Machine Learning. Topic 6: Clustering

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1 Machne Learnng Topc 6: lusterng

2 lusterng Groupng data nto (hopefully useful) sets. Thngs on the left Thngs on the rght

3 Applcatons of lusterng Hypothess Generaton lusters mght suggest natural groups. Hypothess Testng Hypothess: Bg companes nvest abroad. Test: Label companes by amount nvested abroad. luster companes by sze. See f clusters match labels Predcton, based on Groups luster by reacton to a drug therapy For a new patent, fnd the closest cluster to determne the best medcaton to use.

4 Let X be the dataset: Some defntons X { 2 3 x, x, x,... x } 1 n An m-clusterng of X s a partton of X nto m sets (clusters) 1, m such that: 1. lusters are non - empty: {}, 1 m lusters cover all of X : lusters do not overlap : m 1 j X {},f j

5 How many possble clusters? (Strlng numbers) Sze of dataset S( n, m) m 1 ( 1) m! 0 m m n Number of clusters S(15,3) S(20,4) S(100,5) 2,375,101 45,232,115,

6 What does ths mean? We can t try all possble clusterngs. lusterng algorthms look at a small fracton of all parttons of the data. The exact parttons tred depend on the knd of clusterng used.

7 Who s rght? Dfferent technques cluster the same data set DIFFERENTLY. Who s rght? Is there a rght clusterng?

8 Steps n lusterng Select Features Defne a Proxmty Measure Defne lusterng rteron Defne a lusterng Algorthm Valdate the Results Interpret the Results

9 Knds of lusterng Sequental Fast Results depend on data order Herarchcal Start wth many clusters jon clusters at each step ost Optmzaton Fxed number of clusters (typcally) Boundary detecton, Probablstc classfers

10 A Sequental lusterng Method Basc Sequental Algorthmc Scheme (BSAS) S. Theodords and K. Koutroumbas, Pattern Recognton, Academc Press, London England, 1999 Assumpton: The number of clusters s not known n advance. Let: d(x,) Q q be the dstance between feature vector x and cluster. be the threshold of dssmlarty be the maxmum number of clusters

11 BSAS Pseudo ode End End } { Else } { 1 ) and ( ) ), ( ( If ), ( mn ), ( : Fnd 2 to For } { k k m k j j k k x x m m q m x d x d x d n x m Q

12 A ost-optmzaton method K-means clusterng J. B. MacQueen (1967): "Some Methods for classfcaton and Analyss of Multvarate Observatons, Proceedngs of 5-th Berkeley Symposum on Mathematcal Statstcs and Probablty", Berkeley, Unversty of alforna Press, 1: A greedy algorthm Parttons n samples nto k clusters mnmzes the sum of the squared dstances to the cluster centers

13 The K-means algorthm Place K ponts nto the space represented by the objects that are beng clustered. These ponts represent ntal group centrods (means). Assgn each object to the group that has the closest centrod (mean). When all objects have been assgned, recalculate the postons of the K centrods (means). Repeat Steps 2 and 3 untl the centrods no longer move.

14 K-means clusterng The way to ntalze the mean values s not specfed. Randomly choose k samples? Results depend on the ntal means Try multple startng ponts? Results depend on the dstance measure. What s a good one? Assumes K s known. How do we chose ths?

15 12 th MF lusterng Example 25 Blue = Bassoon Red = Tuba

16 Let s look at a demo

17 Greedy Herarchcal lusterng Intalze one cluster for each data pont Untl done Merge the two nearest clusters (What s our measure of dstance between clusters?)

18 Herarchcal lusterng on Strngs Features = contexts n whch strngs appear Adapted from a slde by Doug Downey

19 Algorthms: Summary Sequental clusterng Requres key dstance threshold, senstve to data order K-means clusterng Requres # of clusters, senstve to ntal condtons Specal case of mxture modelng Greedy agglomeratve clusterng Navely takes order of n^2 runtme Hard to tell when you re done

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