cluster analysis i
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DESCRIPTIONCluster Analysis I. 9/28/2012. Outline. Introduction Distance and similarity measures for individual data points A few widely used methods: hierachical clustering, K-means, model-based clustering. Introduction. - PowerPoint PPT Presentation
Cluster Analysis I9/28/2012
OutlineIntroduction Distance and similarity measures for individual data pointsA few widely used methods: hierachical clustering, K-means, model-based clustering
IntroductionTo group or segment a collection of objects into subsets or clusters, such that those within each cluster are more closely related to one another than objects assigned to different clusters.Some times, the goal is to arrange the clusters into a natural hierarchy. Cluster genes: similar expression pattern implies co-regulation.Cluster samples: identify potential sub-classes of disease.
IntroductionAssigning subjects into group.Estimating number of clusters.Assess the strength/confidence of cluster assignments for individual objects
Proximity MatrixAn NxN matrix D (N=number of objects), each element records the proximity (distance) between object i and i.
Most often, we have measurement of p dimension on each object. Then we can define
Two main classes of distance for continuous variables:Distance metric (scale-dependent)1- Correlation coefficients (scale-invariant)
Minkowski distanceFor vectors and of length S, the Minkowski family of distance measures are defined as
Two commonly used special caseManhattan distance (a.k.a. city-block distance, k=1)
Euclidean distance (k=2)
Mahalanobis distanceTaking the correlation structure into account.
When assuming identity covariance matrix, it is the same as Euclidian distance.
Pearson correlation and inner productPearson correlation
After standardization: Sensitive to outliers.
Calculate using the rank of the two vectors (note: sum of the ranks is n(n+1)/2)
When there is no tied observations
Robust to outliers since it is based on ranks of the data.
Standardization of the dataStandardize gene rows to mean 0 and stdev 1.Advantage: makes Euclidean distance and correlation equivalent. Many useful methods require the data to be in Euclidean space.
Clustering methodsClustering algorithms come in two flavors
Hierarchical clustering Produce a tree or dendrogram.They avoid specifying how many clusters are appropriate by providing a partition for each k obtained from cutting the tree at some level. The tree can by built in two distinct waysBottom-up: agglomerative clustering (most used).Top-down: divisive clustering.
Agglomerative MethodsThe most popular hierarchical clustering method.Start with n clusters.At each step, merge the two closest clusters using a measure of between-cluster dissimilarity .
Compute group similarities
Choice of linkage
Comparison of the three methodsSingle-link Elongated clustersIndividual decision, sensitive to outliersComplete-linkCompact clustersIndividual decision, sensitive to outliersAverage-link or centroidIn betweenGroup decision, insensitive to outliers.
Divisive MethodsBegin with the entire data set as a single cluster, and recursively divide one of the existing clusters into two daughter clusters.Do it till each cluster only have one object or all members overlapped with each other. Not as popular as agglomeriative methods.
Divisive AlgorithmsAt each division, other method, e.g. K-means with K=2, could be used.Smith et al. 1965 proposed a method that does not involve other clustering methodStart with 1 cluster G, assign the object that is the furthest from the others (with the highest average pair-wise distance) to cluster H.For the remaining iterations, each time assign the one object in G that is the closest to H (maximum difference between the average pair-wise distance to objects in H and G).Do it till all objects in G is closer to each other than to objects in H.
Hierarchical clusteringThe most overused statistical method in gene expression.Gives us pretty pictures.Results tend to be unstable, sensitive to small changes.
Partitioning methodPartition the data (size N) into a pre-specified number K of mutually excusive and exhaustive groups: a many-to-one mapping, or encoder k=C(i), that assings the ith observation to the kth cluster.
Iteratively reallocate the observations to clusters until some criterion is met, e.g. minimization of a specific loss function
Partitioning methodA natural loss function would be the within cluster point scatter:The total point scatter:
is the between cluster point scatter.Minimizing is equivalent to minimize
Partitioning methodIn principle, we simply need to minimize W or maximize B over all possible assignments of N objects to K clusters. However, the number of distinct assignment, grows rapidly as N and K goes large.
Partitioning methodIn practice, we can only examine a small fraction of all possible encoders. Such feasible strategies are based on iterative greedy descent: An initial partition is specified. At each iterative step, the cluster assignments are changed in such a way that the value of the criterion is improved from its previous value.
K-meansChoose the squared Euclidean distance as dissimilarity measure: .Minimize the within cluster point scatter:
K-means Algorithmclosely related to the EM algorithm for estimating a certainGaussian mixture modelChoose K centroids at random.Make initial partition of objects into k clusters by assigning objects to closest centroid.E step:Calculate the centroid (mean) of each of the k clusters.M step: Reassign objects to the closest centroids.Repeat 3 and 4 until no reallocations occur.
Initial valuesfor K-means.x falls into local minimum.K-means: local minimum problem
K-means: discussionAdvantages:Fast and easyNice relationship with Gaussian mixture model.Disadvantages:Run into local minimum (should start with multiple initials).Need to know the number of clusters (estimation for number of clusters).Does not allow scattered objects (tight clustering).
Mixture model for clustering
Model based clusteringFraley and Raftery (1998) applied a Gaussian mixture model.
The parameters can be estimated by EM algorithm. The cluster membership is decided on the posterior probability of each belong to cluster k.
Review of EM algorithmIt is widely used in solving missing data problem. Here our missing data is the cluster membership.Let us review the EM algorithm with a simple example.
The CML approachIndicators , identifying the mixture component origin for , are treated as unknown parameters. Two CML criteria have been proposed according to the sampling scheme.
Two CMLsRandom sample within each cluster
Random sample from a population of mixture density
-- Classification likelihood:-- Mixture likelihood:Gaussian assumption:
The Classification EM algorithm
Related to K-meansWhen f(x) is assumed to be Gaussian and the covariance matrix is the identical and spherical across all clusters, i.e. for all k, .So maximize C1-CML is equivalent to minimize W.
Model-based methodsAdvantages:Flexibility on cluster covariance structure.Rigorous statistical inference with full model.Disadvantages:Model selection is usually difficult. Data may not fit Gaussian model.Too many parameters to estimate in complex covariance structure.Local minimum problem
ReferencesHastie, T., Tibshirani, R., and Friedman, J. (2009), The Elements of Statistical Learning (2nd ed.), New York: Springer.http://www-stat.stanford.edu/~tibs/ElemStatLearn/
Everitt, B. S., Landau, S., Leese, M., and Stahl, D. (2011), Cluster Analysis (5th ed.), West Sussex, UK: John Wiley & Sons Ltd.Celeux G, Govaert G. A Classification EM algorithm for clustering and two stochastic versions. Computational Statistics & Data Analysis 1992; 14:315-332