The Annals of Applied Statistics

Regularized 3D functional regression for brain image data via Haar wavelets

Xuejing Wang, Bin Nan, Ji Zhu, and Robert Koeppe

Full-text: Open access

Abstract

The primary motivation and application in this article come from brain imaging studies on cognitive impairment in elderly subjects with brain disorders. We propose a regularized Haar wavelet-based approach for the analysis of three-dimensional brain image data in the framework of functional data analysis, which automatically takes into account the spatial information among neighboring voxels. We conduct extensive simulation studies to evaluate the prediction performance of the proposed approach and its ability to identify related regions to the outcome of interest, with the underlying assumption that only few relatively small subregions are truly predictive of the outcome of interest. We then apply the proposed approach to searching for brain subregions that are associated with cognition using PET images of patients with Alzheimer’s disease, patients with mild cognitive impairment and normal controls.

Article information

Source
Ann. Appl. Stat., Volume 8, Number 2 (2014), 1045-1064.

Dates
First available in Project Euclid: 1 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1404229525

Digital Object Identifier
doi:10.1214/14-AOAS736

Mathematical Reviews number (MathSciNet)
MR3262545

Zentralblatt MATH identifier
06333787

Keywords
Alzheimer’s disease brain imaging functional data analysis Haar wavelet Lasso PET image variable selection

Citation

Wang, Xuejing; Nan, Bin; Zhu, Ji; Koeppe, Robert. Regularized 3D functional regression for brain image data via Haar wavelets. Ann. Appl. Stat. 8 (2014), no. 2, 1045--1064. doi:10.1214/14-AOAS736. https://projecteuclid.org/euclid.aoas/1404229525


Export citation

References

  • Bickel, P. J., Ritov, Y. and Tsybakov, A. B. (2009). Simultaneous analysis of lasso and Dantzig selector. Ann. Statist. 37 1705–1732.
  • Bowman, F. D., Caffo, B., Bassett, S. S. and Kilts, C. (2008). A Bayesian hierarchical framework for spatial modeling of fMRI data. NeuroImage 39 146–156.
  • Candes, E. and Tao, T. (2007). The Dantzig selector: Statistical estimation when $p$ is much larger than $n$. Ann. Statist. 35 2313–2351.
  • Cockrell, J. R. and Folstein, M. F. (1988). Mini-Mental State Examination (MMSE). Psychopharmacol. Bull. 24 689–692.
  • Daubechies, I., Defrise, M. and De Mol, C. (2004). An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm. Pure Appl. Math. 57 1413–1457.
  • Fan, J., Guo, S. and Hao, N. (2012). Variance estimation using refitted cross-validation in ultrahigh-dimensional regression. J. R. Stat. Soc. Ser. B Stat. Methodol. 74 37–65.
  • Foster, N. L., Chase, T. N., Mansi, L., Brooks, R., Fedio, P., Patronas, N. J. and Chiro, G. D. (1984). Cortical abnormalities in Alzheimer’s disease. Ann. Neurol. 16 649–654.
  • Friedman, J., Hastie, T., Höfling, H. and Tibshirani, R. (2007). Pathwise coordinate optimization. Ann. Appl. Stat. 1 302–332.
  • Friston, K. J., Poline, J. B., Holmes, A. P., Frith, C. and Frackowiak, R. S. J. (1996). A multivariate analysis of PET activation studies. Hum. Brain Mapp. 4 140–151.
  • Fu, W. J. (1998). Penalized regressions: The bridge versus the lasso. J. Comput. Graph. Statist. 7 397–416.
  • Grimmer, T., Henriksen, G., Wester, H.-J., Förstl, H., Klunk, W. E., Mathis, C. A., Kurz, A. and Drzezga, A. (2009). Clinical severity of Alzheimer’s disease is associated with PIB uptake in PET. Neurobiol. Aging 30 1902–1909.
  • James, G. M., Wang, J. and Zhu, J. (2009). Functional linear regression that’s interpretable. Ann. Statist. 37 2083–2108.
  • Kang, J., Johnson, T. D., Nichols, T. E. and Wager, T. D. (2011). Meta analysis of functional neuroimaging data via Bayesian spatial point processes. J. Amer. Statist. Assoc. 106 124–134.
  • Kerrouche, N., Herholz, K., Mielke, R., Holthoff, V. and Baron, J.-C. (2006). 18FDG PET in vascular dementia: Differentiation from Alzheimer’s disease using voxel-based multivariate analysis. J. Cereb. Blood Flow Metab. 26 1213–1221.
  • Leifer, B. P. (2003). Early diagnosis of Alzheimer’s disease: Clinical and economic benefits. J. Am. Geriatr. Soc. 51 S281–S288.
  • Lovejoy, S. and Schertzer, D. (2012). Haar wavelets, fluctuations and structure functions: Convenient choices for geophysics. Nonlinear Process. Geophys. 19 513–527.
  • Luo, W.-L. and Nichols, T. E. (2003). Diagnosis and exploration of massively univariate neuroimaging models. NeuroImage 19 1014–1032.
  • Meinshausen, N. and Bühlmann, P. (2010). Stability selection. J. R. Stat. Soc. Ser. B Stat. Methodol. 72 417–473.
  • Minoshima, S., Frey, K. A., Koeppe, R. A., Foster, N. L. and Kuhl, D. E. (1995). A diagnostic approach in Alzhimer’s disease using three-dimensional stereotactic surface projections of fluORine-18-FDG PET. J. Nucl. Med. 36 1238–1248.
  • Minoshima, S., Giordani, B., Berent, S., Frey, K. A., Foster, N. L. and Kuhl, D. E. (1997). Metabolic reduction in the posterior cingulate cortex in very early Alzhimer’s disease. Ann. Neurol. 42 85–94.
  • Mueller, S. G., Weiner, M. W., Thal, L. J., Petersen, R. C., Jack, C. R., Jaqust, W., Trojanowski, J. Q., Toga, A. W. and Beckett, L. (2005). Ways toward an early diagnosis in Alzheimer’s disease: The Alzheimer’s Disease Neuroimaging Initiative (ADNI). Alzheimer’s and Dementia: The Journal of the Alzheimer’s Association 1 55–66.
  • Muraki, S. (1992). Approximation and rendering of volume data using wavelet transforms. In Proceedings of Visualization 1992 21–28. IEEE.
  • Nichols, T. E. and Holmes, A. P. (2001). Nonparametric permutation tests for functional neuroimaging: A primer with examples. Hum. Brain Mapp. 15 1–25.
  • Picart, S. S., Butenschön, M. and Shutler, J. D. (2012). Wavelet-based spatial comparison technique for analysing and evaluating two-dimensional geophysical model fields. Geosci. Model Dev. 5 223–230.
  • Plassman, B. L., Langa, K. M., Fisher, G. G., Heeringa, S. G., Weir, D. R., Ofstedal, M. B., Burke, J. R., Hurd, M. D., Potter, G. G., Rodgers, W. L., Steffens, D. C., Willis, R. J. and Wallace, R. B. (2007). Prevalence of dementia in the United States: The aging, demographics, and memory study. Neuroepidemiology 29 125–132.
  • Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis, 2nd ed. Springer, New York.
  • Reiss, P. T. and Ogden, R. T. (2010). Functional generalized linear models with images as predictors. Biometrics 66 61–69.
  • Royston, P. and Sauerbrei, W. (2008). Multivariable Model-Building: A Pragmatic Approach to Regression Analysis Based on Fractional Polynomials for Modelling Continuous Variables, 1st ed. Wiley, Chichester.
  • Sauerbrei, W. and Schumacher, M. (1992). A bootstrap resampling procedure for model building: Application to the Cox regression model. Stat. Med. 11 2093–2109.
  • Shin, J., Lee, S. Y., Kim, S. J., Kim, S. H., Cho, S. J. and Kim, Y. B. (2010). Voxel-based analysis of Alzheimer’s disease PET imaging using a triplet of radiotracers: PIB, FDDNP, and FDG. NeuroImage 52 488–496.
  • Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B 58 267–288.
  • Walker, J. S. (2008). A Primer on Wavelets and Their Scientific Applications, 2nd ed. Chapman & Hall/CRC, Boca Raton, FL.
  • Wand, M. P. and Ormerod, J. T. (2011). Penalized wavelets: Embedding wavelets into semiparametric regression. Electron. J. Stat. 5 1654–1717.
  • Wang, X., Nan, B., Zhu, J., Koeppe, R. and for the Alzheimer’s Disease Neuroimaging Initiative (2014). Supplement to “Regularized 3D functional regression for brain image data via Haar wavelets.” DOI:10.1214/14-AOAS736SUPP.
  • Wu, T. T. and Lange, K. (2008). Coordinate descent algorithms for lasso penalized regression. Ann. Appl. Stat. 2 224–244.
  • Zhao, Y., Todd Ogden, R. and Reiss, P. T. (2012). Wavelet-based LASSO in functional linear regression. J. Comput. Graph. Statist. 21 600–617.
  • Zou, H. (2006). The adaptive lasso and its oracle properties. J. Amer. Statist. Assoc. 101 1418–1429.
  • Zou, H., Hastie, T. and Tibshirani, R. (2007). On the “degrees of freedom” of the lasso. Ann. Statist. 35 2173–2192.

Supplemental materials

  • Supplementary material: Appendix. The online supplementary material contains the technical appendix showing the theoretical results of the proposed approach and an illustrative example showing the desirable feature of Haar wavelets.