Principal component analysis of spectral line data: analytic formulation

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.