  Tensor-based morphometry (TBM) aims at showing local differences in brain
volumes with respect to a common template. TBM images are smooth but they
exhibit (especially in diseased groups) higher values in some brain regions
called lateral ventricles. More specifically, our voxelwise analysis shows both
a mean-variance relationship in these areas and evidence of spatially dependent
skewness. We propose a model for 3-dimensional functional data where mean,
variance, and skewness functions vary smoothly across brain locations. We model
the voxelwise distributions as skew-normal. The smooth effects of age and sex
are estimated on a reference population of cognitively normal subjects from the
Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset and mapped across
the whole brain. The three parameter functions allow to transform each TBM
image (in the reference population as well as in a test set) into a Gaussian
process. These subject-specific normative maps are used to derive indices of
deviation from a healthy condition to assess the individual risk of
pathological degeneration.
