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Periodic Gene Expression Analysis using Functional Data Analysis (FDA)

This article explores the limitations of using PCA for analyzing high-dimensional gene expression data with low sample size. It introduces the concept of Functional Data Analysis (FDA) and demonstrates its application in identifying periodic genes in yeast cell cycle data. Additionally, it discusses adjusting for source and batch effects in breast cancer data using Distance Weighted Discrimination (DWD) technique.

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Periodic Gene Expression Analysis using Functional Data Analysis (FDA)

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  1. Object Orie’d Data Analysis, Last Time • Organizational Matters http://www.unc.edu/~marron/UNCstat322-2005/HomePage.html • Matlab Software • Time Series of Curves • Chemometrics Data • Mortality Data

  2. Data Object Conceptualization Object Space Feature Space Curves Images Manifolds Shapes Tree Space Trees

  3. Functional Data Analysis, Toy EG I

  4. Limitation of PCA • PCA can provide useful projection directions • But can’t “see everything”… • Reason: • PCA finds dir’ns of maximal variation • Which may obscure interesting structure

  5. Limitation of PCA, Toy E.g.

  6. Yeast Cell Cycle Data • “Gene Expression”– Micro-array data • Data (after major preprocessing): Expression “level” of: • thousands of genes (d ~ 1,000s) • but only dozens of “cases” (n ~ 10s) • Interesting statistical issue: High Dimension Low Sample Size data (HDLSS)

  7. Yeast Cell Cycle Data Data from: Spellman, P. T., Sherlock, G., Zhang, M.Q., Iyer, V.R., Anders, K., Eisen, M.B., Brown, P.O., Botstein, D. and Futcher, B. (1998), “Comprehensive Identification of Cell Cycle-regulated Genes of the Yeast Saccharomyces cerevisiae by Microarray Hybridization”, Molecular Biology of the Cell, 9, 3273-3297.

  8. Yeast Cell Cycle Data Analysis here is from: Zhao, X., Marron, J.S. and Wells, M.T. (2004) The Functional Data View of Longitudinal Data, Statistica Sinica, 14, 789-808

  9. Yeast Cell Cycle Data • Lab experiment: • Chemically “synchronize cell cycles”, of yeast cells • Do cDNA micro-arrays over time • Used 18 time points, over “about 2 cell cycles” • Studied 4,489 genes (whole genome) • Time series view of data: 4,489 time series of length 18 • Functional Data View: 4,489 “curves”

  10. Yeast Cell Cycle Data, FDA View Central question: Which genes are “periodic” over 2 cell cycles?

  11. Yeast Cell Cycle Data, FDA View Periodic genes? Naïve approach: Simple PCA

  12. Yeast Cell Cycle Data, FDA View • Central question: which genes are “periodic” over 2 cell cycles? • Naïve approach: Simple PCA • No apparent (2 cycle) periodic structure? • Eigenvalues suggest large amount of “variation” • PCA finds “directions of maximal variation” • Often, but not always, same as “interesting directions” • Here need better approach to study periodicities

  13. Yeast Cell Cycles, Freq. 2 Proj. PCA on Freq. 2 Periodic Component Of Data

  14. Yeast Cell Cycles, Freq. 2 Proj. PCA on periodic component of data • Hard to see periodicities in raw data • But very clear in PC1 (~sin) and PC2 (~cos) • PC1 and PC2 explain 65% of variation (see residuals) • Recall linear combos of sin and cos capture “phase” • since:

  15. Frequency 2 Analysis • Important features of data appear only at frequency 2, • Hence project data onto 2-dim space of sin and cos (freq. 2) • Useful view: scatterplot

  16. Frequency 2 Analysis

  17. Frequency 2 Analysis • Project data onto 2-dim space of sin and cos (freq. 2) • Useful view: scatterplot • Angle (in polar coordinates) shows phase • Colors: Spellman’s cell cycle phase classification • Black was labeled “not periodic” • Within class phases approx’ly same, but notable differences • Later will try to improve “phase classification”

  18. Batch and Source Adjustment • For Stanford Breast Cancer Data • Analysis in Benito, et al (2004) Bioinformatics https://genome.unc.edu/pubsup/dwd/ • Adjust for Source Effects • Different sources of mRNA • Adjust for Batch Effects • Arrays fabricated at different times

  19. Idea Behind Adjustment • Find “direction” from one to other • Shift data along that direction • Details of DWD Direction developed later

  20. Source Batch Adj: Raw Breast Cancer data

  21. Source Batch Adj: Source Colors

  22. Source Batch Adj: Batch Colors

  23. Source Batch Adj: Biological Class Colors

  24. Source Batch Adj: Biological Class Col. & Symbols

  25. Source Batch Adj: Biological Class Symbols

  26. Source Batch Adj: Source Colors

  27. Source Batch Adj: PC 1-2 & DWD direction

  28. Source Batch Adj: DWD Source Adjustment

  29. Source Batch Adj: Source Adj’d, PCA view

  30. Source Batch Adj: Source Adj’d, Class Colored

  31. Source Batch Adj: Source Adj’d, Batch Colored

  32. Source Batch Adj: Source Adj’d, 5 PCs

  33. Source Batch Adj: S. Adj’d, Batch 1,2 vs. 3 DWD

  34. Source Batch Adj: S. & B1,2 vs. 3 Adjusted

  35. Source Batch Adj: S. & B1,2 vs. 3 Adj’d, 5 PCs

  36. Source Batch Adj: S. & B Adj’d, B1 vs. 2 DWD

  37. Source Batch Adj: S. & B Adj’d, B1 vs. 2 Adj’d

  38. Source Batch Adj: S. & B Adj’d, 5 PC view

  39. Source Batch Adj: S. & B Adj’d, 4 PC view

  40. Source Batch Adj: S. & B Adj’d, Class Colors

  41. Source Batch Adj: S. & B Adj’d, Adj’d PCA

  42. Source Batch Adj: Raw Data, Tree View

  43. Source Batch Adj: Raw Data, Array Tree

  44. Source Batch Adj: Raw Array Tree, Source Colored

  45. Source Batch Adj: Raw Array Tree, Batch Colored

  46. Source Batch Adj: Raw Array Tree, Class Colored

  47. Source Batch Adj: DWD Adjusted Data, Tree View

  48. Source Batch Adj: DWD Adjusted Data, Array Tree

  49. Source Batch Adj: DWD Adjusted Data, Source Colored

  50. Source Batch Adj: DWD Adjusted Data, Batch Colored

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