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# Cluster Validation - PowerPoint PPT Presentation

Cluster Validation. Cluster validation Assess the quality and reliability of clustering results. Why validation? To avoid finding clusters formed by chance To compare clustering algorithms To choose clustering parameters e.g., the number of clusters in the K-means algorithm. DBSCAN.

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## PowerPoint Slideshow about ' Cluster Validation' - austine-janus

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Presentation Transcript

• Cluster validation

• Assess the quality and reliability of clustering results.

• Why validation?

• To avoid finding clusters formed by chance

• To compare clustering algorithms

• To choose clustering parameters

• e.g., the number of clusters in the K-means algorithm

K-means

Clusters found in Random Data

Random Points

• Comparing the clustering results to ground truth (externally known results).

• External Index

• Evaluating the quality of clusters without reference to external information.

• Use only the data

• Internal Index

• Determining thereliabilityof clusters.

• To what confidence level, the clusters are not formed by chance

• Statistical framework

• Notation

• N: number of objects in the data set;

• P={P1,…,Pm}: the set of “ground truth” clusters;

• C={C1,…,Cn}: the set of clusters reported by a clustering algorithm.

• The “incidence matrix”

• N  N (both rows and columns correspond to objects).

• Pij = 1 if Oi and Oj belong to the same “ground truth” cluster in P; Pij=0 otherwise.

• Cij = 1 if Oi and Oj belong to the same cluster in C; Cij=0 otherwise.

• A pair of data object (Oi,Oj) falls into one of the following categories

• SS: Cij=1 and Pij=1; (agree)

• DD: Cij=0 and Pij=0; (agree)

• SD: Cij=1 and Pij=0; (disagree)

• DS: Cij=0 and Pij=1; (disagree)

• Rand index

• may be dominated by DD

• Jaccard Coefficient

• “Ground truth” may be unavailable

• Use only the data to measure cluster quality

• Measure the “homogeneity” and “separation” of clusters.

• SSE: Sum of squared errors.

• Calculate the correlation between clustering results and distance matrix.

• Homogeneity is measured by the within cluster sum of squares

Exactly the objective function of K-means.

• Separation is measured by the between cluster sum of squares

Where |Ci| is the size of cluster i, m is the centroid of the whole data set.

• BSS + WSS = constant

• A larger number of clusters tend to result in smaller WSS.

1

m1

2

3

4

m2

5

Sum of Squared Error

K=1 :

K=2 :

K=4:

• Can also be used to estimate the number of clusters.

• SSE curve for a more complicated data set

SSE of clusters found using K-means

• Distance Matrix

• Dij is the similarity between object Oi and Oj.

• Incidence Matrix

• Cij=1 if Oi and Oj belong to the same cluster, Cij=0 otherwise

• Compute the correlation between the two matrices

• Only n(n-1)/2 entries needs to be calculated.

• High correlation indicates good clustering.

• Given Distance Matrix D = {d11,d12, …, dnn } and Incidence Matrix C= { c11, c12,…, cnn } .

• Correlation r between D and C is given by

• Correlation of incidence and proximity matrices for the K-means clusterings of the following two data sets.

Corr = -0.9235

Corr = -0.5810

K-means

Clusters found in Random Data

Random Points

• Order the similarity matrix with respect to cluster labels and inspect visually.

• Clusters in random data are not so crisp

K-means

• Clusters in random data are not so crisp

• Clusters in random data are not so crisp

DBSCAN

• Need a framework to interpret any measure.

• For example, if our measure of evaluation has the value, 10, is that good, fair, or poor?

• Statistics provide a framework for cluster validity

• The more “atypical” a clustering result is, the more likely it represents valid structure in the data.

• Example

• Compare SSE of 0.005 against three clusters in random data

• SSE Histogram of 500 sets of random data points of size 100 distributed over the range 0.2 – 0.8 for x and y values

SSE = 0.005

• Correlation of incidence and distance matrices for the K-means of the following two data sets.

Correlation histogram of random data

Corr = -0.5810

Corr = -0.9235

• Given the total number of genes in the data set associated with term T is M, if randomly draw n genes from the data set N, what is the probability that m of the selected n genes will be associated with T?

Based on Hyper-geometric distribution, the probability of having m genes or fewer associated to T in N can be calculated by summing the probabilities of a random list of N genes having 1, 2, …, m genes associated to T. So the p-value of over-representation is as follows: