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Covariate Adjusted Functional Principal Component Analysis ( FPCA ) for Longitudinal Data

Covariate Adjusted Functional Principal Component Analysis ( FPCA ) for Longitudinal Data. Ci-Ren Jiang & Jane-Ling Wang University of California, Davis National Taiwan University July 9, 2009. TexPoint fonts used in EMF.

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Covariate Adjusted Functional Principal Component Analysis ( FPCA ) for Longitudinal Data

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  1. Covariate Adjusted Functional Principal Component Analysis (FPCA) for Longitudinal Data Ci-Ren Jiang & Jane-Ling Wang University of California, Davis National Taiwan University July 9, 2009 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

  2. Ci-Ren JiangPh. D. Candidate, UC Davis

  3. Outline • Introduction • (Univariate) Covariate adjusted FPCA ? (Multivariate ) Covariate adjusted FPCA • FPCA as a building block for Modeling • Application to PET data

  4. 1. Introduction • Principal Component analysis is a standard dimension reduction tool for multivariate data. It has been extended to functional data and termed functional principal component analysis (FPCA). • Standard FPCA approaches treat functional data as if they are from a single population. • Our goal is to accommodate covariate information in the framework of FPCA for longitudinal data.

  5. Functional vs. Longitudinal Data • A sample of curves, with one curve, X(t), per subject. - These curves are usually considered realizations of a stochastic process in . - dimensional • Functional Data - In reality, X(t) is recorded at a regular and dense time grid high-dimensional. • Longitudinal Data – irregularly sampled X(t). - often sparse, as in medical follow-up studies.

  6. Longitudinal AIDS Data • CD4 counts of 369 patients were recorded. The number , of repeated measurements for subject i, varies with an average of6.44. • This resulted in longitudinal data of uneven no. of measurements at irregular time points.

  7. CD4 Counts of First 25 Patients

  8. Review of FPCA

  9. Review of FPCA • Both longitudinal and functional data may be observed with noise (measurement errors). the observed data for subject i might be

  10. Review of FPCA Functional Data Dauxois, Pousse & Romain (1982) Rice & Silverman (1991) Cardot (2000) Hall & Hosseini-Nasab (2006) Longitudinal Data Shi, Weiss & Taylor(1996) James, Sugar & Hastie(2000) Rice & Wu (2001) Yao, Müller & Wang (2005)

  11. Steps to FPCA TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAA

  12. Estimation of Mean Function Taipei 101

  13. CD4 Counts of First 25 Patients

  14. CD4 Counts of First 25 Patients

  15. Mean Curve: CD4 counts of all patients

  16. Estimation of Covariance Function

  17. Row Covariance Plot:

  18. Row Covariance Plot:

  19. Covariance & Variance

  20. References • Yao, Müller and Wang (2005, JASA) Methods and theory for the mean and covariance functions. • Hall, Müller and Wang (2006, AOS) Theory on eigenfunctions and eigenvalues.

  21. End of Introduction

  22. 2. Covariate adjusted FPCA – Univariate Z • For dense functional data Chiou, Müller & Wang (2003) Cardot (2006) • Their method does not work for sparse dara. • We propose two ways: fFPCA& mFPCA

  23. Covariate adjusted FPCA:Longitudinal Data

  24. Fully Adjusted FPCA (fFPCA)

  25. Mean Adjusted FPCA (mFPCA)

  26. Estimation:Mean Function

  27. Estimation:Mean Function

  28. CD4 Counts of All Patients and Mean Curve

  29. AIDS CD4: Estimated Mean

  30. Estimation:Covariance Function

  31. Estimation:Covariance Function

  32. Example of Covariance Estimates

  33. AIDS CD4: Estimated Covariance

  34. Estimation:Variance of Measurement Errors

  35. Estimation:Variance of Measurement Errors

  36. AIDS: Estimated Covariance + measurement error

  37. Estimation:Eigenvalues and Eigenfunctions

  38. Estimation:Principal Component Scores

  39. Estimation:Principal Component Scores

  40. Theoretical Results

  41. Notations:2D Smoothers

  42. Theoretical Results: 2D Smoothers

  43. Theoretical Results: 2D Smoothers (cont’d)

  44. Mean Function: Nadaraya-Watson Est.

  45. Mean Function: Local Linear Est.

  46. Rate of Convergence • If E(N) < , the rate of convergence for the 2D mean and covariance function is . - This is the optimal rate of convergence for 2D smoothers with independent data. • If E(N) → , the rate of convergence can be as close to as possible but not equal to . • If , the convergence rate is .

  47. Notations: 3D Smoothers

  48. Notations:3D Smoothers (cont’d)

  49. Notations:3D Smoothers (cont’d)

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