Segmentação (Clustering) (baseado nos slides do Han)

SAD Tagus 2004/05 H. Galhardas Non-supervised Learning: Cluster Analysis What is Cluster Analysis? What is Cluster Analysis? Types of Data in Cluster Analysis Types of Data in Cluster Analysis A Categorization of Major Clustering Methods A Categorization of Major Clustering Methods Partitioning Methods Partitioning Methods Hierarchical Methods Hierarchical Methods Summary Summary

SAD Tagus 2004/05 H. Galhardas What is Cluster Analysis? Cluster: a collection of data objects Cluster: a collection of data objects Similar to one another within the same cluster Dissimilar to the objects in other clusters Cluster analysis Cluster analysis Grouping a set of data objects into clusters Clustering is unsupervised classification: no predefined classes Clustering is unsupervised classification: no predefined classes Typical applications Typical applications As a stand-alone tool to get insight into data distribution As a preprocessing step for other algorithms

SAD Tagus 2004/05 H. Galhardas General Applications of Clustering Pattern Recognition Pattern Recognition Spatial Data Analysis Spatial Data Analysis create thematic maps in GIS by clustering feature spaces detect spatial clusters and explain them in spatial data mining Image Processing Image Processing Economic Science (especially market research) Economic Science (especially market research) WWW WWW Document classification Cluster Weblog data to discover groups of similar access patterns

SAD Tagus 2004/05 H. Galhardas Examples of Clustering Applications Marketing: Help marketers discover distinct groups in their customer bases, and then use this knowledge to develop targeted marketing programs 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 Land use: Identification of areas of similar land use in an earth observation database Insurance: Identifying groups of motor insurance policy holders with a high average claim cost Insurance: Identifying groups of motor insurance policy holders with a high average claim cost City-planning: Identifying groups of houses according to their house type, value, and geographical location 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 Earth-quake studies: Observed earth quake epicenters should be clustered along continent faults

SAD Tagus 2004/05 H. Galhardas What Is Good Clustering? A good clustering method will produce high quality clusters with 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 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. The quality of a clustering method is also measured by its ability to discover some or all of the hidden patterns.

SAD Tagus 2004/05 H. Galhardas Requirements of Clustering in Data Mining Scalability Scalability Ability to deal with different types of attributes Ability to deal with different types of attributes Discovery of clusters with arbitrary shape Discovery of clusters with arbitrary shape Minimal requirements for domain knowledge to determine input parameters Minimal requirements for domain knowledge to determine input parameters Able to deal with noise and outliers Able to deal with noise and outliers Insensitive to order of input records Insensitive to order of input records High dimensionality High dimensionality Incorporation of user-specified constraints Incorporation of user-specified constraints Interpretability and usability Interpretability and usability

SAD Tagus 2004/05 H. Galhardas Data Structures Data matrix Data matrix (two modes) Dissimilarity matrix Dissimilarity matrix (one mode)

SAD Tagus 2004/05 H. Galhardas Measure the Quality of Clustering Dissimilarity/Similarity metric: Similarity is expressed in terms of a distance function, which is typically a metric: d(i, j) Dissimilarity/Similarity metric: Similarity is expressed in terms of a distance function, which is typically a metric: d(i, j) There is a separate “quality” function that measures the “goodness” of a cluster. There is a separate “quality” function that measures the “goodness” of a cluster. The definitions of distance functions are usually very different for interval-scaled, boolean, categorical, ordinal and ratio variables. The definitions of distance functions are usually very different for interval-scaled, boolean, categorical, ordinal and ratio variables. Weights should be associated with different variables based on applications and data semantics. Weights should be associated with different variables based on applications and data semantics. It is hard to define “similar enough” or “good enough” It is hard to define “similar enough” or “good enough” the answer is typically highly subjective.

SAD Tagus 2004/05 H. Galhardas Type of data in clustering analysis Interval-scaled variables: Interval-scaled variables: Binary variables: Binary variables: Nominal, ordinal, and ratio variables: Nominal, ordinal, and ratio variables: Variables of mixed types: Variables of mixed types:

SAD Tagus 2004/05 H. Galhardas Interval-valued variables Standardize data Standardize data Calculate the mean absolute deviation: where Calculate the standardized measurement (z-score) Using mean absolute deviation is more robust than using standard deviation Using mean absolute deviation is more robust than using standard deviation

SAD Tagus 2004/05 H. Galhardas Similarity and Dissimilarity Between Objects Distances are normally used to measure the similarity or dissimilarity between two data objects Distances are normally used to measure the similarity or dissimilarity between two data objects Some popular ones include: Minkowski distance Some popular ones include: Minkowski distance where i = (x i1, x i2, …, x ip ) and j = (x j1, x j2, …, x jp ) are two p-dimensional data objects, and q is a positive integer If q = 1, d is Manhattan distance If q = 1, d is Manhattan distance

SAD Tagus 2004/05 H. Galhardas Similarity and Dissimilarity Between Objects (Cont.) If q = 2, d is Euclidean distance: If q = 2, d is Euclidean distance: Properties: d(i,j)  0 d(i,i) = 0 d(i,j) = d(j,i) d(i,j)  d(i,k) + d(k,j) Also, one can use weighted distance, parametric Pearson product moment correlation, or other disimilarity measures Also, one can use weighted distance, parametric Pearson product moment correlation, or other disimilarity measures

SAD Tagus 2004/05 H. Galhardas Partitioning Algorithms: Basic Concept Partitioning method: Construct a partition of a database D of n objects into a set of k clusters Partitioning method: Construct a partition of a database D of n objects into a set of k clusters Given a k, find a partition of k clusters that optimizes the chosen partitioning criterion Given a k, find a partition of k clusters that optimizes the chosen partitioning criterion 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

SAD Tagus 2004/05 H. Galhardas The K-Means Clustering Method Given k, the k-means algorithm is implemented in four steps: Given k, the k-means algorithm is implemented in four steps: 1. Partition objects into k nonempty subsets 2. Compute seed points as the centroids of the clusters of the current partition (the centroid is the center, i.e., mean point, of the cluster) 3. Assign each object to the cluster with the nearest seed point 4. Go back to Step 2, stop when no more new assignment

SAD Tagus 2004/05 H. Galhardas The K-Means Clustering Method Example Example K=2 Arbitrarily choose K object as initial cluster center Assign each objects to most similar center Update the cluster means reassign

SAD Tagus 2004/05 H. Galhardas Comments on the K-Means Method Strength Relatively efficient: O(tkn), where n is # objects, k is # clusters, and t is # iterations. Normally, k, t << n. Comparing: PAM: O(k(n-k) 2 ), CLARA: O(ks 2 + k(n-k)) Comment Often terminates at a local optimum. The global optimum may be found using techniques such as: deterministic annealing and genetic algorithms Weakness 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

SAD Tagus 2004/05 H. Galhardas Variations of the K-Means Method A few variants of the k-means which differ in A few variants of the k-means which differ in Selection of the initial k means Dissimilarity calculations Strategies to calculate cluster means Handling categorical data: k-modes (Huang’98) 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

SAD Tagus 2004/05 H. Galhardas What is the problem of k- Means Method? The k-means algorithm is sensitive to outliers ! 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. 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

SAD Tagus 2004/05 H. Galhardas The K-Medoids Clustering Method Find representative objects, called, in clusters Find representative objects, called medoids, in clusters PAM (Partitioning Around Medoids, 1987) 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

SAD Tagus 2004/05 H. Galhardas Typical k-medoids algorithm (PAM) Total Cost = K=2 Arbitrary choose k object as initial medoids Assign each remainin g object to nearest medoids Randomly select a nonmedoid object,O ramdom Compute total cost of swapping Total Cost = 26 Swapping O and O ramdom If quality is improved. Do loop Until no change

SAD Tagus 2004/05 H. Galhardas PAM (Partitioning Around Medoids) (1987) Use real object to represent the cluster Use real object to represent the cluster 1. Select k representative objects arbitrarily 2. For each pair of non-selected object h and selected object i, calculate the total swapping cost TC ih 3. For each pair of i and h, If TC ih < 0, i is replaced by h Then assign each non-selected object to the most similar representative object 4. Repeat steps 2-3 until there is no change

SAD Tagus 2004/05 H. Galhardas Cost function for k-medoids

SAD Tagus 2004/05 H. Galhardas PAM Clustering j i h t t ih j h i t j t i h j

SAD Tagus 2004/05 H. Galhardas What is the problem with PAM? Pam is more robust than k-means in the presence of noise and outliers because a medoid is less influenced by outliers or other extreme values than a mean 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. 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 cluster

SAD Tagus 2004/05 H. Galhardas Summary Cluster analysis groups objects based on their similarity and has wide applications Cluster analysis groups objects based on their similarity and has wide applications Measure of similarity can be computed for various types of data Measure of similarity can be computed for various types of data Clustering algorithms can be categorized into partitioning methods, hierarchical methods, density-based methods, grid-based methods, and model-based methods Clustering algorithms can be categorized into partitioning methods, hierarchical methods, density-based methods, grid-based methods, and model-based methods Outlier detection and analysis are very useful for fraud detection, etc. and can be performed by statistical, distance-based or deviation-based approaches Outlier detection and analysis are very useful for fraud detection, etc. and can be performed by statistical, distance-based or deviation-based approaches There are still lots of research issues on cluster analysis, such as constraint-based clustering There are still lots of research issues on cluster analysis, such as constraint-based clustering

SAD Tagus 2004/05 H. Galhardas Other Classification Methods k-nearest neighbor classifier k-nearest neighbor classifier case-based reasoning case-based reasoning Genetic algorithm Genetic algorithm Rough set approach Rough set approach Fuzzy set approaches Fuzzy set approaches

SAD Tagus 2004/05 H. Galhardas Instance-Based Methods Instance-based learning: Instance-based learning: Store training examples and delay the processing (“lazy evaluation”) until a new instance must be classified Typical approaches Typical approaches k-nearest neighbor approach Instances represented as points in a Euclidean space. Locally weighted regression Constructs local approximation Case-based reasoning Uses symbolic representations and knowledge- based inference

SAD Tagus 2004/05 H. Galhardas The k-Nearest Neighbor Algorithm All instances correspond to points in the n-dimensional space. All instances correspond to points in the n-dimensional space. The nearest neighbor are defined in terms of Euclidean distance. The nearest neighbor are defined in terms of Euclidean distance. The target function could be discrete- or real- valued. The target function could be discrete- or real- valued.

SAD Tagus 2004/05 H. Galhardas The k-Nearest Neighbor Algorithm For discrete-valued, the k-NN returns the most common value among the k training examples nearest to x q. For discrete-valued, the k-NN returns the most common value among the k training examples nearest to x q. Vonoroi diagram: the decision surface induced by 1-NN for a typical set of training examples. Vonoroi diagram: the decision surface induced by 1-NN for a typical set of training examples.. _ + _ xqxq + _ _ + _ _

SAD Tagus 2004/05 H. Galhardas Discussion (1) The k-NN algorithm for continuous-valued target functions The k-NN algorithm for continuous-valued target functions Calculate the mean values of the k nearest neighbors Distance-weighted nearest neighbor algorithm Distance-weighted nearest neighbor algorithm Weight the contribution of each of the k neighbors according to their distance to the query point x q, giving greater weight to closer neighbors Similarly, for real-valued target functions

SAD Tagus 2004/05 H. Galhardas Discussion (2) Robust to noisy data by averaging k-nearest neighbors Robust to noisy data by averaging k-nearest neighbors Curse of dimensionality: distance between neighbors could be dominated by irrelevant attributes. Curse of dimensionality: distance between neighbors could be dominated by irrelevant attributes. To overcome it, axes stretch or elimination of the least relevant attributes.

SAD Tagus 2004/05 H. Galhardas Bibliografia Data Mining: Concepts and Techniques, J. Han & M. Kamber, Morgan Kaufmann, 2001 (Sect e Cap. 8) Data Mining: Concepts and Techniques, J. Han & M. Kamber, Morgan Kaufmann, 2001 (Sect e Cap. 8)