Kernel smoothing for spatially correlated data

Kernel smoothing is a nonparametric approach for estimating the relationship between a response variable and a set of predictors (or design variables). A major problem for kernel smoothing is the selection of the bandwidth, which controls the amount of smoothing. When data are correlated, former studies on kernel smoothing have been essentially limited to the case of a univariate predictor, with equally spaced design. In this dissertation, we discuss a more general case for correlated data, the case of multivariate predictors with random design. Three types of estimators, the Priestley-Chao estimator, the Nadaraya-Watson estimator, and the local linear estimator, are addressed, with emphasis on the local linear estimator. We will derive formulas for asymptotic mean square, errors of these kernel smoothing estimators, and formulas of asymptotically optimal bandwidth. In the presence of spatially correlated errors, we show that traditional data-driven bandwidth selection methods, such as cross-validation and generalized cross-validation, fail to provide good bandwidth values. We propose several data-driven bandwidth selection methods that account for the presence of spatial correlation. Simulation studies show that these methods are effective when the covariances between the errors are completely known. When the covariances need to be estimated from data, we consider two special cases: spatial data with repeated measurements, and spatial data collected on a grid (with only one realization). For data with repeated measurements, we propose an estimation method based on semi-variogram fitting. For data on a grid, we propose a method based on differencing, with the application of approximate Whittle likelihood estimation. Simulation studies show that these methods can provide reasonably good estimates of the covariances for the purpose of bandwidth selection

Local electronic structures on the superconducting interface LaAlO3/SrTiO3LaAlO_{3}/SrTiO_{3}

Motivated by the recent discovery of superconductivity on the heterointerface $LaAlO_{3}/SrTiO_{3}$, we theoretically investigate its local electronic structures near an impurity considering the influence of Rashba-type spin-orbit interaction (RSOI) originated in the lack of inversion symmetry. We find that local density of states near an impurity exhibits the in-gap resonance peaks due to the quasiparticle scattering on the Fermi surface with the reversal sign of the pairing gap caused by the mixed singlet and RSOI-induced triplet superconducting state. We also analyze the evolutions of density of states and local density of states with the weight of triplet pairing component determined by the strength of RSOI, which will be widely observed in thin films of superconductors with surface or interface-induced RSOI, or various noncentrosymmetric superconductors in terms of point contact tunneling and scanning tunneling microscopy, and thus reveal an admixture of the spin singlet and RSOI-induced triplet superconducting states.Comment: Phys. Rev. B 81, 144504 (2010)

Universal linear-temperature resistivity: possible quantum diffusion transport in strongly correlated superconductors

The strongly correlated electron fluids in high temperature cuprate superconductors demonstrate an anomalous linear temperature ($T$) dependent resistivity behavior, which persists to a wide temperature range without exhibiting saturation. As cooling down, those electron fluids lose the resistivity and condense into the superfluid. However, the origin of the linear-$T$ resistivity behavior and its relationship to the strongly correlated superconductivity remain a mystery. Here we report a universal relation $d\rho/dT=(\mu_0k_B/\hbar)\lambda^2_L$, which bridges the slope of the linear-$T$-dependent resistivity ($d\rho/dT$) to the London penetration depth $\lambda_L$ at zero temperature among cuprate superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ and heavy fermion superconductors CeCoIn$_5$, where $\mu_0$ is vacuum permeability, $k_B$ is the Boltzmann constant and $\hbar$ is the reduced Planck constant. We extend this scaling relation to different systems and found that it holds for other cuprate, pnictide and heavy fermion superconductors as well, regardless of the significant differences in the strength of electronic correlations, transport directions, and doping levels. Our analysis suggests that the scaling relation in strongly correlated superconductors could be described as a hydrodynamic diffusive transport, with the diffusion coefficient ($D$) approaching the quantum limit $D\sim\hbar/m^*$, where $m^*$ is the quasi-particle effective mass.Comment: 8 pages, 2 figures, 1 tabl

Parameter-tuning Networks: Experiments and Active Walk Model

The tuning process of a large apparatus of many components could be represented and quantified by constructing parameter-tuning networks. The experimental tuning of the ion source of the neutral beam injector of HT-7 Tokamak is presented as an example. Stretched-exponential cumulative degree distributions are found in the parameter-tuning networks. An active walk model with eight walkers is constructed. Each active walker is a particle moving with friction in an energy landscape; the landscape is modified by the collective action of all the walkers. Numerical simulations show that the parameter-tuning networks generated by the model also give stretched exponential functions, in good agreement with experiments. Our methods provide a new way and a new insight to understand the action of humans in the parameter-tuning of experimental processes, is helpful for experimental research and other optimization problems.Comment: 4 pages, 5 figure