Strong approximation for cross-covariances of linear variables with long-range dependence

Suppose {_k, −∞ < k < ∞} is an independent, not necessarily identically distributed sequence of random variables, and {c_j}^∞_j=0, {d_j}^∞_j=0 are sequences of real numbers such that Σ_jc²_j < ∞, Σ_jd²_j < ∞. Then, under appropriate moment conditions on {_k, −∞ < k < ∞}, y_k Σ^∞_j=0c_jk-j, z_k Σ^∞_j=0d_jk-j exist almost surely and in ⁴ and the question of Gaussian approximation to S_[t]Σ^[t]_k=1 (y_k z_k − E{y_k z_k}) becomes of interest. Prior to this work several related central limit theorems and a weak invariance principle were proven under stationary assumptions. In this note, we demonstrate that an almost sure invariance principle for S_[t], with error bound sharp enough to imply a weak invariance principle, a functional law of the iterated logarithm, and even upper and lower class results, also exists. Moreover, we remove virtually all constraints on _k for “time” k ≤ 0, weaken the stationarity assumptions on {_k, −∞ < k < ∞}, and improve the summability conditions on {c_j}^∞_j=0, {d_j}^∞_j=0 as compared to the existing weak invariance principle. Applications relevant to this work include normal approximation and almost sure fluctuation results in sample covariances (let d_j = c_j-m for j ≥ m and otherwise 0), quadratic forms, Whittle's and Hosoya's estimates, adaptive filtering and stochastic approximation.     

Least-squares linear estimation of signals from observations with Markovian delays

The least-squares linear estimation of signals from randomly delayed measurements is addressed when the delay is modeled by a homogeneous Markov chain. To estimate the signal, recursive filtering and fixed-point smoothing algorithms are derived, using an innovation approach, assuming that the covariance functions of the processes involved in the observation equation are known. Recursive formulas for filtering and fixed-point smoothing error covariance matrices are obtained to measure the goodness of the proposed estimators.     

Eigenimage-Based Facial Recognition Technique Using Gradient Covariance

In this paper, we propose a novel eigenface-based face recognition approach. First we describe a continuous model for facial feature extraction, involving the covariance operator associated with gradient of the 2-D image function. Then we provide a discretized version of this model. Face identification and verification procedures, using a supervised classification technique, are also proposed.     

Covariance selection by thresholding the sample correlation matrix

This article shows that when the nonzero coefficients of the population correlation matrix are all greater in absolute value than (_C1logp/n)^1/2${(C_{1}logp/n)}^{1/2}$ for some constant _C1$C_1$, we can obtain covariance selection consistency by thresholding the sample correlation matrix. Furthermore, the rate (logp/n)^1/2${(logp/n)}^{1/2}$ is shown to be optimal.     

Student's t Vector Random Fields with Power-Law and Log-Law Decaying Direct and Cross Covariances

This article deals with the Student's t vector random field, which is formulated as a scale mixture of Gaussian vector random fields, and whose finite-dimensional distributions decay in power-law and have heavy tails. There are two classes of Student's t vector random fields, one with second-order moments, and the other without a second-order moment. A Cauchy vector random field is an example of Student's t vector random fields without a first-order moment, and is also an example of Stable vector random fields. A second-order Student's t vector random field allows for any given correlation structure, just as a Gaussian vector random field does. We propose four types of covariance matrix structures for second-order Student's t vector random fields, which decay in power-law or log-law.     

Discovering Sparse Covariance Structures With the Isomap

Regularization of covariance matrices in high dimensions usually either is based on a known ordering of variables or ignores the ordering entirely. This article proposes a method for discovering meaningful orderings of variables based on their correlations using the Isomap, a nonlinear dimension reduction technique designed for manifold embeddings. These orderings are then used to construct a sparse covariance estimator, which is block-diagonal and/or banded. Finding an ordering to which banding can be applied is desirable because banded estimators have been shown to be consistent in high dimensions. We show that in situations where the variables do have such a structure, the Isomap does very well at discovering it, and the resulting regularized estimator performs better for covariance estimation than other regularization methods that ignore variable order, such as thresholding. We also propose a bootstrap approach to constructing the neighborhood graph used by the Isomap, and show it leads to better estimation. We illustrate our method on data on protein consumption, where the variables (food types) have a structure but it cannot be easily described a priori, and on a gene expression dataset. Supplementary materials are available online.     

Bayesian Variable Selection for Logistic Models Using Auxiliary Mixture Sampling

This article presents a Markov chain Monte Carlo algorithm for both variable and covariance selection in the context of logistic mixed effects models. This algorithm allows us to sample solely from standard densities with no additional tuning. We apply a stochastic search variable approach to select explanatory variables as well as to determine the structure of the random effects covariance matrix.

Prior determination of explanatory variables and random effects is not a prerequisite because the definite structure is chosen in a data-driven manner in the course of the modeling procedure. To illustrate the method, we give two bank data examples.     

Nonstationary Positive Definite Tapering On The Plane

A common problem in spatial statistics is to predict a random field f at some spatial location t ₀ using observations f(t ₁), …, f(t_n ) at . Recent work by Kaufman et al. and Furrer et al. studies the use of tapering for reducing the computational burden associated with likelihood-based estimation and prediction in large spatial datasets. Unfortunately, highly irregular observation locations can present problems for stationary tapers. In particular, there can exist local neighborhoods with too few observations for sufficient accuracy, while others have too many for computational tractability. In this article, we show how to generate nonstationary covariance tapers T(s, t) such that the number of observations in {t: T(s, t) > 0} is approximately a constant function of s. This ensures that tapering neighborhoods do not have too many points to cause computational problems but simultaneously have enough local points for accurate prediction. We focus specifically on tapering in two dimensions where quasi-conformal theory can be used. Supplementary materials for the article are available online.     

Numerical simulation and inversion validation of the real correlation random wind field generation for coherent Doppler LiDAR

Simulation of the coherent Doppler LiDAR signal requires accurate computation of homogeneous random wind fields. Based on complex random processes with specified spatial statistics given by the covariance function, an improved real correlation random wind field algorithm is proposed for real random processes, the simulation results are compared with the given covariance function and the real correlation algorithm conforms to the given covariance function quite well.