High Dimensional Semiparametric Scale-Invariant Principal Component Analysis

We propose a new high dimensional semiparametric principal component analysis (PCA) method, named Copula Component Analysis (COCA). The semiparametric model assumes that, after unspecified marginally monotone transformations, the distributions are multivariate Gaussian. COCA improves upon PCA and sparse PCA in three aspects: (i) It is robust to modeling assumptions; (ii) It is robust to outliers and data contamination; (iii) It is scale-invariant and yields more interpretable results. We prove that the COCA estimators obtain fast estimation rates and are feature selection consistent when the dimension is nearly exponentially large relative to the sample size. Careful experiments confirm that COCA outperforms sparse PCA on both synthetic and real-world datasets.Comment: Accepted in IEEE Transactions on Pattern Analysis and Machine Intelligence (TPMAI

Very deep spectroscopy of the bright Saturn Nebula NGC 7009 -- I. Observations and plasma diagnostics

We present very deep CCD spectrum of the bright, medium-excitation planetary nebula NGC 7009, with a wavelength coverage from 3040 to 11000 A. Traditional emission line identification is carried out to identify all the emission features in the spectra, based on the available laboratory atomic transition data. Since the spectra are of medium resolution, we use multi-Gaussian line profile fitting to deblend faint blended lines, most of which are optical recombination lines (ORLs) emitted by singly ionized ions of abundant second-row elements such as C, N, O and Ne. Computer-aided emission-line identification, using the code EMILI developed by Sharpee et al., is then employed to further identify all the emission lines thus obtained. In total about 1200 emission features are identified, with the faintest ones down to fluxes 10^{-4} of H_beta. The flux errors for all emission lines, estimated from multi-Gaussian fitting, are presented. Plots of the whole optical spectrum, identified emission lines labeled, are presented along with the results of multi-Gaussian fits. Plasma diagnostics using optical forbidden line ratios are carried out. Also derived are electron temperatures and densities from the H I, He I and He II recombination spectrum.Comment: 66 pages, 16 figures, 7 tables, paper accepted by MNRAS in Marc