High-dimensional Statistical Analysis and Its Application to an ALMA Map of NGC 253

Abstract

In astronomy, if we denote the dimension of data as d and the number of samples as n, we often find a case with n ≪ d. Traditionally, such a situation is regarded as ill-posed, and there was no choice but to discard most of the information in data dimensions to let d < n. The data with n ≪ d is referred to as the high-dimensional low sample size (HDLSS). To deal with HDLSS problems, a method called high-dimensional statistics has rapidly developed in the last decade. In this work, we first introduce high-dimensional statistical analysis to the astronomical community. We apply two representative methods in the high-dimensional statistical analysis methods, noise-reduction principal component analysis (NRPCA) and automatic sparse principal component analysis (A-SPCA), to a spectroscopic map of a nearby archetype starburst galaxy NGC 253 taken by the Atacama Large Millimeter/submillimeter Array (ALMA). The ALMA map is an example of a typical HDLSS data set. First, we analyzed the original data, including the Doppler shift due to the systemic rotation. High-dimensional PCA can precisely describe the spatial structure of the rotation. We then applied to the Doppler-shift corrected data to analyze more subtle spectral features. NRPCA and R-SPCA were able to quantify the very complicated characteristics of the ALMA spectra. Particularly, we were able to extract information on the global outflow from the center of NGC 253. This method can also be applied not only to spectroscopic survey data, but also to any type of data with a small sample size and large dimension.