Penalized discriminant analysis of [/sup 15/O]-water PET brain images with prediction error selection of smoothness and regularization hyperparameters

The authors propose a flexible, comprehensive approach for analysis of [ ¹⁵O]-water positron emission tomography (PET) brain images using a penalized version of linear discriminant analysis (PDA). They applied it to scans from 20 subjects (eight scans/subject) performing a finger movement task and analyzed: (1) two classes to obtain a covariance-normalized baseline-activation image, and (2) eight classes for the mean within subject temporal structure which contained baseline-activation and time-dependent changes in a two-dimensional canonical subspace. The authors imposed spatial smoothness on the resulting image(s) by expanding it in five tensor-product B-spline (TPS) bases of varying smoothness, and further regularized with a ridge-type penalty on the noise covariance matrix. The discrimination approach of PDA provides a probabilistic framework within which prediction error (PE) estimates are derived. The authors used these to optimize over TPS bases and a ridge hyperparameter (expressed as equivalent degrees of freedom, EDF). They obtained unbiased, low variance PE estimates using modern resampling tools (.632+ Bootstrap and cross validation), and compared PDA of (1) TPS-projected, mean-normalized and unnormalized scans and (2) mean-normalized scans with and without additional presmoothing. By examining the tradeoffs between PE and EDF, as a function of basis selection and image smoothing the authors demonstrate the utility of PDA, the PE framework, and the relationship between singular value decomposition and smooth TPS bases in the analysis of functional neuroimages.