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Estimation of Multivariate Functional Principal Components from Sparse Functional Data

Uche Mbaka, Michelle Carey

Abstract

Traditional Functional Principal Component Analysis typically focuses on densely observed univariate functional data, yet many applications, particularly in longitudinal studies, involve multivariate functional data observed sparsely and irregularly across subjects. A common approach for extracting multivariate functional principal components in such settings relies on an eigen decomposition of univariate functional principal component scores to capture cross-component correlations. We propose a new approach for the estimation of multivariate functional principal components by improving the univariate eigenanalysis through maximum likelihood estimation combined with a modified Gram-Schmidt orthonormalization. The performance of the proposed approach is evaluated against two established methods, and its practical utility is demonstrated through an application to longitudinal cognitive biomarker data from an Alzheimer's disease study and a collection of data on dairy milk yield and milk compositions from research dairy farms in Ireland.

Estimation of Multivariate Functional Principal Components from Sparse Functional Data

Abstract

Traditional Functional Principal Component Analysis typically focuses on densely observed univariate functional data, yet many applications, particularly in longitudinal studies, involve multivariate functional data observed sparsely and irregularly across subjects. A common approach for extracting multivariate functional principal components in such settings relies on an eigen decomposition of univariate functional principal component scores to capture cross-component correlations. We propose a new approach for the estimation of multivariate functional principal components by improving the univariate eigenanalysis through maximum likelihood estimation combined with a modified Gram-Schmidt orthonormalization. The performance of the proposed approach is evaluated against two established methods, and its practical utility is demonstrated through an application to longitudinal cognitive biomarker data from an Alzheimer's disease study and a collection of data on dairy milk yield and milk compositions from research dairy farms in Ireland.
Paper Structure (22 sections, 25 equations, 7 figures, 5 tables)

This paper contains 22 sections, 25 equations, 7 figures, 5 tables.

Table of Contents

  1. Introduction
  2. MFPCA for Sparse Functional Data
  3. Multivariate FPCs and their Basic Properties
  4. The relationship between Univariate and Multivariate FPCA
  5. Univariate Functional Principal Component Analysis for Sparse Functional Data
  6. The Multivariate FPCA for Sparse Data
  7. Simulations
  8. Data Generation
  9. The Covariance
  10. The Eigenfunctions
  11. The Eigenvalues
  12. The Curve Reconstruction
  13. Alzheimer’s disease
  14. Milk Quality in Dairy Cattle
  15. Conclusion
  16. ...and 7 more sections

Figures (7)

  • Figure 1: The empirical correlation between the univariate functional principal component scores.
  • Figure 2: Estimated top two eigenfunctions for the longitudinal marker ADAS-Cog 13 (Disease Assessment Scale-Cognitive 13 items). In the two right-hand panels, the black solid curve represents the overall smoothed mean, which is identical across all cases. The remaining curves show the effect of adding (green, “+”) or subtracting (red, “– –”) an appropriately scaled multiple of the corresponding principal component.
  • Figure 3: Estimated top two eigenfunctions for the longitudinal marker Rey Auditory Verbal Learning Test immediate recall (RAVLT.imme). In the two right-hand panels, the black solid curve represents the overall smoothed mean, which is identical across all cases. The remaining curves show the effect of adding (green, “+”) or subtracting (red, “– –”) an appropriately scaled multiple of the corresponding principal component.
  • Figure 4: Estimated top two eigenfunctions for the longitudinal marker Rey Auditory Verbal Learning Test immediate recall (RAVLT.learn). In the two right-hand panels, the black solid curve represents the overall smoothed mean, which is identical across all cases. The remaining curves show the effect of adding (green, “+”) or subtracting (red, “– –”) an appropriately scaled multiple of the corresponding principal component.
  • Figure 5: Estimated scores vs the last diagnosis of patients.
  • ...and 2 more figures