classi cation. The guideline of striving for high intraclass similarity and low interclass similarity still applies. In data mining, e orts have focused on nding methods for e cient and e ective cluster analysis in large databases. Active themes of research focus on the scalability of clustering methods, the e ectiveness of methods for clustering complex shapes and types of data, high-dimensional clustering techniques, and methods for clustering mixed numerical and categorical data in large databases. Clustering is a challenging eld of research where its potential applications pose their own special requirements. The following are typical requirements of clustering in data mining. 1. Scalability: Many clustering algorithms work well in small data sets containing less than 200 data objects; however, a large database may contain millions of objects. Clustering on a sample of a given large data set may lead to biased results. Highly scalable clustering algorithms are needed. 2. Ability to deal with di erent types of attributes: Many algorithms are designed to cluster interval-based numerical data. However, applications may require clustering other types of data, such as binary, categorical nominal, and ordinal data, or mixtures of these data types. 3. Discovery of clusters with arbitrary shape: Many clustering algorithms determine clusters based on Euclidean or Manhattan distance measures. Algorithms based on such distance measures tend to nd spherical clusters with similar size and density. However, a cluster could be of any shape. It is important to develop algorithms which can detect clusters of arbitrary shape. 4. Minimal requirements for domain knowledge to determine input parameters: Many clustering algorithms require users to input certain parameters in cluster analysis such as the number of desired clusters. The clustering results are often quite sensitive to input parameters. Parameters are often hard to determine, especially for data sets containing high-dimensional objects. This not only burdens users, but also makes the quality of clustering di cult to control. 5. Ability to deal with noisy data: Most real-world databases contain outliers or missing, unknown, or erroneous data. Some clustering algorithms are sensitive to such data and may lead to clusters of poor quality. 6. Insensitivity to the order of input records: Some clustering algorithms are sensitive to the order of input data, e.g., the same set of data, when presented with di erent orderings to such an algorithm, may generate dramatically di erent clusters. It is important to develop algorithms which are insensitive to the order of input. 7. High dimensionality: A database or a data warehouse may contain several dimensions or attributes. Many clustering algorithms are good at handling low-dimensional data, involving only two to three dimensions. Human eyes are good at judging the quality of clustering for up to three dimensions. It is challenging to cluster data objects in high-dimensional space, especially considering that data in high-dimensional space can be very sparse and highly skewed. 8. Constraint-based clustering: Real-world applications may need to perform clustering under various kinds of constraints. Suppose that your job is to choose the locations for a given number of new automatic cash stations ATMs in a city. To decide upon this, you may cluster households while considering constraints such as the city's rivers and highway networks, and customer requirements per region. A challenging task is to nd groups of data with good clustering behavior that satisfy speci ed constraints. 9. Interpretability and usability: Users expect clustering results to be interpretable, comprehensible, and usable. That is, clustering may need to be tied up with speci c semantic interpretations and applications. It is important to study how an application goal may in uence the selection of clustering methods.
8.2. TYPES OF DATA IN CLUSTERING ANALYSIS
With these requirements in mind, our study of cluster analysis proceeds as follows. First, we study di erent types of data and how they can in uence clustering methods. Second, we present a general categorization of clustering methods. We then study each clustering method in detail, including partitioning methods, hierarchical methods, density-based methods, grid-based methods, and model-based methods. We also examine clustering in high-dimensional space and outlier analysis.
8.2 Types of data in clustering analysis
In this section, we study the types of data which often occur in clustering analysis and how to preprocess them for such an analysis. Suppose that a data set to be clustered contains n objects which may represent persons, houses, documents, countries, etc. Main memory-based clustering algorithms typically operate on either of the following two data structures. 1. Data matrix or object-by-variable structure: This represents n objects, such as persons, with p variables also called measurements or attributes, such as age, height, weight, gender, race, and so on. The structure is in the form of a relational table, or n-by-p matrix n objects p variables, as shown in 8.1. 2 3 x11 x1f x1p 6 7 6 6 xi1 xif xip 7 7 8.1 6 4 7 5 xn1 xnf xnp 2. Dissimilarity matrix or object-by-object structure: This stores a collection of proximities that are available for all pairs of n objects. It is often represented by an n-by-n table as shown below, 2 3 0 6 d2; 1 0 7 6 7 6 d3; 1 d3; 2 0 7 8.2 6 . 7 6 . 7 .. .. 4 . 5 . . dn; 1 dn; 2 0 where di; j is the measured di erence or dissimilarity between objects i and j. In general, di; j is a nonnegative number that is close to 0 when objects i and j are highly similar or near" each other, and becomes larger the more they di er. Since di; j = dj; i, and di; i = 0, we have the matrix in 8.2. Measures of dissimilarity are discussed throughout this section. The data matrix is often called a two-mode matrix, whereas the dissimilarity matrix is called a one-mode matrix, since the rows and columns of the former represent di erent entities, while those of the latter represent the same entity. Many clustering algorithms operate on a dissimilarity matrix. If the data are presented in the form of a data matrix, it can rst be transformed into a dissimilarity matrix before applying such clustering algorithms. How can dissimilarity, di; j, be assessed?", you may wonder. In this section, we discuss how object dissimilarity can be computed for objects described by interval-scaled variables, by binary variables, by nominal, ordinal, and ratio-scaled variables, or combinations of these variable types. The dissimilarity data can later be used to compute clusters of objects.
8.2.1 Interval-scaled variables
This section discusses interval-scaled variables and their standardization. It then describes distance measures that are commonly used for computing the dissimilarity of objects described by such variables. These measures include the Euclidean, Manhattan, and Minkowski distances. What are interval-scaled variables?" Interval-scaled variables are continuous measurements of a roughly linear scale. Typical examples include weight and height, latitude and longitude coordinates e.g., when clustering houses, and weather temperature.
CHAPTER 8. CLUSTER ANALYSIS
The measurement unit used can a ect the clustering analysis. For example, changing measurement units from meters to inches for height, or from kilograms to pounds for weight, may lead to a very di erent clustering structure. In general, expressing a variable in smaller units will lead to a larger range for that variable, and thus a larger e ect on the resulting clustering structure. To help avoid dependence on the choice of measurement units, the data should be standardized. Standardizing measurements attempts to give all variables an equal weight. This is particularly useful when given no prior knowledge of the data. However, in some applications, users may intentionally want to give more weight to a certain set of variables than to others. For example, when clustering basketball player candidates, one may like to give more weight to the variable height. How can the data for a variable be standardized?" To standardize measurements, one choice is to convert the original measurements to unitless variables. Given measurements for a variable f, this can be performed as follows. 1. Calculate the mean absolute deviation, sf , 1 8.3 sf = n jx1f , mf j + jx2f , mf j + + jxnf , mf j; 1 where x1f ; ::; xnf are n measurements of f, and mf is the mean value of f, i.e., mf = n x1f + x2f + + xnf . 8.4
2. Calculate the standardized measurement, or z-score, as follows, , zif = xif s mf : f
The mean absolute deviation, sf , is more robust to outliers than the standard deviation, f . When computing the mean absolute deviation, the deviations from the mean i.e., jxif , mf j are not squared, hence the e ect of outliers is somewhat reduced. There are more robust measures of dispersion, such as the median absolute deviation. However, the advantage of using the mean absolute deviation is that the z-scores of outliers do not become too small, hence the outliers remain detectable. Standardization may or may not be useful in a particular application. Thus the choice of whether and how to perform standardization should be left to the user. Methods of standardization are also discussed in Chapter 3 under normalization techniques for data preprocessing. OK", you now ask, once I have standardized the data, how can I compute the dissimilarity between objects?" After standardization, or without standardization in certain applications, the dissimilarity or similarity between the objects described by interval-scaled variables is typically computed based on the distance between each pair of objects. The most popular distance measure is Euclidean distance, which is de ned as q 8.5 di; j = jxi1 , xj 1j2 + jxi2 , xj 2j2 + + jxip , xjp j2; where i = xi1; xi2; : : :; xip, and j = xj 1; xj 2; : : :; xjp, are two p-dimensional data objects. Another well-known metric is Manhattan or city block distance, de ned as di; j = jxi1 , xj 1j + jxi2 , xj 2j + + jxip , xjp j: 8.6 Both the Euclidean distance and Manhattan distance satisfy the following mathematic requirements of a distance function: 1. di; j 0: This states that distance is a nonnegative number; 2. di; i = 0: This states that the distance of an object to itself is 0; 3. di; j = dj; i: This states that distance is a symmetric function; and 4. di; j di; h+ dh; j: This is a triangular inequality which states that going directly from object i to object j in space is no more than making a detour over any other object h.
8.2. TYPES OF DATA IN CLUSTERING ANALYSIS
Minkowski distance is a generalization of both Euclidean distance and Manhattan distance. It is de ned as
di; j = jxi1 , xj 1jq + jxi2 , xj 2jq + + jxip , xjpjq 1=q ; 8.7 where q is a positive integer. It represents the Manhattan distance when q = 1, and Euclidean distance when q = 2. If each variable is assigned a weight according to its perceived importance, the weighted Euclidean distance can be computed as follows. q di; j = w1 jxi1 , xj 1j2 + w2jxi2 , xj 2j2 + + wp jxip , xjpj2 : 8.8 Weighting can also be applied to the Manhattan and Minkowski distances.
8.2.2 Binary variables
This section describes how to compute the dissimilarity between objects described by either symmetric or asymmetric binary variables. A binary variable has only two states: 0 or 1, where 0 means that the variable is absent, and 1 means that it is present. Given the variable smoker describing a patient, for instance, 1 indicates that the patient smokes, while 0 indicates that the patient does not. Treating binary variables as if they are interval-scaled can lead to misleading clustering results. Therefore, methods speci c to binary data are necessary for computing dissimilarities. So, how can I compute the dissimilarity between two binary variables?" One approach involves computing a dissimilarity matrix from the given binary data. If all binary variables are thought of as having the same weight, we have a 2-by-2 contingency table, Table 8.1, where q is the number of variables that equal 1 for both objects i and j, r is the number of variables that equal 1 for object i but that are 0 for object j, s is the number of variables that equal 0 for object i but equal 1 for object j, and t is the number of variables that equal 0 for both objects i and j. The total number of variables is p, where p = q + r + s + t. object j 1 0 sum 1 q r q+r object i 0 s t s+t sum q + s r + t p Table 8.1: A contingency table for binary variables. A binary variable is symmetric if both of its states are equally valuable and carry the same weight, that is, there is no preference on which outcome should be coded as 0 or 1. One such example could be the attribute gender having the states male and female. Similarity that is based on symmetric binary variables is called invariant similarity in that the result does not change when some or all of the binary variables are coded di erently. For invariant similarities, the most well-known coe cient for assessing the dissimilarity between objects i and j is the simple matching coe cient, de ned in 8.9. di; j = q + r + s + t 8.9 r+s A binary variable is asymmetric if the outcomes of the states are not equally important, such as the positive and negative outcomes of a disease test. By convention, we shall code the most important outcome, which is usually the rarest one, by 1 e.g., HIV positive, and the other by 0 e.g., HIV negative. Given two asymmetric binary variables, the agreement of two 1's a positive match is then considered more signi cant than that of two zeros a negative match. Therefore, such binary variables are often considered monary" as if having one state. The similarity based on such variables is called noninvariant similarity. For noninvariant similarities, the most well-known coe cient
What is the di erence between symmetric and asymmetric binary variables?"
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is the Jaccard coe cient, de ned in 8.10, where the number of negative matches, d, is considered unimportant and thus is ignored in the computation. r s 8.10 di; j = q + + + s r When both symmetric and asymmetric binary variables occur in the same data set, the mixed variables approach described in Section 8.2.4 can be applied.
Example 8.1 Dissimilarity between binary variables. Suppose that a patient record table, Table 8.2, contains
name gender fever cough test-1 test-2 test-3 test-4 Jack M Y N P N N N Mary F Y N P N P N Jim M Y P N N N N .. .. .. .. .. .. .. .. . . . . . . . . Table 8.2: A relational table containing mostly binary attributes.
the attributes name, gender, fever, cough, test-1, test-2, test-3, and test-4, where name is an object-id, gender is a symmetric attribute, and the remaining attributes are asymmetric binary.
For asymmetric attribute values, let the values Y and P be set to 1, and the value N be set to 0. Suppose that the distance between objects patients is computed based only on the asymmetric variables. According to the Jaccard coe cient formula 8.10, the distance between each pair of the three patients, Jack, Mary and Jim, should be, 0+1 djack; mary = 2+0+1 = 0:33 8.11 1+1 djack; jim = 1+1+1 = 0:67 8.12 1+2 djim; mary = 1+1+2 = 0:75 8.13 These measurements suggest that Jim and Mary are unlikely to have a similar disease they have the highest dissimilarity value among the three pairs. Of the three patients, Jack and Mary are the most likely to have a similar disease. 2
8.2.3 Nominal, ordinal, and ratio-scaled variables
This section discusses how to compute the dissimilarity between objects described by nominal, ordinal, and ratioscaled variables.
Nominal variables A nominal variable is a generalization of the binary variable in that it can take on more than two states. For
example, map color is a nominal variable that may have, say, ve states: red, yellow, green, pink, and blue. Let the number of states of a nominal variable be M. The states can be denoted by letters, symbols, or a set of integers, such as 1, 2, .. ., M. Notice that such integers are used just for data handling and do not represent any speci c ordering. How is dissimilarity computed between objects described by nominal variables?" The dissimilarity between two objects i and j can be computed using the simple matching approach as in 8.14: di; j = p , m ; 8.14 p where m is the number of matches i.e., the number of variables for which i and j are in the same state, and p is the total number of variables. Weights can be assigned to increase the e ect of m, or to assign greater weight to the matches in variables having a larger number of states.
8.2. TYPES OF DATA IN CLUSTERING ANALYSIS
Nominal variables can be encoded by asymmetric binary variables by creating a new binary variable for each of the M nominal states. For an object with a given state value, the binary variable representing that state is set to 1, while the remaining binary variables are set to 0. For example, to encode the nominal variable map color, a binary variable can be created for each of the ve colors listed above. For an object having the color yellow, the yellow variable is set to 1, while the remaining four variables are set to 0. The dissimilarity coe cient for this form of encoding can be calculated using the methods discussed in Section 8.2.2.
Ordinal variables A discrete ordinal variable resembles a nominal variable, except that the M states of the ordinal value are ordered
in a meaningful sequence. Ordinal variables are very useful for registering subjective assessments of qualities that cannot be measured objectively. For example, professional ranks are often enumerated in a sequential order, such as assistant, associate, and full. A continuous ordinal variable looks like a set of continuous data of an unknown scale, that is, the relative ordering of the values is essential but not their actual magnitude. For example, the relative ranking in a particular sport is often more essential than the actual values of a particular measure. Ordinal variables may also be obtained from the discretization of interval-scaled quantities by splitting the value range into a nite number of classes. The values of an ordinal variable can be mapped to ranks. For example, suppose that an ordinal variable f has Mf states. These ordered states de ne the ranking 1; ::; Mf . How are ordinal variables handled?" The treatment of ordinal variables is quite similar to that of interval-scaled variables when computing the dissimilarity between objects. Suppose that f is a variable from a set of ordinal variables describing n objects. The dissimilarity computation with respect to f involves the following steps. 1. The value of f for the ith object is xif , and f has Mf ordered states, representing the ranking 1; ::; Mf . Replace each xif by its corresponding rank, rif 2 f1; : : :; Mf g. 2. Since each ordinal variable can have a di erent number of states, it is often necessary to map the range of each variable onto 0-1 so that each variable has equal weight. This can be achieved by replacing the rank rif of the i-th object in the f-th variable by rif zif = M , 1 : 8.15 f ,1 3. Dissimilarity can then be computed using any of the distance measures described in Section 8.2.1 for intervalscaled variables, using zif to represent the f value for the ith object.
Ratio-scaled variables A ratio-scaled variable makes a positive measurement on a nonlinear scale, such as an exponential scale, approximately following the formula AeBt or Ae,Bt ; 8.16 where A and B are positive constants. Typical examples include the growth of a bacteria population, or the decay of a radio-active element. How can I compute the dissimilarity between objects described by ration-scaled variables?" There are three methods to handle ratio-scaled variables for computing the dissimilarity between objects. 1. Treat ratio-scaled variables like interval-scaled variables. This, however, is not usually a good choice since it is likely that the scale may be distorted. 2. Apply logarithmic transformation to a ratio-scaled variable f having value xif for object i by using the formula yif = logxif . The yif values can be treated as interval-valued, as described in Section 8.2.1. Notice that for some ratio-scaled variables, one may also apply log-log transformation, or other transformations, depending on the de nition and application. 3. Treat xif as continuous ordinal data and treat their ranks as interval-valued. The latter two methods are the most e ective, although the choice of method used may be dependent on the given application.
CHAPTER 8. CLUSTER ANALYSIS
8.2.4 Variables of mixed types
Sections 8.2.1 to 8.2.3 discussed how to compute the dissimilarity between objects described by variables of the same type, where these types may be either interval-scaled, symmetric binary, asymmetric binary, nominal, ordinal, or ratio-scaled. However, in many real databases, objects are described by a mixture of variable types. In general, a database can contain all of the six variable types listed above. One approach is to group each kind of variables together, performing a separate cluster analysis for each variable type. This is feasible if these analyses derive satisfactory results. However, in real applications, it is unlikely that a separate cluster analysis per variable type will generate satisfactory results. A more preferable approach is to process all variable types together, performing a single cluster analysis. One such technique combines the di erent variables into a single dissimilarity matrix, bringing all of the meaningful variables onto a common scale of the interval 0,1 . Suppose that the data set contains p variables of mixed type. The dissimilarity di; j between objects i and j is de ned as p =1 ; ijf df ij f di; j = 8.17 p f f =1 ij
where the indicator ijf = 0 if either 1 xif or xjf is missing i.e., there is no measurement of variable f for object i or object j, or 2 xif = xjf = 0 and variable f is asymmetric binary; otherwise, ijf = 1. The contribution of variable f to the dissimilarity between i and j, df , is computed dependent on its type: ij
So, how can we compute the dissimilarity between objects of mixed variable types?"
1. If f is binary or nominal: df = 0 if xif = xjf , otherwise df = 1. ij ij
xif 2. If f is interval-based: df = maxhjxhf,xjf j h xhf , where h runs over all nonmissing objects for variable f. ij ,min rif 3. If f is ordinal or ratio-scaled: compute the ranks rif and zif = Mf ,1 , and treat zif as interval-scaled. ,1
Thus, the dissimilarity between objects can be computed even when the variables describing the objects are of di erent types.
8.3 A categorization of major clustering methods
There exist a large number of clustering algorithms in the literature. The choice of clustering algorithm depends both on the type of data available and on the particular purpose and application. If cluster analysis is used as a descriptive or exploratory tool, it is possible to try several algorithms on the same data to see what the data may disclose. In general, major clustering methods can be classi ed into the following categories. 1. Partitioning methods. Given a database of n objects or data tuples, a partitioning method constructs k partitions of the data, where each partition represents a cluster, and k n. That is, it classi es the data into k groups, which together satisfy the following requirements: 1 each group must contain at least one object, and 2 each object must belong to exactly one group. Notice that the second requirement can be relaxed in some fuzzy partitioning techniques. References to such techniques are given in the bibliographic notes. Given k, the number of partitions to construct, a partitioning method creates an initial partitioning. It then uses an iterative relocation technique which attempts to improve the partitioning by moving objects from one group to another. The general criterion of a good partitioning is that objects in the same cluster are close" or related to each other, whereas objects of di erent clusters are far apart" or very di erent. There are various kinds of other criteria for judging the quality of partitions. To achieve global optimality in partitioning-based clustering would require the exhaustive enumeration of all of the possible partitions. Instead, most applications adopt one of two popular heuristic methods: 1 the
8.3. A CATEGORIZATION OF MAJOR CLUSTERING METHODS
k-means algorithm, where each cluster is represented by the mean value of the objects in the cluster; and 2 the k-medoids algorithm, where each cluster is represented by one of the objects located near the center of the cluster. These heuristic clustering methods work well for nding spherical-shaped clusters in small to medium sized databases. For nding clusters with complex shapes and for clustering very large data sets, partitioning-based methods need to be extended. Partitioning-based clustering methods are studied in depth in Section 8.4. Hierarchical methods. A hierarchical method creates a hierarchical decomposition of the given set of data objects. A hierarchical method can be classi ed as being either agglomerative or divisive, based on how the hierarchical decomposition is formed. The agglomerative approach, also called the bottom-up" approach, starts with each object forming a separate group. It successively merges the objects or groups close to one another, until all of the groups are merged into one the topmost level of the hierarchy, or until a termination condition holds. The divisive approach, also called the top-down" approach, starts with all the objects in the same cluster. In each successive iteration, a cluster is split up into smaller clusters, until eventually each object is in one cluster, or until a termination condition holds. Hierarchical methods su er from the fact that once a step merge or split is done, it can never be undone. This rigidity is useful in that it leads to smaller computation costs by not worrying about a combinatorial number of di erent choices. However, a major problem of such techniques is that they cannot correct erroneous decisions. It can be advantageous to combine iterative relocation and hierarchical agglomeration by rst using a hierarchical agglomerative algorithm and then re ning the result using iterative relocation. Some scalable clustering algorithms, such as BIRCH and CURE, have been developed based on such an integrated approach. Hierarchical clustering methods are studied in Section 8.5. Density-based methods. Most partitioning methods cluster objects based on the distance between objects. Such methods can nd only spherical-shaped clusters and encounter di culty at discovering clusters of arbitrary shapes. Other clustering methods have been developed based on the notion of density. Their general idea is to continue growing the given cluster as long as the density number of objects or data points in the neighborhood" exceeds some threshold, i.e., for each data point within a given cluster, the neighborhood of a given radius has to contain at least a minimum number of points. Such a method can be used to lter out noise outliers, and discover clusters of arbitrary shape. DBSCAN is a typical density-based method which grows clusters according to a density threshold. OPTICS is a density-based method which computes an augmented clustering ordering for automatic and interactive cluster analysis. Density-based clustering methods are studied in Section 8.6. Grid-based methods. Grid-based methods quantize the object space into a nite number of cells which form a grid structure. All of the clustering operations are performed on the grid structure i.e., on the quantized space. The main advantage of this approach is its fast processing time which is typically independent of the number of data objects, and dependent only on the number of cells in each dimension in the quantized space. STING is a typical example of a grid-based method. CLIQUE and Wave-Cluster are two clustering algorithms which are both grid-based and density-based. Grid-based clustering methods are studied in Section 8.7. Model-based methods. Model-based methods hypothesize a model for each of the clusters, and nd the best t of the data to the given model. A model-based algorithm may locate clusters by constructing a density function that re ects the spatial distribution of the data points. It also leads to a way of automatically determining the number of clusters based on standard statistics, taking noise" or outliers into account and thus yielding robust clustering methods. Model-based clustering methods are studied in Section 8.8.
Some clustering algorithms integrate the ideas of several clustering methods, so that it is sometimes di cult to classify a given algorithm as uniquely belonging to only one clustering method category. Furthermore, some applications may have clustering criteria which require the integration of several clustering techniques.
CHAPTER 8. CLUSTER ANALYSIS
Algorithm 8.4.1 k-means The k-means algorithm for partitioning based on the mean value of the objects in the cluster. Input: The number of clusters k, and a database containing n objects. Output: A set of k clusters which minimizes the squared-error criterion. Method: 1 arbitrarily choose k objects as the initial cluster centers;
2 3 4 5 repeat reassign each object to the cluster to which the object is the most similar, based on the mean value of the objects in the cluster; update the cluster means, i.e., calculate the mean value of the objects for each cluster; until no change;
Figure 8.1: The k-means algorithm. In the following sections, we examine each of the above ve clustering methods in detail. We also introduce algorithms which integrate the ideas of several clustering methods. Outlier analysis, which typically involves clustering, is described in Section 8.9.
8.4 Partitioning methods
Given a database of n objects, and k, the number of clusters to form, a partitioning algorithm organizes the objects into k partitions k n, where each partition represents a cluster. The clusters are formed to optimize an objective partitioning criterion, often called a similarity function, such as distance, so that the objects within a cluster are similar", whereas the objects of di erent clusters are dissimilar" in terms of the database attributes. Recall that Equation ?? can be used to transform between dissimilarity and similarity coe cients.
8.4.1 Classical partitioning methods: k-means and k-medoids
The most well-known and commonly used partitioning methods are k-means, k-medoids, and their variations.
Centroid-based technique: The k-means method The k-means algorithm takes the input parameter, k, and partitions a set of n objects into k clusters so that the
resulting intra-cluster similarity is high whereas the inter-cluster similarity is low. Cluster similarity is measured in regard to the mean value of the objects in a cluster, which can be viewed as the cluster's center of gravity". How does the k,means algorithm work?" It proceeds as follows. First, it randomly selects k of the objects which initially each represent a cluster mean or center. For each of the remaining objects, an object is assigned to the cluster to which it is the most similar, based on the distance between the object and the cluster mean. It then computes the new mean for each cluster. This process iterates until the criterion function converges. Typically, the squared-error criterion is used, de ned as E = k=1 p2Ci jp , mi j2 ; 8.18 i where E is the sum of square-error for all objects in the database, p is the point in space representing the given object, and mi is the mean of cluster Ci both p and mi are multidimensional. This criterion tries to make the resulting k clusters as compact and as separate as possible. The k-means procedure is summarized in Figure 8.1. The algorithm attempts to determine k partitions that minimize the squared-error function. It works well when the clusters are compact clouds that are rather well separated from one another. The method is relatively scalable and e cient in processing large data sets because the computational complexity of the algorithm is Onkt, where n is the total number of objects, k is the number of clusters, and t is the number of iterations. Normally, k n and t n. The method often terminates at a local optimum. The k-means method, however, can be applied only when the mean of a cluster is de ned. This may not be the case in some applications, such as when data with categorical attributes are involved. The necessity for users to
8.4. PARTITIONING METHODS
+ + b)
+ + c)
Figure 8.2: Clustering of a set of objects based on the k-means method. The mean of each cluster is marked by a +". specify k, the number of clusters, in advance can be seen as a disadvantage. The k-means method is not suitable for discovering clusters with non-convex shapes, or clusters of very di erent size. Moreover, it is sensitive to noise and outlier data points since a small number of such data can substantially in uence the mean value.
Example 8.2 Suppose that there are a set of objects located in space as depicted in the rectangle shown in Figure
8.2a. Let k = 3, that is, the user would like to cluster the objects into three clusters. According to Algorithm 8.4.1, we arbitrarily choose three objects as the three initial cluster centers, where cluster centers are marked by a +". Each object is distributed to a cluster based on the cluster center to which it is the nearest. Such a distribution forms silhouettes encircled by dotted curves, as shown in Figure 8.2a. This kind of grouping will update the cluster centers. That is, the mean value of each cluster is recalculated based on the objects in the cluster. Relative to these new centers, objects are re-distributed to the cluster domains based on which cluster center is the nearest. Such a re-distribution forms new silhouettes encircled by dashed curves, as shown in Figure 8.2b. This process iterates, leading to Figure 8.2c. Eventually, no re-distribution of the objects in any cluster occurs and so the process terminates. The resulting clusters are returned by the clustering process. 2 There are quite a few variants of the k-means method. These can di er in the selection of the initial k means, the calculation of dissimilarity, and the strategies for calculating cluster means. An interesting strategy which often yields good results is to rst apply a hierarchical agglomeration algorithm to determine the number of clusters and to nd an initial classi cation, and then use iterative relocation to improve the classi cation. Another variant to k-means is the k-modes method which extends the k-means paradigm to cluster categorical data by replacing the means of clusters with modes, using new dissimilarity measures to deal with categorical objects, and using a frequency-based method to update modes of clusters. The k-means and the k-modes methods can be integrated to cluster data with mixed numeric and categorical values, resulting in the k-prototypes method. The EM Expectation Maximization algorithm extends the k-means paradigm in a di erent way: Instead of assigning each object to a dedicated cluster, it assigns each object to a cluster according to a weight representing the probability of membership. In other words, there are no strict boundaries between clusters. Therefore, new means are computed based on weighted measures. How can we make the k,means algorithm more scalable?" A recent e ort on scaling the k-means algorithm is based on the idea of identifying three kinds of regions in data: regions that are compressible, regions that must be maintained in main memory, and regions that are discardable. An object is discardable if its membership in a cluster is ascertained. An object is compressible if it is not discardable but belongs to a tight subcluster. A data structure known as a clustering feature is used to summarize objects that have been discarded or compressed. If an object is neither discardable nor compressible, then it should be retained in main memory. To achieve scalability, the iterative clustering algorithm only includes the clustering features of the compressible objects and the objects which must be retained in main memory, thereby turning a secondary-memory based algorithm into a main memory-based algorithm.
CHAPTER 8. CLUSTER ANALYSIS
Figure 8.3: Four cases of the cost function for k-medoids clustering.
Representative object-based technique: The k-medoids method
The k-means algorithm is sensitive to outliers since an object with an extremely large value may substantially distort the distribution of data. How might the algorithm be modi ed to diminish such sensitivity?", you may wonder. Instead of taking the mean value of the objects in a cluster as a reference point, the medoid can be used, which is the most centrally located object in a cluster. Thus the partitioning method can still be performed based on the principle of minimizing the sum of the dissimilarities between each object and its corresponding reference point. This forms the basis of the k-medoids method. The basic strategy of k-medoids clustering algorithms is to nd k clusters in n objects by rst arbitrarily nding a representative object the medoid for each cluster. Each remaining object is clustered with the medoid to which it is the most similar. The strategy then iteratively replaces one of the medoids by one of the non-medoids as long as the quality of the resulting clustering is improved. This quality is estimated using a cost function which measures the average dissimilarity between an object and the medoid of its cluster. To determine whether a non-medoid object, o random, is a good replacement for a current medoid, oj , the following four cases are examined for each of the non-medoid objects, p. Case 1: p currently belongs to medoid oj . If oj is replaced by orandom as a medoid and p is closest to one of oi , i 6= j, then p is re-assigned to oi . Case 2: p currently belongs to medoid oj . If oj is replaced by orandom as a medoid and p is closest to orandom , then p is re-assigned to orandom . Case 3: p currently belongs to medoid oi , i 6= j. If oj is replaced by orandom as a medoid and and p is still closest to oi , then the assignment does not change. Case 4: p currently belongs to medoid oi , i 6= j. If oj is replaced by orandom as a medoid and p is closest to orandom , then p is re-assigned to orandom . Figure 8.3 illustrates the four cases. Each time a re-assignment occurs, a di erence in square-error E is contributed to the cost function. Therefore, the cost function calculates the di erence in square- error value if a current medoid is replaced by a non-medoid object. The total cost of swapping is the sum of costs incurred by all non-medoid objects. If the total cost is negative, then oj is replaced or swapped with orandom since the actual square-error E would be reduced. If the total cost is positive, the current medoid oj is considered acceptable, and nothing is changed in the iteration. A general k-medoids algorithm is presented in Figure 8.4. PAM Partitioning Around Medoids was one of the rst k-medoids algorithms introduced. It attempts to determine k partitions for n objects. After an initial random selection of k medoids, the algorithm repeatedly tries to make a better choice of medoids. All of the possible pairs of objects are analyzed, where one object in each pair is considered a medoid and the other is not. The quality of the resulting clustering is calculated for each such combination. An object, oj , is replaced with the object causing the greatest reduction in square-error. The set of best objects for each cluster in one iteration form the medoids for the next iteration. For large values of n and k, such computation becomes very costly. Which method is more robust | k,means or k,medians?" The k-medoids method is more robust than k-means in the presence of noise and outliers because a medoid is less in uenced by outliers or other extreme values than a mean. However, its processing is more costly than the k-means method. Both methods require the user to specify k, the number of clusters.
8.4.2 Partitioning methods in large databases: from k-medoids to CLARANS
A typical k-medoids partition algorithm like PAM works e ectively for small data sets, but does not scale well for large data sets. To deal with larger data sets, a sampling-based method, called CLARA Clustering LARge Applications can be used.
How e cient is the k,medoids algorithm on large data sets?"
8.5. HIERARCHICAL METHODS
Algorithm 8.4.2 k-medoids A general k-medoids algorithm for partitioning based on medoid or central objects. Input: The number of clusters k, and a database containing n objects. Output: A set of k clusters which minimizes the sum of the dissimilarities of all the objects to their nearest medoid. Method: 1 arbitrarily choose k objects as the initial medoids;
2 3 4 5 6 7 repeat assign each remaining object to the cluster with the nearest medoid; randomly select a non-medoid object, random ; compute the total cost, , of swapping j with random ; if 0 then swap j with random to form the new set of medoids; until no change;
Figure 8.4: The k-medoids algorithm. The idea behind CLARA is as follows: Instead of taking the whole set of data into consideration, a small portion of the actual data is chosen as a representative of the data. Medoids are then chosen from this sample using PAM. If the sample is selected in a fairly random manner, it should closely represent the original data set. The representative objects medoids chosen will likely be similar to those that would have been chosen from the whole data set. CLARA draws multiple samples of the data set, applies PAM on each sample, and returns its best clustering as the output. As expected, CLARA can deal with larger data sets than PAM. The complexity of each iteration now becomes Oks2 +kn , k, where s is the size of the sample, k is the number of clusters, and n is the total number of objects. The e ectiveness of CLARA depends on the sample size. Notice that PAM searches for the best k medoids among a given data set, whereas CLARA searches for the best k medoids among the selected sample of the data set. CLARA cannot nd the best clustering if any sampled medoid is not among the best k medoids. For example, if an object oi is one of the medoids in the best k medoids but it is not selected during sampling, CLARA will never nd the best clustering. This is, therefore, a tradeo for e ciency. A good clustering based on samples will not necessarily represent a good clustering of the whole data set if the sample is biased. How might we improve the quality and scalability of CLARA?" A k,medoids type algorithm called CLARANS Clustering Large Applications based upon RANdomized Search was proposed which combines the sampling technique with PAM. However, unlike CLARA, CLARANS does not con ne itself to any sample at any given time. While CLARA has a xed sample at each stage of the search, CLARANS draws a sample with some randomness in each step of the search. The clustering process can be presented as searching a graph where every node is a potential solution, i.e., a set of k medoids. The clustering obtained after replacing a single medoid is called the neighbor of the current clustering. The number of neighbors to be randomly tried is restricted by a user-speci ed parameter. If a better neighbor is found i.e., having a lower square-error, CLARANS moves to the neighbor's node and the process starts again; otherwise the current clustering produces a local optimum. If the local optimum is found, CLARANS starts with new randomly selected nodes in search for a new local optimum. CLARANS has been experimentally shown to be more e ective than both PAM and CLARA. It can be used to nd the most natural" number of clusters using a silhouette coe cient | a property of an object that speci es how much the object truly belongs to the cluster. CLARANS also enables the detection of outliers. However, the computational complexity of CLARANS is about On2 , where n is the number of objects. Furthermore, its clustering quality is dependent on the sampling method used. The performance of CLARANS can be further improved by exploring spatial data structures, such as R*-trees, and some focusing techniques.
8.5 Hierarchical methods
A hierarchical clustering method works by grouping data objects into a tree of clusters. Hierarchical clustering methods can be further classi ed into agglomerative and divisive hierarchical clustering, depending on whether the hierarchical decomposition is formed in a bottom-up or top-down fashion. The quality of a pure hierarchical clustering
0 step 1 step agglomerative (AGNES) a b c d e 4 step 3 step de cde ab abcde 2 step 3 step 4 step
CHAPTER 8. CLUSTER ANALYSIS
Figure 8.5: Agglomerative and divisive hierarchical clustering on data objects fa; b; c; d; eg. method su ers from its inability to perform adjustment once a merge or split decision has been executed. Recent studies have emphasized the integration of hierarchical agglomeration with iterative relocation methods.
8.5.1 Agglomerative and divisive hierarchical clustering
In general, there are two types of hierarchical clustering methods: 1. Agglomerative hierarchical clustering: This bottom-up strategy starts by placing each object in its own cluster and then merges these atomic clusters into larger and larger clusters, until all of the objects are in a single cluster or until certain termination conditions are satis ed. Most hierarchical clustering methods belong to this category. They di er only in their de nition of inter-cluster similarity. 2. Divisive hierarchical clustering: This top-down strategy does the reverse of agglomerative hierarchical clustering by starting with all objects in one cluster. It subdivides the cluster into smaller and smaller pieces, until each object forms a cluster on its own or until it satis es certain termination conditions, such as a desired number of clusters is obtained or the distance between the two closest clusters is above a certain threshold distance.
Example 8.3 Figure 8.5 shows the application of AGNES AGglomerative NESting, an agglomerative hierarchical clustering method, and DIANA DIvisia ANAlysis, a divisive hierarchical clustering method, to a data set of ve objects, fa; b; c; d; eg. Initially, AGNES places each object into a cluster of its own. The clusters are then merged
step-by-step according to some criterion. For example, clusters C1 and C2 may be merged if an object in C1 and an object in C2 form the minimum Euclidean distance between any two objects from di erent clusters. This is a single-link approach in that each cluster is represented by all of the objects in the cluster, and the similarity between two clusters is measured by the similarity of the closest pair of data points belonging to di erent clusters. The cluster merging process repeats until all of the objects are eventually merged to form one cluster. In DIANA, all of the objects are used to form one initial cluster. The cluster is split according to some principle, such as the maximum Euclidean distance between the closest neighboring objects in the cluster. The cluster splitting process repeats until, eventually, each new cluster contains only a single object. 2 In either agglomerative or divisive hierarchical clustering, one can specify the desired number of clusters as a termination condition. Four widely-used measures for distance between clusters are as follows, where mi is the mean for cluster Ci , ni is the number of objects in Ci , and jp , p0 j is the distance between two objects or points p and p0.
Minimum distance: dminCi ; Cj = minp2Ci;p 2Cj jp , p0j Maximum distance: dmaxCi; Cj = maxp2Ci;p 2Cj jp , p0 j
8.5. HIERARCHICAL METHODS
CF1 CF11 CF12
Figure 8.6: A CF-tree structure.
Mean distance: dmeanCi ; Cj = jmi , mj j Average distance: davg Ci ; Cj = ninj p2Ci p 2Cj
What are some of the di culties with hierarchical clustering?" The hierarchical clustering method, though simple, often encounters di culties regarding the selection of merge or split points. Such a decision is critical because once a group of objects is merged or split, the process at the next step will operate on the newly generated clusters. It will neither undo what was done previously, nor perform object swapping between clusters. Thus merge or split decisions, if not well chosen at some step, may lead to low quality clusters. Moreover, the method does not scale well since the decision of merge or split needs to examine and evaluate a good number of objects or clusters. One promising direction for improving the clustering quality of hierarchical methods is to integrate hierarchical clustering with other clustering techniques for multiple phase clustering. A few such methods are introduced in the following subsections. The rst, called BIRCH, begins by partitioning objects hierarchically using tree structures, and then applies other clustering algorithms to re ne the clusters. The second, called CURE, represents each cluster by a certain xed number of representative objects and then shrinks them toward the center of the cluster by a speci ed fraction. The third, called ROCK, merges clusters based on their inter-connectivity. The fourth, called CHAMELEON, explores dynamic modeling in hierarchical clustering.
8.5.2 BIRCH: Balanced Iterative Reducing and Clustering using Hierarchies
BIRCH Balanced Iterative Reducing and Clustering using Hierarchies is an integrated hierarchical clustering method. It introduces two concepts, that of clustering feature and clustering feature tree CF tree, which
are used to summarize cluster representations. These structures help the clustering method achieve good speed and scalability in large databases. BIRCH is also e ective for incremental and dynamic clustering of incoming objects. Let's have a closer look at the above-mentioned structures. A clustering feature CF is a triplet summarizing information about subclusters of objects. Given N d-dimensional points or objects foi g in a subcluster, then the CF of the subcluster is de ned as ~ CF = N; LS; SS; 8.19 PN o , and SS is the ~ where N is the number of points in the subcluster, LS is the linear sum on N points, i.e., i=1 ~i PN 2 square sum of data points, i.e., i=1 oi . ~ A clustering feature is essentially a summary of the statistics for the given subcluster: the zero-th, rst, and second moments of the subcluster from a statistical point of view. It registers crucial measurements for computing clusters and utilizes storage e ciently since it summarizes the information about the subclusters of objects instead of storing all objects. A CF tree is a height-balanced tree which stores the clustering features for a hierarchical clustering. An example is shown in Figure 8.6. By de nition, a non-leaf node in a tree has descendents or children". The non-leaf nodes store sums of the CFs of their children, and thus, summarize clustering information about their children. A CF tree has two parameters: branching factor, B, and threshold, T. The branching factor speci es the maximum number of children per non-leaf node. The threshold parameter speci es the maximum diameter of subclusters stored at the leaf nodes of the tree. These two parameters in uence the size of the resulting tree. How does the BIRCH algorithm work?" It consists of two phases:
CHAPTER 8. CLUSTER ANALYSIS
Phase 1: BIRCH scans the database to build an initial in-memory CF tree, which can be viewed as a multilevel
compression of the data that tries to preserve the inherent clustering structure of the data. Phase 2: BIRCH applies a selected clustering algorithm to cluster the leaf nodes of the CF-tree. For Phase 1, the CF tree is built dynamically as objects are inserted. Thus, the method is incremental. An object is inserted to the closest leaf entry subcluster. If the diameter of the subcluster stored in the leaf node after insertion is larger than the threshold value, then the leaf node and possibly other nodes are split. After the insertion of the new object, information about it is passed towards the root of the tree. The size of the CF tree can be changed by modifying the threshold. If the size of the memory that is needed for storing the CF tree is larger than the size of the main memory, then a smaller threshold value can be speci ed and the CF tree is rebuilt. The rebuild process is performed by building a new tree from the leaf nodes of the old tree. Thus, the process of rebuilding the tree is done without the necessity of rereading all of the objects or points. This is similar to the insertion and node split in the construction of B+-trees. Therefore, for building the tree, data has to be read just once. Some heuristics and methods have been introduced to deal with outliers and improve the quality of CF trees by additional scans of the data. After the CF tree is built, any clustering algorithm, such as a typical partitioning algorithm, can be used with the CF tree in Phase 2. BIRCH tries to produce the best clusters with the available resources. With a limited amount of main memory, an important consideration is to minimize the time required for I O. BIRCH applies a multiphase clustering technique: A single scan of data set yields a basic good clustering, and one or more additional scans can optionally be used to further improve the quality. The computation complexity of the algorithm is On, where n is the number of objects to be clustered. How e ective is BIRCH?" Experiments have shown the linear scalability of the algorithm with respect to the number of objects, and good quality of clustering of the data. However, since each node in a CF-tree can hold only a limited number of entries due to its size, a CF-tree node does not always correspond to what a user may consider a natural cluster. Moreover, if the clusters are not spherical in shape, BIRCH does not perform well because it uses the notion of radius or diameter to control the boundary of a cluster.
8.5.3 CURE: Clustering Using REpresentatives
Most clustering algorithms either favor clusters with spherical shape and similar sizes, or are fragile in the presence of outliers. CURE Clustering Using REpresentatives is an alternative method which integrates hierarchical and partitioning algorithms. It overcomes the problem of favoring clusters with spherical shape and similar sizes, and is more robust with respect to outliers. CURE employs a novel hierarchical clustering algorithm that adopts a middle ground between the agglomerative bottom-up and divisive top-down approaches. Instead of using a single centroid or object to represent a cluster, a xed number of representative points in space are chosen. The representative points of a cluster are generated by rst selecting well-scattered objects for the cluster and then shrinking" or moving them toward the cluster center by a speci ed fraction, or shrinking factor. At each step of the algorithm, the two clusters with the closest pair of representative points where each point in the pair is from a di erent cluster are merged. Having more than one representative point per cluster allows CURE to adjust well to the geometry of nonspherical shapes. The shrinking or condensing of clusters helps dampen the e ects of outliers. Therefore, CURE is more robust to outliers and identi es clusters having non-spherical shapes and wide variance in size. It scales well for large databases without sacri cing clustering quality. To handle large databases, CURE employs a combination of random sampling and partitioning: a random sample is rst partitioned, and each partition is partially clustered. The partial clusters are then clustered in a second pass to yield the desired clusters. The major steps of the CURE algorithm are brie y outlined as follows: 1. Draw a random sample, S, containing s objects. 2. Partition sample S into p partitions, each of size s=p. 3. Partially cluster the partitions into s=pq clusters, for some q 1.
8.5. HIERARCHICAL METHODS
+ + + + + + + + + + + + + + +
+ + + + + + + + + + + + + + + +
Figure 8.7: Clustering of a set of points or objects by CURE. a A random sample of objects. b The objects are partitioned, and partially clustered. Representative points for each cluster are marked by a +". c The partial clusters are further clustered. For each new cluster, the representative points are shrinked" or moved towards the cluster center. d The nal clusters are of non-spherical shape. 4. Eliminate outliers by random sampling. If a cluster grows too slowly, remove it. 5. Cluster the partial clusters. The representative points falling in each newly-formed cluster are shrinked" or moved towards the cluster center by a user-speci ed fraction, or shrinking factor, . These points then represent and capture the shape of the cluster. 6. Mark the data with the corresponding cluster labels. Let's look at an example of how CURE works.
Example 8.4 Imagine that there are a set of points or objects located in a rectangular region. Suppose that you
would like to cluster the objects into two clusters, therefore p = 2. First, s = 52 objects are sampled as shown in Figure 8.7a. These objects are then distributed into two partitions, each containing 52=2 = 26 points. Suppose q = 2. We partially cluster the partitions into 52=2 2 = 13 clusters based on minimal mean distance. The partial clusters are indicated by dashed curves, as shown in Figure 8.7b. Each cluster representative is marked by a +". The partial clusters are further clustered, resulting in the two clusters illustrated by solid curves in Figure 8.7c. Each new cluster is shrinked" or condensed by moving its representative points towards the cluster center by a fraction, . The representative points capture the shape of each cluster. Thus, the initial objects are partitioned into two clusters, with the outliers excluded, as shown in Figure 8.7d. 2 CURE produces high quality clusters in the existence of outliers, allowing clusters of complex shapes and di erent sizes. The algorithm requires one scan of the entire database. Given n objects, the complexity of CURE is of On.
How sensitive is CURE to its user-speci ed parameters, such as the sample size, number of desired clusters, and shrinking fraction, ?" A sensitivity analysis showed that although some parameters can be varied without impacting
the quality of clustering, the parameter setting in general does have a signi cant in uence on the results. ROCK is an alternative agglomerative hierarchical clustering algorithm that, unlike CURE, is suited for clustering categorical attributes. It measures the similarity of two clusters by comparing the aggregate inter-connectivity of two clusters against a user-speci ed static inter-connectivity model, where the inter-connectivity of two clusters C1 and C2 is de ned by the number of cross links between the two clusters, and linkpi ; pj is the number of common
k-nearest Neighbor Graph Data Set Construct a Sparse Graph Partition the Graph
CHAPTER 8. CLUSTER ANALYSIS
Figure 8.8: CHAMELEON: Hierarchical clustering based on k-nearest neighbors and dynamic modeling. neighbors between two points pi and pj . In other words, cluster similarity is based on the number of points from di erent clusters who have neighbors in common. ROCK rst constructs a sparse graph from a given data similarity matrix using a similarity threshold and the concept of shared neighbors. It then performs a hierarchical clustering algorithm on the sparse graph.
8.5.4 CHAMELEON: A hierarchical clustering algorithm using dynamic modeling
tering process, two clusters are merged if the inter-connectivity and closeness proximity between two clusters are highly related to the internal inter-connectivity and closeness of objects within the clusters. The merge process based on the dynamic model facilitates the discovery of natural and homogeneous clusters, and applies to all types of data as long as a similarity function is speci ed. CHAMELEON is derived based on the observation of the weakness of two hierarchical clustering algorithms: CURE and ROCK. CURE and related schemes ignore information about the aggregate inter-connectivity of objects in two di erent clusters; whereas ROCK and related schemes ignore information about the closeness of two clusters while emphasizing their inter-connectivity. How does CHAMELEON work?" CHAMELEON rst uses a graph partitioning algorithm to cluster the data objects into a large number of relatively small subclusters. It then uses an agglomerative hierarchical clustering algorithm to nd the genuine clusters by repeatedly combining these clusters. To determine the pairs of most similar subclusters, it takes into account both the inter-connectivity as well as the closeness of the clusters, especially the internal characteristics of the clusters themselves. Thus it does not depend on a static, user-supplied model, and can automatically adapt to the internal characteristics of the clusters being merged. Let's look at CHAMELEON in closer detail. As shown in Figure 8.8, CHAMELEON represents its objects based on the commonly used k-nearest neighbor graph approach. Each vertex of the k-nearest neighbor graph represents a data object, and there exists an edge between two vertices objects, if one object is among the k-most similar objects of the other. The k-nearest neighbor graph Gk captures the concept of neighborhood dynamically: the neighborhood radius of an object is determined by the density of the region in which this object resides. In a dense region, the neighborhood is de ned narrowly; in a sparse region, it is de ned more widely. This tends to result in more natural clusters, in comparison with density-based methods such as DBSCAN described in Section 8.6 which instead uses a global neighborhood. Moreover, the density of the region is recorded as the weight of the edges. That is, the edge of a dense region tends to weigh more than that of a sparse region. CHAMELEON determines the similarity between each pair of clusters Ci and Cj according to their relative inter-connectivity RICi ; Cj , and their relative closeness RCCi; Cj . The relative inter-connectivity RICi ; Cj between two clusters Ci and Cj is de ned as the absolute interconnectivity between Ci and Cj , normalized with respect to the internal inter-connectivity of the two clusters Ci and Cj . That is, jECfCi;Cj g j ; 8.20 RICi ; Cj = 1 jECCi j + jECCj j 2 where ECfCi ;Cj g is the edge-cut of the cluster containing both Ci and Cj so that the cluster is broken into Ci and
CHAMELEON is a clustering algorithm which explores dynamic modeling in hierarchical clustering. In its clus-
8.6. DENSITY-BASED METHODS
Cj , and similarly, ECCi or ECCj is the size of its min-cut bisector i.e., the weighted sum of edges that partition the graph into two roughly equal parts. The relative closeness RCCi; Cj between a pair of clusters Ci and Cj is the absolute closeness between Ci and Cj , normalized with respect to the internal closeness of the two clusters Ci and Cj . It is de ned as RCCi; Cj =
jCj j jCi j jCi j+jCj j S EC Ci + jCi j+jCj j S EC Cj ;
S EC Ci ;Cj
where S EC Ci ;Cj is the average weight of the edges that connect vertices in Ci to vertices in Cj , and S EC Ci or S EC Cj is the average weight of the edges that belong to the min-cut bisector of cluster Ci or Cj . It has been shown that CHAMELEON has more power at discovering arbitrarily-shaped clusters of high quality than CURE and DBSCAN. However, the processing cost for high-dimensional data may require On2 time for n objects in the worst case.
8.6 Density-based methods
To discover clusters with arbitrary shape, density-based clustering methods have been developed. These typically regard clusters as dense regions of objects in the data space which are separated by regions of low density representing noise.
8.6.1 DBSCAN: A density-based clustering method based on connected regions with su ciently high density
DBSCAN Density-Based Spatial Clustering of Applications with Noise is a density-based clustering algorithm.
The algorithm grows regions with su ciently high density into clusters, and discovers clusters of arbitrary shape in spatial databases with noise. It de nes a cluster as a maximal set of density-connected points. The basic ideas of density-based clustering involve a number of new de nitions. We intuitively present these de nitions, and then follow up with an example.
1. The neighborhood within a radius of a given object is called the -neighborhood of the object. 2. If the -neighborhood of an object contains at least a minimum number, MinPts, of objects, then the object is called a core object. 3. Given a set of objects, D, we say that an object p is directly density-reachable from object q if p is within the -neighborhood of q, and q is a core object. 4. An object p is density-reachable from object q with respect to and MinPts in a set of objects, D, if there is a chain of objects p1 , . .. , pn, p1 = q and pn = p such that pi+1 is directly density-reachable from pi with respect to and MinPts, for 1 i n, pi 2 D. 5. An object p is density-connected to object q with respect to and MinPts in a set of objects, D, if there is an object o 2 D such that both p and q are density-reachable from o with respect to and MinPts. Density reachability is the transitive closure of direct density reachability, and this relationship is asymmetric. Only core objects are mutually density reachable. Density connectivity, however, is a symmetric relation.
Example 8.5 Consider Figure 8.9 for a given represented by the radius of the circles, and, say, let MinP ts = 3.
Based on the above de nitions: Of the labeled points, M, P, O, and R are core objects since each is in an -neighborhood containing at least three points. M is directly density-reachable from P, and Q is directly density-reachable from M.
CHAPTER 8. CLUSTER ANALYSIS
Q M P S O R
Figure 8.9: Density reachability and density connectivity in density-based clustering. Based on the previous observation, Q is indirectly density-reachable from P . However, P is not densityreachable from Q. Similarly, R and S are density-reachable from O. O, R and S are all density-connected.
A density-based cluster is a set of density-connected objects that is maximalwith respect to density-reachability. Every object not contained in any cluster is considered to be noise. How does DBSCAN nd clusters?" DBSCAN checks the -neighborhood of each point in the database. If the -neighborhood of a point p contains more than MinPts, a new cluster with p as a core object is created. DBSCAN then iteratively collects directly density-reachable objects from these core objects, which may involve the merge of a few density-reachable clusters. The process terminates when no new point can be added to any cluster. The computational complexity of DBSCAN is On log n, where n is the number of database objects. The algorithm is sensitive to the user-de ned parameters. DBSCAN is further discussed in the following section, which compares it to an alternative density-based clustering method called OPTICS.
8.6.2 OPTICS: Ordering Points To Identify the Clustering Structure
Although DBSCAN the density-based clustering algorithm described in Section 8.6.1 can cluster objects given input parameters such as and MinPts, it still leaves the user with the responsibility of selecting parameter values that will lead to the discovery of acceptable clusters. Actually, this is a problem associated with many other clustering algorithms. Such parameter settings are usually empirically set and di cult to determine, especially for real-world, high-dimensional data sets. Most algorithms are very sensitive to such parameter values: slightly di erent settings may lead to very di erent clusterings of the data. Moreover, high-dimensional real data sets often have very skewed distributions. There does not even exist a global parameter setting for which the result of a clustering algorithm may accurately describe the intrinsic clustering structure. To help overcome this di culty, a cluster ordering method called OPTICS Ordering Points To Identify the Clustering Structure was proposed. Rather than produce a data set clustering explicitly, OPTICS computes an augmented cluster ordering for automatic and interactive cluster analysis. This ordering represents the density-based clustering structure of the data. It contains information which is equivalent to density-based clustering obtained from a wide range of parameter settings. By examining DBSCAN, one can easily see that for a constant MinPts-value, density-based clusters with respect to a higher density i.e., a lower value for are completely contained in density-connected sets obtained with respect to a lower density. Recall that the parameter is a distance it is the neighborhood radius. Therefore, in order to produce a set or ordering of density-based clusters, we provide a set of distance parameter values. To construct the di erent clusterings simultaneously, the objects should be processed in a speci c order. This order selects an object that is density-reachable with respect to the lowest value so that clusters with higher density lower will be nished rst. Based on this idea, two values need to be stored for each object core-distance and reachability-distance: The core-distance of an object p is the smallest 0 value that makes p a core object. If p is not a core object, the core-distance of p is unde ned.
8.6. DENSITY-BASED METHODS
The reachability-distance of an object p and another object o is the greater value of the core-distance of p and the Euclidean distance between p and q. If p is not a core object, the reachability-distance between p and q is unde ned.
= 6mm =3mm p p
= 6mm q1 q2
Core distance of p
Reachability-distance (p, q1) = =3mm Reachability-distance (p, q2) = d(p, q2)
Figure 8.10: OPTICS terminology.
Example 8.6 Figure 8.10 illustrates the concepts of core-distance and reachability-distance. Suppose that = 6mm
and MinPts = 5. The core-distance of p is the distance, 0 , between p and the fourth closest data object.
2 How are these values used?" The OPTICS algorithm creates an ordering of the objects in a database, additionally storing the core-distance and a suitable reachability-distance for each object. Such information is su cient for the extraction of all density-based clusterings with respect to any distance 0 that is smaller than the distance used in generating the order. The cluster ordering of a data set can be represented graphically, which helps in its understanding. For example, Figure 8.11 is the reachability plot for a simple 2-dimensional data set, which presents a general overview of how the data are structured and clustered. Methods have also been developed for viewing clustering structures of highdimensional data. Because of the structural equivalence of the OPTICS algorithm to DBSCAN, the OPTICS algorithm has the same run-time complexity as that of DBSCAN, i.e., On log n. Spatial indexing structures can be used to further enhance its performance.
8.6.3 DENCLUE: Clustering based on density distribution functions
DENCLUE DENsity-based CLUstEring is a clustering method based on a set of density distribution functions.
The method is built on the following ideas: 1 the in uence of each data point can be formally modeled using a
Figure 8.11: Cluster ordering in OPTICS.
CHAPTER 8. CLUSTER ANALYSIS
mathematical function, called an in uence function, which describes the impact of a data point within its neighborhood; 2 the overall density of the data space can be modeled analytically as the sum of the in uence function of all data points; and 3 clusters can then be determined mathematically by identifying density attractors, where density attractors are local maxima of the overall density function.
Figure 8.12: Possible density functions for a 2-D data set. Figure is from 14 . Let x and y be objects in F d , a d-dimensional feature space. The in uence function of data object y on x is a y function fB : F d ! R+ , which is de ned in terms of a basic in uence function fB , 0
y fB x = fB x; y:
In principle, the in uence function can be an arbitrary function that can determine the distance between two objects in a neighborhood. It should be re exive and symmetric, such as the Euclidean distance function Section 8.2.1, a square wave in uence function, fSquare x; y = 0 if dx; y 8.23 1 otherwise or a Gaussian in uence function, fGaussx; y = e,
The density function at an object x 2 F d is de ned as the sum of in uence functions of all data points. Given n data objects, D = fx1; : : :; xng F d , the density function at x is de ned as n X x D fB x = fBi x: 8.25
For example, the density function that results from the Gaussian in uence function 8.24 is n X d x;x D fGaussianx = e, i
From the density function, one can de ne the gradient of a function, and the density attractor which is the local maxima of the overall density function. For a continuous and di erentiable in uence function, a hill climbing algorithm guided by the gradient, can be used to determine the density attractor of a set of data points. Figure 8.12 shows square wave and Gaussian density functions for a 2-D data set. Based on these notions, both center-de ned cluster and arbitrary-shape cluster can be formally de ned. A centerde ned cluster is a subset C that is density-extracted, i.e., with its density function value no less than a threshold ; otherwise i.e., if its density function value is less than , it is considered an outlier. An arbitrary-shape cluster is a set of C's, each being density-extracted, with the density function value no less than a threshold , and where
8.7. GRID-BASED METHODS
Figure 8.13: Examples of center-de ned clusters top row and arbitrary-shaped clusters bottom row. Figure is from 14 . there exists a path P from each region to another, and the density function value for each point along the path is no less than . Examples of center-de ned and arbitrary-shape clusters are shown in Figure 8.13. There are several: 1 it has a solid mathematical foundation, and generalizes other clustering methods, including partition-based, hierarchical, and locality-based methods, 2 it has good clustering properties for data sets with large amounts of noise, 3 it allows a compact mathematical description of arbitrarily shaped clusters in high-dimensional data sets, and 4 it uses grid cells yet only keeps information about grid cells that do actually contain data points. It manages these cells in a tree-based access structure, and thus is signi cantly faster than some in uential algorithms, such as DBSCAN by a factor of up to 45. However, the method requires careful selection of the density parameter and noise threshold , as the selection of such parameters may signi cantly in uence the quality of the clustering results.
What major advantages does DENCLUE have in comparison with other clustering algorithms, in general?"
8.7 Grid-based methods
The grid-based clustering approach uses a multiresolution grid data structure. It quantizes the space into a nite number of cells which form a grid structure on which all of the operations for clustering are performed. The main advantage of the approach is its fast processing time which is typically independent of the number of data objects, yet dependent on only the number of cells in each dimension in the quantized space. Some typical example of the grid-based approach include STING, which explores statistical information stored in the grid cells; WaveCluster, which clusters objects using a wavelet transform method; and CLIQUE, which represents a grid- and density-based approach for clustering in high-dimensional data space.
8.7.1 STING: A STatistical INformation Grid approach
STING STatistical INformation Grid is a grid-based multiresolution clustering technique in which the spatial area
CHAPTER 8. CLUSTER ANALYSIS
Figure 8.14: A hierarchical structure for STING clustering. is divided into rectangular cells. There are usually several levels of such rectangular cells corresponding to di erent levels of resolution, and these cells form a hierarchical structure: each cell at a high level is partitioned to form a number of cells at the next lower level. Statistical information regarding the attributes in each grid cell such as the mean, maximum, and minimum values are precomputed and stored. These statistical parameters are useful for query processing, as described below. Figure 8.14 shows a hierarchical structure for STING clustering. Statistical parameters of higher level cells can easily be computed from the parameters of the lower level cells. These parameters include the following: the attribute-independent parameter, count, and attribute-dependent parameters, m mean, s standard deviation, min minimum, max maximum, and the type of distribution that the attribute value in the cell follows, such as normal, uniform, exponential, or none if the distribution is unknown. When the data are loaded into the database, the parameters, count, m, s, min, max, of the bottom level cells are calculated directly from the data. The value of distribution may either be assigned by the user if the distribution type is known beforehand, or obtained by hypothesis tests such as the 2-test. The type of distribution of a higher level cell can be computed based on the majority of distribution types of its corresponding lower level cells in conjunction with a threshold ltering process. If the distributions of the lower level cells disagree with each other and fail the threshold test, the distribution type of the high level cell is set to none. How is this statistical information useful for query-answering?" It can be used in a top-down, grid-based method as follows. First, a layer within the hierarchical structure is determined from which the query-answering process is to start. This layer typically contains a small number of cells. For each cell in the current layer, we compute the con dence interval or estimated range of probability re ecting the cell's relevancy to the given query. The irrelevant cells are removed from further consideration. Processing of the next lower level examines only the remaining relevant cells. This process is repeated until the bottom layer is reached. At this time, if the query speci cation is met, the regions of relevant cells that satisfy the query are returned. Otherwise, the data that fall into the relevant cells are retrieved, and further processed until they meet the requirements of the query. What advantages does STING o er over other clustering methods?" There are several: 1 the grid-based computation is query-independent since the statistical information stored in each cell represents the summary information of the data in the grid cell, independent of the query; 2 the grid structure facilitates parallel processing and incremental updating; and 3 the method's e ciency is a major advantage: STING goes through the database once to compute the statistical parameters of the cells, and hence the time complexity of generating clusters in On, where n is the total number of objects. After generating the hierarchical structure, the query processing time is Og, where g is the total number of grid cells at the lowest level, which is usually much smaller than n. Since STING uses a multiresolution approach to perform cluster analysis, the quality of STING clustering depends on the granularity of the lowest level of the grid structure. If the granularity is very ne, the cost of processing will increase substantially; however, if the bottom level of the grid structure is too coarse, it may reduce the quality of cluster analysis. Moreover, STING does not consider the spatial relationship between the children and their
8.7. GRID-BASED METHODS
neighboring cells for construction of the parent cell. As a result, the shapes of the resulting clusters are isothetic, that is, all of the cluster boundaries are either horizontal or vertical, and no diagonal boundary is detected. This may lower the quality and accuracy of the clusters despite the fast processing time of the technique.
8.7.2 WaveCluster: Clustering using wavelet transformation
WaveCluster is a multiresolution clustering algorithm which rst summarizes the data by imposing a multidimensional grid structure onto the data space. It then uses a wavelet transformation to transform the original feature space, nding dense regions in the transformed space. In this approach, each grid cell summarizes the information of a group of points which map into the cell. This summary information typically ts into main memory for use by the multiresolution wavelet transform and the subsequent cluster analysis.
But what is a wavelet transform?"
A wavelet transform is a signal processing technique that decomposes a signal into di erent frequency subbands. The wavelet model can be applied to n-dimensional signals by applying a one-dimensional wavelet transform n times. In applying a wavelet transform, data are converted from the spatial domain where data are points in a feature or attribute space into the frequency domain showing the data at di erent scales or resolutions so that the natural clusters in the data become more distinguishable. Clusters can then be identi ed by searching for dense regions in the new domain. Wavelet transforms are also discussed in Chapter 3, where they are used for data reduction by compression. Additional references to the technique are given in the bibliographic notes. Why is wavelet transformation useful for clustering?" It o ers the following advantages. First, it provides unsupervised clustering. It uses hat-shape lters which emphasize regions where the points cluster, while at the same time, suppressing weaker information outside of the cluster boundaries. Thus, dense regions in the original feature space act as attractors for nearby points, and as inhibitors for points that are further away. This means that the clusters in the data automatically stand out and clear" the regions around them. Thus, another advantage is that wavelet transformation can automatically result in the removal of outliers.
Figure 8.15: A sample of 2-dimensional feature space. Figure is from 30 . The multiresolution property of wavelet transformations can help in the detection of clusters at varying levels of accuracy. For example, Figure 8.15 shows a sample of 2-dimensional feature space, where each point in the image represents the attribute or feature values of one object in the spatial data set. Figure 8.16 shows the resulting wavelet transformation at di erent resolutions, from a ne scale scale 1 to a coarse scale scale 3. At each level, the four sub-bands into which the original data are decomposed are shown. The sub-band shown at the upper-left quadrant emphasizes the average neighborhood around each data point . The sub-band at the upper-right quadrant emphasizes the horizontal edges of the data. The sub-band at the lower-left quadrant emphasizes the vertical edges, while the sub-band at the lower-right quadrant emphasizes the corners. Wavelet-based clustering is very fast, with a computational complexity of On where n is the number of objects in the database. The algorithm implementation can be made parallel. WaveCluster is a grid-based and density-based algorithm. It conforms with many of the requirements of a good clustering algorithm: It handles large data sets e ciently, discovers clusters with arbitrary shape, successfully handles
CHAPTER 8. CLUSTER ANALYSIS
Figure 8.16: Multiresolution of the feature space in Figure 8.15 at a scale 1 high resolution; b scale 2 medium resolution; c scale 3 low resolution. Figure is from 30 . outliers, is insensitive to the order of input, and does not require the speci cation of input parameters such as the number of clusters or a neighborhood radius. In experimental studies, WaveCluster was found to outperform BIRCH, CLARANS and DBSCAN in terms of both e ciency and clustering quality. However, WaveCluster is only applicable to low-dimensional data.
8.7.3 CLIQUE: Clustering high-dimensional space
The CLIQUE CLustering In QUEst clustering algorithm integrates density-based and grid-based clustering. It is useful for clustering high dimensional data in large databases. CLIQUE is based on the following: Given a large set of multidimensional data points, the data space is usually not uniformly occupied by the data points. CLIQUE's clustering identi es the sparse and the crowded" areas in space or units, thereby discovering the overall distribution patterns of the data set. A unit is dense if the fraction of total data points contained in it exceeds an input model parameter. In CLIQUE, a cluster is de ned as a maximal set of connected dense units.
How does CLIQUE work?" CLIQUE performs multidimensional clustering in two steps:
1. First, CLIQUE partitions the n-dimensional data space into non-overlapping rectangular units, identifying the dense units among these. This is done in 1-D for each dimension. For example, Figure 8.17 shows dense rectangular units found with respect to Age for the dimensions Salary and number of weeks of Vacation. The subspaces representing these dense units are intersected to form a candidate search space in which dense units of higher dimensionality may exist. How does CLIQUE search for dense units of higher dimensionality?" The identi cation of actual dense units within the candidate search space is based on the Apriori property used in association rule mining1. In general, the property employs prior knowledge of items in the search space so that portions of the space can be pruned.
Association rule mining is described in detail in Chapter 6. In particular, the Apriori property is described in Section 6.2.1.
8.8. MODEL-BASED CLUSTERING METHODS
The property, adapted for CLIQUE, states the following: If a k-dimensional unit is dense, then so are its projections in k , 1-dimensional space. That is, given a k-dimensional candidate dense unit, if we check its k , 1-th projection units and nd any that are not dense, then we know that the k-th dimensional unit cannot be dense either. Therefore, we can generate potential or candidate dense units in k,dimensional space from the dense units found in k , 1-dimensional. In general, the resulting space searched is much smaller than the original space. The dense units are then examined in order to determine the clusters. 2. CLIQUE generates a minimal description for each cluster as follows. For each cluster, it determines the maximal region that covers the cluster of connected dense units. It then determines a minimal cover for each cluster.
How e ective is CLIQUE?" CLIQUE automatically nds subspaces of the highest dimensionality such that high density clusters exist in those subspaces. It is insensitive to the order of input tuples and does not presume any canonical data distribution. It scales linearly with the size of input and has good scalability as the number of dimensions in the data is increased. However, the accuracy of the clustering result may be degraded at the expense of the simplicity of the method.
8.8 Model-based clustering methods
Model-based clustering methods attempt to optimize the t between the given data and some mathematical model. Such methods are often based on the assumption that the data are generated by a mixture of underlying probability distributions. Model-based clustering methods follow two major approaches: a statistical approach or a neural network approach. Examples of each approach are described in this section.
8.8.1 Statistical approach
Conceptual clustering is a form of clustering in machine learning which, given a set of unlabeled objects, produces
a classi cation scheme over the objects. Unlike conventional clustering which primarily identi es groups of like objects, conceptual clustering goes one step further by also nding characteristic descriptions for each group, where each group represents a concept or class. Hence, conceptual clustering is a two-step process: rst, clustering is performed, followed by characterization. Here, clustering quality is not solely a function of the individual objects. Rather, it incorporates factors such as the generality and simplicity of the derived concept descriptions. Most methods of conceptual clustering adopt a statistical approach which uses probability measurements in determining the concepts or clusters. Probabilistic descriptions are typically used to represent each derived concept. COBWEB is a popular and simple method of incremental conceptual clustering2 . Its input objects are described by categorical attribute-value pairs. COBWEB creates a hierarchical clustering in the form of a classi cation tree. Figure 8.18 shows a classi cation tree for animal data, based on Fisher 1987. A classi cation tree di ers from a decision tree. Each node in a classi cation tree refers to a concept, and contains a probabilistic description of that concept which summarizes the objects classi ed under the node. The probabilistic description includes the probability of the concept, and conditional probabilities of the form PAi = Vij jCk , where Ai = Vij is an attributevalue pair and Ck is the concept class. Counts are accumulated and stored at each node for computation of the probabilities. This is unlike decision trees, which label branches rather than nodes and use logical rather than probabilistic descriptors3 . The sibling nodes at a given level of a classi cation tree are said to form a partition. To classify an object using a classi cation tree, a partial matching function is employed to descend the tree along a path of `best' matching nodes. COBWEB uses a heuristic evaluation measure called category utility to guide construction of the tree. Category utility CU is de ned as
2 n=1P Ck ij PAi = Vij jCk 2 , i j PAi = Vij k ; n
But, what is a classi cation tree? Is it the same as a decision tree?"
It is incremental in that it takes one object at a time as input, rather than all objects at once. Decision trees are described in Chapter 7 on Classi cation and Prediction.
CHAPTER 8. CLUSTER ANALYSIS
where n is the number of nodes, concepts, or categories" forming a partition fC1; C2; ::; Cng at the given level of the tree. In other words, category utility is the increase in the expected number of attribute values that can be correctly guessed given a partition where this expected number is represented by the rst term of the numerator in Equation 8.27 over the expected number of correct guesses with no such knowledge where this number is the second term of the numerator. Category utility rewards inter-class similarity and intra-class dissimilarity where:
inter-class similarity is the probability PCkjAi = Vij . The larger this value is, the fewer the objects in
contrasting classes that share this attribute-value pair, and the more predictive the pair is of the class. intra-class similarity is the probability PAi = Vij jCk . The larger this value is, the greater the proportion of class members sharing the value, and the more predictable the value is of class members;
Let's have a look at how COBWEB works. COBWEB incrementally incorporates objects into a classi cation tree. Given a new object," you wonder, how does COBWEB decide where to incorporate it into the classi cation tree?" COBWEB descends the tree along an appropriate path, updating counts along the way, in search of the `best host' or node at which to classify the object. This decision is based on temporarily placing the object in each node, and computing the category utility of the resulting partition. The placement which results in the highest category utility should be a good host for the object.
Hmm, but what if the object does not really belong to any of the concepts represented in the tree so far? What if it is better to create a new node for the given object?"
That is a good point. In fact, COBWEB also computes the category utility of the partition that would result if a new node were to be created for the object. This is compared to the above computation based on the existing nodes. The object is then placed in an existing class, or a new class is created for it, based on the partition with the highest category utility value. Notice that COBWEB has the ability to automatically adjust the number of classes in a partition. It does not need to rely on the user to provide such an input parameter. The two operators mentioned above are highly sensitive to the input order of the object. COBWEB has two additional operators which help make it less sensitive to input order. These are merging and splitting. When an object is incorporated, the two best hosts are considered for merging into a single class. Furthermore, COBWEB considers splitting the children of the best host among the existing categories. These decisions are based on category utility. The merging and splitting operators allow COBWEB to perform a bidirectional search | e.g., a merge can undo a previous split. What are the limitations of COBWEB?" The method has several. First, it is based on the assumption that probability distributions on separate attributes are statistically independent of one other. This assumption is, however, not always true since correlation between attributes often exists. Moreover, the probability distribution representation of clusters makes it quite expensive to update and store the clusters. This is especially so when the attributes have a large number of values since their time and space complexities depend not only on the number of attributes, but also on the number of values for each attribute. Furthermore, the classi cation tree is not heightbalanced for skewed input data, which may cause the time and space complexity to degrade dramatically. CLASSIT is an extension of COBWEB for incremental clustering of continuous or real valued data. It stores a continuous normal distribution i.e., mean and standard deviation for each individual attribute in each node and uses a modi ed category utility measure which is an integral over continuous attributes instead of a sum over discrete attributes as in COBWEB. However, it su ers similar problems as COBWEB and thus is not suitable for clustering large database data. Additional research is needed in the application of conceptual clustering methods to data mining.
8.8.2 Neural network approach
The neural network approach to clustering tends to represent each cluster as an exemplar. An exemplar acts as a prototype" of the cluster and does not necessarily have to correspond to a particular data example or object. New objects can be distributed to the cluster whose exemplar is the most similar, based on some distance measure. The attributes of an object assigned to a cluster can be predicted from the attributes of the cluster's exemplar. In this section, we discuss two prominent methods of the neural network approach to clustering. The rst in competitive learning, and the second is self-organizing feature maps, both of which involve competing neural units.
8.9. OUTLIER ANALYSIS
Competitive learning involves an hierarchical architecture of several units or arti cial neurons" which compete in a winner-takes-all" fashion for the object that is currently being presented to the system. Figure 8.19 shows an example of a competitive learning system. Each circle represents a unit. The winning unit within a cluster becomes active indicated by a lled circle, while the others are inactive indicated by empty circles. Connections between layers are excitatory a unit in a given layer can receive inputs from all of the units in the next lower level. The con guration of active units in a layer represents the input pattern to the next higher layer. The units within a cluster at a given layer compete with one another to respond to the pattern that is output from the layer below. Connections within layers are inhibitory so that only one unit in any given cluster may be active. The winning unit adjusts the weights on its connections between other units in the cluster so that it will respond even more strongly to future objects that are the same or similar to the current one. If we view the weights as de ning an exemplar, then new objects are assigned to the cluster with the closest exemplar. The number of clusters, and of units per cluster are input parameters. At the end of the clustering or any clustering, in general, each cluster can be thought of as a new feature" that detects some regularity in the objects. Thus, the resulting clusters can be viewed as a mapping of low-level features to higher-level features. With self-organizing feature maps SOMs, clustering is also performed by having several units compete for the current object. The unit whose weight vector is closest to the current object becomes the winning or active unit. So as to move even closer to the input object, the weights of the winning unit are adjusted, as well as those of its nearest neighbors. SOMs assume that there is some topology or ordering among the input objects, and that the units will eventually take on this structure in space. The organization of units is said to form a feature map. SOMs are believed to resemble processing that can occur in the brain. The neural network approach to clustering has strong theoretical links with actual brain processing. Further research is required in making it readily applicable to large databases due to long processing times and the intricacies of complex data.
8.9 Outlier analysis
Very often, there exist data objects that do not comply with the general behavior or model of the data. Such data objects, which are grossly di erent from or inconsistent with the remaining set of data, are called outliers. Outliers can be caused by measurement or execution error. For example, the display of a person's age as ,999 could be caused by a program default setting of an unrecorded age. Alternatively, outliers may be the result of inherent data variability. The salary of the chief executive o cer of a company, for instance, could naturally stand out as an outlier among the salaries of the other employees in the rm. Many data mining algorithms try to minimize the in uence of outliers, or eliminate them all together. This, however, could result in the loss of important hidden information since one person's noise could be another person's signal". In other words, the outliers themselves may be of particular interest, such as in the case of fraud detection, where outliers may indicate fraudulent activity. Thus, outlier detection and analysis is an interesting data mining task, referred to as outlier mining. Outlier mining has wide applications. As mentioned above, it can be used in fraud detection, e.g., by detecting unusual usage of credit cards or telecommunication services. In addition, it is useful in customized marketing for identifying the spending behavior of customers with extremely low or extremely high incomes, or in medical analysis for nding unusual responses to various medical treatments. Outlier mining can be described as follows: Given a set of n data points or objects, and k, the expected number of outliers, nd the top k objects which are considerably dissimilar, exceptional, or inconsistent with respect to the remaining data. The outlier mining problem can be viewed as two subproblems: 1 de ne what data can be considered as inconsistent in a given data set; and 2 nd an e cient method to mine the outliers so de ned. The problem of de ning outliers is nontrivial. If a regression model is used for data modeling, analysis of the residuals can give a good estimation for data extremeness". The task becomes tricky, however, when nding outliers in time series data as they may be hidden in trend, seasonal, or other cyclic changes. When multidimensional data are analyzed, not any particular one, but rather, a combination of dimension values may be extreme. For non-numeric i.e., categorical data, the de nition of outliers requires special consideration.
What is an outlier?"
CHAPTER 8. CLUSTER ANALYSIS
What about using data visualization methods for outlier detection?", you may wonder. This may seem to be an obvious choice, since human eyes are very fast and e ective at noticing data inconsistencies. However, this does not apply to data containing cyclic plots, where apparently outlying values could be perfectly valid values in reality. Data visualization methods are weak in detecting outliers in data with may categorical attributes, or in data of high-dimensionality since human eyes are good at visualizing numeric data of only two to three dimensions. In this section, we instead examine computer-based methods for outlier detection. These can be categorized into three approaches: the statistical approach, the distance-based approach, and the deviation-based approach, each of which are studied here. Notice that while clustering algorithms discard outliers as noise, they can be modi ed to include outlier detection as a byproduct of their execution. In general, users must check that each outlier discovered by these approaches is indeed a real" outlier.
8.9.1 Statistical-based outlier detection
The statistical approach to outlier detection assumes a distribution or probability model for the given data set e.g., a normal distribution and then identi es outliers with respect to the model using a discordancy test. Application of the test requires knowledge of the data set parameters such as the assumed data distribution, knowledge of distribution parameters such as the mean and variance, and the expected number of outliers. How does the discordancy testing work?" A statistical discordancy test examines two hypotheses: a working hypothesis and an alternative hypothesis. A working hypothesis, H, is a statement that the entire data set of n objects comes from an initial distribution model, F, i.e., H : oi 2 F; where i = 1; 2; : : :; n: The hypothesis is retained if there is no statistically signi cant evidence supporting its rejection. A discordancy test veri es whether an object oi is signi cantly large or small in relation to the distribution F. Di erent test statistics have been proposed for use as a discordancy test, depending on the available knowledge of the data. Assuming that some statistic T has been chosen for discordancy testing, and the value of the statistic for object oi is vi , then the distribution of T is constructed. Signi cance probability SPvi = ProbT vi is evaluated. If some SP vi is su ciently small, then oi is discordant and the working hypothesis is rejected. An alternative hypothesis, H, which states that oi comes from another distribution model, G, is adopted. The result is very much dependent on which F model is chosen since oi may be an outlier under one model, and a perfectly valid value under another. The alternative distribution is very important in determining the power of the test, i.e. the probability that the working hypothesis is rejected when oi is really an outlier. There are di erent kinds of alternative distributions.
Inherent alternative distribution: In this case, the working hypothesis that all of the objects come from
distribution F is rejected in favor of the alternative hypothesis that all of the objects arise from another distribution, G: H : oi 2 G; where i = 1; 2; : : :; n: F and G may be di erent distributions, or di er only in parameters of the same distribution. There are constraints on the form of the G distribution in that it must have potential to produce outliers. For example, it may have a di erent mean or dispersion, or a longer tail. Mixture alternative distribution: The mixture alternative states that discordant values are not outliers in the F population, but contaminants from some other population. In this case, the alternative hypothesis is: H : oi 2 1 , F + G; where i = 1; 2; : : :; n:
Slippage alternative distribution: This alternative states that all of the objects apart from some prescribed
small number arise independently from the initial model F with parameters and 2 , while the remaining objects are independent observations from a modi ed version of F in which the parameters have been shifted. There are two basic types of procedures for detecting outliers: Block procedures: In this case, either all of the suspect objects are treated as outliers, or all of them are accepted as consistent.
8.9. OUTLIER ANALYSIS
Consecutive or sequential procedures: An example of such a procedure is the inside-out procedure. Its
main idea is that the object that is least likely" to be an outlier is tested rst. If it is found to be an outlier, then all of the more extreme values are also considered outliers; otherwise, the next most extreme object is tested, and so on. This procedure tends to be more e ective than block procedures.
How e ective is the statistical approach at outlier detection?" A major drawback is that most tests are for single attributes, yet many data mining problems require nding outliers in multidimensional space. Moreover, the statistical approach requires knowledge about parameters of the data set, such as the data distribution. However, in many cases, the data distribution may not be known. Statistical methods do not guarantee that all outliers will be found for the cases where no speci c test was developed, or the observed distribution cannot be adequately modeled with any standard distribution.