Functional Data Analysis of Big Neuroimaging Data


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1 Functional Data Analysis of Big Neuroimaging Data Hongtu Zhu, Ph.D Department of Biostatistics and Biomedical Research Imaging Center The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA Acknowledgement: Some pictures were copied from multiple resources including Suetens (2009), Fass (2008), Dr. Niethammer, Drs. Lindquist, Rowe Huettel, Wiki, google, etc.
2 Big Neuroimaging Data Is it really big?
3 aims to simulate the complete human brain on Supercomputers to better understand how it functions. The Brain Research through Advancing Innovative Neurotechnologies or BRAIN, aims to reconstruct the activity of every single neuron as they fire simultaneously in different brain circuits, or perhaps even whole brains.
4 Big Neuroimaging Data NIH normal brain development 1000 Functional Connectome Project Alzheimer s Disease Neuroimaging Initiative National Database for Autism Research (NDAR) Human Connectome Project
5 Complex Study Design crosssectional studies; clustered studies including longitudinal and twin/familial studies; ρ 1 5 ρ 5 9 ρ 1 9
6 Complex Data Structure Multivariate Imaging Measures Smooth Functional Imaging Measures Wholebrain Imaging Measures 4DTime Series Imaging Measures
7 Big Data Integration E IS I II IG IS SI G D Selection
8 Models for Big Data Integration ImageonScalar (IS) model Image data as response, clinical variables as predictors. ScalaronImage (SI) model Clinical variables as response, image data as predictors ImageonGenetic (IG) model Image data as response, genetic data as predictors ImageonImage (II) model Image data as response, image data as predictors
9 Noisy Imaging Data Spatial Maps Estimation `Registration Inference `Smoothing Prediction `Correlation `Spatial Heterogeneity
10 ImagingonScalar Regression
11 VBA versus FDA Data {(x i,y i ) :i =1,!, n} Y i = {Y i (d 0 ) : d 0! D 0 } Intrinsic Discrete Approach (VBA) Y i = {Y i (d 0 ) : d 0! D 0 } Intrinsic Functional Approach (FDA) Y i ( ) = {Y i (d) : d! D}
12 n Functional Data Analysis (FDA) y i (d) = x i T B (d)+! i (d)! i ( ) ~ SP(0,!! ) Hotellingtype Test Statistics Y =! Y i / n S Y = (Y i!y )! i=1 Big data Pro: Incorporate spatial smoothness and spatial correlation T n 2 Con: Computational and theoretical difficulties n! (Y i!y ) / n T 2 < Y,u > 2 n = nsup u =1 i=1 < u,s Y u > P(T n 2 =!) =1 S Y!!(S Y ) S Y! diag(s Y )
13 Highdimensional Regression Models Big data Y i = BX i +! i! i ( ) ~ SP(0,!! ) dim(y i ) >> n Hotellingtype Test Statistics T n 2 n n Y =! Y i / n S Y =!(Y i!y )(Y i!y ) T / n T 2 < Y,u > 2 n = nsup u =1 < u,s Y u > i=1 i=1 P(T n 2 =!) =1 (Y,S Y )!!(Y,S Y )???
14 Voxel Based Analysis (VBA) Data {(xi,yi ) : i = 1,!, n} Yi = {Yi (d0 ) : d0! D0 } VBA Stage 0: Gaussian Kernel Smoothing Stage 1: Model Fitting n n! p(yi xi ) =! i=1 i=1! p(y (d ) x,! (d )) i 0 i 0 d"d0 Ignore spatial smoothness Stage 2: Hypothesis Testing H 0 :! (d) =!* (d) for all voxels H1 :! (d)!!* (d) for some voxels Random Field Theory: functional data and local smoothness FDR
15 Cons VBA Potential large smoothing errors. Treat voxels as independent units/images as a collection of independent voxels. Ignore spatial correlation and smoothness in statistical analysis. Inaccurate for both Prediction and Estimation. Decrease statistical power.
16 VBA Bayesian Extensions Bayesian Modeling Spatial smooth prior (MRF) p(!) = p({!(d 0 ) : d 0! D 0 }) n p(! Y )!{" p(y i x i,!)}p(!) = {" " p(y i (d) x i,!(d))}p(!) i=1 i=1 Pro: Computationally straightforward; Bayesian inference based on MCMC samples n d#d 0 Con: Computationally heavy; Lack of understanding for Bayesian inference tools.
17 Varying Coefficient Models Reading materials: 1. Zhu, H. T., Chen, K. H., Yuan, Y. and Wang, J. L. (2014). Functional Mixed Processes Models for Repeated Functional Data. In submission. 2. Liang, J. L., Huang, C., and Zhu, H.T. (2014). Functional singleindex varying coefficient models. In submission. 3. Yuan, Y., Gilmore, J., Geng, X. J., Styner, M., Chen, K. H., Wang, J. L., and Zhu, H.T. (2013). A longitudinal functional analysis framework for analysis of white matter tract statistics. NeuroImage, in press. 4. Yuan, Y., Zhu, H.T., Styner, M., J. H. Gilmore., and Marron, J. S. (2013). Varying coefficient model for modeling diffusion tensors along white matter bundles. Annals of Applied Statistics. 7(1): Zhu, H.T., Li, R. Z., Kong, L.L. (2012). Multivariate varying coefficient models for functional responses. Ann. Stat. 40, Hua, Z.W., Dunson, D., Gilmore, J.H., Styner, M., and Zhu, HT. (2012). Semiparametric Bayesian local functional models for diffusion tensor tract statistics. NeuroImage, 63, Zhu, HT., Kong, L., Li, R., Styner, M., Gerig, G., Lin, W. and Gilmore, J. H. (2011). FADTTS: Functional Analysis of Diffusion Tensor Tract Statistics, NeuroImage, 56, Zhu, H.T., Styner, M., Tang, N.S., Liu, Z.X., Lin, W.L., Gilmore, J.H. (2010). FRATS: functional regression analysis of DTI tract statistics. IEEE Transactions on Medical Imaging, 29, Greven, S., Crainiceanu, C., Caffo, B., Reich, D. (2010). Longitudinal principal component analysis. E.J.Statist. 4, Goodlett, C.B., Fletcher, P. T., Gilmore, J. H., Gerig, G. (2009). Group analysis of dti fiber tract statistics with application to neurodevelopement. NeuroImage, 45, S133S Yushkevich, P. A., Zhang, H., Simon, T., Gee, J. C. (2008). Structurespecific statistical mapping of white matter tracts. NeuroImage, 41, Ramsay, J. O., Silverman, B. W. (2005). Functional Data Analysis, SpringerVerlag, New York.
18 Smoothed Functional Data Covariates (e.g., age, gender, diagnostic)
19 DTI Fiber Tract Data FA Data Diffusion properties (e.g., FA, RA)!! MD Y i (s j ) = (y i,1 (s j ),!, y i,m (s j )) T {s 1,!,s ng } Grids (e) Covariates (e.g., age, gender, diagnostic) x 1,!, x n " 1 " 2 " 3!!!!!!
20 Decomposition: MVCM y i,k (s) = x i T B k (s) + " i,k (s) + # i,k (s) Coefficients Longrange Correlation Shortrange Correlation x 1,!,x n " i,k ( ) ~ SP(0,# " )! i,k ( ) ~ SP(0,!! ), Covariance operator: Zhu, Li, and Kong (2012). AOS! y (s, s') =!! (s, s')+!! (s, s') n{vec( ˆB(d)! B(d)! 0.5O(H 2 L )) : d! D}! "! G(0,! B (d, d '))!
21 Motivation Diffusion Tensor Tract Statistics FA Tensor 2 week 2 week 1 year 1 year 2 year 2 year
22 Longitudinal Data Longitudinal Extensions Spatialtemporal Process t y i (s,t 3 ) y i (s,t 2 ) y i (s,t 1 ) Functional Mixed Effect Models s y i (s,t) = x i (t) T B (s)+ z i (t) T! i (s)+" i (s,t)+# i (s,t) Objectives: Dynamic functional effects of covariates of interest on functional response.
23 Decomposition: FMEM y i (d,t) = x i (t) T B (d)+ z i (t) T! i (d)+! i (d,t)+! i (d,t) Global Noise Components Local Correlated Noise! i (, ) ~ SP(0,!! ), " i ( ) ~ SP(0,!! )! i ( ) ~ SP(0,!! ), n{vec( ˆB(d)! B(d)! 0.5O(H 2 )) : d! D} L! "! G(0,! B (d, d ')) Ying et al. (2014). NeuroImage. Zhu, Chen, Yuan, and Wang (2014). Arxiv.
24 Real Data genu DTImaging parameters: TR/TE = 5200/73 ms Slice thickness = 2mm Inplane resolution = 2x2 mm^2 b = 1000 s/mm^2 One reference scan b = 0 s/mm^2 Repeated 5 times when 6 gradient directions applied.
25 Real Data
26 Real Data Analysis Results
27 Prediction y i,k (s) = f (x i, B k (s)+! i,k (s)) Longrange Correlation +" i,k (s) Smallrange Correlation Missing Big Data??? Hyun,J.W., Li, Y. M., J. H. Gilmore, Z. Lu, M. Styner, H. Zhu (2014) NeuroImage
28 Real Data Prediction Accuracy is much improved
29 Multiscale Adaptive Regression Models Reading materials: 1. Zhu, HT., Fan, J., and Kong, L. (2014). Spatial varying coefficient model and its applications in neuroimaging data with jump discontinuity. JASA, in press. 2. Li, YM, John Gilmore, JA Lin, Shen DG, Martin, S., Weili Lin, and Zhu, HT. (2013). Multiscale adaptive generalized estimating equations for longitudinal neuroimaging data. NeuroImage, 72, Li, YM, John Gilmore, JP Wang, M. Styner, Weili Lin, and Zhu, HT. (2012). Twostage spatial adaptive analysis of twin neuroimaging data. IEEE Transactions on Medical Imaging. 31, Skup, M., Zhu, H.T., and Zhang HP. (2012). Multiscale adaptive marginal analysis of longitudinal neuroimaging data with timevarying covariates. Biometrics, 68(4): Shi, XY, Ibrahim JG, Styner M., Yimei Li, and Zhu, HT. (2011). Twostage adjusted exponential tilted empirical likelihood for neuroimaging data. Annals of Applied Statistics, 5, Li, YM, Zhu HT, Shen DG, Lin WL, Gilmore J, and Ibrahim JG. (2011). Multiscale adaptive regression models for neuroimaging data. JRSS, Series B, 73, Polzehl, Jörg; Voss, Henning U.; Tabelow, Karsten. Structural adaptive segmentation for statistical parametric mapping. NeuroImage, 52 (2010) pp Polzehl, J. and Spokoiny, V. G. (2006). Propagationseparation approach for local likelihood estimation. Probability Theory and Related Fields, 135, J. Polzehl, V. Spokoiny, (2000) Adaptive Weights Smoothing with applications to image restoration, J. R. Stat. Soc. Ser. B Stat. Methodol., 62 pp
30 Piecewise Smooth Data Mathematics. Noisy Piecewise Smooth Functions with Unknown Jumps and Edges Image is the point or set of points in the range corresponding to a designated point in the domain of a given function. " is a compact set. f ( x ) " M # R m x "# $ R k f : " # M $ R m!!! " f (!x) k d!x < # for some k>0!
31 Neuroimaging Data with Discontinuity Noisy Piecewise Smooth Function with Unknown Jumps and Edges Subject1 Subject2 Covariates (e.g., age, gender, diagnostic, stimulus)
32 Decomposition: SVCM y i (d) = f (x i,b(d) +! i (d)) +! i (d),d! D Piecewise Smooth Shortrange Correlation Varying Coefficients Longrange Correlation B(d)! L K Covariance operator: η ij ()~ SP(0, Ση )! y (d,d ') =!! (d,d ') +!! (d,d)! ij ( ) ~ SP(0,!! ), 3D volume/ 2D surface
33 SVCM Cartoon Model Disjoint Partition B k (d) L D = l= 1 Dl and Dl Dl ' = φ Piecewise Smoothness: Lipschitz condition Smoothed Boundary Local Patch Degree of Jumps
34 Challenging Issues Smoothing coefficient images, while preserving unknown boundaries Different patterns in different coefficient images Calculating standard deviation images Asymptotic theory
35 Smoothing Methods PropogationSeperation Method J. Polzehl and V. Spokoiny, (2000,2005) Features Increasing Bandwidth 0 < h < h < < h = S r 0 1 Adaptive Weights 0 Adaptive Estimates
36 Smoothing Methods MARM At each voxel d Increasing Bandwidth Adaptive Weights Adaptive Estimates 0 < h < h < < h r S = 0 ˆµ(d;h 0 ) 1!(d, d ';h 1 )!(d, d ';h 2 ) ˆµ(d;h 1 ) 0!(d, d ';h 2 )!(d, d ';h s ) = K loc ( d! d ' /h s )K st (D µ (d, d ';h s!1 ) / C n ) D µ (d, d ';h s!1 ) = "( ˆµ(d;h s!1 ), ˆµ(d ';h s!1 )) Stopping Rule ˆµ(d;h S )
37 Simulation
38 Simulation
39 Simulation
40 ScalaronImaging Regression
41 ScalaronImage Models Reading materials: 1. Miranda, M. F., Zhu, H.T. and Ibrahim, J. G. (2014). Bayesian partial supervised tensor decomposition with applications in Neuorimaging data analysis. 2. Yang, H., Zhu, H. T., and Ibrahim, J. G. (2014). Multiscale projection model in RKHS. 3. Zhu, H. T. and Shen, D. (2014). Multiscale Weighted PCA for Imaging Prediction. 4. Guo, R.X., Ahye M., and Zhu, H. (2014). Spatially weighted PCA for imaging classification. JCGS. In revision. 5. Zhou, H., Li, L., and Zhu, H. (2013). Tensor regression with applications in Neuorimaging data analysis. JASA. In press. 6. Cuingnet, R., Glaunes, J. A., Chupin, M., Benali, H., Colliot, O., and ADNI. (2012). Spatial and anatomical regularization of SVM: a general framework for neuroimaging data. IEEE PAMI. In press. 7. Cai, T. & Yuan, M. (2012). Minimax and adaptive prediction for functional linear regression. JASA. 107, Fan, J. and Lv, J. (2010). A selective overview of variable selection in highdimensional feature space. Statistica Sinica, 20, Li, B., Kim, M. K., and Altman, N. (2010). On dimension folding of matrix or array valued statistical objects. Ann. Stat., 38, Reiss, P. T., and Ogden, R. T. (2007). Functional principal component regression and functional partial least squares. Journal of the American Statistical Association 102, Bair, E., Hastie, T., Paul, D. and Tibshirani, R. (2006). Prediction by supervised principal components. JASA. 101, Ramsay, J.O. and Silverman, B.W. (2005). Functional Data Analysis, 2nd Edition. Springer, New York. 13.James, G. (2002). Generalized linear models with functional predictor variables. JRSSB. 64, Tibshirani, Robert (1996). Regression shrinkage and selection via the lasso. JRSSB. 58,
42 HRM versus FRM Data {(y i, X i ) :i =1,!, n} X i = {X i (d) : d! D} y i =< X i,! > +" i Strategy 1: Discrete Approach (Highdimension Regression Model (HRM)) Strategy 2: Functional Regression Model (FRM) y i =! 0 +!!(d) X i (d)m(d)+" i D
43 Highdimension Regression Model Approach 1: Regularization Methods Key Conditions: Sparsity of S Restricted nullspace property for design matrix X
44 Highdimension Regression Model Tensor Structure: Ultrahigh dimensionality (256^3) Spatial structure Zhou, Li, and Zhu (2013) Li, Zhou, and Li (2013) CP decomposition Tucker decomposition!
45 ScalaronImage Models Simulations Key Conditions: Tensor Approximation B Restricted space property for X and B
46 ScalaronImage Models ADHD 200
47 ScalaronImage Models Strategy 2: Functional Approach y i =! 0 +!!!(d) X i (d)m(d)+" i D y i =! 0 + "! # k " k (d) X i (d)m(d)+# i k=1 D!!(d) = "! k " k (d) Basis Methods: fixed and datadriven basis functions! K! = {!(d) = "! k " k (d) : (! 1,!) # " 2! } C(d, d ') = Cov(X(d), X(d ')) =! k " k (d)" k (d ') k=1 k=1 " k=1
48 Key Conditions Key Conditions: an excellent set of basis functions Sparsity of {! k : k =1,!} Decay rate of spectral of C(d, d ') = Cov(X(d), X(d '))!(d)! K "! k " k (d) k=1 K << n
49 The ADNI data Mild Cognitive Impairment subjects Interested in predicting the timing of an MCI patient that converts to AD by integrating the imaging data, the clinical variables, and genetic covariates. Full Model: AUC=0.96 Partial Model: AUC=0.82
50 Limitations Is this the right space for statistical inference? {X i (d) : d! D} {!! k (d) X i (d)m(d) : d " D} D
51 Rabbit and Wolf Story {y i :i =1,!, n} y i = f 0 (X i )+! i X i = {X i (d) : d! D} y i = f (G i )+! i G i = G{!X i (d) : d! D 0 }!X i = {! X i (d) : d! D 0 }
52 Feature Space Determination X i = {X i (d) : d! D}!X i = {!X i (d) : d! D 0 } G i = G{! X i (d) : d! D 0 } Splitting/Weighting methods Level sets Factor Models y i =! 0 + "!!(d) X i (d)m(d) +" i y i =! 0 +!!(d) w(d)x i (d)m(d)+" i k Splitting D k D Weighting
53 Spatially Weighted PCA
54 Spatially Weighted PCA
55 Multiscale Factor Prediction Model
56 Multiscale Factor Prediction Model
57 Multiscale Factor Prediction Model
58 ASA: Statistics in Imaging Section SAMSI 2013 Neuroimaging Data Analysis Challenges in Computational Neuroscience
59 Acknowledgement
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