Geometric interpretation of ordinary least squares regression. A vector N (representing the PET scan of an MCI nonconverter) is projected onto a space, C, which is composed of PET scans from MCI patients who converted to AD within 2 years of being scanned. Although C is depicted as being planar, in actuality it has as many dimensions as the number of PET scan vectors that compose it. The projection vector, P, can be computed by means of multiplying a “hat” matrix by the original vector, N. The hat matrix is derived from the matrix C by the equation

, where the −1 superscript represents the matrix inverse and the T superscript represents the matrix transpose. The residual vector, R, is then calculated by subtracting the projection P from N. The residual is orthogonal to all vectors in the column space of C, but retains some similarity to the original vector, N.

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