76,594 research outputs found
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
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