Characterization of Microlensing Planets with Moderately Wide Separations

In future high-cadence microlensing surveys, planets can be detected through a new channel of an independent event produced by the planet itself. The two populations of planets to be detected through this channel are wide-separation planets and free-floating planets. Although they appear as similar short time-scale events, the two populations of planets are widely different in nature and thus distinguishing them is important. In this paper, we investigate the lensing properties of events produced by planets with moderately wide separations from host stars. We find that the lensing behavior of these events is well described by the Chang-Refsdal lensing and the shear caused by the primary not only produces a caustic but also makes the magnification contour elongated along the primary-planet axis. The elongated magnification contour implies that the light curves of these planetary events are generally asymmetric and thus the asymmetry can be used to distinguish the events from those produced by free-floating planets. The asymmetry can be noticed from the overall shape of the light curve and thus can hardly be missed unlike the very short-duration central perturbation caused by the caustic. In addition, the asymmetry occurs regardless of the event magnification and thus the bound nature of the planet can be identified for majority of these events. The close approximation of the lensing light curve to that of the Chang-Refsdal lensing implies that the analysis of the light curve yields only the information about the projected separation between the host star and the planet.Comment: 4 pages, 2 figure

Pairs of Frequency-based Nonhomogeneous Dual Wavelet Frames in the Distribution Space

In this paper, we study nonhomogeneous wavelet systems which have close relations to the fast wavelet transform and homogeneous wavelet systems. We introduce and characterize a pair of frequency-based nonhomogeneous dual wavelet frames in the distribution space; the proposed notion enables us to completely separate the perfect reconstruction property of a wavelet system from its stability property in function spaces. The results in this paper lead to a natural explanation for the oblique extension principle, which has been widely used to construct dual wavelet frames from refinable functions, without any a priori condition on the generating wavelet functions and refinable functions. A nonhomogeneous wavelet system, which is not necessarily derived from refinable functions via a multiresolution analysis, not only has a natural multiresolution-like structure that is closely linked to the fast wavelet transform, but also plays a basic role in understanding many aspects of wavelet theory. To illustrate the flexibility and generality of the approach in this paper, we further extend our results to nonstationary wavelets with real dilation factors and to nonstationary wavelet filter banks having the perfect reconstruction property

ECA: High Dimensional Elliptical Component Analysis in non-Gaussian Distributions

We present a robust alternative to principal component analysis (PCA) --- called elliptical component analysis (ECA) --- for analyzing high dimensional, elliptically distributed data. ECA estimates the eigenspace of the covariance matrix of the elliptical data. To cope with heavy-tailed elliptical distributions, a multivariate rank statistic is exploited. At the model-level, we consider two settings: either that the leading eigenvectors of the covariance matrix are non-sparse or that they are sparse. Methodologically, we propose ECA procedures for both non-sparse and sparse settings. Theoretically, we provide both non-asymptotic and asymptotic analyses quantifying the theoretical performances of ECA. In the non-sparse setting, we show that ECA's performance is highly related to the effective rank of the covariance matrix. In the sparse setting, the results are twofold: (i) We show that the sparse ECA estimator based on a combinatoric program attains the optimal rate of convergence; (ii) Based on some recent developments in estimating sparse leading eigenvectors, we show that a computationally efficient sparse ECA estimator attains the optimal rate of convergence under a suboptimal scaling.Comment: to appear in JASA (T&M

Magnetic structure of our Galaxy: A review of observations

The magnetic structure in the Galactic disk, the Galactic center and the Galactic halo can be delineated more clearly than ever before. In the Galactic disk, the magnetic structure has been revealed by starlight polarization within 2 or 3 kpc of the Solar vicinity, by the distribution of the Zeeman splitting of OH masers in two or three nearby spiral arms, and by pulsar dispersion measures and rotation measures in nearly half of the disk. The polarized thermal dust emission of clouds at infrared, mm and submm wavelengths and the diffuse synchrotron emission are also related to the large-scale magnetic field in the disk. The rotation measures of extragalactic radio sources at low Galactic latitudes can be modeled by electron distributions and large-scale magnetic fields. The statistical properties of the magnetized interstellar medium at various scales have been studied using rotation measure data and polarization data. In the Galactic center, the non-thermal filaments indicate poloidal fields. There is no consensus on the field strength, maybe mG, maybe tens of uG. The polarized dust emission and much enhanced rotation measures of background radio sources are probably related to toroidal fields. In the Galactic halo, the antisymmetric RM sky reveals large-scale toroidal fields with reversed directions above and below the Galactic plane. Magnetic fields from all parts of our Galaxy are connected to form a global field structure. More observations are needed to explore the untouched regions and delineate how fields in different parts are connected.Comment: 10+1 pages. Invited Review for IAU Symp.259: Cosmic Magnetic Fields: From Planets, to Stars and Galaxies (Tenerife, Spain. Nov.3-7, 2009). K.G. Strassmeier, A.G. Kosovichev & J.E. Beckman (eds.

High Dimensional Semiparametric Scale-Invariant Principal Component Analysis

We propose a new high dimensional semiparametric principal component analysis (PCA) method, named Copula Component Analysis (COCA). The semiparametric model assumes that, after unspecified marginally monotone transformations, the distributions are multivariate Gaussian. COCA improves upon PCA and sparse PCA in three aspects: (i) It is robust to modeling assumptions; (ii) It is robust to outliers and data contamination; (iii) It is scale-invariant and yields more interpretable results. We prove that the COCA estimators obtain fast estimation rates and are feature selection consistent when the dimension is nearly exponentially large relative to the sample size. Careful experiments confirm that COCA outperforms sparse PCA on both synthetic and real-world datasets.Comment: Accepted in IEEE Transactions on Pattern Analysis and Machine Intelligence (TPMAI

Magnetic fields of our Galaxy on large and small scales

Magnetic fields have been observed on all scales in our Galaxy, from AU to kpc. With pulsar dispersion measures and rotation measures, we can directly measure the magnetic fields in a very large region of the Galactic disk. The results show that the large-scale magnetic fields are aligned with the spiral arms but reverse their directions many times from the inner-most arm (Norma) to the outer arm (Perseus). The Zeeman splitting measurements of masers in HII regions or star-formation regions not only show the structured fields inside clouds, but also have a clear pattern in the global Galactic distribution of all measured clouds which indicates the possible connection of the large-scale and small-scale magnetic fields.Comment: 9 pages. Invited Talk at IAU Symp.242, 'Astrophysical Masers and their Environments', Proceedings edited by J. M. Chapman & W. A. Baa

Sparse Median Graphs Estimation in a High Dimensional Semiparametric Model

In this manuscript a unified framework for conducting inference on complex aggregated data in high dimensional settings is proposed. The data are assumed to be a collection of multiple non-Gaussian realizations with underlying undirected graphical structures. Utilizing the concept of median graphs in summarizing the commonality across these graphical structures, a novel semiparametric approach to modeling such complex aggregated data is provided along with robust estimation of the median graph, which is assumed to be sparse. The estimator is proved to be consistent in graph recovery and an upper bound on the rate of convergence is given. Experiments on both synthetic and real datasets are conducted to illustrate the empirical usefulness of the proposed models and methods