Model Selection for High Dimensional Quadratic Regression via Regularization

Quadratic regression (QR) models naturally extend linear models by considering interaction effects between the covariates. To conduct model selection in QR, it is important to maintain the hierarchical model structure between main effects and interaction effects. Existing regularization methods generally achieve this goal by solving complex optimization problems, which usually demands high computational cost and hence are not feasible for high dimensional data. This paper focuses on scalable regularization methods for model selection in high dimensional QR. We first consider two-stage regularization methods and establish theoretical properties of the two-stage LASSO. Then, a new regularization method, called Regularization Algorithm under Marginality Principle (RAMP), is proposed to compute a hierarchy-preserving regularization solution path efficiently. Both methods are further extended to solve generalized QR models. Numerical results are also shown to demonstrate performance of the methods.Comment: 37 pages, 1 figure with supplementary materia

Coalescence driven self-organization of growing nanodroplets around a microcap

The coalescence between growing droplets is important for the surface coverage and spatial arrangements of droplets on surfaces. In this work, total internal reflection fluorescence (TIRF) microscopy is utilized to in-situ investigate the formation of nanodroplets around the rim of a polymer microcap, with sub-micron spatial and millisecond temporal resolution. We observe that the coalescence among droplets occurs frequently during their growth by solvent exchange. Our experimental results show that the position of the droplet from two merged droplets is related to the size of the parent droplets. The position of the coalesced droplet and the ratio of parent droplet sizes obey a scaling law, reflecting a coalescence preference based on the size inequality. As a result of droplet coalescence, the angles between the centroids of two neighbouring droplets increase with time, obeying a nearly symmetrical arrangement of droplets at various time intervals. The evolution of the position and number from coalescence of growing droplets is modelled. The mechanism for coalescence driven self-organization of growing droplets is general, applicable to microcaps of different sizes and droplets of different liquids. The understanding from this work may be valuable for positioning nanodroplets by nucleation and growth without using templates.Comment: 10 pages, 9 figure