{"title": "Detrended Partial Cross Correlation for Brain Connectivity Analysis", "book": "Advances in Neural Information Processing Systems", "page_first": 889, "page_last": 897, "abstract": "Brain connectivity analysis is a critical component of ongoing human connectome projects to decipher the healthy and diseased brain. Recent work has highlighted the power-law (multi-time scale) properties of brain signals; however, there remains a lack of methods to specifically quantify short- vs. long- time range brain connections. In this paper, using detrended partial cross-correlation analysis (DPCCA), we propose a novel functional connectivity measure to delineate brain interactions at multiple time scales, while controlling for covariates. We use a rich simulated fMRI dataset to validate the proposed method, and apply it to a real fMRI dataset in a cocaine dependence prediction task. We show that, compared to extant methods, the DPCCA-based approach not only distinguishes short and long memory functional connectivity but also improves feature extraction and enhances classification accuracy. Together, this paper contributes broadly to new computational methodologies in understanding neural information processing.", "full_text": "Detrended Partial Cross Correlation\n\nfor Brain Connectivity Analysis\n\nJaime S Ide\u2217\nYale University\n\nNew Haven, CT 06519\njaime.ide@yale.edu\n\nFabio A Cappabianco\n\nFederal University of Sao Paulo\nS.J. dos Campos, 12231, Brazil\ncappabianco@unifesp.br\n\nFabio A Faria\n\nFederal University of Sao Paulo\nS.J. dos Campos, 12231, Brazil\n\nffaria@unifesp.br\n\nChiang-shan R Li\n\nYale University\nNew Haven, CT\n\nchiang-shan.li-yale.edu\n\nAbstract\n\nBrain connectivity analysis is a critical component of ongoing human connectome\nprojects to decipher the healthy and diseased brain. Recent work has highlighted\nthe power-law (multi-time scale) properties of brain signals; however, there remains\na lack of methods to speci\ufb01cally quantify short- vs. long- time range brain connec-\ntions. In this paper, using detrended partial cross-correlation analysis (DPCCA),\nwe propose a novel functional connectivity measure to delineate brain interactions\nat multiple time scales, while controlling for covariates. We use a rich simulated\nfMRI dataset to validate the proposed method, and apply it to a real fMRI dataset\nin a cocaine dependence prediction task. We show that, compared to extant meth-\nods, the DPCCA-based approach not only distinguishes short and long memory\nfunctional connectivity but also improves feature extraction and enhances classi-\n\ufb01cation accuracy. Together, this paper contributes broadly to new computational\nmethodologies in understanding neural information processing.\n\n1\n\nIntroduction\n\nBrain connectivity is crucial to understanding the healthy and diseased brain states [15, 1]. In recent\nyears, investigators have pursued the construction of human connectomes and made large datasets\navailable in the public domain [23, 24]. Functional Magnetic Resonance Imaging (fMRI) has been\nwidely used to examine complex processes of perception and cognition. In particular, functional\nconnectivity derived from fMRI signals has proven to be effective in delineating biomarkers for many\nneuropsychiatric conditions [15].\nOne of the challenges encountered in functional connectivity analysis is the precise de\ufb01nition of\nnodes and edges of connected brain regions [21]. Functional nodes can be de\ufb01ned based on activation\nmaps or with the use of functional or anatomical atlases. Once nodes are de\ufb01ned, the next step is to\nestimate the weights associated with the edges. Traditionally, these functional connectivity weights\nare measured using correlation-based metrics. Previous simulation studies have shown that they can\nbe quite successful, outperforming higher-order statistics (e.g. linear non-gaussian acyclic causal\nmodels) and lag-based approaches (e.g. Granger causality) [20].\nOn the other hand, very few studies have investigated the power-law cross-correlation properties\n(equivalent to multi-time scale measures) of brain connectivity. Recent research suggested that fMRI\n\n\u2217Corresponding author: Department of Psychiatry, 34 Park St. S110. New Haven CT 06519.\n\n31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.\n\n\fsignals have power-law properties (e.g. their power-spectrum follows a power law) [8, 3] and that the\ndeviations from the typical range of power-exponents have been noted in neuropsychiatric disorders\n[11]. For instance, in [3], using wavelet-based multivariate methods, authors observed that scale-free\nproperties are characteristic not only of univariate fMRI signals but also of pairwise cross-temporal\ndynamics. Moreover, they found an association between the magnitude of scale-free dynamics and\ntask performance. We hypothesize that power-law correlation measures may capture additional\ndimensions of brain connectivity not available from conventional analyses and thus enhance clinical\nprediction.\nIn this paper, we aim to answer three key open questions: (i) whether and how brain networks are\ncross-correlated at different time scales with long-range dependencies (\u201clong-memory\u201d process,\nequivalent to power-law in the frequency domain); (ii) how to extract the intrinsic association between\ntwo regions controlling for the in\ufb02uence of other interconnected regions; and (iii) whether multi-time\nscale connectivity measures can improve clinical prediction. We address the \ufb01rst two questions\nby using the detrended partial cross-correlation analsyis (DPCCA) coef\ufb01cient [25], a measure that\nquanti\ufb01es correlations on multiple time scales between non-stationary time series, as is typically the\ncase with task-related fMRI signals. DPCCA is an extension of detrended cross-correlation analysis\n[17, 13], and has been successfully applied to analyses of complex systems, including climatological\n[26] and \ufb01nancial [18] data. Unlike methods based on \ufb01ltering particular frequency bands, DPCCA\ndirectly informs correlations across multiple time scales, and unlike wavelet-based approaches (e.g.\ncross wavelet transformation and wavelet transform coherence [2]), DPCCA has the advantage of\nestimating pairwise correlations controlling for the in\ufb02uence of other regions. This is critical because\nbrain regions and thus fMRI signals thereof are highly interconnected. To answer the third question,\nwe use the correlation pro\ufb01les, generated from DPCCA, as input features for different machine\nlearning methods in classi\ufb01cation tasks and compare the performance of DPCCA-based features with\nall other competing features.\nIn Section 2, we describe the simulated and real data sets used in this study, and show how features\nof the classi\ufb01cation task are extracted from the fMRI signals. In Section 3, we provide further details\nabout DPCCA (Section 3.1), and present the proposed multi-time scale functional connectivity mea-\nsure (Section 3.2). In Section 4, we describe core experiments designed to validate the effectiveness\nof DPCCA in brain connectivity analysis and clinical prediction. We demonstrate that DPCCA\n(i) detects connectivity at multiple-time scales while controlling for covariates (Sections 4.1 and\n4.3), (ii) accurately identi\ufb01es functional connectivity in well-known gold-standard simulated data\n(Section 4.2), and (iii) improves classi\ufb01cation accuracy of cocaine dependence with fMRI data of\nseventy-\ufb01ve cocaine dependent and eighty-eight healthy control individuals (Section 4.4). In Section\n5, we conclude by highlighting the signi\ufb01cance of the study as well as the limitations and future\nwork.\n\n2 Material and Methods\n\n2.1 Simulated dataset: NetSim fMRI data\n\nWe use fMRI simulation data - NetSim [20] - previously developed for the evaluation of network\nmodeling methods. Simulating rich and realistic fMRI time series, NetSim is comprised of twenty-\neight different brain networks, with different levels of complexity. These signals are generated\nusing dynamic causal modeling (DCM [6]), a generative network model aimed to quantify neuronal\ninteractions and neurovascular dynamics, as measured by the fMRI signals. NetSim graphs have 5\nto 50 nodes organized with \u201csmall-world\u201d topology, in order to re\ufb02ect real brain networks. NetSim\nsignals have 200 time points (mostly) sampled with repetition time (TR) of 3 seconds. For each\nnetwork, 50 separate realizations (\u201csubjects\u201d) are generated. Thus, we have a total of 1400 synthetic\ndataset for testing. Finally, once the signals are generated, white noise of standard deviation 0.1-1%\nis added to reproduce the scan thermal noise.\n\n2.2 Real-world dataset: Cocaine dependence prediction\n\nSeventy-\ufb01ve cocaine dependent (CD) and eighty-eight healthy control (HC) individuals matched in\nage and gender participated in this study. CD were recruited from the local, greater New Haven area\nin a prospective study and met criteria for current cocaine dependence, as diagnosed by the Structured\nClinical Interview for DSM-IV. They were drug-free while staying in an inpatient treatment unit.\n\n2\n\n\fThe Human Investigation committee at Yale University School of Medicine approved the study, and\nall subjects signed an informed consent prior to participation. In the MR scanner, they performed a\nsimple cognitive control paradigm called stop-signal task [14]. FMRI data were collected with 3T\nSiemens Trio scanner. Each scan comprised four 10-min runs of the stop signal task. Functional\nblood oxygenation level dependent (BOLD) signals were acquired with a single-shot gradient echo\necho-planar imaging (EPI) sequence, with 32 axial slices parallel to the AC-PC line covering the\nwhole brain: TR=2000 ms, TE=25 ms, bandwidth=2004 Hz/pixel, \ufb02ip angle=85\u25e6, FOV=220\u00d7220\nmm2, matrix=66\u00d764, slice thickness=4 mm and no gap. A high-resolution 3D structural image\n(MPRAGE; 1 mm resolution) was also obtained for anatomical co-registration. Three hundred\nimages were acquired in each session. Functional MRI data was pre-processed with standard pipeline\nusing Statistical Parametric Mapping 12 (SPM12) (Wellcome Department of Imaging Neuroscience,\nUniversity College London, U.K.).\n\n2.2.1 Brain activation\n\nWe constructed general linear models and localized brain regions responding to con\ufb02ict (stop signal)\nanticipation (encoded by the probability P(stop)) at the group level [10]. The regions responding to\nP(stop) comprised the bilateral parietal cortex, the inferior frontal gyrus (IFG) and the right middle\nfrontal gyrus (MFG); and regions responding to motor slowing bilateral insula, the left precentral\ncortex (L.PC), and the supplementary motor area (SMA) (Fig. 1(a))2. These regions of interest (ROIs)\nwere used as masks to extract average activation time courses for functional connectivity analyses.\n\n2.2.2 Functional connectivity\n\nWe analyzed the frontoparietal circuit involved in con\ufb02ict anticipation and response adjustment using\na standard Pearson correlation analysis and multivariate Granger causality analysis or mGCA [19]. In\nFig. 1(b), we illustrate \ufb01fteen correlation coef\ufb01cients derived from the six ROIs for each individual\nCD and HC as shown in Fig. 1(a). According to mGCA, connectivities from bilateral parietal to L.PC\nand SMA were disrupted in CD (Fig. 1(b)). These \ufb01ndings offer circuit-level evidence of altered\ncognitive control in cocaine addiction.\n\nFigure 1: Disrupted frontoparietal circuit\nin cocaine addicts. The frontoparietal\ncircuit included six regions responding\nto Bayesian con\ufb02ict anticipation (\u201cS\u201d)\nand regions of motor slowing (\u201cRT\u201d): (a)\nCD and HC shared connections (orange\narrows). (b) Connectivity strengths be-\ntween nodes in the frontoparietal circuit.\nWe show connectivity strengths between\nnodes for each individual subject in CD\n(red line) and HC (blue line) groups.\n\n(a)\n\n(b)\n\n3 A Novel Measure of Brain Functional Connectivity\n\n3.1 Detrended partial cross-correlation analysis (DPCCA)\n\nDetrended partial cross-correlation is a novel measure recently proposed by [25]. DPCCA combines\nthe advantages of detrended cross-correlation analysis (DCCA) [17] and standard partial correlation.\nGiven two time series {x(a)},{x(b)} \u2208 Xt, where Xt \u2208 IRm, t = 1, 2, ..., N time points, DPCCA is\ngiven by Equation 1:\n\n\u03c1DP CCA(a, b; s) =\n\n,\n\n(1)\n\n(cid:112)Ca,a(s).Cb,b(s)\n\n\u2212Ca,b(s)\n\n2Peak MNI coordinates for IFG:[39,53,-1], MFG:[42,23,38],bilateral insula:[-33,17,8] and [30,20,2], L.PC:[-\n\n36,-13,56], and SMA:[-9,-1,50] in mm.\n\n3\n\n\fXt =(cid:80)t\n\ni=1 xi and Yt =(cid:80)t\n\nThe partial sums (pro\ufb01les) are obtained with sliding windows across the integrated time series\ni=1 yi. For each time window j with size s, detrended covariances\n\nand variances are computed according to Equations 5-6:\n\nf 2\nDCCA(s, j) =\n\n,\n\n(5)\n\n(cid:80)j+s\u22121\n\nt=j\n\n(Xt \u2212 (cid:91)Xt,j)(Yt \u2212 (cid:100)Yt,j)\n(cid:80)j+s\u22121\n\ns \u2212 1\n(Xt \u2212 (cid:91)Xt,j)2\ns \u2212 1\n\nt=j\n\n,\n\nwhere s is the time scale and each term Ca,b(s) is obtained by inverting the matrix \u03c1(s), e.g.\nC(s) =\u03c1\u22121(s). The coef\ufb01cient \u03c1a,b \u2208 \u03c1(s) is the so called DCCA coef\ufb01cient [13]. The DCCA\ncoef\ufb01cient is an extension of the detrented cross correlation analysis [17] combined with detrended\n\ufb02uctuation analysis (DFA) [12].\nGiven two time series {x}, {y} \u2208 Xt (indices omitted for the sake of simplicity) with N time\npoints and time scale s, DCCA coef\ufb01cient is given by Equation 2:\n\n\u03c1(s) =\n\nF 2\n\nDCCA(s)\n\nFDF A,x(s)FDF A,y(s)\n\n,\n\n(2)\n\nwhere the numerator and denominator are the average of detrended covariances and variances of the\nN \u2212 s + 1 windows (partial sums), respectively, as described in Equations 3-4:\n\n(cid:80)N\u2212s+1\n(cid:80)N\u2212s+1\n\nj=1\n\nj=1\n\nF 2\n\nDCCA(s) =\n\nF 2\n\nDF A,x(s) =\n\nf 2\nDCCA(s, j)\nN \u2212 s\n\nf 2\nDF A,x(s, j)\nN \u2212 s\n\n(3)\n\n(4)\n\n.\n\nf 2\nDF A,x(s, j) =\n\nwhere (cid:91)Xt,j and (cid:100)Yt,j are polynomial \ufb01ts of time trends. We used a linear \ufb01t as originally proposed\n\n[13], but higher order \ufb01ts could also be used [25]. DCCA can be used to measure power-law\ncross-correlations. However, we focus on DCCA coef\ufb01cient as a robust measure to detect pairwise\ncross-correlation in multiple time scales, while controlling for covariates. Importantly, DPCCA\nquanti\ufb01es correlations among time series with varying levels of non-stationarity [13].\n\n(6)\n\n3.2 DPCCA for functional connectivity analysis\n\nIn this section, we propose the use of DPCCA as a novel measure of brain functional connectivity.\nFirst, we show in simulation experiments that the measure satis\ufb01es desired connectivity properties.\nFurther, we de\ufb01ne the proposed connectivity measure. Although these properties are expected by\nmathematical de\ufb01nition of DPCCA, it is critical to con\ufb01rm its validity on real fMRI data. Additionally,\nit is necessary to establish the statistical signi\ufb01cance of the computed measures at the group level.\n\n3.2.1 Desired properties\n\nGiven real fMRI signals, the measure should accurately detect the time scale in which the pairwise\nconnections occur, while controlling for the covariates. To verify this, we create synthetic data by\ncombining real fMRI signals and sinusoidal waves (Fig. 2). To simplify, we assume additive property\nof signals and sinusoidal waves re\ufb02ecting the time onset of the connections. For each simulation, we\nrandomly sample 100 sets of time series or \u201csubjects\u201d.\na) Distinction of short and long memory connections. Given two fMRI signals {xA}, {xB}, we\nderive three pairs with known connectivity pro\ufb01les: short-memory {XA = xA + sin(T1) + e},\n{XB = xB + sin(T1) + e}, long-memory {XA = xA + sin(T2) + e}, {XB = xB +\nsin(T2) + e} and mixed {XA = xA + sin(T1) + sin(T2) + e}, {XB = xB + sin(T1) +\nsin(T2) + e}, where T1 << T2 and e is a Gaussian signal to simulate measurement noise. We\nhypothesize that the two nodes A and B are functionally connected at time scales T1 and T2.\n\n4\n\n\fb) Control for covariates. Given three fMRI signals {xA}, {xB}, {xC}, we derive three signals\nwith known connectivity {XAC = xA+xC +sin(T )+e}, {XBC = xB +xC +sin(T )+e},\n{XC = xC + e}, where e is the measurement noise. We hypothesize that the two nodes A and B\nare functionally connected mostly at scale T, once the mutual in\ufb02uence of node C is controlled.\n\nFigure 2: Illustration of synthetic fMRI sig-\nnals generated by combining real fMRI sig-\nnals and sinusoidal waves. (a) Original fMRI\nsignals, (b) original signals with sin(T =\n10s) and sin(T = 30s) waves added.\n\n(a)\n\n(b)\n\n3.2.2 Statistical signi\ufb01cance\n\nGiven two nodes and their time series, we assume that they are functionally connected if the\nmax |\u03c1DP CCA|, within a time range srange, is signi\ufb01cantly greater than the null distribution.\nEmpirical null distributions are estimated from the original data by randomly shuf\ufb02ing time series\nacross different subjects and nodes, as proposed in [20]. In this way, we generate realistic distributions\nof connectivity weights occurring by chance. Since we have a multivariate measure, the null dataset\nis always generated with the same number of nodes as the tested network. Multiple comparisons are\ncontrolled by estimating the false discovery rate. Importantly, the null distribution is also computed on\nmax |\u03c1DP CCA| within the time range srange. We use a srange from 6 to 18 seconds, assuming\nthat functional connections transpire in this range. Thus, we allow connections with different\ntime-scales. We use this binary de\ufb01nition of functional connectivity for the current approach to be\ncomparable with other methods, but it is also possible to work with the whole temporal pro\ufb01le of\n\u03c1DP CCA(s), as is done in the classi\ufb01cation experiment (Section 4.4). To keep the same statistical\ncriteria, we also generate null distributions for all the other connectivity measures.\n\n3.2.3 DPCCA + Canonical correlation analysis\n\nAs further demonstrated by simulation results (Table 1), DPCCA alone has lower true positive rate\n(TPR) compared to other competing methods, likely because of its restrictive statistical thresholds. In\norder to increase the sensitivity of DPCCA, we augmented the method by including an additional\ncanonical correlation analysis (CCA) [7]. CCA was previously used in fMRI in different contexts\nto detect brain activations [5], functional connectivity [27], and for multimodal information fusion\n[4]. In short, given two sets of multivariate time series {XA(t) \u2208 IRm, t = 1, 2, ..., N} and\n{XB(t) \u2208 IRn, t = 1, 2, ..., N}, where m and n are the respective dimensions of the two sets\nA and B, and N is the number of time points, CCA seeks the linear transformations u and v so that\nthe correlation between the linear combinations XA(t)u and XB(t)v is maximized. In this work,\nwe propose the use of CCA to de\ufb01ne the existence of a true connection, in addition to the DPCCA\nconnectivity results. The proposed method is summarized in Algorithm 1. With CCA (Lines 8-14),\nwe identify the nodes that are strongly connected after linear transformations. In Line 18, we use\nCCA to inform DPCCA in terms of positive connections.\n\n4 Experiments and Results\n\n4.1 Connectivity properties: Controlling time scales and covariates\n\nIn Figure 3, we observe that DPCCA successfully captured the time scales of the correlations\nbetween time series {XA}, {XB}, despite the noisy nature of fMRI signals. For instance, it\ndistinguished between short and long-memory connections, represented using T1 = 10s and\nT2 = 30s, respectively (Figs. 3a-c). Importantly, it clearly detected the peak connection at 10s after\ncontrolling for the in\ufb02uence of covariate signal XC (Fig. 3f). Further, unlike DPCCA, the original\nDCCA method did not rule out the mutual in\ufb02uence of XC with peak at 30s (Fig. 3e).\n\n5\n\n\fthe number of time points; time range srange with k values\n\nfor pair of vectors {x(a)}, {x(b)} \u2208 Xt do\n\nAlgorithm 1 DPCCA+CCA\nInput: Time series {Xt \u2208 IRm, t = 1, 2, ..., N}, where m is the number of vectors and N is\nOutput: Connectivity matrix F C : [m \u00d7 m] and associated matrices\n1: Step: DPCCA(Xt)\n2:\n3:\n4:\n5:\n6:\n7:\n8: Step: CCA(Xt)\n9:\n10:\n11:\n\nfor s in srange do\nF C[a, b] \u2190 max |\u03c1DP CCA| in srange\nP [a, b] \u2190 statistical signi\ufb01cance of F C[a, b] given the null empirical distribution\n\nrCCA[a, b] \u2190 (1\u2212 CCA between {x(a)}, {x(c)}, c (cid:54)= a, b)\n\n(cid:46) Matrix of connection weights and p-values\n(cid:46) Compute CCA connectivity\n\nreturn F C and P\nfor x(a) \u2208 Xt do\n\nfor x(b) \u2208 Xt, b (cid:54)= a do\n\nCompute the coef\ufb01cient \u03c1DP CCA(a, b; s)\n\n(cid:46) Compute pairwise DPCCA\n\nindexcon \u2190 k-means(rCCA[a])\nCCA[a, indexcon] \u2190 1\n\nexcluding node b\n\n12:\n13:\nreturn CCA\n14:\n15: Step: DPCCA+CCA(P,CCA)\nfor pair of nodes {a, b} do\n16:\n17:\n18:\n19:\n\nreturn F C\u2217, F C and P\n\nF C\u2217[a, b] \u2190 1, if P [a, b] < 0.05\nF C\u2217[a, b] \u2190 max(F C\u2217[a, b], CCA[a, b])\n\n(cid:46) Equation(1)\n\n(cid:46) Effect of\n\n(cid:46) Split connections into binary groups\n\n(cid:46) CCA is a binary connectivity matrix\n(cid:46) Augment DPCCA with CCA results\n\n(cid:46) DPCCA signi\ufb01cant connections\n(cid:46) Fill missing connections\n(cid:46) F C\u2217 is a binary matrix\n\nFigure 3: DPCCA temporal pro\ufb01les\namong the synthetic signals (details\nin Section 3.2.1). (a)-(c): DPCCA\nwith peak at T=10s and T=30s, and\nmixed.\n(d) DPCCA of the origi-\nnal fMRI signals used to generate\nthe synthetics signals. (e) Temporal\npro\ufb01le obtained with DCCA with-\nout partial correlation. (f) DPCCA\npeak at T=10s after controlling for\nXC. Dashed lines are the 95% con-\n\ufb01dence interval of DPCCA for the\nempirical null distribution.\n\n4.2 Simulated networks: Improved connectivity accuracy\n\nThe goal of this experiment is to validate the proposed methods in an extensive dataset designed\nto test functional connectivity methods. In this dataset, ground truth networks are known with the\narchitectures aimed to re\ufb02ect real brain networks. We use the full NetSim dataset comprised of\n28 different brain circuits and 50 subjects. For each sample of time series, we compute the partial\ncorrelation (parCorr) and the regularized inverse covariance (ICOV), reported as the best performers\nin [20], as well as the proposed DPCCA and DPCCA+CCA methods. For each measure, we construct\nempirical null distributions, as described in Section 3.2.2, and generate the binary connectivity matrix\nusing threshold \u03b1 = 0.05. To evaluate their connectivity accuracy, given the ground truth networks,\nwe compute the true positive and negative rates (TPR and TNR, respectively) and the balanced\naccuracy BAcc= (T P R+T N R)\nUsing NetSim fMRI data as the testing benchmark, we observed that the proposed DPCCA+CCA\nmethod provided more accurate functional connectivity results than the best methods reported in the\noriginal paper [20]. Results are summarized in Table 1. Here we use the balanced accuracy (BAcc)\n\n2\n\n.\n\n6\n\n\fas the evaluation metric, since it is a straightforward way to quantify both true positive and negative\nconnections.\n\nTable 1: Comparison of functional connectivity methods using NetSim dataset. Mean and standard\ndeviation of balanced accuracy (BAcc), true positive rate (TPR) and true negative rate (TNR) are\nreported. ParCorr: partial correlation, ICOV: regularized inverse covariance, DPCCA: detrended\ncross correlation analysis, DPCCA+CCA: DPCCA augmented with CCA. DPCCA+CCA balanced\naccuracy is signi\ufb01cantly higher than the best competing method ICOV (Wilcoxon signed paired test,\nZ=3.35 and p=8.1e-04).\n\nMetrics\n\nMean\nStd\n\nParCorr\n\nTPR\n0.866\n0.129\n\nBAcc\n0.834\n0.096\n\nTNR\n0.804\n0.188\n\nBAcc\n0.841\n0.095\n\nFunctional connectivity measures\nICOV\nDPCCA\nTPR\n0.866\n0.131\n\nTNR\n0.817\n0.181\n\nBAcc\n0.846\n0.095\n\nTPR\n0.835\n0.150\n\nTNR\n0.855\n0.177\n\nDPCCA+CCA\n\nBAcc\n0.859\n0.091\n\nTPR\n0.893\n0.081\n\nTNR\n0.824\n0.169\n\n4.3 Real-world dataset: Learning connectivity temporal pro\ufb01les\n\nWe use unsupervised methods to (i) learn representative temporal pro\ufb01les of connectivity from\nDPCCAF ull, and (ii) perform dimensionality reduction. The use of temporal pro\ufb01les may capture\nadditional information (such as short- and long-memory connectivity). However, it increases the\nfeature set dimensionality, imposing additional challenges on classi\ufb01er training, particularly with\nsmall dataset. The \ufb01rst natural choice for this task is principal component analysis (PCA), which can\nrepresent original features by their linear combination. Additionally, we use two popular non-linear\ndimensionality reduction methods Isomap [22] and autoencoders [9]. With Isomap, we attempt to\nlearn the intrinsic geometry (manifold) of the temporal pro\ufb01le data. With autoencoders, we seek to\nrepresent the data using restricted Boltzmann machines stacked into layers.\nIn Figure 4, we show some representative correlation pro\ufb01les obtained by computing DPPCA\namong frontoparietal regions (circuit presented in Fig. 1), and the \ufb01rst three principal components.\nInterestingly, PCA seemed to learn some of the characteristic temporal pro\ufb01les. For instance, as\nexpected, the \ufb01rst components captured the main trend, while the second components captured some\nof the short (task-related) and long (resting-state) memory connectivity trends (Figs.4a-b).\n\nFigure 4: Illustration of some\nDPCCA pro\ufb01les and their prin-\ncipal components. IFG: infe-\nrior frontal gyrus, SMA: sup-\nplementary motor area, PC:\npremotor cortex. Explained\nvariances of the components\nare also reported.\n\n4.4 Real-world dataset: Cocaine dependence prediction\n\nThe classi\ufb01cation task consists of predicting the class membership, cocaine dependence (CD) and\nhealthy control (HC), given each individual\u2019s fMRI data. After initial preprocessing (Section 2.2), we\nextract average time series within the frontoparietal circuit of 6 regions 3 (Figure 1), and compute\nthe different cross-correlation measures. These coef\ufb01cients are used as features to train and test\n(leave-one-out cross-validation) a set of popular classi\ufb01ers available in scikit-learn toolbox [16]\n(version 0.18.1), including k-nearest neighbors (kNN), support vector machine (SVM), multilayer\nperceptron (MLP), Gaussian processes (GP), naive Bayes (NB) and the ensemble method Adaboost\n(Ada). For the DPCCA coef\ufb01cients, we test both peak values DPCCAmax as well as the rich\ntemporal pro\ufb01les DPCCAF ull. Finally, we also include the brain activation maps (Section 2.2.1) as\nfeature set, thus allowing comparison with popular fMRI classi\ufb01cation softwares such as PRONTO\n(http://www.mlnl.cs.ucl.ac.uk/pronto/). Features are summarized in Table 2.\n\n3Although these regions are obtained from the whole-group, no class information is used to avoid in\ufb02ated\n\nclassi\ufb01cation rates.\n\n7\n\n\fTable 2: Features used in the cocaine dependence classi\ufb01cation task.\n\nDescription\n\nType\n\nActivation\n\nConnectivity\n\nName\nP(stop)\nUPE\nCorr\n\nParCorr\nICOV\n\nDPCCAmax\nDPCCAF ull\nDPCCAIso\n\nDPCCAAutoE\nDPCCAP CA\n\nSize\n1042\n1042\n15\n15\n15\n15\n270\n\n135-180\n30-45\n135-180\n\nBrain regions responding to anticipation of stop signals\n\nBrain regions responding to unsigned prediction error of P(stop)\nPearson cross-correlation among the six frontoparietal regions\nPartial cross-correlation among the six frontoparietal regions\n\nRegularized inverse covariance among the six frontoparietal regions\n\nMaximum DPCCA within the range 6-40 seconds\n\nTemporal pro\ufb01le of DPCCA within the range 6-40 seconds\n\nIsomap with 9-12 components and 30 neighbors\n\nAutoencoders with 2-3 hidden layers, 5-20 neurons, batch=100, epoch=1000\n\nPCA with 9-12 components\n\nClassi\ufb01cation results are summarized in Table 3 and Figure 5. We used the area under curve (AUC)\nas an evaluation metric in order to consider both sensitivity and speci\ufb01city of the classi\ufb01ers, as well as\nbalanced accuracy (BAcc). Here we tested all features described in Table 2, including the DPCCA full\npro\ufb01les after dimensionality reduction (Isomap, autoencoders and PCA). Activation maps produced\npoor classi\ufb01cation results (P(stop): 0.525\u00b10.048 and UPE: 0.509\u00b10.032), comparable to the results\nobtained with PRONTO software using the same features (accuracy 0.556).\n\nFeatures\n\nCorr\n\nParCorr\nICOV\n\nTop classi\ufb01er\n(AUC / BAcc)\n\nDPCCAAutoE\nDPCCAP CA\n\nDPCCAmax\nDPCCAF ull\nDPCCAIso\n\nGP / NB\nGP / Ada\nGP / SVM\nGP / Ada\nGP / GP\nGP / MLP\n\nMean BAcc\n(\u00b1 std)\n0.674 (\u00b1 0.037)\n0.848 (\u00b1 0.025)\n0.838 (\u00b1 0.023)\n0.831 (\u00b1 0.022)\n0.820 (\u00b1 0.052)\n0.827 (\u00b1 0.068)\n0.813 (\u00b1 0.106)\n0.844 (\u00b1 0.064)\n\nMean AUC\n(\u00b1 std)\n0.757 (\u00b1 0.041)\n0.901 (\u00b1 0.034)\n0.900 (\u00b1 0.030)\n0.906 (\u00b1 0.019)\n0.899 (\u00b1 0.028)\n0.902 (\u00b1 0.030)\n0.815 (\u00b1 0.149)\n0.928 (\u00b1 0.035)\n\nAccuracy\n(AUC / BAcc)\n0.794 / 0.710\n0.948 / 0.875\n0.948 / 0.858\n0.929 / 0.857\n0.957 / 0.874\n0.954 / 0.894\n0.939 / 0.863\n0.963 / 0.911\nTable 3: Comparison of classi\ufb01cation results for different\nfeatures. The DPCCA features combined with PCA produced\nthe top classi\ufb01ers according to both criteria (0.963/0.911).\nHowever, DPCCAP CA is not statistically better than ParCorr\nor ICOV (Wilcoxon signed paired test, p>0.05). See Figure 5\nfor accuracy across different classi\ufb01cation methods.\n\nSVM / kNN5\n\nAda / NB\n\nFigure 5: Comparison of classi-\n\ufb01cation results for different fea-\ntures and methods (described in\nSection 4.4).\n\n5 Conclusions\n\nIn summary, as a multi-time scale approach to characterize brain connectivity, the proposed method\n(DPCCA+CCA) (i) identi\ufb01ed connectivity peak-times (Fig. 3), (ii) produced higher connectivity\naccuracy than the best competing method ICOV (Table 1), and (iii) distinguished short/long memory\nconnections between brain regions involved in cognitive control (IFC&SMA and SMA&PC) (Fig.\n4). Second, using the connectivity weights as features, DPCCA measures combined with PCA\nproduced the highest individual accuracies (Table 3). However, it was not statistically different\nfrom the second best feature (ParCorr) across different classi\ufb01ers. Further separate test set would be\nnecessary to identify the best classi\ufb01ers. We performed extensive experiments with a large simulated\nfMRI dataset to validate DPCCA as a promising functional connectivity analytic. On the other\nhand, our conclusions on clinical prediction (classi\ufb01cation task) are still limited to one case. Finally,\nfurther optimization of Isomap and autoencoders methods could improve the learning of connectivity\ntemporal pro\ufb01les produced by DPCCA.\n\nAcknowledgments\n\nSupported by FAPESP (2016/21591-5), CNPq (408919/2016-7), NSF (BCS1309260) and NIH\n(AA021449, DA023248).\n\nReferences\n[1] DS Bassett and ET Bullmore. Human Brain Networks in Health and Disease. Current opinion in neurology,\n\n22(4):340\u2013347, 2009.\n\n8\n\n\f[2] C Chang and GH Glover. 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