  Principal component analysis is a powerful statistical system to investigate
the structure and dynamics of the molecular interstellar medium, with
particular emphasis on the study of turbulence, as revealed by spectroscopic
imaging of molecular line emission. To-date, the method to retrieve the power
law index of the velocity structure function or power spectrum has relied on an
empirical calibration and testing with model turbulent velocity fields, while
lacking a firm theoretical basis. In this paper, we present an analytic
formulation that reveals the detailed mechanics of the method and confirms
previous empirical calibrations of its recovery of the scale dependence of
turbulent velocity fluctuations.
