10601 Machine Learning. Hierarchical clustering. Reading: Bishop: 9-9.2
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1 161 Machine Learning Hierarchical clustering Reading: Bishop: 9-9.2
2 Second half: Overview Clustering - Hierarchical, semi-supervised learning Graphical models - Bayesian networks, HMMs, Reasoning under uncertainty Putting it together - Model / feature selection, Boosting, dimensionality reduction Advanced classification - SVM
3 What is Clustering? Organizing data into clusters such that there is high intra-cluster similarity low inter-cluster similarity Informally, finding natural groupings among objects. Why do we want to do that? Any REAL application?
4 Example: clusty
5 Example: clustering genes Microarrays measures the activities of all genes in different conditions Clustering genes can help determine new functions for unknown genes An early killer application in this area The most cited (12,9) paper in PNAS!
6 Unsupervised learning Clustering methods are unsupervised learning techniques - We do not have a teacher that provides examples with their labels We will also discuss dimensionality reduction, another unsupervised learning method later in the course
7 Outline Distance functions Hierarchical clustering Number of clusters
8 What is Similarity? The quality or state of being similar; likeness; resemblance; as, a similarity of features. Webster's Dictionary Similarity is hard to define, but We know it when we see it The real meaning of similarity is a philosophical question. We will take a more pragmatic approach.
9 Defining Distance Measures Definition: Let O 1 and O 2 be two objects from the universe of possible objects. The distance (dissimilarity) between O 1 and O 2 is a real number denoted by D(O 1,O 2 ) gene1 gene
10 gene1 gene2 d('', '') = d(s, '') = d('', s) = s -- i.e. length of s d(s1+ch1, s2+ch2) = min( d(s1, s2) + if ch1=ch2 then else 1 fi, d(s1+ch1, s2) + 1, d(s1, s2+ch2) + 1 ) Inside these black boxes: some function on two variables (might be simple or very complex) A few examples: Euclidian distance Correlation coefficient s(x,y) d(x,y) (x i y i ) 2 i i (x i x )(y i y ) x y Similarity rather than distance Can determine similar trends
11 Outline Distance measure Hierarchical clustering Number of clusters
12 Desirable Properties of a Clustering Algorithm Scalability (in terms of both time and space) Ability to deal with different data types Minimal requirements for domain knowledge to determine input parameters Interpretability and usability Optional - Incorporation of user-specified constraints
13 Two Types of Clustering Partitional algorithms: Construct various partitions and then evaluate them by some criterion Hierarchical algorithms: Create a hierarchical decomposition of the set of objects using some criterion (focus of this class) Bottom up or top down Hierarchical Top down Partitional
14 (How-to) Hierarchical Clustering The number of dendrograms with n leafs = (2n -)!/[(2 (n -2) ) (n -2)!] Number Number of Possible of Leafs Dendrograms ,9,2 Bottom-Up (agglomerative): Starting with each item in its own cluster, find the best pair to merge into a new cluster. Repeat until all clusters are fused together.
15 We begin with a distance matrix which contains the distances between every pair of objects in our database D(, ) = 8 D(, ) = 1 1
16 Bottom-Up (agglomerative): Starting with each item in its own cluster, find the best pair to merge into a new cluster. Repeat until all clusters are fused together. Consider all possible merges Choose the best
17 Bottom-Up (agglomerative): Starting with each item in its own cluster, find the best pair to merge into a new cluster. Repeat until all clusters are fused together. Consider all possible merges Choose the best Consider all possible merges Choose the best
18 Bottom-Up (agglomerative): Starting with each item in its own cluster, find the best pair to merge into a new cluster. Repeat until all clusters are fused together. Consider all possible merges Choose the best Consider all possible merges Choose the best Consider all possible merges Choose the best
19 Bottom-Up (agglomerative): Starting with each item in its own cluster, find the best pair to merge into a new cluster. Repeat until all clusters are fused together. Consider all possible merges Consider all possible merges Choose the best But how do we compute distances between clusters rather than objects? Choose the best Consider all possible merges Choose the best
20 Computing distance between clusters: Single Link cluster distance = distance of two closest members in each class - Potentially long and skinny clusters
21 Computing distance between clusters: : Complete Link cluster distance = distance of two farthest members + tight clusters
22 Computing distance between clusters: Average Link cluster distance = average distance of all pairs the most widely used measure Robust against noise
23 Example: single link
24 Example: single link (1,2) (1,2) min{9,8} }, min{ 9 min{1,9} }, min{ min{6,} }, min{ 2, 1, (1,2), 2, 1, (1,2), 2, 1, (1,2), d d d d d d d d d
25 Example: single link 7 (1,2,) (1,2,) (1,2) (1,2) min{8,} }, min{ 7 min{9,7} }, min{, (1,2), (1,2,),, (1,2), (1,2,), d d d d d d
26 Example: single link 7 (1,2,) (1,2,) (1,2) (1,2) }, min{ (1,2,), (1,2,), 1,2,),(,) ( d d d
27 Single linkage 7 6 Height represents distance between objects / clusters Average linkage
28 Summary of Hierarchal Clustering Methods No need to specify the number of clusters in advance. Hierarchical structure maps nicely onto human intuition for some domains They do not scale well: time complexity of at least O(n 2 ), where n is the number of total objects. Like any heuristic search algorithms, local optima are a problem. Interpretation of results is (very) subjective.
29 But what are the clusters? In some cases we can determine the correct number of clusters. However, things are rarely this clear cut, unfortunately.
30 One potential use of a dendrogram is to detect outliers The single isolated branch is suggestive of a data point that is very different to all others Outlier
31 Example: clustering genes Microarrays measures the activities of all genes in different conditions Clustering genes can help determine new functions for unknown genes
32 Partitional Clustering Nonhierarchical, each instance is placed in exactly one of K non-overlapping clusters. Since the output is only one set of clusters the user has to specify the desired number of clusters K.
33 expression in condition 2 K-means Clustering: Finished! Re-assign and move centers, until no objects changed membership. k k 2 k 1 2 expression in condition 1
34 Gaussian mixture clustering
35 Clustering methods: Comparison Hierarchical K-means GMM Running time naively, O(N ) Assumptions requires a similarity / distance measure Input parameters none fastest (each iteration is linear) strong assumptions K (number of clusters) fast (each iteration is linear) strongest assumptions K (number of clusters) Clusters subjective (only a tree is returned) exactly K clusters exactly K clusters
36 Outline Distance measure Hierarchical clustering Number of clusters
37 How can we tell the right number of clusters? In general, this is a unsolved problem. However there are many approximate methods. In the next few slides we will see an example
38 When k = 1, the objective function is
39 When k = 2, the objective function is
40 When k =, the objective function is
41 Objective Function We can plot the objective function values for k equals 1 to 6 The abrupt change at k = 2, is highly suggestive of two clusters in the data. This technique for determining the number of clusters is known as knee finding or elbow finding. 1.E+ 9.E+2 8.E+2 7.E+2 6.E+2.E+2.E+2.E+2 2.E+2 1.E+2.E+ k Note that the results are not always as clear cut as in this toy example
42 Cross validation We can also use cross validation to determine the correct number of classes Recall that GMMs is a generative model. We can compute the likelihood of the left out data to determine which model (number of clusters) is more accurate n k p(x 1 x n ) p(x j C i)w i j1 i1
43 Cross validation
44 What you should know Why is clustering useful What are the different types of clustering algorithms What are the assumptions we are making for each, and what can we get from them Unsolved issues: number of clusters, initialization, etc.
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