Data Mining and Data Warehousing Henryk Maciejewski Data Mining Clustering


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1 Data Mining and Data Warehousing Henryk Maciejewski Data Mining Clustering
2 Clustering Algorithms Contents Kmeans Hierarchical algorithms Linkage functions Vector quantization
3 Clustering Formulation Objects Attributes Find groups of similar points (observations) in multidimensional space No target variable (unsupervised learning) Model
4 Methods of Clustering  Overview Variety of methods: Hierarchical clustering create hierarchy of clusters (one cluster entirely contained within another cluster) Nonhierarchical methods create disjoint clusters Overlapping clusters (objects can belong to >1 cluster simultaneously) Fuzzy clusters (defined by the probability (grade) of membership of each object in each cluster) Useful data preprocessing prior to clustering: PCA (Principal Components Analysis) to reduce dimensionality of data Data standarization (transform data to reduce large influence of variables with larger variance on results of clustering)
5 Introductory Example 97 countries described by 3 attributes: Birth, Death, InfantDeath rate (given as number per 1000, data from year 1995)
6 Analysis I Clustering raw data Kmeans algorithm Result: 3 clusters (no. of obs. in each cluster: 13, 32, 52) Example cntd.
7
8 Example Profiles of Clusters
9 Example Profiles of Clusters Notice: data clustered based on InfantDeath Rate only!
10 Example Standarization of Data Analysis II Data standarized prior to clustering (variables divided by their standard deviation) Result: 3 clusters (with 35, 46, 16 obs.) Data clustered based on InfantDeath and Death Analysis II Analysis I Observe that data with largest variance have largest influence on results of clustering
11 Example Profiles of Clusters Analysis II: profiles of clusters
12 Methods of Clustering Nonhierarchical methods Kmeans clustering Nondeterministic O(n), n  number of observations Hierarchical methods Aglomerative (join small clusters) Divisive (split big clusters) Deterministic methods O(n 2 ) O(n 3 ), depending on the clustering method (i.e. definition of intercluster distance)
13 Methods of Clustering  Remarks Clustering large datasets Kmeans If results of hierarchical clustering needed first use Kmeans yielding e.g. 50 clusters, followed by hierarchical clustering on results of Kmeans Consensus clustering Discover real clusters in data analyze stability of results with noise injected
14 Kmeans Algorithm Kmeans clustering Select k points (centroids of initial clusters; select randomly) Assign each observation to the nearest centroid (nearest cluster) For each cluster find the new centroid Repeat step 2 and 3 until no change occurs in cluster assignments
15 Kmeans Algorithm Result: k separate clusters Algorithm requires that the correct number of clusters k is specified in advance (difficult problem: how to know the real number of clusters in data )
16 Hierarchical Clustering Notation x i observations, i=1..n C k clusters G current number of clusters D KL distance between clusters C K and C L Betweencluster distance D KL linkage function (various definitions available, results of clustering depend on D KL ) C L C K D KL
17 Hierarchical Clustering Algorithm (agglomerative hierarchical clustering) C k = {x k }, k=1..n, G=n Find K, L such that D KL = min D IJ, 1<=I,J<=G Replace clusters C K and C L by cluster C K C L, G=G1 Repeat steps 2 and 3 while G>1 C L D KL C K Result: hierarchy of clusters dendrogram
18 Hierarchy of Clusters  Dendrogram
19 Definitions of Distance Between Clusters Different definitions of distance between clusters Average linkage Single linkage Complete linkage Density linkage Ward s minimum variance method (SAS CLUSTER procedure accepts 11 different definitions of intercluster distance)
20 Notation x i observations, i=1..n Average Linkage d(x,y) distance between observations (Euclidean distance assumed from now on) C k clusters N K number of observations in cluster C K D KL distance between clusters C K and C L mean CK mean observation in cluster C K W K = x i mean CK 2 x i C K variance in cluster Average linkage Tends to join clusters with small variance Resulting clusters tend to have similar variance
21 Notation x i observations, i=1..n Complete Linkage d(x,y) distance between observations C k clusters N K number of observations in cluster C K D KL distance between clusters C K and C L mean CK mean observation in cluster C K W K = x i mean CK 2 x i C K variance in cluster Complete linkage Resulting clusters tend to have similar diameter
22 Notation x i observations, i=1..n Single Linkage d(x,y) distance between observations C k clusters N K number of observations in cluster C K D KL distance between clusters C K and C L mean CK mean observation in cluster C K W K = x i mean CK 2 x i C K variance in cluster Single linkage Tends to produce elongated clusters, irregular in shape
23 Ward s Minimum Variance Method Notation x i observations, i=1..n d(x,y) distance between observations C k clusters N K number of observations in cluster C K D KL distance between clusters C K and C L mean CK mean observation in cluster C K W K = x i mean CK 2 x i C K variance in cluster B KL =W M W K W L where C M =C K C L Ward s minimum variance method Tends to join small clusters Tends to produce clusters with similar number of observations
24 Density Linkage Notation x i observations, i=1..n d(x,y) distance between observations r a fixed constant f(x) proportion of observations within sphere centered at x with radius r divided by the volume of the sphere (measure of density of points near observation x) Density linkage We realize single linkage using the measure d* Capable of discovering clusters of irregular shape
25 Example Average Linkage Elongated clusters in data
26 Elongated clusters in data Example Kmeans
27 Example Density Linkage Elongated clusters in data
28 Nonconvex clusters in data Example Kmeans
29 Example Centroid Linkage Nonconvex clusters in data
30 Example Density Linkage Nonconvex clusters in data
31 Clusters of unequal size Example True Clusters
32 Clusters of unequal size Example Kmeans
33 Example Ward s Method Clusters of unequal size
34 Example Average Linkage Method: average linkage
35 Example Centroid Linkage Clusters of unequal size
36 Example Single Linkage Clusters of unequal size
37 Example Well Separated Data Any method will work
38 Example Poorly Separated Data True clusters
39 Example Poorly Separated Data Method: Kmeans
40 Example Poorly Separated Data Ward s method
41 Clustering Methods Final Remarks Standarization of variables prior to clustering Often necessary, otherwise variables with large variance tend to have large influence on clustering Often standarized measurement z ij is computed as the zscore: where x ij original measurement in observation i and variable j, µ j mean value of variable j, s j mean absolute deviation of variable j (or its standard deviation) Other ideas: divide variable by its range, max value or standard deviation
42 Clustering Methods Final Remarks The number of clusters No satisfactory theory to determine the right number of clusters in data Various criteria can be observed to help determine the right number of clusters, e.g. criteria based on variance accounted for by clusters R 2 =1P G /T or semipartial R 2 =B KL /T where T total variance of observations; P G = W K over G clusters B KL =W M W K W L where C M =C K C L Cubic Clustering Criterion (CCC) Often data visualization useful for determining the number of clusters Scatterplot for 23 dimensional data In high dimensions apply PCA transformation (or similar) visualize data in 23 dimensional space of first principal components
43 Example 2 R, Semipartial 2 R
44 Example Number of Clusters Useful Checks PST2: 3 or 6 or 9 (one before peak in value) PSF: 9 (peak in value) CCC: 18 (CCC around 3)
45 Kohonen VQ (Vector Quantization) Algorithm similar to kmeans Idea of VQ algorithm: Select k points (initial cluster centroids) For observation x i find nearest centroid (winning seed) denoted by S n Modify S n according to the formula: S n =S n (1L)+x i L, where L learning constant (decresing during learning process) Repeat steps 2 and 3 over all training observations Repeat steps 24 given number of iterations
46 Kohonen SOM (Self Organizing Maps) Idea of the SOM algorithm Select k initial points (cluster centroids), represent them on a 2D map For observation x i find winning seed S n Modify all centroids : S j =S j (1K(j,n)L)+x i K(j,n)L, where L learning constant (decreasing during training) K(j,n) function decreasing with increasing distance on the 2D map between S j i S n centroids (K(j,j)=1) Repeat steps 2 and 3 over all training observations
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