Multivariate methods for the treatment of high dimensional complex neuroimaging genetics data

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As research encompassing neuroimaging and genetics gains momentum, extraordinary information can be uncovered on the genetic architecture of the human brain. Indeed, modern neuroimaging technology offers an unprecedented opportunity to search for genetic and environmental influences among thousands of voxels in individual high resolution brain images. In this context, collecting brain and genetics data on a population of twins can lead to a powerful approach for studying the relative contributions of genetic (i.e., heritable) and environmental influences on variability in brain morphometry and function. Diffusion MRI measures white matter microstructure in the brain. Using this technique, we can enable reconstruction of orientation distribution functions giving the probability of directional water diffusion within each voxel of the brain. It then becomes possible to examine genetic influences on the geometries of axon fiber pathways. It is also desirable to study the extent to which individual differences in neural activity, as measured by functional Magnetic Resonance Imaging, are influenced by genetic or environmental factors. This latter technique involves a sequence of magnetic resonance images, each consisting of roughly 100,000 uniformly spaced voxels that partition the brain into equally sized boxes. The total amount of data that needs to be analyzed is staggering and exhibit a complicated temporal and spatial noise structure with a relatively weak signal. Statistics plays a crucial role in understanding the nature of the data, and in obtaining relevant results that can be used and interpreted by geneticists/neuroscientists. Inversely, problems raised in new studies and the huge size of the data open new and very exciting opportunities for theoretical research developments for statisticians. In this proposal, new powerful innovative statistical methods will be developed to treat these challenging data, mainly in order to relate genetic variants to the structure and/or function of the brain, but also to validate pre-suppositions of the data. Several statistical areas will be involved in this work: time series analysis, goodness-of-fit tests, multiplicity of tests, multivariate analysis (e.g., Independent Component Analysis or Partial Least Square regression), development of R packages, sparsity, complex-valued models, dependence measures, bootstrap. For example, new goodness-of-fit tests of the complex normal distribution, an assumption often made for the noise in fMRI data, will be developed. These results, relying on the empirical characteristic process, will then be used to develop an analogue of the Student t-test for complex-valued random variables, which will also be extended to regression models. In the same vein, an extension to complex-valued Vector Auto Regressive (VAR) models of tests of normality which we have already developed for ARMA models could be considered. These models are often used to fit fMRI data. Partial Least Squares (PLS) regression appears to be a good candidate to look for associations between two blocks of data (brain images and genetic matrices), as it extracts pairs of correlated latent variables (one linear combination of the variables for each block). Another approach called parallel Independent Component Analysis has also been recently developed to combine functional MRI data and SNPs from candidate regions. Nevertheless, all of these multivariate methods encounter critical over-fitting issues due to very high dimensionality of the data. In this proposal, several extensions of these methods will be proposed, using sparsity and more general dependence measures, and we will use some strategies of regularization to propose a sparse generalization of PLS and ICA.