Making Time: Pseudo Time-Series for the Temporal Analysis of Cross-Section Data

Emma Peeling, Allan TuckerCentre for Intelligent Data AnalysisBrunel UniversityWest London

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

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

Longitudinal Study

Cross-Section vs Longitudinal

Onset

Cross SectionStudy

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

Multi-Dimensional Scaling

Can be used to visualise distance between data points and pathways Here we use classic MDS

Metric-based – Euclidean Distance

Minimum Spanning Tree Connects all nodes in graph Links contain minimal weights

Weighted Graph MST

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

Dynamic Bayesian Network Classifiers

DBNCs are used to calculate: P(C|Xt, Xt-1)

Here, we use the DBNC to model the Pseudo Time-Series for classifying data

Pseudo Time-Series Models In Summary:

1: Input: Cross-section data2: Construct weighted graph and MST3: Construct PQ tree from MST4: Derive Pseudo Time-Series from PQ-tree

using hill-climb search on P-nodes tominimise sequence length

5: Build DBNC model using pseudo temporal ordering of samples

6: Output: Temporal model of cross-section data

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

B-Cell: MDS & Pseudo Time-Series

Plots show discovered path in 3D Classification of B-Cell data in 2D

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

Expert KnowledgeOrdering 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

Visual Field: MDS & Pseudo Time-Series

Plots show Path found for VF data in 3D Classification of VF data in 2D

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

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

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

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

Multiple Trajectories

Acknowledgements Thanks to:

David Garway-Heath, Moorifield’s Eye Hospital, London

Paul Kellam, University College London