Probability-based imputation method for fuzzy cluster analysis of gene expression microarray data

1. Probability-based imputation method for fuzzy cluster analysis of gene expression microarray data Thanh Le, Tom Altman and KatheleenGardiner University of Colorado Denver April 16, 2012

2. Overview • Introduction • Data clustering with missing values • Current approaches • Proposed method: fzPBI • Data clustering using Fuzzy C-Means • Imputation using probability model • Datasets • Artificial and real datasets for testing fzPBI • Experimental results • Discussion

3. Clustering with missing values • Data points & missing values x(x1, x2, …, xn-1, xn) Data points with missing values, x(x1, ?, …, ?, xn) XM = { ? }; XP = X \ XM • Problem • Cluster analysis is based on dissimilarity • Distance is computed using every attribute of data objects. • Improper distance measurement provides incorrect clustering results.

4. Current approaches • Data preprocess to predict missing values • Remove data points with missing values • Imputation of missing values • During the clustering process • Application of clustering model • Missing values are estimated and used • Popular clustering methods, • Expectation-Maximization (EM), Model based clustering • K-Means Crisp membership • Fuzzy C-Means (FCM) Fuzzy membership, soft cluster boundaries Each data point can belong to multiple clusters, more relationship information provided

5. Current approaches’ issues • Heuristic methods • Imputation using nearest data points • Heuristics, data distribution is not used • EM based methods • Model based imputation of missing values • Model assumptions, slow convergence • Missing values impact parameter estimation • FCM based methods • Distance based imputation of missing values • Fast convergence, maybe the best approach • Data distribution is omitted

6. Probability-based imputation - fzPBI • Data clustering using FCM • Possibility to probability transformation • Application of the central limit theory into creation of the probability model of data distribution • Application of the probability model into missing value imputation • Repeat steps 1-4 until convergence

7. Fuzzy C-Means algorithm • Objective function • Model parameters estimation:

8. Distance measurement • p: Data space dimensions • Each missing value, xij, is used with confidence, wj, which is, • 0 at the beginning • 1 at the end

9. Probability model • Central limit theory application, • Cluster is the mean of different distribution models that describe the cluster’s members. It can be approximated using the normal distribution model. • Possibility to probability transformation • {uki}i=1..n - possibility distribution of X at vk • {pki}i=1..n - probability distribution of X at vk, • Create the probability model at vk using {pki} • Missing value imputation using probability model

10. Datasets • Artificial datasets • A dataset generated using finite mixture model • A non-uniform dataset manually created • Clusters differ in size • Cluster distances are different • Real datasets • Iris, Wine datasets at UC Irvine Machine Learning Repository • RCNS (Rat central nervous system), Serum, Yeast and Yeast-MIPS gene expression datasets. • Incomplete datasets were generated using different percentages of missing values

11. Performance measures • Root mean square error – RMSE • Misclassification error - ME • Compare the cluster label of each data object with its actual class label

12. Uniform dataset • fzPBI- Probability based method • OCS- optimal complete strategy • NPS- nearest prototype strategy • FCMimp- FCM based impute • CIAO- Alternating Optimization • FCMGOimp- FCM & GO based impute

13. Non-uniform dataset

14. Iris dataset

15. RCNS gene expression dataset

16. Yeast gene expression dataset

17. Serum gene expression dataset

18. The advantages of fzPBI • Approximate the data distribution using probability model • Apply the model into missing value imputation • Inherit the advantages of FCM and model based methods, and the application of the central limit theory

19. Future work • Combine fzPBI with biological knowledge: protein-protein-interaction, Gene ontology • Internal measures using the data • External measures using the biological knowledge • Internal measures at missing values are adjusted using external measures.

20. Thank you! Questions? • We acknowledge the support from Vietnamese Ministry of Education and Training, the 322 scholarship program.