Evaluating stationarity via change-point alternatives with applications to fMRI data. (English) Zbl 1257.62072

Summary: Functional magnetic resonance imaging (fMRI) is now a well-established technique for studying the brain. However, in many situations, such as when data are acquired in a resting state, it is difficult to know whether the data are truly stationary or if level shifts have occurred. To this end, change-point detection in sequences of functional data is examined where the functional observations are dependent and where the distributions of change-points from multiple subjects are required. Of particular interest is the case where the change-point is an epidemic change-a change occurs and then the observations return to baseline at a later time.
The case where the covariance can be decomposed as a tensor product is considered with particular attention to the power analysis for detection. This is of interest in the applications to fMRI, where the estimation of a full covariance structure for the three-dimensional image is not computationally feasible. Using the developed methods, a large study of resting state fMRI data is conducted to determine whether the subjects undertaking the resting scan have nonstationarities present in their time courses. It is found that a sizeable proportion of the subjects studied are not stationary. The change-point distribution for those subjects is empirically determined, as well as its theoretical properties examined.


62H35 Image analysis in multivariate analysis
92C55 Biomedical imaging and signal processing
92C20 Neural biology
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)


Full Text: DOI arXiv Euclid


[1] Aston, J. A. D. and Gunn, R. N. (2005). Statistical estimation with Kronecker products in positron emission tomography. Linear Algebra Appl. 398 25-36. · Zbl 1062.92041 · doi:10.1016/j.laa.2003.11.018
[2] Aston, J. A. D. and Kirch, C. (2012a). Supplement to “Evaluating stationarity via change-point alternatives with applications to fMRI data.” . · Zbl 1257.62072
[3] Aston, J. A. D. and Kirch, C. (2012b). Detecting and estimating changes in dependent functional data. J. Multivariate Anal. 109 204-220. · Zbl 1241.62121 · doi:10.1016/j.jmva.2012.03.006
[4] Aston, J. A. D., Gunn, R. N., Hinz, R. and Turkheimer, F. E. (2005). Wavelet variance components in image space for spatiotemporal neuroimaging data. Neuroimage 25 159-168. · Zbl 1062.92041 · doi:10.1016/j.laa.2003.11.018
[5] Aue, A., Gabrys, R., Horváth, L. and Kokoszka, P. (2009). Estimation of a change-point in the mean function of functional data. J. Multivariate Anal. 100 2254-2269. · Zbl 1176.62025 · doi:10.1016/j.jmva.2009.04.001
[6] Beckmann, C. F. and Smith, S. M. (2005). Tensorial extensions of independent component analysis for multisubject FMRI analysis. Neuroimage 25 294-311.
[7] Beckmann, C. F., DeLuca, M., Devlin, J. T. and Smith, S. M. (2005). Investigations into resting-state connectivity using independent component analysis. Philosophical Transactions of the Royal Society B : Biological Sciences 360 1001-1013.
[8] Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: A practical and powerful approach to multiple testing. J. Roy. Statist. Soc. Ser. B 57 289-300. · Zbl 0809.62014
[9] Berkes, I., Gabrys, R., Horváth, L. and Kokoszka, P. (2009). Detecting changes in the mean of functional observations. J. R. Stat. Soc. Ser. B Stat. Methodol. 71 927-946. · doi:10.1111/j.1467-9868.2009.00713.x
[10] Biswal, B. B. et al. (2010). Toward discovery science of human brain function. Proc. Natl. Acad. Sci. USA 107 4734-4739.
[11] Bosq, D. (2000). Linear Processes in Function Spaces : Theory and Applications. Lecture Notes in Statistics 149 . Springer, New York. · Zbl 0962.60004 · doi:10.1007/978-1-4612-1154-9
[12] Botev, Z. I., Grotowski, J. F. and Kroese, D. P. (2010). Kernel density estimation via diffusion. Ann. Statist. 38 2916-2957. · Zbl 1200.62029 · doi:10.1214/10-AOS799
[13] Bullmore, E., Fadili, J., Breakspear, M., Salvador, R., Suckling, J. and Brammer, M. (2003). Wavelets and statistical analysis of functional magnetic resonance images of the human brain. Stat. Methods Med. Res. 12 375-399. · Zbl 1121.62581 · doi:10.1191/0962280203sm339ra
[14] Cole, D. M., Smith, S. M. and Beckmann, C. F. (2010). Advances and pitfalls in the analysis and interpretation of resting-state FMRI data. Frontiers in System Neuroscience 4 1-15.
[15] Damoiseaux, J. S., Rombouts, S. A. R. B., Barkhof, F., Scheltens, P., Stam, C. J., Smith, S. M. and Beckmann, C. F. (2006). Consistent resting-state networks across healthy subjects. Proc. Natl. Acad. Sci. USA 103 13848-13853.
[16] Diggle, P., Rowlingson, B. and Su, T.-L. (2005). Point process methodology for on-line spatio-temporal disease surveillance. Environmetrics 16 423-434. · doi:10.1002/env.712
[17] Doucet, G., Naveau, M., Petit, L., Zago, L., Crivello, F., Jobard, G., Delcroix, N., Mellet, E., Tzourio-Mazoyer, N., Mazoyer, B. and Joliot, M. (2012). Patterns of hemodynamic low-frequency oscillations in the brain are modulated by the nature of free thought during rest. Neuroimage 59 3194-3200.
[18] Dryden, I. L., Bai, L., Brignell, C. J. and Shen, L. (2009). Factored principal components analysis, with applications to face recognition. Stat. Comput. 19 229-238. · doi:10.1007/s11222-008-9087-6
[19] Dutilleul, P. (1999). The MLE algorithm for the matrix normal distribution. J. Stat. Comput. Simul. 64 105-123. · Zbl 0960.62056 · doi:10.1080/00949659908811970
[20] Ferraty, F. and Vieu, P. (2006). Nonparametric Functional Data Analysis : Theory and Practice . Springer, New York. · Zbl 1119.62046 · doi:10.1007/0-387-36620-2
[21] Friston, K. J., Frith, C. D., Liddle, P. F. and Frackowiak, R. S. J. (1993). Functional connectivity: The principal-component analysis of large (PET) data sets. Journal of Cerebral Blood Flow and Metabolism 13 5-14.
[22] Fuentes, M. (2006). Testing for separability of spatial-temporal covariance functions. J. Statist. Plann. Inference 136 447-466. · Zbl 1077.62076 · doi:10.1016/j.jspi.2004.07.004
[23] Genton, M. G. (2007). Separable approximations of space-time covariance matrices. Environmetrics 18 681-695. · doi:10.1002/env.854
[24] Gohberg, I., Goldberg, S. and Kaashoek, M. A. (2003). Basic Classes of Linear Operators . Birkhäuser, Basel. · Zbl 1065.47001
[25] Härdle, W. and Simar, L. (2007). Applied Multivariate Statistical Analysis , 2nd ed. Springer, Berlin. · Zbl 1115.62057
[26] Hörmann, S. and Kokoszka, P. (2010). Weakly dependent functional data. Ann. Statist. 38 1845-1884. · Zbl 1189.62141 · doi:10.1214/09-AOS768
[27] Horváth, L. and Kokoszka, P. (2012). Inference for Functional Data with Applications . Springer, New York. · Zbl 1279.62017
[28] Hušková, M. and Kirch, C. (2008). Bootstrapping confidence intervals for the change-point of time series. J. Time Series Anal. 29 947-972. · Zbl 1194.62063 · doi:10.1111/j.1467-9892.2008.00589.x
[29] Hušková, M. and Kirch, C. (2010). A note on Studentized confidence intervals for the change-point. Comput. Statist. 25 269-289. · Zbl 1223.62147 · doi:10.1007/s00180-009-0175-7
[30] Jenkinson, M., Bannister, P. R., Brady, J. M. and Smith, S. M. (2002). Improved optimisation for the robust and accurate linear registration and motion correction of brain images. NeuroImage 17 825-841.
[31] Kirch, C. (2006). Resampling methods for the change analysis of dependent data. Ph.D. thesis, Univ. Cologne, Cologne. Available at . · Zbl 1189.62078
[32] Kirch, C. (2007). Block permutation principles for the change analysis of dependent data. J. Statist. Plann. Inference 137 2453-2474. · Zbl 1274.62320 · doi:10.1016/j.jspi.2006.09.026
[33] Kirch, C. and Politis, D. N. (2011). TFT-bootstrap: Resampling time series in the frequency domain to obtain replicates in the time domain. Ann. Statist. 39 1427-1470. · Zbl 1220.62107 · doi:10.1214/10-AOS868
[34] Kokoszka, P. and Leipus, R. (1998). Change-point in the mean of dependent observations. Statist. Probab. Lett. 40 385-393. · Zbl 0935.62097 · doi:10.1016/S0167-7152(98)00145-X
[35] Lahiri, S. N. (2003). Resampling Methods for Dependent Data . Springer, New York. · Zbl 1028.62002
[36] Lindquist, M. A., Waugh, C. and Wager, T. D. (2007). Modeling state-related fMRI activity using change-point theory. Neuroimage 35 1125-1141.
[37] Long, C. J., Purdon, P. L., Temereanca, S., Desai, N. U., Hämäläinen, M. S. and Brown, E. N. (2011). State-space solutions to the dynamic magnetoencephalography inverse problem using high performance computing. Ann. Appl. Stat. 5 1207-1228. · Zbl 1223.62160 · doi:10.1214/11-AOAS483
[38] Mitchell, M. W., Genton, M. G. and Gumpertz, M. L. (2005). Testing for separability of space-time covariances. Environmetrics 16 819-831. · doi:10.1002/env.737
[39] Morris, J. S., Baladandayuthapani, V., Herrick, R. C., Sanna, P. and Gutstein, H. (2011). Automated analysis of quantitative image data using isomorphic functional mixed models, with application to proteomics data. Ann. Appl. Stat. 5 894-923. · Zbl 1454.62367 · doi:10.1214/10-AOAS407
[40] Nam, C. F. H., Aston, J. A. D. and Johansen, A. M. (2012). Quantifying uncertainty in change points. J. Time Series Anal. · Zbl 1281.62174 · doi:10.1111/j.1467-9892.2011.00777.x
[41] Ogawa, S., Lee, T. M., Kay, A. R. and Tank, D. W. (1990). Brain magnetic resonance imaging with contrast dependent on blood oxygenation. Proc. Natl. Acad. Sci. USA 87 9868-9872.
[42] Page, E. S. (1954). Continuous inspection schemes. Biometrika 41 100-115. · Zbl 0056.38002 · doi:10.1093/biomet/41.1-2.100
[43] Politis, D. N. (2011). Higher-order accurate, positive semidefinite estimation of large-sample covariance and spectral density matrices. Econometric Theory 27 703-744. · Zbl 1219.62144 · doi:10.1017/S0266466610000484
[44] Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis , 2nd ed. Springer, New York. · Zbl 1079.62006
[45] Robinson, L. F., Wager, T. D. and Lindquist, M. A. (2010). Change point estimation in multi-subject fMRI studies. NeuroImage 49 1581-1592.
[46] Ruttimann, U. E., Unser, M., Rawlings, R. R., Rio, D., Ramsey, N. F., Mattay, V. S., Hommer, D. W., Frank, J. A. and Weinberger, D. R. (1998). Statistical analysis of functional MRI data in the wavelet domain. IEEE Trans. Med. Imaging 17 142-154.
[47] Van Loan, C. F. and Pitsianis, N. (1993). Approximation with Kronecker products. In Linear Algebra for Large Scale and Real-Time Applications ( Leuven , 1992) (M. S. Moonen and G. H. Golub, eds.). NATO Advanced Science Institutes Series E : Applied Sciences 232 293-314. Kluwer Academic, Dordrecht. · Zbl 0813.65078
[48] Vanhaudenhuyse, A., Demertzi, A., Schabus, M., Noirhomme, Q., Bredart, S., Boly, M., Phillips, C., Soddu, A., Luxen, A., Moonen, G. and Laureys, S. (2010). Two distinct neuronal networks mediate the awareness of environment and of self. J. Cogn. Neurosci. 23 570-578.
[49] Werner, K., Jansson, M. and Stoica, P. (2008). On estimation of covariance matrices with Kronecker product structure. IEEE Trans. Signal Process. 56 478-491. · Zbl 1390.94472 · doi:10.1109/TSP.2007.907834
[50] Worsley, K. J., Liao, C. H., Aston, J. A. D., Petre, V., Duncan, G. H., Morales, F. and Evans, A. C. (2002). A general statistical analysis for fMRI data. Neuroimage 15 1-15.
[51] Zipunnikov, V., Caffo, B., Crainiceanu, C., Yousem, D. M., Davatzikos, C. and Schwartz, B. S. (2011). Multilevel functional principal component analysis for high-dimensional data. J. Comput. Graph. Statist. 20 852-873. · doi:10.1198/jcgs.2011.10122
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