The Annals of Applied Statistics

Longitudinal high-dimensional principal components analysis with application to diffusion tensor imaging of multiple sclerosis

Vadim Zipunnikov, Sonja Greven, Haochang Shou, Brian S. Caffo, Daniel S. Reich, and Ciprian M. Crainiceanu

Full-text: Open access

Abstract

We develop a flexible framework for modeling high-dimensional imaging data observed longitudinally. The approach decomposes the observed variability of repeatedly measured high-dimensional observations into three additive components: a subject-specific imaging random intercept that quantifies the cross-sectional variability, a subject-specific imaging slope that quantifies the dynamic irreversible deformation over multiple realizations, and a subject-visit-specific imaging deviation that quantifies exchangeable effects between visits. The proposed method is very fast, scalable to studies including ultrahigh-dimensional data, and can easily be adapted to and executed on modest computing infrastructures. The method is applied to the longitudinal analysis of diffusion tensor imaging (DTI) data of the corpus callosum of multiple sclerosis (MS) subjects. The study includes 176 subjects observed at 466 visits. For each subject and visit the study contains a registered DTI scan of the corpus callosum at roughly 30,000 voxels.

Article information

Source
Ann. Appl. Stat., Volume 8, Number 4 (2014), 2175-2202.

Dates
First available in Project Euclid: 19 December 2014

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1419001739

Digital Object Identifier
doi:10.1214/14-AOAS748

Mathematical Reviews number (MathSciNet)
MR3292493

Zentralblatt MATH identifier
06408774

Keywords
Principal components linear mixed model diffusion tensor imaging brain imaging data multiple sclerosis

Citation

Zipunnikov, Vadim; Greven, Sonja; Shou, Haochang; Caffo, Brian S.; Reich, Daniel S.; Crainiceanu, Ciprian M. Longitudinal high-dimensional principal components analysis with application to diffusion tensor imaging of multiple sclerosis. Ann. Appl. Stat. 8 (2014), no. 4, 2175--2202. doi:10.1214/14-AOAS748. https://projecteuclid.org/euclid.aoas/1419001739


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References

  • 3D-Slicer (2011). http://www.slicer.org/.
  • Aston, J. A. D., Chiou, J.-M. and Evans, J. P. (2010). Linguistic pitch analysis using functional principal component mixed effect models. J. R. Stat. Soc. Ser. C. Appl. Stat. 59 297–317.
  • Bigelow, J. L. and Dunson, D. B. (2009). Bayesian semiparametric joint models for functional predictors. J. Amer. Statist. Assoc. 104 26–36.
  • Budavari, T., Wild, V., Szalay, A. S., Dobos, L. and Yip, C.-W. (2009). Reliable eigenspectra for new generation surveys. Monthly Notices of the Royal Astronomical Society 394 1496–1502.
  • Crainiceanu, C. M., Staicu, A.-M. and Di, C.-Z. (2009). Generalized multilevel functional regression. J. Amer. Statist. Assoc. 104 1550–1561.
  • Crainiceanu, C. M., Caffo, B. S., Luo, S., Zipunnikov, V. M. and Punjabi, N. M. (2011). Population value decomposition, a framework for the analysis of image populations. J. Amer. Statist. Assoc. 106 775–790.
  • Demmel, J. W. (1997). Applied Numerical Linear Algebra. SIAM, Philadelphia, PA.
  • Di, C., Crainiceanu, C. M. and Jank, W. S. (2010). Multilevel sparse functional principal component analysis. Stat. 3 126–143.
  • Di, C.-Z., Crainiceanu, C. M., Caffo, B. S. and Punjabi, N. M. (2009). Multilevel functional principal component analysis. Ann. Appl. Stat. 3 458–488.
  • Everson, R. and Roberts, S. (2000). Inferring the eigenvalues of covariance matrices from limited, noisy data. IEEE Trans. Signal Process. 48 2083–2091.
  • Goldsmith, J., Crainiceanu, C. M., Caffo, B. S. and Reich, D. S. (2011). Penalized functional regression analysis of white-matter tract profiles in multiple sclerosis. NeuroImage 57 431–439.
  • Golub, G. H. and Van Loan, C. F. (1996). Matrix Computations, 3rd ed. Johns Hopkins Univ. Press, Baltimore, MD.
  • Greven, S., Crainiceanu, C., Caffo, B. and Reich, D. (2010). Longitudinal functional principal component analysis. Electron. J. Stat. 4 1022–1054.
  • Guo, W. (2002). Functional mixed effects models. Biometrics 58 121–128.
  • Hall, P., Müller, H.-G. and Yao, F. (2008). Modelling sparse generalized longitudinal observations with latent Gaussian processes. J. R. Stat. Soc. Ser. B Stat. Methodol. 70 703–723.
  • Harville, D. (1976). Extension of the Gauss–Markov theorem to include the estimation of random effects. Ann. Statist. 4 384–395.
  • Hua, Z. W., Dunson, D. B., Gilmore, J. H., Styner, M. and Zhu, H. T. (2012). Semiparametric Bayesian local functional models for diffusion tensor tract statistics. NeuroImage 63 460–474.
  • Karhunen, K. (1947). Über lineare Methoden in der Wahrscheinlichkeitsrechnung. Annales Academie Scientiarum Fennicae 37 1–79.
  • Li, Y., Zhu, H., Shen, D., Lin, W., Gilmore, J. H. and Ibrahim, J. G. (2011). Multiscale adaptive regression models for neuroimaging data. J. R. Stat. Soc. Ser. B Stat. Methodol. 73 559–578.
  • Loève, M. (1978). Probability Theory II, 4th ed. Springer, New York.
  • McCulloch, C. E. and Searle, S. R. (2001). Generalized, Linear, and Mixed Models. Wiley, New York.
  • Minka, T. P. (2000). Automatic choice of dimensionality for PCA. Adv. Neural Inf. Process. Syst. 13 598–604.
  • MIPAV (2011). http://mipav.cit.nih.gov.
  • Mohamed, A. and Davatzikos, C. (2004). Medical Image Computing and Computer-Assisted Intervention. Springer, Berlin.
  • Mori, S. (2007). Introduction to Diffusion Tensor Imaging. Elsevier, Amsterdam.
  • Morris, J. S. and Carroll, R. J. (2006). Wavelet-based functional mixed models. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 179–199.
  • Morris, J. S., Baladandayuthapani, V., Herrick, R. C., Sanna, P. and Gutstein, H. (2011). Automated analysis of quantitative image data using isomorphic functional mixed models, with application to proteomics data. Ann. Appl. Stat. 5 894–923.
  • Pujol, S. (2010). 3D-Slicer (tutorial). National Alliance for Medical Image Computing (NA-MIC).
  • Raine, C. S., McFarland, H. and Hohlfeld, R. (2008). Multiple Sclerosis: A Comprehensive Text. Saunders, Philadelphia, PA.
  • Reich, D. S., Ozturk, A., Calabresi, P. A. and Mori, S. (2010). Automated vs conventional tractography in multiple sclerosis: Variablity and correlation with disability. NeuroImage 49 3047–3056.
  • Reiss, P. T. and Ogden, R. T. (2008). Functional generalized linear models with applications to neuroimaging. In Poster presentation Workshop on Contemporary Frontiers in High-Dimensional Statistical Data Analysis, Isaac Newton Institute, University of Cambridge, UK.
  • Reiss, P. T. and Ogden, R. T. (2010). Functional generalized linear models with images as predictors. Biometrics 66 61–69.
  • Reiss, P. T., Ogden, R. T., Mann, J. and Parsey, R. V. (2005). Functional logistic regression with PET imaging data: A voxel-level clinical diagnostic tool. Journal of Cerebral Blood Flow & Metabolism 25 s635.
  • Rodríguez, A., Dunson, D. B. and Gelfand, A. E. (2009). Bayesian nonparametric functional data analysis through density estimation. Biometrika 96 149–162.
  • Roweis, S. (1997). EM algorithms for PCA and SPCA. Adv. Neural Inf. Process. Syst. 10 626–632.
  • Shinohara, R., Crainiceanu, C., Caffo, B., Gaita, M. I. and Reich, D. S. (2011). Population wide model-free quantification of blood-brain-barrier dynamics in multiple sclerosis. NeuroImage 57 1430–1446.
  • Shou, H., Zipunnikov, V., Crainiceanu, C. and Greven, S. (2013). Structured functional principal component analysis. Available at arXiv:1304.6783.
  • Staicu, A.-M., Crainiceanu, C. M. and Carroll, R. J. (2010). Fast analysis of spatially correlated multilevel functional data. Biostatistics 11 177–194.
  • Weng, J., Zhang, Y. and Hwang, W.-S. (2003). Candid covariance-free incremental principal component analysis. IEEE Transactions on Pattern Analysis and Machine Intelligence 25 1034–1040.
  • Xiao, L., Ruppert, D., Zipunnikov, V. and Crainiceanu, C. (2013). Fast covariance estimation for high-dimensional functional data. Available at arXiv:1306.5718.
  • Yuan, Y., Gilmore, J. H., Geng, X., Styner, M., Chen, K., Wang, J. L. and Zhu, H. (2014). Fmem: Functional mixed effects modeling for the analysis of longitudinal white matter tract data. NeuroImage 84 753–764.
  • Zhao, H., Yuen, P. C. and Kwok, J. T. (2006). A novel incremental principal component analysis and its application for face recognition. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 36 873–886.
  • Zhu, H., Brown, P. J. and Morris, J. S. (2011). Robust, adaptive functional regression in functional mixed model framework. J. Amer. Statist. Assoc. 106 1167–1179.
  • Zipunnikov, V., Caffo, B., Yousem, D. M., Davatzikos, C., Schwartz, B. S. and Crainiceanu, C. (2011a). Multilevel functional principal component analysis for high-dimensional data. J. Comput. Graph. Statist. 20 852–873.
  • Zipunnikov, V., Caffo, B., Yousem, D. M., Davatzikos, C., Schwartz, B. S. and Crainiceanu, C. M. (2011b). Functional principal component models for high dimensional brain volumetrics. NeuroImage 58 772–784.
  • Zipunnikov, V., Greven, S., Shou, H., Caffo, B., Reich, D. S. and Crainiceanu, C. (2014). Supplement to “Longitudinal high-dimensional principal components analysis with application to diffusion tensor imaging of multiple sclerosis.” DOI:10.1214/14-AOAS748SUPP.

Supplemental materials

  • Supplementary material: Supplement to “Longitudinal high-dimensional principal components analysis with application to diffusion tensor imaging of multiple sclerosis”. We provide extra figures and tables summarizing the results of simulation studies and the analysis of DTI images of MS patients.