Large-scale Augmented Granger Causality (lsAGC) for Connectivity Analysis in Complex Systems: From Computer Simulations to Functional MRI (fMRI)
LLarge-scale Augmented Granger Causality (lsAGC) forConnectivity Analysis in Complex Systems: From ComputerSimulations to Functional MRI (fMRI)
Axel Wism¨uller, a,b,c,d and M. Ali Vosoughi aa Department of Electrical and Computer Engineering, University of Rochester, NY, USA b Department of Imaging Sciences, University of Rochester, NY, USA c Department of Biomedical Engineering, University of Rochester, NY, USA d Faculty of Medicine and Institute of Clinical Radiology, Ludwig Maximilian University,Munich, Germany
ABSTRACT
We introduce large-scale Augmented Granger Causality (lsAGC) as a method for connectivity analysis in complexsystems. The lsAGC algorithm combines dimension reduction with source time-series augmentation and usespredictive time-series modeling for estimating directed causal relationships among time-series. This method is amultivariate approach, since it is capable of identifying the influence of each time-series on any other time-seriesin the presence of all other time-series of the underlying dynamic system. We quantitatively evaluate the perfor-mance of lsAGC on synthetic directional time-series networks with known ground truth. As a reference method,we compare our results with cross-correlation, which is typically used as a standard measure of connectivity inthe functional MRI (fMRI) literature. Using extensive simulations for a wide range of time-series lengths and twodifferent signal-to-noise ratios of 5 and 15 dB, lsAGC consistently outperforms cross-correlation at accuratelydetecting network connections, using Receiver Operator Characteristic Curve (ROC) analysis, across all testedtime-series lengths and noise levels. In addition, as an outlook to possible clinical application, we perform apreliminary qualitative analysis of connectivity matrices for fMRI data of Autism Spectrum Disorder (ASD) pa-tients and typical controls, using a subset of 59 subjects of the Autism Brain Imaging Data Exchange II (ABIDEII) data repository. Our results suggest that lsAGC, by extracting sparse connectivity matrices, may be usefulfor network analysis in complex systems, and may be applicable to clinical fMRI analysis in future research, suchas targeting disease-related classification or regression tasks on clinical data.
Further author information: (Send correspondence to Ali Vosoughi)Ali Vosoughi: E-mail: [email protected] a r X i v : . [ q - b i o . N C ] J a n eywords: machine learning, resting-state fMRI, large-scale Augmented Granger Causality, functional connec-tivity, autism spectrum disorder
1. INTRODUCTION
Currently, the quantification of directed information transfer between interacting brain areas is one of the mostchallenging methodological problems in computational neuroscience. A fundamental problem is identifying con-nectivity in very high-dimensional systems. A common practice has been to transform a high-dimensional systeminto a simplified representation, e.g. by clustering, Principal, or Independent Component Analysis. The draw-back of such methodology is that an identified interaction between such simplified components cannot readilybe transferred back into the original high-dimensional space. Thus, directed interactions between the originalnetwork nodes can no longer be revealed. Although this significantly limits the interpretation of brain net-work activities in physiological and disease states, surprisingly little effort has been devoted to circumvent theinevitable information loss induced by the aforementioned frequently employed techniques.Various methods have been proposed to obtain directional relationships in multivariate time-series data, e.g.,transfer entropy [1] and mutual information [2]. However, as the multivariate problem’s dimensions increase,computation of the density function becomes computationally expensive [3, 4]. Under the Gaussian assumption,transfer entropy is equivalent to Granger causality [5]. However, the computation of multivariate Grangercausality for short time series in large-scale problems is challenging [6, 7]. To address these problems, we havepreviously proposed a method for multivariate Granger causality analysis using linear multivariate auto-regressive(MVAR) modeling, which simultaneously circumvents the drawbacks of above mentioned simplification strategiesby introducing an invertible dimension reduction followed by a back-projection of prediction residuals into theoriginal data space (large-scale Granger Causality, lsGC) [8]. We have also demonstrated the applicability ofthis approach to resting-state fMRI analysis [9]. Recently, we have also presented an alternative multivariateGranger causality analysis method, large-scale Extended Granger Causality (lsXGC), that uses an augmenteddimension-reduced time-series representation for predicting target time-series in the original high-dimensionalsystem directly, i.e., without inverting the dimensionality reduction step [10].In this paper, we introduce a hybrid of both methods, large-scale Augmented Granger Causality (lsAGC)that combines both invertible dimension reduction and time-series augmentation. It first uses an augmenteddimension-reduced time-series representation for prediction in the low-dimensional space, followed by an inversionof the initial dimension reduction step. In the following, we explain the lsAGC algorithm and present quantitativeresults on synthetic time-series data with known connectivity ground truth. Finally, as an outlook to possibleclinical application, we perform a preliminary qualitative analysis of connectivity matrices for fMRI data ofutism Spectrum Disorder (ASD) patients and typical controls, using a subset of the Autism Brain ImagingData Exchange II (ABIDE II) data repository.This work is embedded in our group’s endeavor to expedite artificial intelligence in biomedical imaging by meansof advanced pattern recognition and machine learning methods for computational radiology and radiomics, e.g.,[ 11–68 ].
2. DATA2.1 Synthetic Networks with Known Ground Truth for Quantitative Analysis
We quantitatively evaluate the performance of lsAGC for network structure recovery using synthetic networkswith known ground truth. We constructed ground truth networks with N = 50 nodes, each containing 5 modulesof 10 nodes with high (low) probability for the existence of directed intra- (inter-) module connections. Wesimulated two values of additive white Gaussian noise, with signal-to-noise ratios (SNR) of 15 dB and 5 dB,and repeated the experiment 100 times with different noise seed. Networks were realized as noisy stationarymultivariate auto-regressive (MVAR) processes of model order p = 2 in each of T = 1000 temporal processiterations. The network structure was adapted from [69] and [56].
The following explanation of participants and data in this section follows the description in [70]: The AutismBrain Imaging Data Exchange II (ABIDE II) initiative has made publicly available MRI data from multi-ple sites. We used resting-state fMRI data from 59 participants of the online ABIDE II repository (http://fcon 1000.projects.nitrc.org/indi/abide) in this analysis, namely the data from the Olin Neuropsychiatry Re-search Center, Institute of Living at Hartford Hospital [71].This data set consists of 24 ASD subjects (18-31 years) and 35 typical controls (18-31 years). Autism diagnosiswas based on the ASD cutoff of the Autism Diagnostic Observation Schedule-Generic (ADOS-G). The typicalcontrols (TC) were screened for autism using the ADOS-G and any psychiatric disorder based on the StructuredClinical Interview for DSM-IV Axis I Disorders-Research Version (SCID-I RV) [71].In this data set, resting-state fMRI scans were obtained from all subjects using Siemens Magnetom SkyraSyngo MR D13. The study protocol included: (i) High-resolution structural imaging using T1-weighted magnetization-prepared rapid gradient-echo (MPRAGE) sequence, TE = 2.88 ms, TR = 2200 ms, isotropic voxel size 1 mm,flip angle 13 ◦ . (ii) Resting-state fMRI scans with TE = 30 ms, TR = 475 ms, flip angle 60 ◦
3. ALGORITHM
Large-scale Augmented Granger Causality (lsAGC) has been developed based on 1) the principle of originalGranger causality, which quantifies the causal influence of time-series x s on time-series x t by quantifying theamount of improvement in the prediction of x t in presence of x s . 2) the idea of dimension reduction, whichresolves the problem of the tackling a under-determined system, which is frequently faced in fMRI analysis, sincethe number of acquired temporal samples usually is not sufficient for estimating the model parameters. Consider the ensemble of time-series
X ∈ R N × T , where N is the number of time-series (Regions Of Interest– ROIs) and T the number of temporal samples. Let X = ( x , x , . . . , x N ) T be the whole multidimensionalsystem and x i ∈ R × T a single time-series with i = 1 , , . . . , N , where x i = ( x i (1) , x i (2) , . . . , x i ( T )). In orderto overcome the under-determined problem, first X will be decomposed into its first p high-variance principalcomponents Z ∈ R p × T using Principal Component Analysis (PCA), i.e., Z = W X , (1)where W ∈ R p × N represents the PCA coefficient matrix. Subsequently, the dimension-reduced time-seriesensemble Z is augmented by one original time-series x s yielding a dimension-reduced augmented time-seriesensemble Y ∈ R ( p +1) × T for estimating the influence of x s on all other time-series.Following this, we locally predict the dimension-reduced representation Z of the original high-dimensionalsystem X at each time sample t , i.e. Z ( t ) ∈ R p × by calculating an estimate ˆ Z x s ( t ). To this end, we fit an affinemodel based on a vector of m vector of m time samples of Y ( τ ) ∈ R ( p +1) × ( τ = t − , t − , . . . , t − m ), which is y ( t ) ∈ R m. ( p +1) × , and a parameter matrix A ∈ R p × m. ( p +1) and a constant bias vector b ∈ R p × ,ˆ Z x s ( t ) = A y ( t ) + b , t = m + 1 , m + 2 , . . . , T. (2)ubsequently, we use the prediction ˆ Z x s ( t ) to calculate an estimate of X at time t , i.e. X ( t ) ∈ R N × byinverting the PCA of equation (2), i.e. X = W † Z , (3)where W † ∈ R N × p represents the inverse of the PCA coefficient matrix W , which is calculated as the Moore–Penrose pseudoinverse of W .Now ˆ X \ x s ( t ), which is the prediction of X ( t ) without the information of x s , will be estimated. The estimationprocesses is identical to the previous one, with the only difference being that we have to remove the augmentedtime-series x s and its corresponding column in the PCA coefficient matrix W .The computation of a lsAGC index is based on comparing the variance of the prediction errors obtained withand without consideration of x s . The lsAGC index f x s −→ x t , which indicates the influence of x s on x t , can becalculated by the following equation: f x s −→ x t = log var( e s )var( e \ s ) , (4)where e \ s is the error in predicting x t when x s was not considered, and e s is the error, when x s was used. Basedon preliminary analyses, in this study, we set p = 7 and m = 3.
4. RESULTS
Quantitative Analysis of Synthetic Networks with Known Ground Truth:
Network reconstructionresults for the synthetic networks with known ground truth, using the Area Under the Curve (AUC) for ReceiverOperating Characteristic (ROC) analysis, are shown in Fig. 1. For each time-series length and each noise level,we performed 100 simulations. As can be seen from Fig. 1, lsAGC consistently outperforms cross-correlationin its ability to accurately recover network structure over a wide range of time-series-lengths in both high- andlow-noise scenarios, with a mean AUC for lsAGC for a time-series length of 1000 temporal samples equal to 98.9%and 97.1% for signal-to-noise values of 15 and 5 dB, respectively. On the other hand, cross-correlation performsquite poorly compared to lsAGC with its mean AUC ranging around 0.5 for all examined time-series lengths andnoise levels, equivalent to the quality of randomly guessing the presence or absence of network connections.
Qualitative Analysis of Connectivity Matrices Extracted from fMRI Data:
Averaged connectivitymatrices, which were extracted using lsAGC and cross-correlation, are shown in Fig. 2 for both healthy controlsand ASD patients. These matrices were obtained by calculating and then averaging over the connectivity matricesof the 24 ASD patients and 35 typical controls, using the proposed lsAGC algorithm as well as conventional cross-correlation analysis. Visual inspection of the mean connectivity matrices in Fig. 2 reveals subtle differencesbetween ASD patients and healthy controls for both methods, which may be exploited for classification among
00 400 600 800 10000.40.60.81.0
Correlation, Low noise
200 400 600 800 10000.40.60.81.0 lsAGC, Low noise
200 400 600 800 10000.40.60.81.0
Correlation, High noise
200 400 600 800 10000.40.60.81.0 lsAGC, High noise
Figure 1. Quantitative performance comparison of cross-correlation and lsAGC for recovery of synthetic networks. Thevertical axis is the Area Under the Curve (AUC) for Receiver Operating Characteristics (ROC) analysis, where an AUC= 1 indicates a perfect network recovery and AUC = 0.5 random assignment. Whiskers are related to the 95% confidenceinterval, green diamonds represent outliers, orange lines represent medians, and boxes are drawn from the first quartileto the third quartile. It is clearly seen that lsAGC outperforms cross-correlation over all tested time-series lengths andnoise levels. he two cohorts in future research. We also find from visual inspection of Fig. 2 that the features extractedby the two methods are likely different, where the mean connectivity matrices for lsAGC appear to be more“sparse” than for cross-correlation. To quantify this qualitative visual impression, we calculated the entropyof the connectivity matrix elements for each of the 59 subjects as a surrogate for matrix “sparseness”. Themean entropy for healthy controls with lsAGC and correlation was 1.66 ± ± ± ± p < − ). We conclude thatlsAGC may be useful for disease-related classification or regression tasks on clinical fMRI data, because it mayextract relevant features potentially not captured by cross-correlation, which is currently used as the mainstayof fMRI connectivity analysis. This hypothesis can be further investigated in future research.
5. CONCLUSIONS
In this work, we have introduced large-scale Augmented Granger Causality (lsAGC) as a method for connectiv-ity analysis in complex systems. The lsAGC algorithm combines dimension reduction with source time-seriesaugmentation and uses multivariate predictive time-series modeling for estimating directed causal relationshipsamong time-series. We quantitatively evaluated the performance of lsAGC on synthetic directional time-seriesnetworks with known ground truth. Using simulations for a wide range of time-series lengths and differentsignal-to-noise ratios, we compared lsAGC with cross-correlation, which is currently used as the clinical standardfor fMRI connectivity analysis. We found that lsAGC consistently outperformed cross-correlation at accuratelydetecting network connections. In addition, we performed a preliminary qualitative analysis of connectivitymatrices for fMRI data of Autism Spectrum Disorder (ASD) patients and typical controls, using a subset of theABIDE II data repository. Our results suggest that lsAGC, by extracting sparse connectivity matrices, maybe useful for network analysis in complex systems, and may be applicable to clinical fMRI analysis in futureresearch, such as targeting disease-related classification or regression tasks on clinical data.
ACKNOWLEDGMENTS
This research was funded by Ernest J. Del Monte Institute for Neuroscience Award from the Harry T. MangurianJr. Foundation. This work was conducted as a Practice Quality Improvement (PQI) project related to AmericanBoard of Radiology (ABR) Maintenance of Certificate (MOC) for Prof. Dr. Axel Wism¨uller. This work is notbeing and has not been submitted for publication or presentation elsewhere. ealthy controls using lsAGC ASD patients using lsAGCHealthy controls using cross-correlation ASD patients using cross-correlation
Figure 2. Averaged connectivity matrices: top left: average connectivity matrix of healthy control subjects using lsAGC,top right: average connectivity matrix of ASD patients using lsAGC, bottom left: average connectivity matrix of healthycontrol subjects using cross-correlation, and bottom right: average connectivity matrix of ASD patients using cross-correlation. Note that the different methods capture different connectivity features, and that there are slight differencesof connectivity patterns between healthy subjects and ASD patients. Also, the lsAGC connectivity matrices appear tobe significantly “sparser” than cross-correlation matrices. This observation is quantitatively confirmed by calculating theentropy over the matrix elements, as explained in the text.
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