1. A validation method for fuzzy clustering A biological problem of gene expression data
Thanh Le, Katheleen J. Gardiner
University of Colorado Denver
July 18th, 2011

2. Overview Introduction fzBLE Datasets Experimental results Discussion:
Data clustering: approaches and current challenges
fzBLE
a novel method for validation of clustering results
Datasets
artificial and real datasets for testing fzBLE
Experimental results
Discussion:
Advantages and limitations of fzBLE

3. Clustering problem Genes are clustered based on
Similarity
Dissimilarity
Clusters are described by
Boundaries & overlaps
Number of clusters
Compactness within clusters
Separation between clusters

4. Clustering approaches
Hierarchical approach
Partitioning approach
Hard clustering approach
Crisp cluster boundaries
Crisp cluster membership
Soft/Fuzzy clustering approach
Overlapping cluster boundaries
Soft/Fuzzy membership
Appropriate for many real-world problems

5. Fuzzy C-Means algorithm
The model
Features:
Fuzzy membership, soft cluster boundaries,
One gene can belong to multiple clusters & be assigned to multiple biological processes

6. Fuzzy C-Means (contd.)
Possibility-based model
Model parameters estimated using an iteration process
Rapid convergence
Most appropriate for gene expression data
Challenges:
Determining the number of clusters
Avoiding local optima
The goodness-of-fit to validate clustering results

7. Methods for fuzzy clustering validation
Methods based on compactness and separation
Problem:
Over-fit - the larger the number of cluster is, the better the cluster index is.
No rationale for how to scale the two factors in the model
Methods based on goodness of fit
Statistics approach
Expectation-Maximization (EM) method
Slowly convergent, particularly at cluster boundaries because of the exponential function.
Inappropriate to real dataset because of the model assumption of data distributions: Gaussian, chi-squared…

8. The fzBLE method for cluster validation
Cluster using Fuzzy C-Means clustering algorithm
Validate using the goodness-of-fit (the log likelihood estimator) and Bayesian approach

9. Cluster validation: Goodness-of-fit & fuzzy clustering
Convert the possibility model into a probability model
Use Bayesian approach to compute the statistics.
Apply the Central Limit Theory
To effectively represent the data distribution
Model selection based on goodness-of-fit

10. Datasets Artificial datasets Real datasets
Finite mixture model based datasets
Real datasets
Iris, Wine and Glass datasets at UC Irvine Machine Learning Repository
Gene datasets which are more complex
Yeast cell cycle gene expression (Yeast)
Yeast gene functional annotations (Yeast-MIPS)
Rat Central Nervous System (RCNS) gene expression

11. Experimental results on artificial datasets
Correctness Ratios in determining the number of clusters
# clusters
fzBLE PC PE FS XB
CWB
PBMF BR CF 3
1.00
0.42
0.83
0.00 4
0.92 5
0.75 6
0.58 7
0.67 8 9
0.33
PC-partition coefficient, PE-partition entropy, FS-Fukuyama-Sugeno, XB-Xie and Beni, CWB-Compose Within and Between scattering, PBMF-Pakhira, Bandyopadhyay and Maulik Fuzzy, BR-Rezaee B., CF-Compactness factor; loop=5, #cluster range=[2,12]

12. Experimental results on Glass dataset
Algorithm Cluster Validity Scores and Decisions (highlighted in yellow)
# clusters
fzble PC PE FS XB
CWB
PBMF BR CF 2
0.8884
0.1776
0.3700
0.7222
0.3732
1.9817
0.5782 3
0.8386
0.2747
0.1081
0.7817
0.4821
1.5004
0.4150 4
0.8625
0.2515
0.6917
0.4463
1.0455
0.3354 5
0.8577
0.2698
0.6450
0.4610
0.8380
0.2818 6
0.8004
0.3865
1.4944
0.3400
0.8371
0.2430 7
0.8183
0.3650
1.3802
0.3891
0.6914
0.2214 8
0.8190
0.3637
1.4904
0.6065
0.5916
0.2108 9
0.8119
0.3925
1.7503
0.3225
0.5634
0.1887 10
0.8161
0.3852
1.7821
0.3909
0.4926
0.1758 11
0.8259
0.3689
1.6260
0.3265
0.4470
0.1704 12
0.8325
0.3555
1.4213
0.5317
0.3949
0.1591 13
0.8317
0.3556
1.4918
0.6243
0.3544
0.1472

13. Experimental results on RCNS - more complex dataset; two-factor scaling issue
Algorithm Cluster Validity Scores and Decisions (highlighted in yellow)
#clusters
fzble PC PE FS XB
CWB
PBMF BR CF 2
0.9942
0.0121
0.0594
5.5107
4.2087
1.1107 3
0.9430
0.0942
0.4877
4.1309
4.2839
1.6634 4
0.9142
0.1470
0.9245
6.1224
3.3723
1.3184 5
0.8900
0.1941
1.3006
9.4770
2.6071
1.1669 6
0.8695
0.2387
2.5231
1.9499
1.1026 7
0.8707
0.2386
2.1422
2.8692
0.7875 8
0.8925
0.2078
1.7245
2.5323
0.5894 9
0.8863
0.2192
1.6208
2.6041
0.5019 10
0.8847
0.2241
1.1897
3.4949
0.3918
112 genes during RCNS development at 9 time points
6 clusters, 4 of which are functionality-annotated (Somogyi et al. 1995, Wen et al. 1998)

14. Discussion: The advantages of fzBLE
Performs better than other approaches on 3 levels of data.
Compactness-separation approaches
Solves the over-fit problem using goodness-of-fit.
Eliminates need for two scaling factors
Mixture model with EM approach
Rapid convergence
No assumption on data distribution
The approach of scaling the two factors: compactness and separation is similar to that of scaling gene expression by within condition before clustering. The problem is that:
The number of genes in each chip is known while we are not sure the number of clusters
The values in multiple experimental conditions are consistent (fc, log of fc,…) while the values of the two factor are not.

15. Discussion: The limitations of fzBLE
Depends on internal validity
External validities are needed
Biological validity
GO terms, Pathways, PPI
Future work on gene expression:
Distance definition based on biological context
Combine fzBLE with biological homology and stability indices

16. Thank you! Questions? We acknowledge the support from
National Institutes of Health
Linda Crnic Institute
Vietnamese Ministry of Education and Training