Penalized Clustering of Large Scale Functional Data with Multiple Covariates

In this article, we propose a penalized clustering method for large scale data with multiple covariates through a functional data approach. In the proposed method, responses and covariates are linked together through nonparametric multivariate functions (fixed effects), which have great flexibility in modeling a variety of function features, such as jump points, branching, and periodicity. Functional ANOVA is employed to further decompose multivariate functions in a reproducing kernel Hilbert space and provide associated notions of main effect and interaction. Parsimonious random effects are used to capture various correlation structures. The mixed-effect models are nested under a general mixture model, in which the heterogeneity of functional data is characterized. We propose a penalized Henderson's likelihood approach for model-fitting and design a rejection-controlled EM algorithm for the estimation. Our method selects smoothing parameters through generalized cross-validation. Furthermore, the Bayesian confidence intervals are used to measure the clustering uncertainty. Simulation studies and real-data examples are presented to investigate the empirical performance of the proposed method. Open-source code is available in the R package MFDA

Exploration of Resonant Continuum and Giant Resonance in the Relativistic Approach

Single-particle resonant-states in the continuum are determined by solving scattering states of the Dirac equation with proper asymptotic conditions in the relativistic mean field theory (RMF). The regular and irregular solutions of the Dirac equation at a large radius where the nuclear potentials vanish are relativistic Coulomb wave functions, which are calculated numerically. Energies, widths and wave functions of single-particle resonance states in the continuum for ^{120}Sn are studied in the RMF with the parameter set of NL3. The isoscalar giant octupole resonance of ^{120}Sn is investigated in a fully consistent relativistic random phase approximation. Comparing the results with including full continuum states and only those single-particle resonances we find that the contributions from those resonant-states dominate in the nuclear giant resonant processes.Comment: 16 pages, 2 figure