We then sought to create a normative model that would characterize the joint probability density function (PDF) of our features across the entire cortex in our healthy volunteer population, allowing for characterization of the expected range of normal cortical variability. Inspection of the histograms and joint scatter plots of the PCA components, although they were uncorrelated with zero mean and unit variance, demonstrated a fair amount of residual structure in the form of variable degrees of skew and kurtosis (overview fig). To allow for more straightforward estimation of outlierness, as well as similarities and differences between cortical vertices, we applied an invertible non-linear transformation to map the features into a latent representation where the PDF follows an approximately multivariate normal distribution using the Rotation-based Iterative Gaussianization (RBIG) procedure \citep{LapaEtal2011}. RBIG is a fast iterative procedure consisting of a sequence of pairs of transformations: 1) a non-linear transformation applied to each of the columns (marginals) of the data matrix, and 2) a linear transformation applied to the entire data matrix. The non-linear column-wise operation is a univariate Gaussianization that converts percentile scores computed using the rank transformation to standard scores (scikit-learn QuantileTransformer). The orthogonal transformation is performed using PCA, with all components retained after each iteration (figure \ref{fig:gaussianize}). Ten iterations were performed. Following this procedure, the histograms and joint scatter plots of each of the resulting components now had an approximately Gaussian distribution, as expected (overview fig).
Although similar to features used in other imaging applications REFSS, this set of features has not been used to our knowledge to describe normal cortical variability or to detect cortical abnormalities. We therefore wished to determine whether our feature set could be used to predict the values obtained using some more commonly used metrics such as surface-based measurements of curvature, sulcal depth, cortical thickness, and gray/white contrast (as calculated using FreeSurfer), or volumetric measures of myelination as in \cite{Glasser_2011}, calculated by dividing the T1 intensity by the T2 intensity sampled onto the surface. Using our feature set as the input, linear regression models (ordinary least-squares) were estimated for each target metric using scikit-learn. Model training and testing was performed in the group of healthy volunteers. Each model was evaluated using a 10-fold cross validation procedure. Each fold was trained on all cortical vertices from 25 randomly selected healthy volunteers and tested on all cortical vertices from the remaining 5 subjects. Performance was evaluated for each model using the coefficient of determination \(r{^2}\) (regression figure goes here), and effect size was reported as in \cite{cohen1988statistical}: \(r=0.1\) as small, \(r=0.3\) as medium, and \(r=0.5\) as large.