By the work of Griffiths, the cohomology of a family of complex projective manifolds determines a period map from the base of the family to the quotient of a flag domain D. In the case where D is hermitian symmetric, these quotients admit a number of partial compactifications including the Baily-Borel and toroidal AMRT compactifications. I describe recent work with Matt Kerr on computing the Mumford-Tate group of the analogs of the ARMT boundary components of a degeneration of Hodge structure arbitrary weight.