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Lecture 10: Clustering Artificial Intelligence CS-GY-6613-I Julian Togelius / Catalina Jaramillo julian.togelius@nyu.edu / cmj383@nyu.edu Types of learning Supervised learning Learning to predict or classify labels based on labeled input data Unsupervised learning Finding patterns in unlabeled data Reinforcement learning Learning well-performing behavior from state observations and rewards Clustering Cluster: a collection of data objects Similar to one another within the same cluster Dissimilar to the objects in other clusters Cluster analysis: Finding similarities between data according to the characteristics found in the data and grouping similar data objects into clusters Unsupervised learning: no predefined classes 4 Applications As a stand-alone tool to get insight into data distribution As a preprocessing step for other algorithms 5 Marketing: Help marketers discover distinct groups in their customer bases, and then use this knowledge to develop targeted marketing programs Land use: Identification of areas of similar land use in an earth observation database Insurance: Identifying groups of motor insurance policy holders with similar behavior patterns City-planning: Identifying groups of houses according to their house type, value, and geographical location Earth-quake studies: Observed earth quake epicenters should be clustered along continent faults Games: identify player groups / archetypes What is good clustering A good clustering method will produce high quality clusters with high intra-class similarity low inter-class similarity The quality of a clustering result depends on both the similarity measure used by the method and its implementation The quality of a clustering method is also measured by its ability to discover some or all of the hidden patterns Data Mining: Concepts and Techniques 8 Similarity and Dissimilarity Between Objects ! Distances are normally used to measure the similarity or dissimilarity between two data objects ! Some popular ones include: Minkowski distance: where i = (xi1, xi2, …, xip) and j = (xj1, xj2, …, xjp) are two p-dimensional data objects, and q is a positive integer ! If q = 1, d is Manhattan distance Data Mining: Concepts and Techniques 9 Similarity and Dissimilarity Between Objects (Cont.) ! If q = 2, d is Euclidean distance: ! Also, one can use weighted distance, parametric Pearson product moment correlation, or other dissimilarity measures Some requirements… Scalability Ability to deal with different types of attributes Ability to handle dynamic data Discovery of clusters with arbitrary shape Able to deal with noise and outliers Insensitive to order of input records High dimensionality Incorporation of user-specified constraints Clustering approaches Partitioning approach: Construct various partitions and then evaluate them by some criterion, e.g., minimizing the sum of square errors Typical methods: k-means, k-medoids, CLARANS Hierarchical approach: Create a hierarchical decomposition of the set of data (or objects) using some criterion Typical methods: Diana, Agnes, BIRCH, ROCK, CAMELEON Density-based approach: Based on connectivity and density functions Typical methods: DBSCAN, OPTICS, DenClue 11 Grid-based approach: based on a multiple-level granularity structure Typical methods: STING, WaveCluster, CLIQUE Model-based: A model is hypothesized for each of the clusters and tries to find the best fit of that model to each other Typical methods: EM, SOM, COBWEB Frequent pattern-based: Based on the analysis of frequent patterns Typical methods: pCluster User-guided or constraint-based: Clustering by considering user- specified or application-specific constraints Typical methods: COD (obstacles), constrained clustering 12 In this lecture Partitioning approaches Hierarchical approaches Measuring cluster quality Partitioning algorithms Partitioning method: Construct a partition of a database D of n objects into a set of k clusters, s.t., min sum of squared distance between each point in the cluster i (p) and the center point of the cluster (ci) for all clusters (k) in D Given a k, find a partition of k clusters that optimizes the chosen partitioning criterion Which is the simplest possible clustering algorithm Partitioning algorithms Global optimal: exhaustively enumerate all partitions Heuristic methods: k-means and k-medoids algorithms k-means (MacQueen’67): Each cluster is represented by the center of the cluster k-medoids or PAM (Partition around medoids) (Kaufman & Rousseeuw’87): Each cluster is represented by one of the objects in the cluster 15 k-means Given k, the k-means algorithm is implemented in four steps: 1. Arbitrarily choose k objects from D as the initial cluster center 2. Assign each object to the cluster to which the object is the most similar, based on the cluster center value 3. Update the cluster center, that is, calculate the mean value of the objects for each cluster 4. Go back to Step 2, stop when no more new assignment or max number of iterations is reached K=2 Arbitrarily choose K object as initial cluster center Assign each objects to most similar center Update the cluster means Update the cluster means reassign k-means 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 10 reassign 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 100 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 10 Strength: Relatively efficient: O(tkn), where n is # objects, k is # clusters, and t is # iterations. Normally, k, t << n. Comparing: PAM: O(k(n-k)2 ), CLARA: O(ks2 + k(n-k)) Comment: Often terminates at a local optimum. The global optimum may be found using techniques such as: deterministic annealing and genetic algorithms Weaknesses: Applicable only when mean is defined, then what about categorical data Need to specify k, the number of clusters, in advance Unable to handle noisy data and outliers Not suitable to discover clusters with non-convex shapes 18 Variations A few variants of the k-means which differ in Selection of the initial centroids Dissimilarity calculations Strategies to calculate cluster means Class Exercise, Q1 20 Handling categorical data Handling categorical data: k-modes (Huang’98) Replacing means of clusters with modes Using new dissimilarity measures to deal with categorical objects Using a frequency-based method to update modes of clusters A mixture of categorical and numerical data: k- prototype method 21 A problem with k-means The k-means algorithm is sensitive to outliers ! Since an object with an extremely large value may substantially distort the distribution of the data. k-medoids Instead of taking the mean value of the object in a cluster as a reference point, medoids can be used, which is the most centrally located object in a cluster. 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 k-medoids Find representative objects, called medoids, in clusters PAM (Partitioning Around Medoids, 1987) starts from an initial set of medoids and iteratively replaces one of the medoids by one of the non-medoids if it improves the total distance of the resulting clustering PAM works effectively for small data sets, but does not scale well for large data sets CLARA (Kaufmann & Rousseeuw, 1990) CLARANS (Ng & Han, 1994): Randomized sampling Focusing + spatial data structure (Ester et al., 1995) 24 PAM (Kaufman and Rousseeuw, 1987) Use real object to represent the cluster 1. Select k representative objects arbitrarily 2. Assign each non-selected object to the cluster with the nearest representative object 3. Select a random non representative object, h 4. For each pair of representative object (i) and h, calculate the total swapping cost TCih 5. For each pair of i and h, if TCih < 0, i is replaced by h 6. repeat steps 2-5 until there is no change or max number of iterations is reached PAM Total Cost = 20 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 K=2 Arbitrary choose k object as initial medoids Assign each remaining object to nearest medoids Randomly select a nonmedoid object,Oramdom Compute total cost of swapping Total Cost = 16 Swapping O and Oramdom If quality is improved. Do loop Until no change 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 PAM problem Pam is more robust than k-means in the presence of noise and outliers because a medoid is less influenced by outliers or other extreme values than a mean Pam works efficiently for small data sets but does not scale well for large data sets. O(k(n-k)2 ) for each iteration where n is # of data, k is # of clusters Hierarchical clustering Use distance matrix as clustering criteria. This method does not require the number of clusters k as an input, but needs a termination condition Step 0 Step 1 Step 2 Step 3 Step 4 b d c e a a d c d a b c d e Step 4 Step 3 Step 2 Step 1 Step 0 agglomerative (AGNES) divisive (DIANA) AGNES (Agglomerative Nesting) Use the Single-Link method (distance between cluster a and b)=distance between closest members of clusters a and b) and the dissimilarity matrix.** Merge nodes that have the least dissimilarity Go on in a non-descending fashion Eventually all nodes belong to the same cluster 0 3 5 8 10 0 3 5 8 10 0 3 5 8 10 0 3 5 8 10 0 3 5 8 10 0 3 5 8 10 29 Class Exercise, Q2 30 Dendrogram Where do you cut Where do you cut Single-linkage Distance between nearest clusters is bigger than min distance threshold Complete-linkage Max distance between nearest clusters exceeds a maximum threshold A given number of clusters DIANA (Divisive Analysis) Inverse order of AGNES Eventually each node forms a cluster on its own 0 3 5 8 10 0 3 5 8 10 0 3 5 8 10 0 3 5 8 10 0 3 5 8 10 0 3 5 8 10 High-dimensional data Clustering high-dimensional data Many applications: text documents, DNA micro-array data Major challenges: Many irrelevant dimensions may mask clusters Distance measure becomes meaningless—due to equi- distance Clusters may exist only in some subspaces High-dimensional data Methods Feature transformation: only effective if most dimensions are relevant PCA & SVD useful only when features are highly correlated/ redundant Feature selection: wrapper or filter approaches Useful to find a subspace where the data have nice clusters Subspace-clustering: find clusters in all the possible subspaces CLIQUE, ProClus, and frequent pattern-based clustering 35 The curse of dimensionality Data in only one dimension is relatively packed Adding a dimension “stretches” the points across that dimension, making them further apart Adding more dimensions will make the points further apart—high dimensional data is extremely sparse Distance measure becomes meaningless—due to equi-distance Measuring clustering quality Dissimilarity/Similarity metric: Similarity is expressed in terms of a distance function, typically metric: d(i, j) There is a separate “quality” function that measures the “goodness” of a cluster. The definitions of distance functions are usually very different for interval-scaled, boolean, categorical, ordinal ratio, and vector variables. It is hard to define “similar enough” or “good enough”; the answer is typically highly subjective. Silhouette coefficient Where i is an instance, a(i) is the instance’s average similarity to all other points in its own cluster, and b(i) is the average similarity of the instance to all points in the closest other cluster Silhouette of a clustering: average s(i) of all points Value between -1 and 1. A low value is related with too few of to many clusters 38 Elbow Method How to identify the best number of clusters k Uses the change in explained variation by adding/removing a cluster Measure of variance examples: ratio between-group variance to total variance or ratio between-group variance to within-group variance Unsupervised learning in general Maybe most of what we do is unsupervised learning, all the time Learning the relations between actions and consequences in the world Source: Yann LeCun