An Introduction to Cluster Analysis. Zhaoxia Yu Department of Statistics Vice Chair of Undergraduate Affairs
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1 An Introduction to Cluster Analysis Zhaoxia Yu Department of Statistics Vice Chair of Undergraduate Affairs 1
2 What can you say about the figure? signal C subjects Two measurements per subject signal T 2
3 signal C signal T 3
4 CC signal C CT TT signal T 4
5 Cluster Analysis Seeks rules to group data Large between-cluster difference Small within-cluster difference Exploratory Aims to understand/ learn the unknown substructure of multivariate data 5
6 Cluster Analysis vs Classification Data are unlabeled The number of clusters are unknown Unsupervised learning Goal: find unknown structures The labels for training data are known The number of classes are known Supervised learning Goal: allocate new observations, whose labels are unknown, to one of the known classes 6
7 The Iris Data It was collected by F. A. Fisher A famous data set that has been widely used in textbooks Four features: sepal length in cm sepal width in cm petal length in cm petal width in cm
8 The Iris Data Three types: Setosa Versicolor Virginica 8
9 The Iris Data Sepal L. Sepal W. Petal L. Petal W. [1,] [2,] [3,] [4,] [5,] [6,] [7,] [8,] [9,] [45,] [46,] [47,] [48,] [49,] [50,] Iris Setosa Sepal L. Sepal W. Petal L. Petal W. [1,] [2,] [3,] [4,] [5,] [6,] [7,] [8,] [9,] [45,] [46,] [47,] [48,] [49,] [50,] Iris Versicolor Sepal L. Sepal W. Petal L. Petal W. [1,] [2,] [3,] [4,] [5,] [6,] [7,] [8,] [9,] [45,] [46,] [47,] [48,] [49,] [50,] Iris Virginica
10 The Iris Data Sepal L. Sepal W. Petal L. Petal W. Setosa Versicolor Virginica 10
11 Clustering Methods Model-free: Nonhierarchical clustering. K-means. Hierarchical clustering. Based on similarity measures Model-based clustering 11
12 Model-Free Clustering Nonhierarchical Clustering: K-Means 12
13 K-Means Assign each observation to the cluster with the nearest mean Nearest is usually defined based on Euclidean distance 13
14 K-Means: Algorithm Step 0: Preprocess data. Standardize data if appropriate Step 1: Partition the observations into K initial clusters. Step 2 2.a (update step): Calculate the centroids. 2.b (assignment step): Assign each observation to its nearest cluster. Repeat step 2 until no more changes in assignments 14
15 From An Introduction to Statistical Learning 15
16 Remarks Before convergence, each step is guaranteed to decrease the within-cluster sum of squares objective Within a finite number of steps, the algorithm might converge to a (local) minimum Use different and random initial values 16
17 Different Initial Values From An Introduction to Statistical Learning 17
18 Example: Cluster Analysis of Iris Data (Petal L & W) Pretend that the iris types of the observations are unknown => cluster analysis As an example, and for illustration purpose, we will use petal length and width Choose K=3 K-means 18
19 K-Mean Clustering: Iris (Petal L & W) Note: the animation in the figure doesn t work appropriately on MAC. 19
20 K-Mean Clustering: Iris (Petal L & W) 20
21 K-Mean Clustering: Iris (Petal L & W) Iteration= 1 Iteration= 2 Iteration= Setosa Versicolor Virginica Setosa Versicolor Virginica Setosa Versicolor Virginica Iteration= 4 Iteration= 5 Iteration= Setosa Versicolor Virginica Setosa Versicolor Virginica Setosa Versicolor Virginica Iteration= 7 Iteration= 8 Iteration= Setosa Versicolor Virginica Setosa Versicolor Virginica Setosa Versicolor Virginica 21
22 K-Mean Clustering: Iris (Petal L & W) Setosa Versicolor Virginica Note: the animation in the figure doesn t work appropriately on MAC. 22
23 Model-Free Clustering: Hierarchical Clustering 23
24 Hierarchical Clustering The number of clusters is not required Gives a tree-based representation of observations - dendrogram Each leaf represents an observation Leaves similar with each other are fused to branches Leaves/branches similar with each other are fused to branches 24
25 Setosa Virginica Versicolor 25
26 Hierarchical Clustering To grow a tree, we need to define dissimilarities (distances) between leaves/branches Two leaves: easy. One can use a dissimilarity measure A leaf and a branch: there are different options Two branches: similar to a leaf and a branch, there are different options 26
27 Distance between Two Branches/Clusters Single linkage Complete linkage Average linkage Many other options! 27
28 Model-Based Clustering 28
29 Model-Based Clustering: Mixture Model Consider a random variable X. We say it follows a mixture of K distributions if its distribution can be represented using K distributions: The weights p k, k=1,,k are nonnegative numbers and they add up to 1 29
30 Cluster Analysis Based on Mixture Model I present a frequentist version Choose an appropriate model. E.g., A Gaussian mixture model with K=2 clusters Write down the likelihood function Find the maximum likelihood estimate of the parameters Calculate the Pr(cluster k observation x i ) for i=1,,n, k=1,2 30
31 The Maximum Likelihood Estimate (MLE) of the Parameters An easy-to-implement algorithm to find the MLEs is called the Expectation and Maximization (EM) algorithm Initialize parameters E step: calculate conditional expectation. conditional means conditional on current estimate of the parameters This step involves calculating prob(cluster k obs I, current estimate of para), k=1,,k, i=1,,n This step is similar to the assignment step in an K- means algorithm 31
32 The Maximum Likelihood Estimate (MLE) of the Parameters The M step: find the set of values that maximize the conditional expectation calculated in the E step. This step updates the parameter values Repeat the E and M steps until convergence 32
33 EM vs K-Mean EM Step 1: initialization E: Calculate conditional probabilities M step: Find optimal values for parameters Repeat the E and M steps until convergence Allows clusters to have different shapes K-Mean Step 1: initialization Step 2a: guess cluster membership Step 2b: find cluster centers Repeat 2a-2b until convergence 33
34 Example: Gaussian Mixture Model Observed data (simulated from two normal distributions) Assuming K=2 Parameters: μ 1, μ 0, σ 1, σ 0, p 34
35 Example: simulated data Group 1 Group 2 ~ N(1,1) ~ N(3,1) 35 Note: the animation in the figure doesn t work appropriately on MAC.
36 Example: Cluster Analysis of Iris Data Using Petal Length Setosa Versicolor Note: the animation in the figure doesn t work appropriately on MAC. 36
37 R Package: MCLUST Developed by Adrian Raftery and colleagues Gaussian mixture model EM Clustering, classification, density estimation Please try it out! 37
38 Clustering Analysis For Multidimensional Data 38
39 Multidimensional Data Human faces, images 3D objects Text documents Brain imaging 39
40 Whole Brain Connectivity task 1 task 2 task 3 rest 1 rest 2 rest 3 Sub 1 Sub 2 Sub 3 Sub 4 40
41 Brain Connectivity vs Fingerprint Subject ID 41
42 task 1 task 2 task 3 rest 1 rest 2 rest 3 42
43 Some Technical Details? 43
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