Bayesian Bi-Cluster Change-Point Model for Exploring Functional Brain Dynamics

Transcription

1 Int'l Conf. Bioinformatics and Computational Biology BIOCOMP'18 85 Bayesian Bi-Cluster Change-Point Model for Exploring Functional Brain Dynamics Bing Liu 1*, Xuan Guo 2, and Jing Zhang 1** 1 Department of Mathematics and Statistics, Georgia State University, Atlanta, GA, U.S.A. 2 Department of Computer Science and Engineering, University of North exas, Denton, X, U.S.A. * Previous Georgia State University, currently employed at Eli Lilly and Company, Indianapolis, IN, U.S.A. ** Correspondence should be addressed to Jing Zhang (jzhang47@gsu.edu) Regular Research Paper Abstract - Human brain's functional dynamics have been demonstrated with recent studies. Detecting the functional connectivity change points of a single subject is also carried out by different studies. However, the clustering of multiple subjects with finding change points at the same time is still very challenging. o contribute in this area, we present a novel Bayesian Bi-Cluster Change-Point Model (BBCCPM). his model simultaneously infers the dynamics of functional brain interactions as well as the cluster of different subjects based on the boundaries of temporally quasi-stable blocks. he proposed model analyzes the joint probabilities among multiple subjects with whole brain ROIs between different time-periods and applies the MCMC scheme to sample the posterior probability distribution of each time-point as being a change-point as well as different subject cluster scheme. Finding the change points can help investigate temporal functional brain dynamics and grouping them can help us distinguish the differences of the brain dynamics among multiple subjects and lead to further research on finding the reasons behind it. he BBCCPM has been evaluated and validated on several experimental datasets, and good results are obtained. he code of the proposed method is available at Keywords: Brain Connectivity, Dynamics, Bi-clustering, Change-point, Bayesian Inference 1 Introduction In recent years, there are lots of neuroimaging studies on functional Magnetic Resonance Imaging (fmri) data. Particularly, various Bayesian-inference-based methods have been designed to detect magnitude or functional connectivity change points and furthermore led to infer the interaction patterns based on the temporal blocks [1]. As discussed in Lindquist M.A., et al. (2007) [2], the detection of functional brain state-related change points without known timing information has been a significant consideration, because psychological processes could not be specified in advance. At the same time, the onset time and the duration of the response may vary considerably across subjects. In addition to functional connectivity, the study on functional network connectivity, which focuses on the interactions in network level, estimates the clusters of brain regions having similar functionalities and has been applied to many diseases to examine brain network differences between healthy and diseased brains [3] [4] [5]. However, we are also very interested in clustering of multiple subjects with finding change points at the same time, and this is still very challenging. In previous studies, multivariate graphical causal models based on Bayesian networks are more robust and reliable in estimating functional interactions and less sensitive to noise in the fmri signals [6]. Lian et al. have developed a novel Bayesian Connectivity Change Point Model (BCCPM) [7] to detect the change points by finding the boundaries of temporal blocks via a unified Bayesian framework via the analysis of the dynamics of multivariate functional interactions. In this work, we present a novel Bayesian bi-cluster change-point model (BBCCPM). his model simultaneously infers the dynamics of functional brain interactions as well as the cluster of different subjects based on the boundaries of temporally quasi-stable blocks. he proposed model analyzes the joint probabilities among multiple subjects with whole brain ROIs between different time-periods and applies the MCMC scheme to sample the posterior probability distribution of each time-point as being a change-point as well as various subject cluster scheme. 2 Methodology 2.1 Bayesian bi-cluster connectivity changepoint model Motivated by the importance of investigating the dynamics of functional brain interactions (Gilbert and Sigman 2007 [8]; Majeed et al., 2011 [9]; Lindquist et al., 2007 [2]; Chang and Glover 2010 [10]), we developed a novel approach of BBCCPM to simultaneously infer the dynamics of functional brain interactions as well as the cluster of different subjects based on the boundaries of temporally quasi-stable blocks. he proposed model analyzes the joint probabilities among multiple subjects with whole brain ROIs between

2 86 Int'l Conf. Bioinformatics and Computational Biology BIOCOMP'18 different time-periods and applies the MCMC scheme to sample the posterior probability distribution of each timepoint as being a change-point as well as different subject cluster scheme. herefore, the boundaries of quasi-stable blocks are detected as the change-points that separate the temporal segments exhibiting substantial differences in functional interactions from each other, and multiple subjects are clustered into different groups based on different dynamics of functional brain interaction pattern Bayesian bi-cluster connectivity change-point model Given an r ROI data matrix, in which is the number of observations and r is the number of ROIs as shown in the sample matrix (Fig. 1), we are interested in if there are some differences in the joint probabilities within the r ROIs between different time periods and the locations of change points from each other Fundamentals of Bayesian inference For vectors i.i.d. (independent and identically distributed) from r-dimensional multivariate normal distribution is the number of vectors, r is the dimension of vector y t, µ is the r-dimensional mean vector, and is the r r covariance matrix. he prior distribution of (µ, ), which is conjugate, is the N Inv Wishart (, /,, ) (Gelman et al., 2003 [11]): ~ N(, / ), ~ Inv Wishart (, ) Fig. 1 Data matrix of Y and block indicator vector I, y t are the values of all ROIs at time t (the t-th column in the matrix), Y j are the values of the j-th ROI at all times (the j-th row in the matrix) and I t is a block indicator (identifying the change-points) for time t. Now we define a block indicator vector: he posterior distribution of (µ, ) given the data is the same type of N Inv Wishart (, /,, ), with: y,,, ( )( ), 0 0 S y 0 y 0 0 ( t )( t ) t 1 S y y y y Here, S is a r r matrix. hen we calculate the probability of as follows (Gelman et al., 2003 [11]): in which It 1 if the t-th observation y t is a change-point, I 0 otherwise. hen the observations would be divided t into t 1I t blocks, in which the starting time-point I 1 is always considered as a change-point. he marginal likelihood of the data matrix can be represented as follows: I i p( Y I ) p( Y b ) (2) b1 Y b are the observations belonging to b-th block and p( Y b ) ca be calculated according to Eq. (1). We assume statistical independence among the blocks. herefore, the posterior distribution of pi ( Y) can be easily obtained: r is the multivariate gamma function: r( r 1)/4 r r ( z) ( z (1 j) / 2). j1 (1) t1 p( I Y) p( I ) p( Y I ) p( I ) p( I ) and pi ( ) is ~Bern (0.5). Likelihood of multiple subjects t t (1) (2) ( ) Now say for N subjects data matrix Y ( Y, Y,..., Y N ), ( n) Y is r ROI data matrix for the n-th subject. Based on

3 Int'l Conf. Bioinformatics and Computational Biology BIOCOMP'18 87 equation (2), assuming all the subjects follow the same block partition, the marginal likelihood of Y is, ( Ii N ( n) ) ( b ) b1n1 p Y I p Y (3) Y s are the observations belonging to b-th block of n b ( n) the n-th subject and p( Y ) can be calculated according to Eq. (1). Clustering of subjects based on block partition b he main purpose of the proposed method is to cluster multiple subjects into different groups, and subjects within each group follow the same dynamics of functional brain interaction (i.e. the same block partition). Please note that the total number of groups of subjects S is unknown but between 1 and N. Fig. 2 illustrate an example of the proposed idea, the first two subjects belong to one group and the third subject belongs to a different group. Remember that we totally have S (unknown) groups, thus we define a block indicator matrix I ( IG1, IG2,... IG S ) IG s is a block indicator vector for s-th clustering groups G s as defined previously. herefore, in each clustering group G s, all the subjects are segmented into multiple blocks by the block indicator vector I Gs. Given a subject indicator vector J and a block indicator matrix I, the marginal likelihood of the data (1) (2) ( ) Y ( Y, Y,..., Y N ) is: S p( Y J, I) p( Y Gs J, IGs) (4) s1 Y Gs is the data of all the subjects belongs to s-th clustering group Gs and the likelihood p( Y Gs J, IGs) can be calculated by Eq. (3) given the block indicator vector I Gs.Different clustering groups are independent. hus the posterior distribution is p( J, I Y) p( J, I) p( Y J, I ) (5) Fig. 2 hree subjects in two clustering groups. We define a specific representation of clustering group structures: Each subject receives a unique label: 1, 2,, N. Each clustering group of subjects also obtains a unique label which is the lowest label among all the subjects it contains. For example, given the clustering group structure illustrated in Fig. 2, the Group 1 (G 1) containing subjects 1 and 2 is labeled 1 and the Group 2 (G 2) consisting of subject 3 is labeled 3. here is no the group label 2 for such grouping structure. his representation of group labels makes sure each distinct group structure will receive a unique labeling. herefore, the same clustering group structure cannot be labeled in two different ways. Based on the above representation of group labels, we define a subject indicator vector J ( J1,..., J N ) J n denotes the label of grouping which the n-th subject belongs to. Each subject only belongs to one group, meaning there is no overlapping between different clustering groups. Still taking the above clustering group structure of 3 subjects in Fig. 2 as an example, its subject indicator vector is J (1,1,3) according to the above clustering. he 3 subjects are clustered into two groups. p( J, I) p( J) p( I ) p( J) p( IGs ), and we use S s1 independent uniform priors for pj ( ) and pi ( G s ). By substituting Eq. (4) into Eq. (5), we have S p( J, I Y) p( J, I) p( Y Gs J, IGs) (6) s1 2.2 wo-level MCMC scheme In this section, a two-level algorithm with MCMC scheme [12] is proposed to sample from the posterior distribution of the cluster grouping structures and block boundaries within each clustering group, as illustrated in Figure 3. he lower level MCMC scheme samples from the posterior distribution of the block boundaries within each clustering group, and the higher level MCMC samples from the posterior distribution of the clustering group structures. he posterior distribution of the configuration can be evaluated using the formula (Eq. (3) and (6)) described previously. In the higher level MCMC, three proposals for updating subject indicator vector J are: 1) randomly selecting one clustering group and dividing it into two smaller clustering groups; 2) randomly selecting two clustering groups and merging them together; 3) randomly selecting two subjects and switching their clustering group memberships. In the lower level MCMC, within each clustering group, there are also three proposals for updating block indicator vector I G s : 1) randomly selecting one block and dividing it into two

4 88 Int'l Conf. Bioinformatics and Computational Biology BIOCOMP'18 smaller blocks; 2) selecting two consecutive blocks and merging them together; 3) randomly selecting one block and shift its boundary to the left or the right. In the higher level MCMC, after proposing a new subject indicator vector, we need to re-label the subject indicator to satisfy our specific representation of clustering groups. For example, given J (1,1,3), we randomly select two subjects 1 and 3, and switch their clustering group memberships and have J (3,1,1). After re-labeling, a new group indicator vector * J (1,2,2) is generated. dependent with a chain structure (2 1 3) with correlation coefficient ρ=0.8. Some other simulation methods can be also found in Lian et al [7]. Fig. 4 Experiment 1 design: Five subjects with two clusters and their change point distributions. he repetition of the lower MCMC is set at 500 and the repetition of higher MCMC is set at 200 (larger numbers for these two repetitions may be expected for more complicated change point distributions and group clusters to converge, in fact it converges very fast as shown in Fig.5). he results are very good as the proposed method detects the change points correctly, and at the same time, clusters the five subjects into two groups. Figure 5 shows the overall convergence trace plot of the 2-level MCMC. Figures 6-9 shows the convergence trace plots of the lower MCMC in the last repetition of the higher MCMC, and the corresponding change point detection results in Group 1 and Group 2. Fig. 3 Workflow of two-level algorithm with MCMC scheme. 3 Experimental Results In this section, we are going to validate our proposed Bayesian bi-cluster change point model on three experimental datasets. Different clustering and change point distributions are taken into consideration and all the designed experimental datasets obtained excellent outcomes with both correct clustering results and correct change point detection results. he code of the proposed method is available at this website Experiment 1 he first experimental dataset includes five subjects with two clustering groups: Subject 1 and 2 in one group, and Subjects 3, 4, and 5 in another group. Figure 4 shows the change point distributions and clustering structure. he first group has 3 temporal blocks with change points at 101 and 201, and the second group has 2 temporal blocks with change point at 151; both groups have 3 ROI's and 300 time-points. For example, in Group 2 (shown in Fig. 4), each subject has 3 ROIs R1, R2, and R3 with two 150 time-point blocks, in the first block (from time point 1 to 150), all three ROIs are independent of each other, with a normal distribution N(0, Σ), ; in the second block (151 to 300), three ROIs are Fig. 5 Overall convergence curve for Experiment 1. Fig. 6 Convergence curve for lower level MCMC in Group 1 (Subjects 1,2) for Experiment 1. Fig. 7 Change point detection result for Group 1 (Subjects 1,2) for Experiment 1.

5 Int'l Conf. Bioinformatics and Computational Biology BIOCOMP'18 89 Fig. 8 Convergence curve for lower level MCMC in Group 2 (Subjects 3,4,5) for Experiment 1. very good as the proposed method detects the change points correctly, and at the same time, clusters the ten subjects into two groups. Figure 11 shows the overall convergence trace plot of the 2-level MCMC. Figures show the change point detection results in groups 1 and 2. (he convergence trace plots of the lower MCMC in the last repetition of the higher MCMC are similar to previous experiments and thus omitted for simplicity, available upon request.) Fig. 11 Overall convergence curve for Experiment 2. Fig. 9 Change point detection result for Group 2 (Subjects 3,4,5) for Experiment 1. he result of the subject indicator vector is J (1,1,3,3,3), which correctly clusters the five subjects into two groups, the first two in Group 1 and the later three subjects in Group 2. he probability of getting this subject indicator is calculated as 100% without burn-in period. 3.2 Experiment 2 he second experimental dataset includes ten subjects with two clustering groups: Subjects 1-6 in one group, and Subjects 7-10 in another group. Figure 10 shows the change point distributions and clustering structure. he first group has 3 temporal blocks with change points at 271 and 541, and the second group has 8 temporal blocks with change point at 101, 201, 301, 401, 501, 601 and 701; both groups have 4 ROI's and 800 time points. Fig. 12 Change point detection result for Group 1 (Subjects 1-6) for Experiment 2. Fig. 13 Change point detection result for Group 2 (Subjects 7-10) for Experiment 2. he result of the subject indicator vector is J (1,1,1,1,1,1, 7, 7, 7, 7), which correctly clusters the ten subjects into two groups, the first six subjects in Group 1 and the later four subjects in Group 2. he probability of getting this subject indicator is calculated as 100% without burn-in period. 3.3 Experiment 3 Fig. 10 Experiment 2 design: en subjects with two clusters and their change point distributions. he repetition of the lower MCMC is set at 1000 and the repetition of higher MCMC is set at 100. he results are also he third experimental dataset has more complicated clustering structure. It includes ten subjects with three clustering groups: Subjects 1-3 in the first group, Subjects 4-7 in the second group, and Subjects 8 10 in the third group. Figure 14 shows the change point distributions and clustering structure. he first group has 3 temporal blocks

6 90 Int'l Conf. Bioinformatics and Computational Biology BIOCOMP'18 with change points at 271 and 541, the second group has 8 temporal blocks with change point at 101, 201, 301, 401, 501, 601 and 701, and the third group has 2 temporal blocks with change points at 151 and 651; all the three groups have 4 ROI's and 800 time points. Fig. 17 Change point detection result for Group 2 (Subjects 4-7) for Experiment 3. Fig. 14 Experiment 3 design: en subjects with three clusters and their change point distributions. he repetition of the lower MCMC is set at 1000 and the repetition of higher MCMC is set at 150. he results are also very good as the proposed method detects the change points correctly, and at the same time, clusters the ten subjects into three groups as designed. Figure 15 shows the overall convergence trace plot of the 2-level MCMC. Figures show the corresponding change point detection results in groups 1, 2 and 3. (he convergence trace plots of the lower MCMC in the last repetition of the higher MCMC are similar to previous experiments and thus omitted for simplicity, available upon request.) Fig. 15 Overall convergence curve for Experiment 3. Fig. 16 Change point detection result for Group 1 (Subjects 1-3) for Experiment 3. Fig. 18 Change point detection result for Group 3 (Subjects 8-10). he result of the subject indicator vector is J (1,1,1, 4, 4, 4, 4,8,8,8), which correctly clusters the ten subjects into three groups, the first three subjects in Group 1, the next four subjects in Group 2, and the last three subjects in Group 3. he probability of getting this subject indicator is calculated as 100% without burn-in period. o better illustrate the clustering results in the three experiments, Fig. 19 displays the clustering progress for Experiment 1, 2 and 3. he progresses are in accordance to the convergence plots. For example, in the far-left plot for Experiment 1, there are some variations in the cluster belongings for Subjects 2 5 at the beginning of our algorithm but after the very short fluctuation period, the clustering result becomes clear that the first two subjects are in one group (in light blue), and the last three subjects are in another group (in light green). hat is, say at iteration 8 (as an example), the subject indicator vector is (1,2,2,2,2), but after about iteration 15 when the posterior converges, the subject indicator vector becomes (1,1,3,3,3), and it lasts through all the rest iterations until 200. Similar results can be found in Experiment 2 as we observe fluctuation before about iteration 53, then the subject indication becomes (1,1,1,1,1,1,7,7,7,7) and remains to be so until the last iteration 100. Likewise, in Experiment 3, we see fluctuation before about iteration 70, then the subject indicator vector becomes (1,1,1,4,4,4,4,8,8,8) and remains to be so until the last iteration 150. Another observation is that as the number of clusters increases, it takes more iterations to find the correct clustering results, which is expected. Color code for subject cluster indicator is also shown in Fig. 19 in different experiments.

7 Int'l Conf. Bioinformatics and Computational Biology BIOCOMP'18 91 no. 2, pp , [5] D. Zhu, K. Li, D. P. erry, A. Nicholas Puente, L. Wang, D. Shen, L. S. Miller and. Liu, "Connectome-scale assessments of structural and functional connectivity in MCI," Human Brain Mapping, vol. 35, no. 7, pp , [6] J. D. Ramsey, S. J. Hanson and C. Glymour, "Multisubject search correctly identifies causal connections and most causal directions in the DCM models of the Smith et al. simulation study," NeuroImage, vol. 58, pp , Fig. 19 Clustering progress for the three experiments. 4 Conclusions In this paper, we present a novel Bayesian bi-cluster connectivity change-point model (BBCCPM), which can simultaneously infer the dynamics of functional brain interactions as well as the cluster of different subjects based on the boundaries of temporally quasi-stable blocks. he method has been evaluated on three sets of simulated datasets, and excellent results are obtained. In the future, the proposed method will be applied to study the brain dynamics in groups of people and has the potential to classify healthy and diseased patients according to their brain dynamics. As this method utilizes a two-level MCMC scheme and it takes much more time than one-level MCMC regarding the convergence rate, so a modified genetic algorithm [13] [14] could also be employed to reduce the computational cost without losing accuracy. 5 References [1] X. Guo, B. Liu, L. Chen, G. Chen, Y. Pan and J. Zhang, "Bayesian Inference for Functional Dynamics Exploring in fmri Data," Computational and Mathematical Methods in Medicine, [2] M. A. Lindquist, C. Waugh and. D. Wager, "Modeling state-related fmri activity using change-point theory," NeuroImage, vol. 35, pp , [3] V. D. Calhoun, K. A. Kiehl and G. D. Pearlson, "Modulation of temporally coherent brain networks estimated using ICA at rest and during cognitive tasks," Human Brain Mapping, vol. 29, no. 7, pp , [4] M. Assaf, K. Jagannathan, V. Calhoun, M. Kraut, J. Hart Jr. and G. Pearlson, "emporal sequence of hemispheric network activation during semantic processing: a functional net-work connectivity analysis," Brain and Cognition, vol. 70, [7] Z. Lian, X. Li, J. Xing, J. Lv, X. Jiang, D. Zhu, S. Zhang, J. Xu, M. N. Potenza,. Liu and J. Zhang, "Exploring functional brain dynamics via a Bayesian connectivity change point model," in Proceedings of the 11th International Symposium on Biomedical Imaging (ISBI '14), Beijing, China, [8] C. D. Gilbert and M. Sigman, "Brain states: top-down influences in sensory processing," Neuron, vol. 54, no. 5, pp , [9] W. Majeed, M. Magnuson, W. Hasenkamp, H. Schwarb, E. H. Schumacher, L. Barsalou and S. D. Keilholz, "Spatiotemporal dynamics of low frequency BOLD fluctuations in rats and humans," NeuroImage, vol. 54, no. 2, pp , [10] C. Chang and G. H. Glover, "ime-frequency dynamics of resting-state brain connectivity measured with fmri," NeuroImage, vol. 50, no. 1, pp , [11] A. Gelman, J. B. Carlin,. H. S. Stern and D. B. Rubin, Bayesian Data Analysis, 2nd edition, Chapman & Hall/CRC, [12] Z. Lian, X. Li, Y. Pan, X. Guo, L. Chen, G. Chen, Z. Wei,. Liu and J. Zhang, "Dynamic Bayesian brain network partition and connectivity change point detection," in 2015 IEEE 5th International Conference on Computational Advances in Bio and Medical Sciences, Miami, FL, [13] X. Xiao, B. Liu, J. Zhang, X. Xiao and Y. Pan, "Detecting Change Points in fmri Data via Bayesian Inference and Genetic Algorithm Model," in ISBRA 2017: Bioinformatics Research and Applications, Springer, Cham, 2017, pp [14] X. Xiao, B. Liu, J. Zhang, X. Xiao and Y. Pan, "An Optimized Method for Bayesian Connectivity Change Point Model," Journal of Computational Biology, vol. ahead of print