1. Making Time: Pseudo Time-Series for the Temporal Analysis of Cross-Section Data Emma Peeling, Allan Tucker Centre for Intelligent Data Analysis Brunel University West London

2. Cross-Section Data Studies often involve data sampled from a cross-section of a population Especially in biological and medical studies Collecting medical information on patients suffering from a particular disease and controls (healthy) Essentially these studies show a “snapshot” of the disease process

3. Cross-Section Data Many processes are inherently temporal in nature Previously healthy people can develop a disease over time going through different stages of severity If we want to model the development of such processes, usually require longitudinal data

4. Longitudinal Study Cross-Section vs Longitudinal Onset Cross Section Study

5. Pseudo Time-Series Models In this presentation we explore: Ordering data based upon Minimum Spanning Trees & PQ-Trees (Rifkin et al. 2000) Treating this ordered data as “Pseudo Time- Series” Using Pseudo Time-Series to build temporal models Test using a dynamic Bayesian network model for classifying: Medical Data Gene Expression Data

6. Multi-Dimensional Scaling Can be used to visualise distance between data points and pathways Here we use classic MDS Metric-based – Euclidean Distance

7. Minimum Spanning Tree Connects all nodes in graph Links contain minimal weights Weighted Graph MST

8. PQ-Tree PQ-Trees are used to encode partial orderings on variables P nodes: children can be in any order Q nodes: children order can only be reversed

9. Dynamic Bayesian Network Classifiers DBNCs are used to calculate: P(C|X t, X t-1 ) Here, we use the DBNC to model the Pseudo Time-Series for classifying data

10. Pseudo Time-Series Models In Summary: 1: Input: Cross-section data 2: Construct weighted graph and MST 3: Construct PQ tree from MST 4: Derive Pseudo Time-Series from PQ-tree using hill-climb search on P-nodes to minimise sequence length 5: Build DBNC model using pseudo temporal ordering of samples 6: Output: Temporal model of cross-section data

11. The Datasets B-Cell Microarray Data 3 classes of B-Cell data A number of patients Pre-ordered into expert pseudo time-series Visual Field Test Data One large cross-section study Healthy and Glaucomatous eyes One longitudinal study for testing the models

12. B-Cell: MDS & Pseudo Time-Series Plots show discovered path in 3D Classification of B-Cell data in 2D

13. B-Cell Accuracy Plot shows mean accuracy and variance over Cross-Validation with repeats

14. Expert Knowledge Ordering Sequence length Biologist = 512.0506: 1-26 PQ-tree: = 528.9907: 1-6,7,9,8,11,10,12-18,26,19,21,20,22-25 PQ-tree and hill-climb = 521.1865: 1-18,26,19-25

15. Visual Field: MDS & Pseudo Time-Series Plots show Path found for VF data in 3D Classification of VF data in 2D

16. VF Accuracy Plot shows mean accuracy and variance over Train / Test data with repeats

17. Related Work Semi-Supervised Methods Some datapoints are labelled with classes These are used to assist classification of others in an incremental manner Pseudo MTS imposes an order on the data as well as a distance between data Allows for the prediction of future states

18. Conclusions Cross Section data usually models snapshot of a process Longitudinal data usually needed to model temporal nature Here we use ordering methods to create Pseudo Time-Series models Early results on medical and biological data are promising

19. Future Work Dealing with outliers in dataspace Multiple trajectories (e.g. in VF data) Normalisation (rather than discretisation) Combining a number of longitudinal and cross-section studies

20. Multiple Trajectories

21. Acknowledgements Thanks to: David Garway-Heath, Moorifield’s Eye Hospital, London Paul Kellam, University College London