Fast computation of cross-validated properties in full linear leave-many-out procedures

A general theorem which allows the fast and direct computation of predicted properties in a full multiple linear leave-many-out procedure is demonstrated by induction. The result allows the description of a general algorithm which only requires a single multiple linear regression calculation. From the data generated by this fitting, in a full leave-n-out procedure involving a set of m objects, the resolution of (^m_n) linear systems of equations of dimension n × n suffices to obtain all the sets of cross-validated properties.