Regularized Principal Component Analysis for Spatial Data

In many atmospheric and earth sciences, it is of interest to identify dominant spatial patterns of variation based on data observed at $p$ locations and $n$ time points with the possibility that $p>n$. While principal component analysis (PCA) is commonly applied to find the dominant patterns, the eigenimages produced from PCA may exhibit patterns that are too noisy to be physically meaningful when $p$ is large relative to $n$. To obtain more precise estimates of eigenimages, we propose a regularization approach incorporating smoothness and sparseness of eigenimages, while accounting for their orthogonality. Our method allows data taken at irregularly spaced or sparse locations. In addition, the resulting optimization problem can be solved using the alternating direction method of multipliers, which is easy to implement, and applicable to a large spatial dataset. Furthermore, the estimated eigenfunctions provide a natural basis for representing the underlying spatial process in a spatial random-effects model, from which spatial covariance function estimation and spatial prediction can be efficiently performed using a regularized fixed-rank kriging method. Finally, the effectiveness of the proposed method is demonstrated by several numerical example

Hedgehog spin texture and competing orders associated with strains on the surface of a topological crystalline insulator

We have investigated spin reorientation phenomena and interaction driven effects under the presence of applied strains on the (001) surface of Pb$_{1-x}$Sn$_x$(Te, Se) topological crystalline insulators, which host multiple Dirac cones. Our analysis is based on a four-band $k\cdot p$ model, which captures the spin and orbital textures of the surface states at low energies around the $\bar{X}$ and $\bar{Y}$ points, including the Lifshitz transition. Even without breaking the time-reversal symmetry, we find that certain strains which break the mirror symmetry can induce hedgehog-like spin texture associated with gap formation at the Dirac points. The Chern number of the gapped surface ground state is shown to be tunable through the interplay of strains and a perpendicular Zeeman field. We also consider effects of strain in the presence of interactions in driving competing orders, and obtain the associated phase diagram at the mean-field level. Potential applications of our results for low power consuming electronics are discussed.Comment: 11 pages, 9 figures, and two table

Spatial modeling using graphical Markov models and wavelets

Graphical Markov models use graphs to represent possible dependencies among random variables. This class of models is extremely rich and includes inter alia causal Markov models and Markov random fields. In this dissertation, we develop a very efficient optimal-prediction algorithm for graphical Markov models. The algorithm is a generalization of the Kalman-filter algorithm for temporal processes, and it can in principle be applied to any Gaussian undirected graphical model and any Gaussian acyclic directed graphical model;We also propose a new class of multiscale models for stochastic processes in terms of scale-recursive dynamics defined on acyclic directed graphs. The models are an extension of multiscale tree-structured models. The optimal prediction can be obtained using the newly developed generalized Kalman-filter algorithm referred to above, and the parameters can be estimated by maximum likelihood via the EM algorithm. A subclass of these models are multiscale wavelet models, for which we show that the optimal predictors of hidden state variables can be obtained by a level-dependent (scale-dependent) wavelet shrinkage rule;In a series of papers, D. Donoho and I. Johnstone develop wavelet shrinkage methods to solve statistical problems. We propose a new rationale for wavelet shrinkage, based on the assumption that the underlying process can be decomposed into a large-scale deterministic trend plus a small-scale Gaussian process. Our approach has several advantages over current shrinkage methods. It takes the dependencies of empirical wavelet coefficients, both within scales and across scales, into account. Moreover, it does not rely on asymptotic properties for its justification so that it is also appropriate when the sample size is small;Finally, we introduce partially ordered Markov models, which are acyclic directed graphical models for spatial problems. The model can be regarded as a Markov random field with neighborhood structures derivable from an associated partially ordered set. We use a martingale approach to derive the asymptotic properties of maximum (composite) likelihood estimators for partially ordered Markov models. We prove that the maximum (composite) likelihood estimators are consistent, asymptotically normal, and also asymptotically efficient under checkable conditions

FFTPL: An Analytic Placement Algorithm Using Fast Fourier Transform for Density Equalization

We propose a flat nonlinear placement algorithm FFTPL using fast Fourier transform for density equalization. The placement instance is modeled as an electrostatic system with the analogy of density cost to the potential energy. A well-defined Poisson's equation is proposed for gradient and cost computation. Our placer outperforms state-of-the-art placers with better solution quality and efficiency

Numerical earthquake models of the 2013 Nantou, Taiwan, earthquake series: Characteristics of source rupture processes, strong ground motions and their tectonic implication

On 27 March and 2 June 2013, two large earthquakes with magnitudes of ML 6.2 and ML 6.5, named the Nantou earthquake series, struck central Taiwan. These two events were located at depths of 15–20 km, which implied that the mid-crust of central Taiwan is an active seismogenic area even though the subsurface structures have not been well established. To determine the origins of the Nantou earthquake series, we investigated both the rupture processes and seismic wave propagations by employing inverse and forward numerical simulation techniques. Source inversion results indicated that one event ruptured from middle to shallow crust in the northwest direction, while the other ruptured towards the southwest. Simulations of 3-D wave propagation showed that the rupture characteristics of the two events result in distinct directivity effects with different amplified shaking patterns. From the results of numerical earthquake modeling, we deduced that the occurrence of the Nantou earthquake series may be related to stress release from the easternmost edge of a preexistent strong basement in central Taiwan