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Statistical analysis of fMRI using wavelets: Big Data, denoising, large‐p‐small‐n matrices

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Abstract Over the past decade functional Magnetic Resonance Imaging (fMRI) has been intensively used to study the complex functional network organization of the human brain and how it changes in time. An fMRI machine produces 3D time‐course cerebral images that contain hundreds of thousands of voxels and each voxel is scanned for hundreds of times. This potentially allows the researchers to explore functional connectivity on a voxel‐to‐voxel level, and also yields a number of serious statistical complications. First of all, the high‐dimension property of fMRI data turns it into Big Data. Furthermore, the study of functional brain network for so many voxels involves the problem of estimation of and simultaneous inference for large‐p‐small‐n cross‐covariance matrices. Furthermore, all problems should be solved in the presence of notoriously large fMRI noise which often forces statisticians to average signals over large areas instead of considering a network between individual voxels. An attractive alternative to the averaging, discussed in the paper, is a multiresolution wavelet analysis complemented by special procedures of estimating noise and estimation and simultaneous inference for cross‐covariance and cross‐correlation matrices for hundreds of thousands pairs of voxels, and it is fair to say that if wavelets have not been already known, fMRI applications would necessitates their creation. Both task and resting‐state fMRI are considered, and lessons from the wavelet analysis of ultra‐fast and conventional neuroplasticity fMRI experiments are presented. The article is self‐contained and does not require familiarity with wavelets or fMRI. This article is categorized under: Algorithms and Computational Methods > Numerical Methods Applications of Computational Statistics > Signal and Image Processing and Coding
Wavelet multiresolution decomposition of fMRI signal. Diagram (a) shows data for a particular voxel which is scanned with the period of 2 s. Diagram (b) shows data for the same voxel with the period of 50 ms. Diagrams (c–g) exhibit results of the wavelet multiresolution analysis of the signal shown in diagram (b). Diagram (c) shows fMRI noise, diagram (d) shows cardiac signal, diagram (e) shows respiratory signal, diagram (f) shows the hemodynamic response to visual stimuli, and diagram (g) shows the fMRI drift. The vertical dotted lines in the diagram (f) show times of the visual stimulus activation
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The number of active interhemispheric neural pathways for each participant in the neuroplasticity experiment during pre‐ and post‐training. A neural pathway between two voxels in different hemispheres is declared as active if, with the simultaneous 0.95 degree of confidence over all considered neural pathways, the cross‐correlation exceeds the threshold level 0.6
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Algorithms and Computational Methods
Applications of Computational Statistics > Signal and Image Processing and Coding

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