Spatio Temporal Tensor Analysis for Whole Brain fMRI

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spa b Spatial volume of voxels, a Spatial temporal structure of fMRI data c Time series of voxels. Figure 1 Example of fMRI brain images which are inherently coupled with sophisticated spatio temporal structure Voxels. are highly correlated with surrounding voxels in the spatial volume and their signals are often very noisy in the time series. classification However as shown in Fig 1 c the signal To deal with the above challenges in this paper. in each voxel of the brain volume changes along with we propose a Spatio Temporal Tensor Kernel STTK. time If the time series is averaged the varying trend framework for whole brain fMRI image analysis Specif. in the time series that reflects the brain activity will be ically we first perform time series extraction to reduce. lost Some studies 13 14 have focused on analyzing the noise and filter out the less informative time points. the fMRI time series of each individual voxel while in the original volumetric time series Then we utilize. ignoring structural information in the spatial domain the shifted CANDECOMP PARAFAC SCP 15 fac. For instance the multilinear decomposition model 13 torization for feature extraction of the spatio temporal. analyzes the time profile of the voxel vector converted data Finally spatio temporal structure mapping is per. from the 3D tensor in the spatial domain formed for kernel generation Empirical studies on real. Although leveraging the spatio temporal informa world resting state fMRI brain images demonstrate that. tion is desired in building a predictive kernel method it our proposed approach can significantly boost the fMRI. is very challenging due to the following three reasons classification performance on divergent disease diagno. Noisy fMRI time series analysis Due to hard sis i e Alzheimer s disease ADHD and HIV. ware reasons and subject factors e g thermal motion. of electrons there are often various nuisance compo 2 Preliminaries. nents and random noise in fMRI signals leading to a low In this section we define some necessary notions and no. signal to noise ratio SNR 11 Since fMRI data has tations related to tensors and then present the problem. low temporal resolution the signal of each voxel would formulation Before proceeding we introduce some ba. not discriminatively change within a session of several sic notations that will be used throughout this paper. time points limiting ability to identify brain events in Tensors i e multidimensional arrays are denoted by. time frame Furthermore time shifts delays which calligraphic letters A B C matrices by boldface. occur naturally during the fMRI image acquisition pro capital letters A B C vectors by boldface low. cess should be taken into account while analyzing the ercase letters a b c and scalars by lowercase. data How to filter the noise and extract discriminative letters a b c The columns of a matrix are de. information from the time series is critical in fMRI time noted by boldface lower letters with a subscript e g. series analysis ai is the ith column of matrix A The elements of a. Spatio temporal feature extraction Since matrix or a tensor are denoted by lowercase letters with. fMRI data reflect brain activity from the spatial domain subscripts i e the i1 in element of an n th order. and temporal domain a good feature extraction method tensor A is denoted by ai1 in Z is denoted by the. should be able to extract a compact and informative rep set of positive integers Additionally we will often use. resentation from both domains while considering their Gothic letters A B C to denote general sets or. correlations Note that the time shift factor discussed spaces regardless of their specific nature. previously should also be taken into consideration. Kernel modeling As discussed above the exist 2 1 Tensor Algebra. ing works do not differentiate the spatial domain and. temporal domain How to incorporate both the corre Definition 1 Tensor An nth order tensor is an. lation and the discrepancy between both domains into element of the tensor product of n vector spaces each. knowledge encoding is crucial for kernel modeling of which has its own coordinate system. Definition 2 Tensor product Given order n and V is loss function that indicates how differences be. and m tensors A RI1 In and B RI1 Im tween yi and f Xi should be penalized. their tensor product A B is a tensor of order n m The attractiveness of kernel methods lies in its. with the elements elegant treatment of nonlinear problems and its effi. ciency in high dimension Different kernel methods or. 2 1 A B i1 in i0 i0 ai1 in bi01 i0m kernel machines arise from using different loss func. tions In this paper we use the hinge loss function. Note that a rank one tensor of order n is the tensor max 0 1 y f X for support vector machine SVM. product of n vectors Clearly an important operation. applicable to our analysis is the tensor product also 3 Kernel Modeling. called the outer product The tensor product general. Two components of kernel methods need to be dis, izes from the Kronecker product but results in another. tinguished the kernel machine and the kernel func, tensor rather than a block matrix which naturally en. tion The kernel machine encapsulates the learning task. dows tensor with the structure of tensor product rep. which usually can be formulated as an optimization. resentations and tensor product spaces The space is. problem The kernel function encapsulates the hypoth. equipped with inner product and norm, esis language i e how to perform data transformation. Definition 3 Inner product The inner product and knowledge encoding By restricting to positive def. of two same sized tensors A B RI1 In is defined inite kernel functions the optimization problem will be. as the sum of the products of their elements convex and solution will be unique Throughout the. paper we take valid to mean positive definite, 2 2 hA Bi ai1 in bi1 in Definition 5 Positive Definite Kernel A.
i1 1 in 1 symmetric function X X R is a pos,itive definite kernel on X if for all n Z. Clearly for rank one tensors A a 1 a n and X1 Xn X and c1 cn R it follows. B b 1 b n it holds that P,that i j 1 n ci cj Xi Xj 0. 2 3 hA Bi ha 1 b 1 i ha n b n i A kernel function corresponds to the inner prod. uct in some feature space a Hilbert space which is. Qm For i brevity we denote x 1 x m by in general different from the representation space of the. i 1 x instances The computational attractiveness of kernel. methods comes from the fact that quite often a closed. Definition 4 Norm The norm of a tensor A is form of feature space inner products exists 17 In. defined to be the square root of the sum of all elements stead of mapping the data explicitly the kernel can be. of the tensor squared i e calculated directly According to Mercer s theorem 18. any valid kernel corresponds to an inner product in some. p uX X feature space and we can verify whether a kernel func. 2 4 kAk hA Ai t ai1 in ai1 in,tion is valid by the following Theorem 19. Theorem 1 A function defined on X X is a, 2 2 Problem Formulation In a typical fMRI clas positive definite kernel of H if and only if there exists. sification task we are given a collection of n training a feature mapping function X 7 H such that. examples Xi yi ni 1 X Y where Xi RI J K T,3 6 X Y h X Y i.
is the input fMRI sample with 3D space time tensor. form and yi is the class label of Xi The goal is to find for any X Y X X. a function f X Y that accurately predicts the label In particular an important property of positive definite. of an unseen example in X In the kernel learning sce kernels is that they are closed under sum multiplication. nario this problem can be formulated into the following by a scalar and product 20. optimization task By the representer theorem 21 the solutions of. C X 2 Eq 2 5 can be given by,2 5 f arg min V yi f Xi kf kH n. f H n i 1 X,3 7 f X ci Xi X, where C controls the trade off between the empirical i 1. risk and the regularization term kf kH H is a set of func where ci R are suitable coefficients and is a valid. tions forming a Hilbert space the hypothesis space kernel of H. 4 Spatio Temporal Tensor Kernel framework nique for single voxel time series extraction and noise. From the above discussions it is clear that a good kernel removal 14 Let pt xi j k pt t 1 Tp and. should be data dependent As noted in the introduction qt xi j k qt t 1 Tq be the maxima series and. fMRI data are inherently coupled with spatio temporal minima series of xi j k t where pt and qt are the time in. tensor structure involving time shift and have very low dexes Tp and Tq are the number of maxima and minma. temporal resolution and SNR To facilitate kernel learn repsectively Then for each voxel xi j k t we measure its. ing for fMRI data we propose a spatio temporal tensor energy by. kernel STTK framework that takes both the correla 1. tion and discrepancy between spatial and temporal do 4 8 E t xi j k t 1 if t qt. mains into account This framework consists of three. 0 otherwise, steps 1 volumetric time series extraction for extract. ing discriminative information from the time series 2 where the values of 1 and 1 mean importance and 0. spatio temporal feature extraction for obtaining a more means no importance. compact and informative representation and 3 tensor Volumetric Energy Measurement We measure. structure mapping for kernel generation the energy of each volume by summing up the energies of. all the voxels in it In particular we separately consider. 4 1 Volumetric Time Series Extraction In fMRI the maxima and minima voxels by. time series extraction a key issue is to determine the X. energy level for different time points Most of existing 4 9 Emax t X t max E t xi j k t 0. work focus on single voxel analysis 13 while they i j k. ignore the spatial correlations between voxels which X. 4 10 Emin t X t max E t xi j k t 0, may lead to suboptimal outcomes In this section we. develop a volumetric time series extraction approach. for fMRI time series In particular we show how Volumetric Time Series Extraction We ex. the volumetric spatial correlations and the temporal tract the time series from measured volumes based on. varying properties can contribute to the energy levels Emax and Emin Let ERt be the time series extrac. Given an fMRI example X RI J K T let tion rate defined by N T where N is the number of. xi j k t xi j k t 1 T be a T element time extracted time points Given an extraction rate ERt. series of voxel xi j k and X t is a volume of X at we first rank all the volumetric time points according to. time point t E t X t E t xi j k t is the energy Emax and Emin respectively Then we select the top k. function of time point t where E is separated by volume time points from each of the two ranked time point sets. and voxel for computational purposes and E t X t and concatenate them which forms the extracted time. Emin t X t Emax t X t correspond to the minima series where k equals to ERt T 2. and the maxima to be defined later As an illustration Fig 2 shows the time series of. The choice of energy function plays a critical role a voxel with different time series extraction techniques. in explaining how the knowledge transforms into mean From the original time series a we can see that it. ings and contexts The success of time series extraction is nontrivial to distinguish activation fluctuations from. strongly depends on the data knowledge encoded into the background noise if no time series extraction is per. the energy function Two important points must be formed Comparing with the time series b extracted. emphasized First in order to reduce the noise present using single voxel technique the time series c with. in the measurement new features should be used to the same amount of sampling time points as b ex. describe voxels rather than using the noisy voxel in tracted by our volumetric approach can better capture. tensities as features Second due to the low temporal the significant changes of signal over time For example. resolution each voxel signal would not experience a dis during the time interval 70 105 between red lines. criminative changing within a short measurement time the signals in c experience notable irregular changes. period It is necessary to make a discriminant analysis which can also be observed from a Comparatively. along time prior to the volume measurements Based b only captures the most distinct changes within this. on these two points we propose the following three step period For the period 0 70 the original series shows. procedure slightly fluctuated changes which can also be reflected. Voxel Energy Measurement We first ex by c while the time series b has much more changes. tract the maxima and minima extrema points for This is majorly because the single voxel technique. each voxel s time series using the extrema extraction chooses time points only based on the extrema of the. method 22 which is an effective and efficient tech single voxel time series In contrast our approach. a Matrix Factorization b CP Factorization c Shifted CP Factorization. Figure 3 CP factorization is a generalization of matrix factorization to tensors The SCP model allows shifts to occur over. the second mode such that for each index of the third mode the component of the second mode is shifted a given amount. ysis but it cannot well capture the structural informa. tion of a spatio temporal tensor Recently it was found. that shifted CP SCP factorization 13 is particularly. Spatio Temporal Tensor Analysis for Whole Brain fMRI Classi cation Guixiang Ma 1 Lifang He2 Chun Ta Lu Philip S Yu1 3 Linlin Shen2y and Ann B Ragin 4 1University of Illinois at Chicago Chicago IL USA fgma4 clu29g uic edu 2Shenzhen University Shenzhen China flifanghe llsheng szu edu cn 3Tsinghua University Beijing China psyu uic edu 4Northwestern University Chicago IL USA

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