×

A focused information criterion for graphical models in fMRI connectivity with high-dimensional data. (English) Zbl 1397.62466

Summary: Connectivity in the brain is the most promising approach to explain human behavior. Here we develop a focused information criterion for graphical models to determine brain connectivity tailored to specific research questions. All efforts are concentrated on high-dimensional settings where the number of nodes in the graph is larger than the number of samples. The graphical models may include autoregressive times series components, they can relate graphs from different subjects or pool data via random effects. The proposed method selects a graph with a small estimated mean squared error for a user-specified focus. The performance of the proposed method is assessed on simulated data sets and on a resting state functional magnetic resonance imaging (fMRI) data set where often the number of nodes in the estimated graph is equal to or larger than the number of samples.

MSC:

62P10 Applications of statistics to biology and medical sciences; meta analysis
62B10 Statistical aspects of information-theoretic topics
62H12 Estimation in multivariate analysis
05C90 Applications of graph theory
92C55 Biomedical imaging and signal processing
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] Abegaz, F. and Wit, E. (2013). Sparse time series chain graphical models for reconstructing genetic networks. Biostatistics 14 586-599.
[2] Achard, S., Salvador, R., Whitcher, B., Suckling, J. and Bullmore, E. (2006). A resilient, low-frequency, small-world human brain functional network with highly connected association cortical hubs. J. Neurosci. 26 63-72.
[3] Allen, E. A., Damaraju, E., Plis, S. M., Erhardt, E. B., Eichele, T. and Calhoun, V. D. (2014). Tracking whole-brain connectivity dynamics in the resting state. Cereb. Cortex 24 663-676.
[4] Banerjee, O., El Ghaoui, L. and d’Aspremont, A. (2008). Model selection through sparse maximum likelihood estimation for multivariate Gaussian or binary data. J. Mach. Learn. Res. 9 485-516. · Zbl 1225.68149
[5] Bassett, D. S., Bullmore, E., Verchinski, B. A., Mattay, V. S., Weinberger, D. R. and Meyer-Lindenberg, A. (2008). Hierarchical organization of human cortical networks in health and schizophrenia. J. Neurosci. 28 9239-9248.
[6] Buckner, R. L., Andrews-Hanna, J. R. and Schacter, D. L. (2008). The brain’s default network: Anatomy, function, and relevance to disease. Ann. N. Y. Acad. Sci. 1124 1-38.
[7] Bühlmann, P. (2013). Statistical significance in high-dimensional linear models. Bernoulli 19 1212-1242. · Zbl 1273.62173 · doi:10.3150/12-BEJSP11
[8] Bühlmann, P. and van de Geer, S. (2011). Statistics for High-Dimensional Data : Methods , Theory and Applications . Springer, Heidelberg. · Zbl 1273.62015 · doi:10.1007/978-3-642-20192-9
[9] Bullmore, E. and Sporns, O. (2009). Complex brain networks: Graph theoretical analysis of structural and functional systems. Nat. Rev. Neurosci. 10 186-198.
[10] Bunea, F., She, Y., Ombao, H., Gongvatana, A., Devlin, K. and Cohen, R. (2011). Penalized least squares regression methods and applications to neuroimaging. Neuroimage 55 1519-1527.
[11] Cai, T., Liu, W. and Luo, X. (2011). A constrained \(\ell_{1}\) minimization approach to sparse precision matrix estimation. J. Amer. Statist. Assoc. 106 594-607. · Zbl 1232.62087 · doi:10.1198/jasa.2011.tm10155
[12] Cammoun, L., Gigandet, X., Meskaldji, D., Thiran, J. P., Sporns, O., Do, K. Q., Maeder, P., Meuli, R. and Hagmann, P. (2012). Mapping the human connectome at multiple scales with diffusion spectrum MRI. J. Neurosci. Methods 203 386-397.
[13] Chai, X. J., Whitfield-Gabrieli, S., Shinn, A. K., Gabrieli, J. D. E., Castañón, A. N., McCarthy, J. M., Cohen, B. M. and Ongür, D. (2011). Abnormal medial prefrontal cortex resting-state connectivity in bipolar disorder and schizophrenia. Neuropsychopharmacology 36 2009-2017.
[14] Claeskens, G. (2012). Focused estimation and model averaging with penalization methods: An overview. Stat. Neerl. 66 272-287. · doi:10.1111/j.1467-9574.2012.00514.x
[15] Claeskens, G. and Hjort, N. L. (2003). The focused information criterion. J. Amer. Statist. Assoc. 98 900-945. · Zbl 1045.62003 · doi:10.1198/016214503000000819
[16] Craven, P. and Wahba, G. (1978/79). Smoothing noisy data with spline functions. Estimating the correct degree of smoothing by the method of generalized cross-validation. Numer. Math. 31 377-403. · Zbl 0377.65007 · doi:10.1007/BF01404567
[17] Cribben, I., Haraldsdottir, R., Atlas, L. Y., Wager, T. D. and Lindquist, M. A. (2012). Dynamic connectivity regression: Determining state-related changes in brain connectivity. NeuroImage 61 907-920.
[18] Dahlhaus, R. and Eichler, M. (2003). Causality and graphical models in time series analysis. In Highly Structured Stochastic Systems. Oxford Statist. Sci. Ser. 27 115-144. Oxford Univ. Press, Oxford.
[19] Dempster, A. P. (1972). Covariance selection. Biometrics 28 157-175.
[20] Deshpande, G., Santhanam, P. and Hu, X. (2011). Instantaneous and causal connectivity in resting state brain networks derived from functional MRI data. Neuroimage 54 1043-1052.
[21] Desikan, R. S., Sègonne, F., Fischl, B., Quinn, B. T., Dickerson, B. C., Blacker, D., Buckner, R. L., Dale, A. M., Maguire, R. P., Hyman, B. T., Albert, M. S. and Killiany, R. J. (2006). An automated labeling system for subdividing the human cerebral cortex on MRI scans into gyral based regions of interest. NeuroImage 31 968-980.
[22] Fan, J., Feng, Y. and Wu, Y. (2009). Network exploration via the adaptive lasso and SCAD penalties. Ann. Appl. Stat. 3 521-541. · Zbl 1166.62040 · doi:10.1214/08-AOAS215
[23] Fan, J. and Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. J. Amer. Statist. Assoc. 96 1348-1360. · Zbl 1073.62547 · doi:10.1198/016214501753382273
[24] Fan, T., Yao, L. and Wu, X. (2012). Independent component analysis of the resting-state brain functional MRI study in adults with bipolar depression. In Proceedings of 2012 International Conference on Complex Medical Engineering 38-42. IEEE.
[25] Foygel, R. and Drton, M. (2010). Extended Bayesian information criteria for Gaussian graphical models. In Advances in Neural Information Processing Systems 23 (J. Lafferty, C. K. I. Williams, J. Shawe-Taylor, R. S. Zemel and A. Culotta, eds.) 604-612. MIT Press, Cambridge, MA.
[26] Frank, M. J. (2011). Computational models of motivated action selection in corticostriatal circuits. Curr. Opin. Neurobiol. 21 381-386.
[27] Friedman, J., Hastie, T. and Tibshirani, R. (2008). Sparse inverse covariance estimation with the graphical lasso. Biostatistics 9 432-441. · Zbl 1143.62076 · doi:10.1093/biostatistics/kxm045
[28] Friston, K. J., Kahan, J., Biswal, B. and Razi, A. (2014). A DCM for resting state fMRI. Neuroimage 94 396-407.
[29] Fu, W. J. (1998). Penalized regressions: The bridge versus the lasso. J. Comput. Graph. Statist. 7 397-416.
[30] Gao, W. and Tian, Z. (2010). Latent ancestral graph of structure vector autoregressive models. J. Syst. Eng. Electron. 21 233-238.
[31] Gerhard, S., Daducci, A., Lemkaddem, A., Meuli, R., Thiran, J.-P. and Hagmann, P. (2011). The connectome viewer toolkit: An open source framework to manage, analyze, and visualize connectomes. Front Neuroinform 5 3.
[32] Hagmann, P., Cammoun, L., Gigandet, X., Meuli, R., Honey, C. J., Wedeen, J. V. and Sporns, O. (2008). Mapping the structural core of human cerebral cortex. PLoS Biology 6 e159.
[33] Honey, C. J., Sporns, O., Cammoun, L., Gigandet, X., Thiran, J. P., Meuli, R. and Hagmann, P. (2009). Predicting human resting-state functional connectivity from structural connectivity. Proc. Natl. Acad. Sci. USA 106 2035-2040.
[34] Humphries, M. D. and Gurney, K. (2008). Network “small-world-ness”: A quantitative method for determining canonical network equivalence. PLoS ONE 3 e0002051.
[35] Humphries, M. D., Gurney, K. and Prescott, T. J. (2006). The brainstem reticular formation is a small-world, not scale-free, network. Proceedings of the Royal Society B 273 503-511.
[36] Hunter, D. R. and Li, R. (2005). Variable selection using MM algorithms. Ann. Statist. 33 1617-1642. · Zbl 1078.62028 · doi:10.1214/009053605000000200
[37] Isoda, M. and Hikosaka, O. (2007). Switching from automatic to controlled action by monkey medial frontal cortex. Nat. Neurosci. 10 240-248.
[38] Jahfari, S., Waldorp, L. J., van den Wildenberg, W. P. M., Scholte, H. S., Ridderinkhof, K. R. and Forstmann, B. U. (2011). Effective connectivity reveals important roles for both the hyperdirect (fronto-subthalamic) and the indirect (fronto-striatal-pallidal) fronto-basal ganglia pathways during response inhibition. J. Neurosci. 31 6891-6899.
[39] Jahfari, S., Verbruggen, F., Frank, M. J., Waldorp, L. J., Colzato, L., Ridderinkhof, K. R. and Forstmann, B. U. (2012). How preparation changes the need for top-down control of the basal ganglia when inhibiting premature actions. J. Neurosci. 32 10870-10878.
[40] James, W. and Stein, C. (1961). Estimation with quadratic loss. In Proc. 4 th Berkeley Sympos. Math. Statist. and Prob. , Vol. I 361-379. Univ. California Press, Berkeley, CA. · Zbl 1281.62026
[41] James, G. A., Kelley, M. E., Craddock, R. C., Holtzheimer, P. E., Dunlop, B., Nemeroff, C. and Hu, X. P. (2009). Exploratory structural equation modeling of resting-state fMRI: Applicability of group models to individual subjects. Neuroimage 45 778-787.
[42] Jenkinson, M. and Smith, S. (2001). A global optimisation method for robust affine registration of brain images. Med. Image Anal. 5 143-156.
[43] Jenkinson, M., Bannister, P., Brady, M. and Smith, S. (2002). Improved optimization for the robust and accurate linear registration and motion correction of brain images. Neuroimage 17 825-841.
[44] Kolar, M., Song, L., Ahmed, A. and Xing, E. P. (2010). Estimating time-varying networks. Ann. Appl. Stat. 4 94-123. · Zbl 1189.62142 · doi:10.1214/09-AOAS308
[45] Koyama, M. S., Martino, A. D., Zuo, X.-N., Kelly, C., Mennes, M., Jutagir, D. R., Castellanos, F. X. and Milham, M. P. (2011). Resting-state functional connectivity indexes reading competence in children and adults. J. Neurosci. 31 8617-8624.
[46] Krishnamurthy, V., Ahipaşaoğlu, S. D. and d’Aspremont, A. (2012). A pathwise algorithm for covariance selection. In Optimization for Machine Learning (S. Sra, S. Nowozin and S. J. Wright, eds.) 479-494. MIT Press, Cambridge, MA.
[47] Laird, N., Lange, N. and Stram, D. (1987). Maximum likelihood computations with repeated measures: Application of the EM algorithm. J. Amer. Statist. Assoc. 82 97-105. · Zbl 0613.62063 · doi:10.2307/2289129
[48] Lauritzen, S. L. (1996). Graphical Models. Oxford Statistical Science Series 17 . The Clarendon Press, Oxford Univ. Press, New York. · Zbl 0907.62001
[49] Lei, Y., Tong, L. and Yan, B. (2013). A mixed L2 norm regularized HRF estimation method for rapid event-related fMRI experiments. Comput. Math. Methods Med. 2013 643129. · Zbl 1275.92049 · doi:10.1155/2013/643129
[50] Leonardi, N., Richiardi, J., Gschwind, M., Simioni, S., Annoni, J. M., Schluep, M. and Van De Ville, D. (2013). Principal components of functional connectivity: A new approach to study dynamic brain connectivity during rest. NeuroImage 83 937-950.
[51] Li, L. and Toh, K.-C. (2010). An inexact interior point method for \(L_{1}\)-regularized sparse covariance selection. Math. Program. Comput. 2 291-315. · Zbl 1208.90131 · doi:10.1007/s12532-010-0020-6
[52] Li, X., Zhao, T. and Liu, H. (2013). camel: Calibrated machine learning. R package version 0.2.0.
[53] Lindquist, M. A. (2008). The statistical analysis of fMRI data. Statist. Sci. 23 439-464. · Zbl 1329.62296 · doi:10.1214/09-STS282
[54] Liu, H. and Wang, L. (2012). TIGER: A tuning-insensitive approach for optimally estimating large undirected graphs. Technical report.
[55] Mazumder, R. and Hastie, T. (2012). The graphical lasso: New insights and alternatives. Electron. J. Stat. 6 2125-2149. · Zbl 1295.62066 · doi:10.1214/12-EJS740
[56] McLachlan, G. J. and Krishnan, T. (2008). The EM Algorithm and Extensions , 2nd ed. Wiley, Hoboken, NJ. · Zbl 1165.62019 · doi:10.1002/9780470191613
[57] Meinshausen, N. and Bühlmann, P. (2006). High-dimensional graphs and variable selection with the lasso. Ann. Statist. 34 1436-1462. · Zbl 1113.62082 · doi:10.1214/009053606000000281
[58] Mohammadi, A. and Wit, E. (2015). Bayesian structure learning in sparse Gaussian graphical models. Bayesian Anal. 10 109-138. · Zbl 1335.62056 · doi:10.1214/14-BA889
[59] Moussa, M. N., Steen, M. R., Laurienti, P. J. and Hayasaka, S. (2012). Consistency of network modules in resting-state FMRI connectome data. PLoS ONE 7 e44428.
[60] O’Neil, E. B., Hutchison, R. M., McLean, D. A. and Köhler, S. (2014). Resting-state fMRI reveals functional connectivity between face-selective perirhinal cortex and the fusiform face area related to face inversion. Neuroimage 92 349-355.
[61] Pircalabelu, E., Claeskens, G. and Waldorp, L. (2015). A focused information criterion for graphical models. Stat. Comput. 25 1071-1092. · Zbl 1331.62057 · doi:10.1007/s11222-014-9504-y
[62] Raichle, M. E., MacLeod, A. M., Snyder, A. Z., Powers, W. J., Gusnard, D. A. and Shulman, G. L. (2001). A default mode of brain function. Proc. Natl. Acad. Sci. USA 98 676-682.
[63] Ravikumar, P. D., Raskutti, G., Wainwright, M. J. and Yu, B. (2008). Model selection in Gaussian graphical models: High-dimensional consistency of \(l_{1}\)-regularized MLE. In Proceedings of the 22 nd Annual Conference on Neural Information Processing Systems (D. Koller, D. Schuurmans, Y. Bengio and L. Bottou, eds.) 1329-1336. MIT Press, Cambridge, MA.
[64] Ridderinkhof, K. R., Ullsperger, M., Crone, E. A. and Nieuwenhuis, S. (2004). The role of the medial frontal cortex in cognitive control. Science 306 443-447.
[65] Ryali, S., Supekar, K., Abrams, D. A. and Menon, V. (2010). Sparse logistic regression for whole-brain classification of fMRI data. Neuroimage 51 752-764.
[66] Ryali, S., Chen, T., Supekar, K. and Menon, V. (2012). Estimation of functional connectivity in fMRI data using stability selection-based sparse partial correlation with elastic net penalty. Neuroimage 59 3852-3861.
[67] Scheinberg, K. and Rish, I. (2010). Learning sparse Gaussian Markov networks using a greedy coordinate ascent approach. In Proceedings of the 2010 European Conference on Machine Learning and Knowledge Discovery in Databases : Part III 196-212. Springer, Berlin.
[68] Schmidt, M., Niculescu-Mizil, A. and Murphy, K. (2007). Learning graphical model structure using \(\ell _{1}\)-regularization paths. In Proceedings of the 22 nd National Conference on Artificial Intelligence 2 1278-1283. AAAI Press, Menlo Park, CA.
[69] Smith, S. M. (2002). Fast robust automated brain extraction. Hum. Brain Mapp. 17 143-155.
[70] Sporns, O. and Honey, C. J. (2006). Small worlds inside big brains. Proc. Natl. Acad. Sci. USA 103 19219-19220.
[71] Thompson, P. M., Cannon, T. D., Narr, K. L., van Erp, T., Poutanen, V. P., Huttunen, M., Lönnqvist, J., Standertskjöld-Nordenstam, C. G., Kaprio, J., Khaledy, M., Dail, R., Zoumalan, C. I. and Toga, A. W. (2001). Genetic influences on brain structure. Nat. Neurosci. 4 1253-1258.
[72] Wainwright, M. J., Ravikumar, P. and Lafferty, J. D. (2007). High-dimensional graphical model selection using \(\ell _{1}\)-regularized logistic regression. In Advances in Neural Information Processing Systems 19 (B. Schölkopf, J. Platt and T. Hoffman, eds.) 1465-1472. MIT Press, Cambridge, MA.
[73] Waldorp, L. J. (2009). Robust and unbiased variance of GLM coefficients for misspecified autocorrelation and hemodynamic response models in fMRI. Int. J. Biomed. Imaging 2009 1-11.
[74] Weeda, W. D., Waldorp, L. J., Christoffels, I. and Huizenga, H. M. (2010). Activated region fitting: A robust high-power method for fMRI analysis using parameterized regions of activation. Hum. Brain Mapp. 30 2595-2605.
[75] Wink, A. M. and Roerdink, J. B. T. M. (2006). BOLD noise assumptions in fMRI. Int. J. Biomed. Imaging 2006 1-11.
[76] Witten, D. M., Friedman, J. H. and Simon, N. (2011). New insights and faster computations for the graphical lasso. J. Comput. Graph. Statist. 20 892-900. · doi:10.1198/jcgs.2011.11051a
[77] Woodward, N. D., Rogers, B. and Heckers, S. (2011). Functional resting-state networks are differentially affected in schizophrenia. Schizophr. Res. 130 86-93.
[78] Worsley, K. J. (2001). Statistical analysis of activation images. In Functional MRI : An Introduction to Methods (P. Jezzard, P. Matthews and S. M. Smith, eds.) 251-270. Oxford Univ. Press, London.
[79] Yuan, M. and Lin, Y. (2007). Model selection and estimation in the Gaussian graphical model. Biometrika 94 19-35. · Zbl 1142.62408 · doi:10.1093/biomet/asm018
[80] Zhang, X. and Liang, H. (2011). Focused information criterion and model averaging for generalized additive partial linear models. Ann. Statist. 39 174-200. · Zbl 1209.62088 · doi:10.1214/10-AOS832
[81] Zhao, T., Liu, H., Roeder, K., Lafferty, J. and Wasserman, L. (2012). The huge package for high-dimensional undirected graph estimation in R. J. Mach. Learn. Res. 13 1059-1062. · Zbl 1283.68311
[82] Zhou, S., Lafferty, J. and Wasserman, L. (2010). Time varying undirected graphs. Mach. Learn. 80 295-319. · Zbl 1475.62174 · doi:10.1007/s10994-010-5180-0
[83] Zou, H. (2006). The adaptive lasso and its oracle properties. J. Amer. Statist. Assoc. 101 1418-1429. · Zbl 1171.62326 · doi:10.1198/016214506000000735
[84] Zou, H. and Hastie, T. (2005). Regularization and variable selection via the elastic net. J. R. Stat. Soc. Ser. B Stat. Methodol. 67 301-320. · Zbl 1069.62054 · doi:10.1111/j.1467-9868.2005.00503.x
[85] Zou, H. and Li, R. (2008). One-step sparse estimates in nonconcave penalized likelihood models. Ann. Statist. 36 1509-1533. · Zbl 1142.62027 · doi:10.1214/009053607000000802
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.