Regularized Multivariate Regression Models with Skew-\u3cem\u3et\u3c/em\u3e Error Distributions

We consider regularization of the parameters in multivariate linear regression models with the errors having a multivariate skew-t distribution. An iterative penalized likelihood procedure is proposed for constructing sparse estimators of both the regression coefficient and inverse scale matrices simultaneously. The sparsity is introduced through penalizing the negative log-likelihood by adding L1-penalties on the entries of the two matrices. Taking advantage of the hierarchical representation of skew-t distributions, and using the expectation conditional maximization (ECM) algorithm, we reduce the problem to penalized normal likelihood and develop a procedure to minimize the ensuing objective function. Using a simulation study the performance of the method is assessed, and the methodology is illustrated using a real data set with a 24-dimensional response vector

Quasi-B-mode generated by high-frequency gravitational waves and corresponding perturbative photon fluxes

Interaction of very low-frequency primordial(relic) gravitational waves(GWs) to cosmic microwave background(CMB) can generate B-mode polarization. Here, for the first time we point out that the electromagnetic(EM) response to high-frequency GWs(HFGWs) would produce quasi-B-mode distribution of the perturbative photon fluxes, and study the duality and high complementarity between such two B-modes. Based on this quasi-B-mode in HFGWs, it is shown that the distinguishing and observing of HFGWs from the braneworld would be quite possible due to their large amplitude, higher frequency and very different physical behaviors between the perturbative photon fluxes and background photons, and the measurement of relic HFGWs may also be possible though face to enormous challenge.Comment: 22 pages, 6 figures, research articl

A homogeneous high precision direct integration based on Chebyshev interpolation

Based on Chebyshev’s interpolation theory, the non-homogeneous term of the second-order linear differential equations is interpolated, and a precise integration algorithm with easy programming, high computational efficiency and precision design is realized. The method does not involve inverse operation, and does not need to additionally calculate the matrix index on the integration point, and can control the error boundary based on different precision requirements, so it has high stability and controllability. Numerical examples of periodic loads common in vibration engineering show the effectiveness of the method

Topics on Regularization of Parameters in Multivariate Linear Regression

My dissertation mainly focuses on the regularization of parameters in the multivariate linear regression under different assumptions on the distribution of the errors. It consists of two topics where we develop iterative procedures to construct sparse estimators for both the regression coefficient and scale matrices simultaneously, and a third topic where we develop a method for testing if the skewness parameter in the skew-normal distribution is parallel to one of the eigenvectors of the scale matrix. In the first project, we propose a robust procedure for constructing a sparse estimator of a multivariate regression coefficient matrix that accounts for the correlations of the response variables. Robustness to outliers is achieved using heavy-tailed t distributions for the multivariate response, and shrinkage is introduced by adding to the negative log-likelihood l1 penalties on the entries of both the regression coefficient matrix and the precision matrix of the responses. Taking advantage of the hierarchical representation of a multivariate t distribution as the scale mixture of normal distributions and the EM algorithm, the optimization problem is solved iteratively where at each EM iteration suitably modified multivariate regression with covariance estimation (MRCE) algorithms proposed by Rothman, Levina and Zhu are used. We propose two new optimization algorithms for the penalized likelihood, called MRCEI and MRCEII, which differ from MRCE in the way that the tuning parameters for the two matrices are selected. Estimating the degrees of freedom when penalizing the entries of the matrices presents new computational challenges. A simulation study and real data analysis demonstrate that the MRCEII, which selects the tuning parameter of the precision matrix of the multiple responses using the Cp criterion, generally does the best among all methods considered in terms of the prediction error, and MRCEI outperforms the MRCE methods when the regression coefficient matrix is less sparse. The second project is motivated by the existence of the skewness in the data for which the symmetric distribution assumption on the errors does not hold. We extend the procedure we have proposed to the case where the errors in the multivariate linear regression follow a multivariate skew-normal or skew-t distribution. Based on the convenient representation of skew-normal and skew-t as well as the EM algorithm, we develop an optimization algorithm, called MRST, to iteratively minimize the negative penalized log-likelihood. We also carry out a simulation study to assess the performance of the method and illustrate its application with one real data example. In the third project, we discuss the asymptotic distributions of the eigenvalues and eigenvectors for the MLE of the scale matrix in a multivariate skew-normal distribution. We propose a statistic for testing whether the skewness vector is proportional to one of the eigenvectors of the scale matrix based on the likelihood ratio. Under the alternative, the likelihood is maximized numerically with two different ways of parametrization for the scale matrix: Modified Cholesky Decomposition (MCD) and Givens Angle. We conduct a simulation study and show that the statistic obtained using Givens Angle parametrization performs well and is more reliable than that obtained using MCD

Experimental demonstrations of high-Q superconducting coplanar waveguide resonators

We designed and successfully fabricated an absorption-type of superconducting coplanar waveguide (CPW) resonators. The resonators are made from a Niobium film (about 160 nm thick) on a high-resistance Si substrate, and each resonator is fabricated as a meandered quarter-wavelength transmission line (one end shorts to the ground and another end is capacitively coupled to a through feedline). With a vector network analyzer we measured the transmissions of the applied microwave through the resonators at ultra-low temperature (e.g., at 20 mK), and found that their loaded quality factors are significantly high, i.e., up to 10^6. With the temperature increases slowly from the base temperature (i.e., 20 mK), we observed the resonance frequencies of the resonators are blue shifted and the quality factors are lowered slightly. In principle, this type of CPW-device can integrate a series of resonators with a common feedline, making it a promising candidate of either the data bus for coupling the distant solid-state qubits or the sensitive detector of single photons.Comment: Accepted by Chinese Science Bulleti

A genome-wide association scan for rheumatoid arthritis data by Hotelling's T2 tests

We performed a genome-wide association scan on the North American Rheumatoid Arthritis Consortium (NARAC) data using Hotelling's T2 tests, i.e., TH based on allele coding and TG based on genotype coding. The objective was to identify associations between single-nucleotide polymorphisms (SNPs) or markers and rheumatoid arthritis. In specific candidate gene regions, we evaluated the performance of Hotelling's T2 tests. Then Hotelling's T2 tests were used as a tool to identify new regions that contain SNPs showing strong associations with disease. As expected, the strongest association evidence was found in the region of the HLA-DRB1 locus on chromosome 6. In the region of the TRAF1-C5 genes, we identified two SNPs, rs2900180 and rs3761847, with the largest and the second largest TH and TG scores among all SNPs on chromosome 9. We also identified one SNP, rs2476601, in the region of the PTPN22 gene that had the largest TH score and the second largest TG score among all SNPs on chromosome 1. In addition, SNPs with the largest TH score on each chromosome were identified. These SNPs may be located in the regions of genes that have modest effects on rheumatoid arthritis. These regions deserve further investigation