Unbiased prediction risk

We use a method that implements the Unbiased Prediction Risk criterion [13] to provide a data-driven approach for the selection of the regularization parameter. The equality constraints are handled with LQ factorization [14] and an iterative method suggested by Villalobos and Wahba [15] is used to incorporate the inequality constraints [10]. The method is well suited for the relatively large-scale problem associated with analyzing each image voxel as no user intervention is required and all the voxels can be analyzed in parallel. 

In general, a regularization parameter should be chosen for each voxel. Since there may be thousands of voxels, the use of graphical or other methods requiring intervention is prohibitive. In the present work, an automatic, data-driven method is utilized to obtain a reliable estimate of the regularization parameter for each voxel. It is based on nonparametric statistical theory, which can incorporate a number of performance criteria, including unbiased prediction risk (UBPR),9 cross-validation (CV),15 and generalized cross-validation (GCV).16... 

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