A Sparse Bayesian Method for Determination of Flexible ... - IEEE Xplore

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 52, NO. 12, DECEMBER 2005

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A Sparse Bayesian Method for Determination of Flexible Design Matrix for fMRI Data Analysis Huaien Luo and Sadasivan Puthusserypady, Senior Member, IEEE

Abstract—The construction of a design matrix is critical to the accurate detection of activation regions of the brain in functional magnetic resonance imaging (fMRI). The design matrix should be flexible to capture the unknown slowly varying drifts as well as robust enough to avoid overfitting. In this paper, a sparse Bayesian learning method is proposed to determine a suitable design matrix for fMRI data analysis. Based on a generalized linear model, this learning method lets the data itself determine the form of the regressors in the design matrix. It automatically finds those regressors that are relevant to the generation of the fMRI data and discards the others that are irrelevant. The proposed approach integrates the advantages of currently employed methods of fMRI data analysis (the model-driven and the data-driven methods). Results from the simulation studies clearly reveal the superiority of the proposed scheme to the conventional -test method of fMRI data analysis. Index Terms—Design matrix, functional magnetic resonance imaging (fMRI), generalized linear model, receiver operating characteristic (ROC) curve, sparse Bayesian learning.

I. INTRODUCTION

F

UNCTIONAL magnetic resonance imaging (fMRI) is a neuroimaging technique with the power to map neural activities of the brain with high spatial resolution. An fMRI system is basically an advanced MRI system that is programmed to detect a functional signal rather than a structural signal. Since different tissues have different nuclear magnetic resonance (NMR) relaxation time constants, images could be formed based on these variations of the time constants. fMRI measures the blood oxygenation level dependent (BOLD) signal in the brain. When neural activity occurs in the brain, both the cerebral blood flow (CBF) and the cerebral metabolic rate of oxygen (CMRO ) increase. However, the increase of CMRO is much less than the increase of CBF. This imbalance results in a drop in oxygen extraction as compared to oxygen supply and a corresponding decrease of deoxyhemoglobin content in the blood. The deoxyhemoglobin is paramagnetic, which interacts with and distorts the applied magnetic field and changes the time constants as well [1]. Thus, for the activated regions of the brain, the MR signal increases due to a longer time constant: this is known as the BOLD effect [2]. The MRI scanner detects this signal changes and produces a sequence of three-dimensional (3-D) images, covering the whole brain repeatedly with a repetition time (TR). Manuscript received January 27, 2005; revised July 30, 2005. This paper was recommended by Guest Editor Y. Lian. The authors are with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576 (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TCSI.2005.857083

Though the BOLD effect is the basis of fMRI [3], the exact mechanism of the BOLD effect is still unknown. Furthermore, the small signal change due to BOLD effect is affected by noises and drifts [4]. Drifts and trends are slowly varying interferences which may be due to the instability of the magnetic field, physiological processes such as respiration, cardiac process and so on. The noises may come from genuine random physiological noise and scanner noise. The analysis methods could also induce a residual noise due to an imperfect model. Recently, researchers discovered that the dynamics underlying the brain activity are believed to be nonlinear [5]. These complexities of the BOLD response as well as the noisy and nonlinear nature of the fMRI signals pose the biggest challenges to its analysis. Many methods have been proposed to analyze the fMRI data. Generally, these methods stem from either the model-driven or data-driven ideas. The model-driven methods fit the fMRI data to an assumed model and through the analysis of the parameters of the model, a conclusion could be drawn. These methods include the well known general linear model (GLM) implemented in the statistical parametric mapping (SPM) [6]. The data-driven methods, on the contrary, instead of fitting the data to an assumed model, let the fMRI data explain themselves. These methods include blind source separation (BSS) and clustering. The BSS method aims at finding an unmixing matrix to decompose the fMRI data into different sources, among which some are important to evaluate the activation of the brain. These methods include principal component analysis (PCA) [7] and independent component analysis (ICA) [8]. The clustering method [9] tries to classify thousands of the voxels (volume element—the smallest box-shaped element of a 3-D image) into a few groups whose prototype profiles are predominantly different from each other. The model-driven and data-driven methods have their own advantages. The data-driven methods are flexible especially when an appropriate data generation model is not available. However, the necessity to explore the whole data sets leads to high computational demand and difficulties in interpreting the results. The model-driven methods on the other hand make an assumption of the underlying model and take into account the information of the experimental paradigm; they often require less computation and lead to an easy interpretation of the results. Due to the complexity of the fMRI data, these methods may impose an improper model on the data and thus result in misinterpretation. Especially, the BOLD response and the interferences may vary for different subjects and different regions of the brain. The complexity of the fMRI data requires that the assumed method be flexible enough to accommodate

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the variations of the signal and adaptable to different noises and interferences. Recent work has suggested that the combination of data-driven and model-driven methods could give better results for fMRI data analysis. In [10], the author proposed a hybrid approach which uses the results of ICA to form the regressors in the GLM. In [11], the ICA with reference (ICA-R) which uses the information of the input stimuli as reference was proposed. In [12], a semi-blind ICA of fMRI to incorporate prior information about paradigm time courses was introduced. These methods are based on the powerful data-driven techniques and use some information of the experimental paradigm to guide the analysis. These methods help to understand the fMRI data analysis problem from data-driven methods toward model-driven methods. In this paper, a mixed model which starts with model-driven methods and utilizes the data-driven methods to guide the selection of the regressors is proposed. This method assumes a generalized linear model. However, instead of specifying the whole design matrix before analysis (as in the General Linear Model method), only the BOLD response regressor in the design matrix is specified and the data automatically determines the remaining regressors through sparse Bayesian learning. Furthermore, some evidence about the existence of the regressors in the design matrix is obtained through the learning procedure and the irrelevant regressors are discarded to avoid overfitting. This new method integrates the advantages of the data-driven and model-driven methods and gives full flexibility to let the data itself determine its regressors in the design matrix. II. METHODOLOGY The aim of fMRI data analysis is to determine the activation regions of the brain (i.e., to determine which voxel embodies the properties of activation). Normally, it is the temporal properties that determine whether a voxel is activated or not. Hence, the fMRI data is processed voxel by voxel and the time series at each voxel is investigated [13]. A temporal model is imposed to this time series. The parameters of the model are then tested to draw a conclusion on the activation of the voxel. A. General Linear Model The GLM is the core of the SPM [6] method. It is the most fundamental method to analyze fMRI data sets. This method assumes a general linear model to the time series of each voxel. Let denote the vector of observed/measured intensity changes in one voxel of the fMRI dataset, with dimen, where is the total length of the time series or sion indexes of observations (e.g., scans). GLM is then given by (1) in which is a vector of parameters with dimension which could be found by the least square methods. is the vector of error terms which is assumed to be normal , is the autocorrelation matrix of the time series. The where has dimension with each row corredesign matrix sponding to one time point (scan) of the regressors, and the

columns corresponding to the different explanatory variables or regressors in the model. As shown in (1), GLM assumes that the measured time series at each voxel is the weighted linear combination of different regressors in the design matrix. in (1) is not white ( , where Normally, the error is the identity matrix) because of the temporal autocorrelation in the noise of fMRI data [14], [15]. Many methods such as coloring, prewhitening and so on were proposed to deal with this issue [16], [17]. In our work, prewhitening is performed to make the error to be white. This needs a square filtering matrix to filter both sides of (1) as follows: (2) where the error after prewhitening is and hence white. In the following section, to simplify our discussion, we assume the errors to be white. For temporally autocorrelated errors, once prewhitening is performed, the proposed method of determination of flexible design matrix is still valid. In GLM, the design matrix is specified before analysis and will not change during the analysis procedure. The regressors (or the column vectors of the design matrix) often consist of a canonical BOLD response (and its derivatives), a vector of constant value 1 representing mean value and several discrete cosine transform (DCT) basis functions representing the highpass filter to remove the unwanted low-frequency components from the data [18]. The design matrix in GLM is not flexible and may cause problems. The number of the DCT waveforms to be included in the design matrix should be determined carefully before analysis. In SPM, this is implemented by specifying the highpass cutoff. Generally, too many basis functions would lead to an overfitting problem, while too few basis functions may not filter out the slowly varying interference efficiently. So, the selection of the number of basis functions is a very tricky issue. Furthermore, the inclusion of the canonical BOLD response regressors into the design matrix may result in a deviated model if the voxel is not activated. For accurate detection, an efficient method is required, which could switch the BOLD response regressor ON and OFF according to the measured data during the data learning procedure. In the method proposed, the initial design matrix includes the BOLD response regressor, the vector of constant value 1 and a set of general nonlinear functions (instead of DCT basis fucntions) that account for the slowly varying drifts and trends. Compared to the design matrix used in the GLM, the proposed model is much more flexible due to the use of general nonlinear functions which in fact construct a flexible subspace and could capture signal variations more efficiently. Besides, different from GLM in which the design matrix will not change during the whole analysis, the proposed method could adapt/adjust the design matrix according to the data. This is implemented under the Bayesian framework. By using the sparse Bayesian learning, the regressors could be learned from the data automatically to best account for the observed signal. The unwanted regressors are then removed from the design matrix to avoid the overfitting problem.

LUO AND PUTHUSSERYPADY: SPARSE BAYESIAN METHOD FOR DETERMINATION OF FLEXIBLE DESIGN MATRIX

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B. Sparse Bayesian Learning is as

Suppose the initial number of regressors in the design matrix and the design matrix is , which is defined

(3) where the first regressors in the design matrix are flexible radial basis functions which is further defined as Gaussian

Fig. 1. Illustration of a block design and its square waveform representation.

Maximizing the left-hand side of (7) is equivalent to maximizing the two probabilities on the right hand side. The first probability on the right-hand side is also a Gaussian distribution given by

(4) where are the time points (or scan indices), is th regressor is the the width of the basis function. The vector of canonical BOLD response which is the convolution of the experimental paradigm and the haemodynamic response th regressor is a vector of confunction (HRF) [18]; the stant value 1. The aim of sparse Bayesian learning is to determine a suitable (where ) regressors and its corresponding set of in weighting coefficients . Since the noise (1) are assumed as independent samples of Gaussian noise with zero mean and variance (for temporally correlated errors, the prewhitening procedure is required), the output signal is also Gaussian distributed with variance . With the assumption of the independence of the output , the likelihood of the whole data set can be derived as (5) and is the design matrix. For the initial settings, is equal to and has the form defined in (3). To avoid overfitting, some of the parameters are constrained by defining prior probability distribution over them. Especially, is treated as a random variable with the weight vector ) Gaussian prior probability (with zero-mean and variance [19] where

(6) hyperparameters. This prior in (6) is where is a vector of known as an Automatic Relevance Determination (ARD) prior [19], [20]. This prior could also be set up using the general parametric empirical Bayesian (PEB) framework [21] by assigning to each regression coefficients a covariance basis function . This prior means, at this time, the best guess about the value is 0, and represents the uncertainty about this guess. of Furthermore, the ’s and the noise variance are defined with uniform distributions. The basic idea of Bayesian learning is to maximize the posterior probability over the weights and the hyperparameters and given the data , i.e., maximizing . This posterior is further decomposed as (7)

(8) with the posterior covariance

and mean

being (9) (10)

. where The maximum is clearly at the mean, i.e., the best estimate of the weights is (11) Maximizing second probability of the right-hand side of (7) is further decomposed as maximization of with respect to and . For uniform prior distributions, the following update equations are derived [19], [20]: (12) (13) (14) is the i-th element of the posterior mean vector in where is the i-th diagonal element of the posterior covari(11) and ance matrix in (9). and By iterative updating of (12) to (14), together with from (9) to (11), this updating algorithm converges and the solution is then found. In practice, many of the ’s approach infinity, which means becomes infinitely large the probability density at zero. This shows that it is certain to some extent that the should be zero given the data at hand. Thus, particular the corresponding regressor functions could be “pruned” and fewer regressors are kept to construct a suitable design matrix. These regression vectors are called Relevance Vectors in [19]. Specifically, the drifts and trends will be modeled by automatically selecting a minimum number of flexible basis functions which could capture the slow variations. The canonical BOLD response regressor may be “pruned” as well given the fMRI data. Thus, a method is achieved to determine a flexible design matrix in fMRI data analysis. This method could capture the underlying slowly varying drift in the fMRI data and avoid

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Fig. 2. Simulated BOLD signal corrupted by drift and noise (Type 3) is decomposed by the proposed approach into different sources. (a) Simulated noisy fMRI signal. (b) BOLD response. (c) Constant mean value. (d) Slowly varying drift. (e) Noise.

overfitting. There are some other methods such as the Bayesian information criterion (BIC), Akaike information criterion (AIC), minimum description length (MDL), and cross-validation that also try to prevent overfitting. However, most of these traditional model selection criteria may not work well when the number of training examples is small. As for the fMRI time series, the number of available data points is limited. The proposed method works well under such situations. Furthermore, besides the estimation of the weights , Bayesian learning also provides the estimate of the additive noise level . This term is important in the statistical evaluation of the weights , which are evaluated by defining a contrast vector and forming a -test value [2] (15) These -test values obtained at each voxel are then used to form a statistical parameter map and a threshold is determined to find the activation regions of the brain. III. RESULTS The proposed approach was first tested on simulated data with the total number of time points (or scans) equal to 100. These signals simulate block-task related fMRI signals. Fig. 1 shows a sketch of block-task experiment and its square waveform representation. When the stimulant is applied (ON), the representing

waveform has the value 1; while when the stimulant is absent (OFF), the representing waveform has the value 0. A comparison between the proposed approach and the conventional -test method is also given. Then, this new approach is applied on the real fMRI data. The results are discussed in detail in the following sections. A. Simulated Data The aim of this simulation is to investigate whether the proposed approach could capture the underlying slowly varying drift and can give some evidence to include the BOLD response regressor in the design matrix or not. Particularly, four types of signals are simulated and tested: ; Type 1) Type 2) ; Type 3) ; . Type 4) The BOLD response is simulated by assuming the brain and MR-scanner as a linear system. Thus, BOLD responses are generated as the convolution of the experimental paradigm and the haemodynamic response function (HRF) [18]. For the block-task related fMRI experiment, the experimental paradigm is simulated by a square waveform (or boxcar function) as in Fig. 1. The HRF is chosen as the difference between two gamma functions [22]. The mean value is a randomly


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Fig. 3. Simulated signals and their reconstruction. (a) Type 1: BOLD response corrupted by noise. (b) Type 2: No BOLD response, only noise. (c) Type 4: No BOLD response, only noise and drift.

generated constant representing the grey level of the specified voxel in the brain image. The drift is simulated by a slowly varying sine wave. The Gaussian noise with zero mean and variance 1 is added to the simulated signal, which results in the dB. signal-to-noise ratio (SNR) of about Fig. 2 shows the result of analyzing the Type 3 signal. Fig. 2(a) is the simulated BOLD signal corrupted by the drift and noise. This noisy signal is decomposed into the pure BOLD response [Fig. 2(b)], the constant mean value [Fig. 2(c)], the slowly varying drift [Fig. 2(d)], and the noise [Fig. 2(e)] after fitting the simulated data to the model learned by the proposed method. Initially, the design matrix has the . Through sparse Bayesian dimension 100 102 learning, the proposed approach discards the irrelevant columns in the design matrix and only retains the relevant columns. The number of final regressors in the design matrix for this simulated signal is reduced to 4 (i.e., the final design matrix has the dimension 100 4), with one BOLD response re), one constant mean gressor (with ) and two value regressor (with basis functions to account for the slowly varying drift (with

TABLE I OF DIFFERENT t-VALUE THRESHOLDS FOR DIFFERENT TYPES OF SIGNALS

ERROR RATE

and ). The values of ’s (different from zero) and ’s (not too large) give us some evidence that these four regressors are reasonably relevant regressors and the design matrix formed by these four regressors is suitable for the above simulated signal. The -test value of this signal is 4.145. Fig. 3 is the result of using the proposed model determination method to analyze the other three types of the simulated signals. The simulated noisy signal is denoted in dashed line, while the reconstructed signal is denoted in solid line. The initial design matrix is formed as in (3), however, during the learning process, the BOLD response regressor as well as some of the

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Fig. 4. ROC curves for simulated noisy data (2D plus time).

other regressors in the design matrix are discarded since they are irrelevant to the generation of the data. In Fig. 3(a), when there is BOLD response in the simulated signals, the corresponding regressor column is kept in the design matrix after the learning. While in Fig. 3(b) and (c), when the signal contains no BOLD response and only consists of either drift and noise or only noise, the proposed method will automatically switch OFF the BOLD response regressor column in the design matrix through learning. In this simulation, for Type 2 and Type 4 signals, in most cases (around 90%), the BOLD response regressor is discarded through learning and the -test values are thus 0; but in a few cases(around 10%), the BOLD regressor is not discarded in the design matrix, resulting in the -test value greater than zero. However, these -test values are normally small. Table I summarizes the error rates under different -value thresholds for the four types of simulated signals. For this simulation, 10 000 realizations of each type of signals were generated and processed using the proposed method. The -test was carried out under the null hypothesis that there is no BOLD response in the simulated signal. Both Type I error (rejecting the null hywhen it is true) and Type II error (not rejecting the pothesis when it is false) [23] are computed. In this null hypothesis table, the errors are displayed under different signal types and different -test values. For Type 1 and Type 3 signals, the error rates are of Type II error, while for Type 2 and Type 4 signals, the error rates are of Type I error. From this table, it could be seen that the proposed method can make correct decisions with small error rates. A comparison of the detection ability of the proposed method and the conventional -test method is also investigated by using receiver operator characteristic (ROC) analysis [24]. ROC method reflects the ability of different processing methods to detect most of the real activations while minimizing the

detection of false activations. For ROC analysis, two values were computed: the true positive ratio (proportion of correctly detected voxels to all voxels with added activations) and the false positive ratio (proportion of voxels that were incorrectly recognized as active to all voxels without added activation). The ROC curve is a plot of true positive ratio versus false positive ratio under different threshold values. In this simulation, one slice from the fMRI data set was used to form the background and Gaussian noise was added to construct the two-dimensional (2-D) time series. Simulated BOLD responses were added to specific areas to simulate the active brain areas. For the simulated activated voxels, the signal is Type 1, while for inactivated voxels, the signal is Type 2. In this simulation, the drift is not added. The conventional -test (which compares signal intensities between different paradigm conditions) and the proposed method were applied to this simulated data. Fig. 4 shows the ROC curves under both the conventional -test method and the proposed Bayesian learning method. Observing the ROC curves of Fig. 4, it can be seen that under the same false positive ratio, the proposed method could actually detect more real activations. This clearly proves the better performance of the proposed method compared to the conventional -test method. B. Real fMRI Data The proposed method is also validated on a block-design real fMRI data. This real fMRI experiment was designed for visuospatial processing task—judgement of line orientation. During the activation condition, the subjects were shown two stimulus lines in the top half of the screen, and an exemplar consisting of nine radial lines arranged in a semi-circle in the bottom of the screen. The two stimulus lines can be any two of the nine radial lines in the exemplar. The subjects had to decide if these two


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REFERENCES

Fig. 5. Results of fMRI data analysis to a visuospatial processing task. (a) Conventional t-test (t > 3:8). (b) Proposed method with sparse Bayesian learning (t > 6:3).

stimulus lines matched the two highlighted lines on the exemplar. During the baseline condition, the subjects were asked to ascertain if the two stimulus lines were on the same level. The details of the experiment can be found in [25]. The total number of data points for this experiment is 100, with the experimental block length and control block length 10. The first ten volumes effects in the initial scan of of the data are discarded due to an fMRI time series acquisition with low RT (Here RT is the time duration between scans). Fig. 5 shows the results of both the conventional -test and the proposed method. From Fig. 5, we could see that the proposed method could detect the activation regions with less erratic points and higher -threshold value. This clearly confirms that the proposed method is more robust than the conventional -test methods. IV. CONCLUSION In fMRI data analysis, especially in SPM, the construction of design matrix is very important to the data analysis. A flexible design matrix that can account for the measured data while not being too flexible to induce interference is desired. In this paper, the sparse Bayesian learning is applied to determine the regressors in the design matrix. The initial design matrix is a very flexible one which may induce the overfitting problem. Through sparse Bayesian learning, some confidence about the existence of the regressors is obtained and those regressors that are unlikely present and irrelevant to the measured fMRI data are discarded. Thus, a method to determine a flexible design matrix is achieved. This method could capture any unknown underlying slowly varying drift and avoid overfitting problem. It imposes a flexible model to the data, and lets the data itself determine what the model should be like. This new method integrates the advantages of the data-driven and model-driven methods and gives full flexibility to let the data itself determine its regressors in the design matrix. Validation results from both simulated and real fMRI data show that the proposed method provides a much better performance than the conventional -test method and enhances the ability of brain activity detection.

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Huaien Luo received the B.E. and M.E. degrees in electronics and information engineering from Huazhong University of Science and Technology, Wuhan, China, in 2000 and 2003, respectively. He is currently working toward the Ph.D. degree at the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. His research interests include statistical signal processing, especially biomedical signal processing and data analysis.

Sadasivan Puthusserypady (M’00–SM’05) received the B.Tech. degree in electrical engineering and the M.Tech. degree in instrumentation and control systems engineering from University of Calicut, Calicut, India, in 1986 and 1989, respectively, and the Ph.D. degree in electrical communication engineering from the Indian Institute of Science, Bangalore, India, in 1995. During 1993–1996, he was a Research Associate in the Department of Psychopharmacology, National Institute of Mental Health and Sciences (NIMHANS), Bangalore, India. He was a Postdoctoral Research Fellow in the Communications Research Laboratory, McMaster University, Hamilton, ON, Canada, from 1996 to 1998. From 1998 to 2000, he was a Senior Systems Engineer at Raytheon Systems Canada Ltd., Waterloo, ON, Canada. He is currently an Assistant Professor in the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. His research interests are in multiuser detection, chaotic communication systems, biomedical signal processing, and neural networks.