Unsupervised Learning : Clustering

Things to be Addressed Traditional Learning Models. Cluster Analysis K-means Clustering Algorithm Drawbacks of traditional clustering algorithms. Clustering as a complex multi-modal optimization problem.

What is Cluster Analysis? Finding groups of objects such that the objects in a group will be similar (or related) to one another and different from (or unrelated to) the objects in other groups Intra-cluster distances are minimized Inter-cluster distances are maximized

However, Clustering Similarity is hard to define, and. We know it when we see it The real meaning of similarity is a philosophical question. We will take a more pragmatic approach. PC: http://www.funnyphotos.net.au/dog-owners/

Notion of a Cluster can be Ambiguous How many clusters? Six Clusters Two Clusters Four Clusters

Types of Clustering A clustering is a set of clusters Important distinction between hierarchical and partitional sets of clusters Partitional Clustering A division data objects into non-overlapping subsets (clusters) such that each data object is in exactly one subset Hierarchical clustering A set of nested clusters organized as a hierarchical tree

Other Distinctions Between Sets of Clusters Exclusive versus non-exclusive In non-exclusive clustering, points may belong to multiple clusters. Can represent multiple classes or border points Fuzzy versus non-fuzzy In fuzzy clustering, a point belongs to every cluster with some weight (membership) between 0 and 1 Weights must sum to 1 Each cluster is a fuzzy set. Partial versus complete In some cases, we only want to cluster some of the data Heterogeneous versus homogeneous Cluster of widely different sizes, shapes, and densities

K-MEANS CLUSTERING The k-means algorithm is an algorithm to cluster n objects based on attributes into k partitions, where k < n. It is similar to the expectation-maximization algorithm for mixtures of Gaussians in that they both attempt to find the centers of natural clusters in the data. It assumes that the object attributes form a vector space.

An algorithm for partitioning (or clustering) N data points into k disjoint subsets S j containing data points so as to minimize the sum-of-squares criterion k 2 ( 1, 2,..., k ) ( i, j ) j= 1 Z C SSE ICS C C C d Z V = = Z i is a data point in cluster C j and V j is the representative point (most often the mean) for cluster C j. Given two clusters, we can choose the one with the smallest error One easy way to reduce SSE is to increase k, the number of clusters A good clustering with smaller k can have a higher SSE than a poor clustering with higher k i i

Simply speaking k-means clustering is an algorithm to classify or to group the objects based on attributes/features into K number of group. K is positive integer number. The grouping is done by minimizing the sum of squares of distances between data and the corresponding cluster centroid.

How the K-Mean Clustering algorithm works?

Step 1: Begin with a decision on the value of k = number of clusters. Step 2: Put any initial partition that classifies the data into k clusters. You may assign the training samples randomly,or systematically as the following: 1.Take the first k training sample as singleelement clusters 2. Assign each of the remaining (N-k) training sample to the cluster with the nearest centroid. After each assignment, recompute the centroid of the gaining cluster.

Step 3: Take each sample in sequence and compute its distance from the centroid of each of the clusters. If a sample is not currently in the cluster with the closest centroid, switch this sample to that cluster and update the centroid of the cluster gaining the new sample and the cluster losing the sample. Step 4. Repeat step 3 until convergence is achieved, that is until a pass through the training sample causes no new assignments.

A Simple example showing the implementation of k-means algorithm (using k=2)

Step 1: Initialization: Randomly we choose following two centroids (k=2) for two clusters. In this case the 2 centroid are: m1=(1.0,1.0) and m2=(5.0,7.0).

Step 2: Thus, we obtain two clusters containing: {1,2,3} and {4,5,6,7}. Their new centroids are:

Step 3: Now using these centroids we compute the Euclidean distance of each object, as shown in table. Therefore, the new clusters are: {1,2} and {3,4,5,6,7} Next centroids are: m1=(1.25,1.5) and m2 = (3.9,5.1)

Step 4 : The clusters obtained are: {1,2} and {3,4,5,6,7} Therefore, there is no change in the cluster. Thus, the algorithm comes Thus, the algorithm comes to a halt here and final result consist of 2 clusters {1,2} and {3,4,5,6,7}.

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(with K=3) Step 1 Step 2

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Real-Life Numerical Example of K-Means Clustering We have 4 medicines as our training data points object and each medicine has 2 attributes. Each attribute represents coordinate of the object. We have to determine which medicines belong to cluster 1 and which medicines belong to the other cluster. Object Medicine A Attribute1 (X): weight index Attribute 2 (Y): ph 1 1 Medicine B 2 1 Medicine C 4 3 Medicine D 5 4

Step 1: Initial value of centroids : Suppose we use medicine A and medicine B as the first centroids. Let and c 1 and c 2 denote the coordinate of the centroids, then c 1 =(1,1) and c 2 =(2,1)

Objects-Centroids distance : we calculate the distance between cluster centroid to each object. Let us use Euclidean distance, then we have distance matrix at iteration 0 is Each column in the distance matrix symbolizes the object. The first row of the distance matrix corresponds to the distance of each object to the first centroid and the second row is the distance of each object to the second centroid. For example, distance from medicine C = (4, 3) to the first centroid is, and its distance to the second centroid is, is etc.

Step 2: Objects clustering : We assign each object based on the minimum distance. Medicine A is assigned to group 1, medicine B to group 2, medicine C to group 2 and medicine D to group 2. The elements of Group matrix below is 1 if and only if the object is assigned to that group.

Iteration-1, Objects-Centroids distances : The next step is to compute the distance of all objects to the new centroids. Similar to step 2, we have distance matrix at iteration 1 is

Iteration-1, Objects clustering:based on the new distance matrix, we move the medicine B to Group 1 while all the other objects remain. The Group matrix is shown below Iteration 2, determine centroids: Now we repeat step 4 to calculate the new centroids coordinate based on the clustering of previous iteration. Group1 and group 2 both has two members, thus the new centroids are and

Iteration-2, Objects-Centroids distances : Repeat step 2 again, we have new distance matrix at iteration 2 as

Iteration-2, Objects clustering: Again, we assign each object based on the minimum distance. We obtain result that. Comparing the grouping of last iteration and this iteration reveals that the objects does not move group anymore. Thus, the computation of the k-mean clustering has reached its stability and no more iteration is needed..

Two different K-means Clustering 3 2.5 2 Original Points 1.5 y 1 0.5 0-2 -1.5-1 -0.5 0 0.5 1 1.5 2 x 3 3 2.5 2.5 2 2 1.5 1.5 y y 1 1 0.5 0.5 0 0-2 -1.5-1 -0.5 0 0.5 1 1.5 2 x Optimal Clustering -2-1.5-1 -0.5 0 0.5 1 1.5 2 x Sub-optimal Clustering

Importance of Choosing Initial Centroids 3 Iteration 12 34 56 2.5 2 1.5 y 1 0.5 0-2 -1.5-1 -0.5 0 0.5 1 1.5 2 x

Importance of Choosing Initial Centroids 3 Iteration 1 3 Iteration 2 3 Iteration 3 2.5 2.5 2.5 2 2 2 1.5 1.5 1.5 y y y 1 1 1 0.5 0.5 0.5 0 0 0-2 -1.5-1 -0.5 0 0.5 1 1.5 2 x -2-1.5-1 -0.5 0 0.5 1 1.5 2 x -2-1.5-1 -0.5 0 0.5 1 1.5 2 x 3 Iteration 4 3 Iteration 5 3 Iteration 6 2.5 2.5 2.5 2 2 2 1.5 1.5 1.5 y y y 1 1 1 0.5 0.5 0.5 0 0 0-2 -1.5-1 -0.5 0 0.5 1 1.5 2 x -2-1.5-1 -0.5 0 0.5 1 1.5 2 x -2-1.5-1 -0.5 0 0.5 1 1.5 2 x

Importance of Choosing Initial Centroids 3 Iteration 12 34 5 2.5 2 1.5 y 1 0.5 0-2 -1.5-1 -0.5 0 0.5 1 1.5 2 x

Types of Clusters Well-separated clusters Center-based clusters Contiguous clusters Density-based clusters Property or Conceptual Described by an Objective Function

Types of Clusters: Well-Separated Well-Separated Clusters: A cluster is a set of points such that any point in a cluster is closer (or more similar) to every other point in the cluster than to any point not in the cluster. 3 well-separated clusters

Types of Clusters: Center-Based Center-based A cluster is a set of objects such that an object in a cluster is closer (more similar) to the center of a cluster, than to the center of any other cluster The center of a cluster is often a centroid, the average of all the points in the cluster, or a medoid, the most representative point of a cluster 4 center-based clusters

Types of Clusters: Contiguity-Based Contiguous Cluster (Nearest neighbor or Transitive) A cluster is a set of points such that a point in a cluster is closer (or more similar) to one or more other points in the cluster than to any point not in the cluster. 8 contiguous clusters

Types of Clusters: Density-Based Density-based A cluster is a dense region of points, which is separated by low-density regions, from other regions of high density. Used when the clusters are irregular or intertwined, and when noise and outliers are present. 6 density-based clusters

Types of Clusters: Conceptual Clusters Shared Property or Conceptual Clusters Finds clusters that share some common property or represent a particular concept.. 2 Overlapping Circles

Types of Clusters: Objective Function Clusters Defined by an Objective Function Finds clusters that minimize or maximize an objective function. Enumerate all possible ways of dividing the points into clusters and evaluate the `goodness' of each potential set of clusters by using the given objective function. (No. of feasible partitions could be very large and the problem is NP Hard) Can have global or local objectives. Hierarchical clustering algorithms typically have local objectives Partitional algorithms typically have global objectives A variation of the global objective function approach is to fit the data to a parameterized model. Parameters for the model are determined from the data. Mixture models assume that the data is a mixture' of a number of statistical distributions.

Types of Clusters: Objective Function Map the clustering problem to a different domain and solve a related problem in that domain Proximity matrix defines a weighted graph, where the nodes are the points being clustered, and the weighted edges represent the proximities between points Clustering is equivalent to breaking the graph into connected components, one for each cluster. Want to minimize the edge weight between clusters and maximize the edge weight within clusters

Evaluating K-means Clusters Most common measure is Sum of Squared Error (SSE) or summed Intra Cluster Spread (ICS) For each point, the error is the distance to the nearest cluster center To get SSE, we square these errors and sum them. SSE = ICS( C ( Z, m k 2 1, C2,..., Ck ) = d i j j = 1 Z C Z i is a data point in cluster C i and m i is the representative point for cluster C i can show that m i corresponds to the center (mean) of the cluster Given two clusters, we can choose the one with the smallest error One easy way to reduce SSE is to increase k, the number of clusters A good clustering with smaller k can have a lower SSE than a poor clustering with higher k i i )

Problems with Selecting Initial Points skip If there are K real clusters then the chance of selecting one centroid from each cluster is small. Chance is relatively small when K is large If clusters are the same size, n, then For example, if K = 10, then probability = 10!/10 10 = 0.00036 Sometimes the initial centroids will readjust themselves in right way, and sometimes they don t

Solutions to Initial Centroids Problem Multiple runs Helps, but probability is not on your side Sample and use hierarchical clustering to determine initial centroids Select more than k initial centroids and then select among these initial centroids Select most widely separated Postprocessing Bisecting K-means Not as susceptible to initialization issues

Handling Empty Clusters Basic K-means algorithm can yield empty clusters Several strategies Choose the point that contributes most to SSE Choose a point from the cluster with the highest SSE If there are several empty clusters, the above can be repeated several times.

Limitations of K-means K-means has problems when clusters are of differing Sizes Densities Non-globular shapes K-means has problems when the data contains outliers.

Limitations of K-means: Differing Sizes Original Points K-means (3 Clusters)

Limitations of K-means: Differing Density Original Points K-means (3 Clusters)

Limitations of K-means: Non-globular and linearly non-separable Shapes Original Points K-means (2 Clusters)

Overcoming K-means Limitations Original Points K-means Clusters One solution is to use many clusters. Find parts of clusters, but need to put together.

Cluster Validity For supervised classification we have a variety of measures to evaluate how good our model is Accuracy, precision, recall For cluster analysis, the analogous question is how to evaluate the goodness of the resulting clusters? But clusters are in the eye of the beholder! Then why do we want to evaluate them? To avoid finding patterns in noise To compare clustering algorithms To compare two sets of clusters To compare two clusters

Clusters found in Random Data 1 1 0.9 0.9 0.8 0.8 Random Points y 0.7 0.6 0.5 y 0.7 0.6 0.5 DBSCAN 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0.2 0.4 0.6 0.8 1 x 0 0 0.2 0.4 0.6 0.8 1 x 1 1 0.9 0.9 K-means 0.8 0.7 0.6 0.8 0.7 0.6 Complete Link y 0.5 y 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0.2 0.4 0.6 0.8 1 x 0 0 0.2 0.4 0.6 0.8 1 x

Different Aspects of Cluster Validation 1. Determining the clustering tendency of a set of data, i.e., distinguishing whether non-random structure actually exists in the data. 2. Comparing the results of a cluster analysis to externally known results, e.g., to externally given class labels. 3. Evaluating how well the results of a cluster analysis fit the data without reference to external information. - Use only the data 4. Comparing the results of two different sets of cluster analyses to determine which is better. 5. Determining the correct number of clusters. For 2, 3, and 4, we can further distinguish whether we want to evaluate the entire clustering or just individual clusters.

Landscape for Clustering Problem The k-means objective function ICS( C ( Z k 2 1, C2,..., Ck ) = d i, m j j= 1 Z C i i ) Example of an extremely simple one-dimensional dataset Fitness function (Reciprocal of ICS) plot of the hard c-means algorithm for above dataset

Landscape for Clustering Problem (Contd.) The previous data but now With some noise points Fitness function (Reciprocal of ICS) plot of the hard c-means algorithm for above dataset In addition to two global minima, we have local minima at: approximately (0, 10), (5, 10), (10, 0), (10, 5). For these local minima one prototype covers one of the data clusters, while the other prototype is mislead to the noise cluster.

K Nearest Neighbor Classifiers