Unsupervised: no target value to predict
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1 Clustering Unsupervised: no target value to predict Differences between models/algorithms: Exclusive vs. overlapping Deterministic vs. probabilistic Hierarchical vs. flat Incremental vs. batch learning Problem: Evaluation? usually by inspection But: If treated as density estimation problem, clusters can be evaluated on test data! 87
2 Hierarchical clustering Bottom up Start with single-instance clusters At each step, join the two closest clusters Design decision: distance between clusters Top down E.g.two closest instances in clusters vs. distance between means Start with one universal cluster Find two clusters Proceed recursively on each subset Can be very fast Both methods produce a dendrogram g a c i e d k b j f h 88
3 The k-means algorithm To cluster data into k groups: (k is predefined) 1. Choose k cluster centers e.g. at random. Assign instances to clusters based on distance to cluster centers 3. Compute centroids of clusters 4. Go to step 1 until convergence 89
4 Discussion Result can vary significantly based on initial choice of seeds Can get trapped in local minimum Example: initial cluster centres instances To increase chance of finding global optimum: restart with different random seeds 90
5 Incremental clustering Heuristic approach (COBWEB/CLASSIT) Form a hierarchy of clusters incrementally Start: tree consists of empty root node Then: add instances one by one update tree appropriately at each stage to update, find the right leaf for an instance May involve restructuring the tree Base update decisions on category utility 91
6 9 Clustering weather data N M L K J I H G F E D C B A ID Rainy Hot Overcast Overcast Rainy Cool Cool Overcast Cool Rainy Cool Rainy Rainy Hot Overcast Hot Hot Windy Humidity Temp. Outlook 1 3
7 Clustering weather data ID Outlook Temp. Humidity Windy 4 A Hot B Hot C Overcast Hot D Rainy E F Rainy Rainy Cool Cool 5 G H Overcast Cool Merge best host and runner-up I Cool J K Rainy 3 L Overcast M Overcast Hot N Rainy Consider splitting the best host if merging doesn t help 93
8 Final hierarchy ID Outlook Temp. Humidity Windy A Hot B Hot C Overcast Hot D Rainy Oops! a and b are actually very similar 94
9 Example: the iris data (subset)
10 Clustering with cutoff 96
11 Category utility Category utility: quadratic loss function defined on conditional probabilities: CU ( C 1, C,..., C k! C!! Pr[ l ] (Pr[ ai = vij Cl ] " Pr[ ai = l i j ) = k v ij ] ) Every instance in different category numerator becomes! Pr[ a i vij ] maximum m = number of attributes 97
12 Numeric attributes Assume normal distribution: f ( a# µ ) 1 ( a) = e! "! Then: " Pr[ ai = vij ] # f ( ai ) dai = j! 1 %$ i Thus CU becomes! C!! Pr[ l ] (Pr[ ai = vij Cl ] " Pr[ ai = l i j = k 1 ' 1 1 $! Pr[ C! l ] % ( " * &) il ) # k l i i CU = v ij ] ) Prespecified minimum variance acuity parameter 98
13 Probability-based clustering Problems with heuristic approach: Division by k? Order of examples? Are restructuring operations sufficient? Is result at least local minimum of category utility? Probabilistic perspective seek the most likely clusters given the data Also: instance belongs to a particular cluster with a certain probability 99
14 Finite mixtures Model data using a mixture of distributions One cluster, one distribution governs probabilities of attribute values in that cluster Finite mixtures : finite number of clusters Individual distributions are normal (usually) Combine distributions using cluster weights 100
15 Two-class mixture model data A 51 A 43 B 6 B 64 A 45 A 4 A 46 A 45 A 45 B 6 A 47 A 5 B 64 A 51 B 65 A 48 A 49 A 46 B 64 A 51 A 5 B 6 A 49 A 48 B 6 A 43 A 40 A 48 B 64 A 51 B 63 A 43 B 65 B 66 B 65 A 46 A 39 B 6 B 64 A 5 B 63 B 64 A 48 B 64 A 48 A 51 A 48 B 64 A 4 A 48 A 41 model µ A =50, σ A =5, p A =0.6 µ B =65, σ B =, p B =
16 Using the mixture model Probability that instance x belongs to cluster A: Pr[ A x] = Pr[ x A]Pr[ A] = Pr[ x] f ( x; µ A,! A ) p Pr[ x] A with f ( x# µ ) 1 ( x; µ,! ) = e! "! Likelihood of an instance given the clusters: Pr[ x the distributions] =! i Pr[ x cluster ]Pr[cluster ] i i 10
17 Learning the clusters Assume: we know there are k clusters Learn the clusters determine their parameters I.e. means and standard deviations Performance criterion: likelihood of training data given the clusters EM algorithm finds a local maximum of the likelihood 103
18 EM algorithm EM = Expectation-Maximization Generalize k-means to probabilistic setting Iterative procedure: E expectation step: Calculate cluster probability for each instance M maximization step: Estimate distribution parameters from cluster probabilities Store cluster probabilities as instance weights Stop when improvement is negligible 104
19 105 More on EM Estimate parameters from weighted instances Stop when log-likelihood saturates Log-likelihood: n n n A w w w x w x w x w!!! =... ) (... ) ( ) ( µ µ µ " n n n A w w w x w x w w x = µ ]) Pr[ ] Pr[ ( log B x p A x p i B i A i!
20 Extending the mixture model More then two distributions: easy Several attributes: easy assuming independence! Correlated attributes: difficult Joint model: bivariate normal distribution with a (symmetric) covariance matrix n attributes: need to estimate n n (n1)/ parameters 106
21 More mixture model extensions Nominal attributes: easy if independent Correlated nominal attributes: difficult Two correlated attributes v 1 v parameters Missing values: easy Can use other distributions than normal: log-normal if predetermined minimum is given log-odds if bounded from above and below Poisson for attributes that are integer counts Use cross-validation to estimate k! 107
22 Bayesian clustering Problem: many parameters EM overfits Bayesian approach : give every parameter a prior probability distribution Incorporate prior into overall likelihood figure Penalizes introduction of parameters Eg: Laplace estimator for nominal attributes Can also have prior on number of clusters! Implementation: NASA s AUTOCLASS 108
23 Discussion Can interpret clusters by using supervised learning post-processing step Decrease dependence between attributes? pre-processing step E.g. use principal component analysis Can be used to fill in missing values Key advantage of probabilistic clustering: Can estimate likelihood of data Use it to compare different models objectively 109
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