Local Polynomial Kernel Smoothing with Correlated Errors

Kernel smoothing method is one of the most widely used nonparametric regression meth- ods. The smoothing methods impose few assumptions about the shape of the mean function, and it is a highly flexible, data-driven regression method. Though we do not need to assume any parametric form of the mean function, we need to choose an appropriate bandwidth when we use kernel smoothing, and most of the time, the bandwidth will have a huge impact on the final prediction or estimation. When the errors in the regression model are independent and identically distributed, cross- validation method is often used to select the bandwidth for kernel smoothing and it will, in general, produce decent results. However, when errors are correlated, the cross-validation method will fail to give good bandwidths in most cases. Many methods are proposed and studied trying to solve the bandwidth selection problem for correlated data, and most of them, if not all of them, choose to impose assumptions on the correlation structure of the errors. In contrast, in this thesis, we consider to keep the very best of nonparametric regression and choose a way that is able to give us more flexibility in correlation structure. We will discuss a new bandwidth selection method that does not require any parametric assumption on the correlation structure of the errors. First, we will start with a fixed design situation, and then extend it to a more complex partially linear model. Then, we will develop the asymptotic theorems showing that under some conditions, the new method will also work for random design. Finally, we will discuss possible ways of further extending the results to spatial regression

Atomic Force Microscopy Tip-enhanced Laser Ablation

In the present work, an apertureless atomic force microscope (AFM) tip-enhanced laser ablation (TELA) system was developed and investigated. An AFM was coupled to an optical parametric oscillator (OPO) wavelength tunable laser for sample ablation with a submicron sampling size. The AFM was used to image the surface and hold the AFM tip 10 nm above the sample surface. The AFM tip is coated with a layer of gold with a thickness of 35 nm. The incident laser wavelength was tuned in the visible and near-infrared (IR) region and focused on the AFM tip. With the tip-enhancement effect, ablation craters on the surface with a submicron size were obtained. The mechanism of TELA was investigated using anthracene and three laser dyes: rhodamine B, methylene blue, and IR 797 chloride. All samples were prepared in thin films and the laser energy was set just below their far-field ablation threshold. The wavelength was tuned from 450 to 1100 nm to cover the visible and near-IR range. It was found that ablation is independent of the absorption of the compounds. The ablation crater volume was measured and found to have a maximum at 500 nm and an approximately linear drop to 800 nm. Craters could not be produced between 800 and 1200 nm and were slightly smaller at 450 nm compared to 500 nm. Apertureless TELA was then performed to sample plasmid DNA with 532 nm, which resulted in a sampling volume of 0.14 μm3 with 12% in variation. The captured DNA was amplified and the amount of sample transferred from each ablation crater was quantitated at 20 ag/spot