Principal component analysis for second-order stationary vector time series

We extend the principal component analysis (PCA) to second-order stationary vector time series in the sense that we seek for a contemporaneous linear transformation for a $p$-variate time series such that the transformed series is segmented into several lower-dimensional subseries, and those subseries are uncorrelated with each other both contemporaneously and serially. Therefore those lower-dimensional series can be analysed separately as far as the linear dynamic structure is concerned. Technically it boils down to an eigenanalysis for a positive definite matrix. When $p$ is large, an additional step is required to perform a permutation in terms of either maximum cross-correlations or FDR based on multiple tests. The asymptotic theory is established for both fixed $p$ and diverging $p$ when the sample size $n$ tends to infinity. Numerical experiments with both simulated and real data sets indicate that the proposed method is an effective initial step in analysing multiple time series data, which leads to substantial dimension reduction in modelling and forecasting high-dimensional linear dynamical structures. Unlike PCA for independent data, there is no guarantee that the required linear transformation exists. When it does not, the proposed method provides an approximate segmentation which leads to the advantages in, for example, forecasting for future values. The method can also be adapted to segment multiple volatility processes.Comment: The original title dated back to October 2014 is "Segmenting Multiple Time Series by Contemporaneous Linear Transformation: PCA for Time Series

High dimensional stochastic regression with latent factors, endogeneity and nonlinearity

We consider a multivariate time series model which represents a high dimensional vector process as a sum of three terms: a linear regression of some observed regressors, a linear combination of some latent and serially correlated factors, and a vector white noise. We investigate the inference without imposing stationary conditions on the target multivariate time series, the regressors and the underlying factors. Furthermore we deal with the endogeneity that there exist correlations between the observed regressors and the unobserved factors. We also consider the model with nonlinear regression term which can be approximated by a linear regression function with a large number of regressors. The convergence rates for the estimators of regression coefficients, the number of factors, factor loading space and factors are established under the settings when the dimension of time series and the number of regressors may both tend to infinity together with the sample size. The proposed method is illustrated with both simulated and real data examples

Estimation of subgraph density in noisy networks

While it is common practice in applied network analysis to report various standard network summary statistics, these numbers are rarely accompanied by uncertainty quantification. Yet any error inherent in the measurements underlying the construction of the network, or in the network construction procedure itself, necessarily must propagate to any summary statistics reported. Here we study the problem of estimating the density of an arbitrary subgraph, given a noisy version of some underlying network as data. Under a simple model of network error, we show that consistent estimation of such densities is impossible when the rates of error are unknown and only a single network is observed. Accordingly, we develop method-of-moment estimators of network subgraph densities and error rates for the case where a minimal number of network replicates are available. These estimators are shown to be asymptotically normal as the number of vertices increases to infinity. We also provide confidence intervals for quantifying the uncertainty in these estimates based on the asymptotic normality. To construct the confidence intervals, a new and non-standard bootstrap method is proposed to compute asymptotic variances, which is infeasible otherwise. We illustrate the proposed methods in the context of gene coexpression networks

Testing for high-dimensional white noise using maximum cross-correlations

We propose a new omnibus test for vector white noise using the maximum absolute autocorrelations and cross-correlations of the component series. Based on an approximation by the L∞-norm of a normal random vector, the critical value of the test can be evaluated by bootstrapping from a multivariate normal distribution. In contrast to the conventional white noise test, the new method is proved to be valid for testing departure from white noise that is not independent and identically distributed. We illustrate the accuracy and the power of the proposed test by simulation, which also shows that the new test outperforms several commonly used methods, including the Lagrange multiplier test and the multivariate Box–Pierce portmanteau tests, especially when the dimension of the time series is high in relation to the sample size. The numerical results also indicate that the performance of the new test can be further enhanced when it is applied to pre-transformed data obtained via the time series principal component analysis proposed by J. Chang, B. Guo and Q. Yao (arXiv:1410.2323). The proposed procedures have been implemented in an R package

Confidence regions for entries of a large precision matrix

We consider the statistical inference for high-dimensional precision matrices. Specifically, we propose a data-driven procedure for constructing a class of simultaneous confidence regions for a subset of the entries of a large precision matrix. The confidence regions can be applied to test for specific structures of a precision matrix, and to recover its nonzero components. We first construct an estimator for the precision matrix via penalized node-wise regression. We then develop the Gaussian approximation to approximate the distribution of the maximum difference between the estimated and the true precision coefficients. A computationally feasible parametric bootstrap algorithm is developed to implement the proposed procedure. The theoretical justification is established under the setting which allows temporal dependence among observations. Therefore the proposed procedure is applicable to both independent and identically distributed data and time series data. Numerical results with both simulated and real data confirm the good performance of the proposed method

Measuring the black hole masses in accreting X-ray binaries by detecting the Doppler orbital motion of their accretion disk wind absorption lines

So far essentially all black hole masses in X-ray binaries have been obtained by observing the companion star's velocity and light curves as functions of the orbital phase. However a major uncertainty is the estimate of the orbital inclination angle of an X-ray binary. Here we suggest to measure the black hole mass in an X-ray binary by measuring directly the black hole's orbital motion, thus obtaining the companion to black hole mass ratio. In this method we assume that accretion disk wind moves with the black hole and thus the black hole's orbital motion can be obtained from the Doppler velocity of the absorption lines produced in the accretion disk wind. We validate this method by analyzing the Chandra/HETG observations of GRO J1655-40, in which the black hole orbital motion with line of sight velocity of 90.8 (+-11.3) km/s, inferred from the Doppler velocity of disk-wind absorption lines, is consistent with the prediction from its previously measured system parameters. We obtain the black hole mass of 5.41 (+0.98, -0.57) solar masses and system inclination of 72.0 (+7.8, -7.5) degrees in GRO J1655-40. Additional observations of this source covering more orbital phases can improve estimates on its system parameters substantially. We then apply the method to the black hole X-ray binary LMC X-3 observed with HST/COS near orbital phase 0.75. We find that the disk-wind absorption lines of CIV doublet were shifted to about 50 km/s, which yields a companion-to-black-hole mass ratio of 0.6 for an assumed disk wind velocity of -400 km/s. Additional observations covering other orbital phases (0.25 in particular) are crucial to ease this assumption and then to directly constrain the mass ratio. This method in principle can also be applied to any accreting compact objects with detectable accretion disk wind absorption line features.Comment: 8 pages, 4 figures, 1 table. Accepted for publication in MNRA

Confidence regions for entries of a large precision matrix

We consider the statistical inference for high-dimensional precision matrices. Specifically, we propose a data-driven procedure for constructing a class of simultaneous confidence regions for a subset of the entries of a large precision matrix. The confidence regions can be applied to test for specific structures of a precision matrix, and to recover its nonzero components. We first construct an estimator for the precision matrix via penalized node-wise regression. We then develop the Gaussian approximation to approximate the distribution of the maximum difference between the estimated and the true precision coefficients. A computationally feasible parametric bootstrap algorithm is developed to implement the proposed procedure. The theoretical justification is established under the setting which allows temporal dependence among observations. Therefore the proposed procedure is applicable to both independent and identically distributed data and time series data. Numerical results with both simulated and real data confirm the good performance of the proposed method

Bending-induced isostructural transitions in ultrathin layers of van der Waals ferrielectrics

Using Landau-Ginzburg-Devonshire (LGD) phenomenological approach we analyze the bending-induced re-distribution of electric polarization and field, elastic stresses and strains inside ultrathin layers of van der Waals ferrielectrics. We consider a CuInP2S6 (CIPS) thin layer with fixed edges and suspended central part, the bending of which is induced by external forces. The unique aspect of CIPS is the existence of two ferrielectric states, FI1 and FI2, corresponding to big and small polarization values, which arise due to the specific four-well potential of the eighth-order LGD functional. When the CIPS layer is flat, the single-domain FI1 state is stable in the central part of the layer, and the FI2 states are stable near the fixed edges. With an increase of the layer bending below the critical value, the sizes of the FI2 states near the fixed edges decreases, and the size of the FI1 region increases. When the bending exceeds the critical value, the edge FI2 states disappear being substituted by the FI1 state, but they appear abruptly near the inflection regions and expand as the bending increases. The bending-induced isostructural FI1-FI2 transition is specific for the bended van der Waals ferrielectrics described by the eighth (or higher) order LGD functional with consideration of linear and nonlinear electrostriction couplings. The isostructural transition, which is revealed in the vicinity of room temperature, can significantly reduce the coercive voltage of ferroelectric polarization reversal in CIPS nanoflakes, allowing for the curvature-engineering control of various flexible nanodevices.Comment: 26 pages, 7 figures and Appendices A-

Short- and long-term effects of manganese, zinc and copper ions on nitrogen removal in nitritation-anammox process

This study provided a deep insight into the impacts of trace elements (Mn2+, Zn2+ and Cu2+) on nitritation-anammox process. For short-term exposure, all the three elements could improve the nitrogen removal rate (NRR) and the optimal concentrations were 2.0 mg/L, 2.0 mg/L and 0.5 mg/L for Mn2+, Zn2+ and Cu2+, respectively. Accordingly, the NRR were enhanced 54.62%, 45.93% and 44.09%. The long-term experiments were carried out in lab-scale sequencing batch reactors. The surprising results showed that only Mn2+ addition could enhance the long-term nitritation-anammox process, and the NRR increased from 0.35 ± 0.01 kg N/m3/d (control, no extra trace element addition) to 0.49 ± 0.03 kg N/m3/d. Vice versa, the amendment of Zn2+ reduced the NRR to 0.28 ± 0.02 kg N/m3/d, and Cu2+ had no significant effect on the NRR (0.36 ± 0.01 kg N/m3/d). From the analysis of microbial community structure, it was explained by the increasing abundance of anaerobic ammonium oxidizing bacteria (AnAOB) only in Mn2+ treatment, whereas Zn2+ predominantly promoted ammonium oxidizing bacteria (AOB). Additionally, the majority of Mn2+ was identified inside AnAOB cells, and Zn2+ and Cu2+ were mainly located in AOB. Our results indicated the synergistic effects of trace elements on nitritation-anammox, both short-term encouraging activities of AnAOB and long-term altering microbial community structure. This work implies the importance of trace elements addition in nitritation-anammox process