non-Gaussian Vector Research Articles

Modeling non-Gaussian time-varying vector autoregressive processes by particle filtering

We present a novel and general methodology for modeling time-varying vector autoregressive processes which are widely used in many areas such as modeling of chemical processes, mobile communication channels and biomedical signals. In the literature, most work utilize multivariate Gaussian models for the mentioned applications, mainly due to the lack of efficient analytical tools for modeling with non-Gaussian distributions. In this paper, we propose a particle filtering approach which can model non-Gaussian autoregressive processes having cross-correlations among them. Moreover, time-varying parameters of the process can be modeled as the most general case by using this sequential Bayesian estimation method. Simulation results justify the performance of the proposed technique, which potentially can model also Gaussian processes as a sub-case.

Block Simulation of Multiple Correlated Variables

Numerical representations of multivariate natural phenomena, including characteristics of mineral deposits, petroleum reservoirs and geo-environmental attributes, need to consider and reproduce the spatial relationships between correlated attributes of interest. There are, however, only a few methods that can practically jointly simulate large size multivariate fields. This paper presents a method for the conditional simulation of a non-Gaussian vector random field directly on block support. The method is derived from the group sequential simulation paradigm and the direct block simulation algorithm which leads to the efficient joint simulation of large multivariate datasets jointly and directly on the block support. This method is a multistage process. First, a vector random function is orthogonalized with minimum/maximum autocorrelation factors (MAF). Blocks are then simulated by performing LU simulation on their discretized points, which are later back-rotated and averaged to yield the block value. The internal points are then discarded and only the block value is stored in memory to be used for further conditioning through a joint LU, resulting in the reduction of memory requirements. The method is termed direct block simulation with MAF or DBMAFSIM. A proof of the concept using an exhaustive data set demonstrates the intricacies and the performance of the proposed approach.

Multivariate Distributions with Specified Marginals: Applications to Wind Engineering

Bounds on and approximations of the distribution of a non-Gaussian vector X specified partially by the marginal distributions Fi of its coordinates Xi or the marginal distributions Fi and some information on the correlation structure of X are developed. These results are used to: (1) bound and approximate the distribution of the largest coordinate of X, that is, the random variable Y=maxi{Xi}; and (2) evaluate current practice in wind engineering based on models for directional and directionless wind speeds (that is, yearly maximum wind speeds recorded in several directions and irrespective of direction) that are postulated independently of each other. It is shown that, although current directional and directionless wind speed models are inconsistent with probability theory, they can deliver similar design wind speeds if calibrated to very long wind speed records but their predictions can differ significantly if based on typically available records.

Asymptotic minimaxity of false discovery rate thresholding for sparse exponential data

We apply FDR thresholding to a non-Gaussian vector whose coordinates Xi, i=1, …, n, are independent exponential with individual means μi. The vector μ=(μi) is thought to be sparse, with most coordinates 1 but a small fraction significantly larger than 1; roughly, most coordinates are simply ‘noise,’ but a small fraction contain ‘signal.’ We measure risk by per-coordinate mean-squared error in recovering log(μi), and study minimax estimation over parameter spaces defined by constraints on the per-coordinate p-norm of log(μi), $\frac{1}{n}\sum_{i=1}^{n}\,\log^{p}(\mu_{i})\leq \eta^{p}$. We show for large n and small η that FDR thresholding can be nearly minimax. The FDR control parameter 0<q<1 plays an important role: when q≤1/2, the FDR estimator is nearly minimax, while choosing a fixed q>1/2 prevents near minimaxity. These conclusions mirror those found in the Gaussian case in Abramovich et al. [Ann. Statist. 34 (2006) 584–653]. The techniques developed here seem applicable to a wide range of other distributional assumptions, other loss measures and non-i.i.d. dependency structures.

Optimality of KLT for High-Rate Transform Coding of Gaussian Vector-Scale Mixtures: Application to Reconstruction, Estimation, and Classification

The Karhunen-Loeacuteve transform (KLT) is known to be optimal for high-rate transform coding of Gaussian vectors for both fixed-rate and variable-rate encoding. The KLT is also known to be suboptimal for some non-Gaussian models. This paper proves high-rate optimality of the KLT for variable-rate encoding of a broad class of non-Gaussian vectors: Gaussian vector-scale mixtures (GVSM), which extend the Gaussian scale mixture (GSM) model of natural signals. A key concavity property of the scalar GSM (same as the scalar GVSM) is derived to complete the proof. Optimality holds under a broad class of quadratic criteria, which include mean-squared error (MSE) as well as generalized f-divergence loss in estimation and binary classification systems. Finally, the theory is illustrated using two applications: signal estimation in multiplicative noise and joint optimization of classification/reconstruction systems

Extremes of Gaussian and non-Gaussian vector processes: a geometric approach

An alternative approach to direct application of the out-crossing rate for Gaussian and non-Gaussian vector processes is considered. Multivariate extreme-value distributions are established, and the properties of these distributions are investigated. The feasibility of the “expected extreme” ellipsoids are considered for the same categories of vector-processes in relation to load combination problems. Focus is presently on two-dimensional vector processes, but higher-dimensional extensions are also established. Examples of extreme hyper-ellipsoids in three dimensions are presented for both Gaussian and non-Gaussian processes.

Asymptotic Expansion of the Distribution of a Homogeneous Functional of a Strictly Stable Random Vector. II

Let u be a strictly stable non-Gaussian vector with the exponent of stability $\alpha\ge 1$, taking on values in a separable Banach space B. Let $h\cl B\to\bR$ be a smooth homogeneous functional and let F be the distribution function of the random variable $h(u)$. For the function $1-F(x)$ we obtain an asymptotic expansion of the form $\sum_{k=1}^n c_kx^{-k\alpha}+O(x^{-(n+1)\alpha})$, $x\to\iy$ (n~is determined by the smoothness of h). To establish the expansion we use a new approach which is based on the decomposition of the distribution into the sum of linear functionals.

Simulation of homogeneous nonGaussian stochastic vector fields

A spectral representation-based simulation methodology is proposed to generate sample functions of a multi-variate, multi-dimensional, nonGaussian stochastic vector field, according to a prescribed cross-spectral density matrix and prescribed (nonGaussian) marginal probability distribution functions. The proposed methodology starts by generating a Gaussian vector field that is then transformed into the desired nonGaussian one using a memoryless nonlinear transformation in conjunction with an iterative scheme. The generation of the Gaussian vector field is performed taking advantage of the Fast Fourier Transform technique for great computational efficiency. The special case of simulation of nonGaussian vector fields modeling material properties is examined, mainly from the point of view of certain simplifying assumptions that can be made for such random media. Finally, a numerical example involving a tri-variate, two-dimensional, nonGaussian stochastic vector field is presented in order to demonstrate the capabilities and the efficiency of the proposed methodology.

Testing cointegration in infinite order vector autoregressive processes

This paper studies test procedures which can be used to determine the cointegrating rank in infinite order vector autoregressive processes. The considered tests are analogs or close versions of previous likelihood ratio tests obtained for finite-order Gaussian vector autoregressive processes. It is shown that the use of the likelihood ratio tests is justified even when the data are generated by an infinite order non-Gaussian vector autoregressive process. New tests are also developed for cases where intercept terms are included in the cointegrating relations. These tests are based on a new approach of estimating the intercept terms. They have the property that, under the null hypothesis, the same asymptotic distribution theory applies as in the case where the values of the intercept terms are a priori known and not estimated. A limited simulation study indicates that the new tests can be considerably more powerful than their previous counterparts.

Identification and deconvolution of multichannel linear non-Gaussian processes using higher order statistics and inverse filter criteria

This paper is concerned with the problem of estimation and deconvolution of the matrix impulse response function of a multiple-input multiple-output (MIMO) system given only the measurements of the vector output of the system. The system is assumed to be driven by a temporally i.i.d. and spatially independent non-Gaussian vector sequence (which is not observed). An iterative, inverse filter criteria-based approach is developed using the third-order or the fourth-order normalized cumulants of the inverse filtered data at zero lag. Stationary points of the proposed cost functions are investigated. The approach is input iterative, i.e., the input sequences are extracted and removed one by one. The matrix impulse response is then obtained by cross correlating the extracted inputs with the observed outputs. Identifiability conditions are analyzed. The strong consistency of the proposed approach is also briefly discussed. Computer simulation examples are presented to illustrate the proposed approaches.

Nonparametric Approach for Non-Gaussian Vector Stationary Processes

Suppose that {z(t)} is a non-Gaussian vector stationary process with spectral density matrixf(λ). In this paper we consider the testing problemH:∫π−πK{f(λ)}dλ=cagainstA:∫π−πK{f(λ)}dλ≠c, whereK{·} is an appropriate function andcis a given constant. For this problem we propose a testTnbased on ∫π−πK{f(λ)}dλ=c, wheref(λ) is a nonparametric spectral estimator off(λ), and we define an efficacy ofTnunder a sequence of nonparametric contiguous alternatives. The efficacy usually depnds on the fourth-order cumulant spectraf4Zofz(t). If it does not depend onf4Z, we say thatTnis non-Gaussian robust. We will give sufficient conditions forTnto be non-Gaussian robust. Since our test setting is very wide we can apply the result to many problems in time series. We discuss interrelation analysis of the components of {z(t)} and eigenvalue analysis off(λ). The essential point of our approach is that we do not assume the parametric form off(λ). Also some numerical studies are given and they confirm the theoretical results.

Discriminant analysis for non-gaussian vector stationary processes

This paper is concerned with spectral discrimination for non-Gaussian vector stationary time series. The usual discriminant rule is to maximize the (quasi) likelihood of observations derived on Gaussian assumptions, although no such distributional assumptions are made. In this paper, we introduce an alternative approach based on the disparity measure between spectral density matrices.

A generalization of the entropy power inequality with applications

The authors prove the following generalization of the entropy power inequality: h(ax)>or=h(Ax) where h(.) denotes (joint-) differential-entropy x=x/sub 1/...x/sub n/, is a random vector with independent components, x=x...x/sub n/, is a Gaussian vector with independent components such that h(x/sub i/)=h(x/sub i/), i=1...n, and A is any matrix. This generalization of the entropy-power inequality is applied to show that a non-Gaussian vector with independent components becomes closer to Gaussianity after a linear transformation, where the distance to Gaussianity is measured by the information divergence. Another application is a lower bound, greater than zero, for the mutual-information between nonoverlapping spectral components of a non-Gaussian white process. They also describe a dual generalization of the Fisher information inequality. >

Rotary Cross-Bispectra and Energy Transfer Functions Between Non-Gaussian Vector Processes. II. Winds and Currents off the Oregon Coast

Abstract Application of rotary cross-bispectra and energy transfer functions to a set of wind and current data measured from the Totem research buoy shows these elements to be non-Gaussian and that nonlinear interactions do occur between the wind stress and current at 14 m depth. Such transfers account for 35% of the total energy estimates and for 100% of the estimates at the inertial frequency. For the latter, the most effective set of nonlinear interacting components include half-day and quarter-day components of wind stress. No energy transfers linearly. The effect of nonlinear interaction in broadening the current response spectrum is noted.

Rotary Cross-Bispectra and Energy Transfer Functions Between Non-Gaussian Vector Processes I. Development and Example

Abstract Bispectrum and cross-bispectrum analyses of the rotary components of stationary random vector processes are more easily interpreted than similar analyses of their scalar components, and have the advantage that the bispectral estimates are invariant to coordinate rotation. Application to some wind-ocean current data shows these to be non-Gaussian and subject to significant nonlinear coupling over a wide range of interacting triplets of rotary components. A set of complex-valued energy transfer functions are developed by which the magnitudes of the linear and quadratic interactions may be compared.

Monte Carlo study of the optimal non-linear estimator: linear systems with non-gaussian initial states †

The optimal, in the mean-square sense, estimate of state vector of a linear discrete system that is excited by white zero mean gaussian noise and that has non-gaussian initial state vector is presented. Both the optimal estimate and the corresponding error covariance matrix are given. It is shown that the optimal estimator consists of two parts : a linear estimator which is a Kalman filter and a non-linear part which is a parameter estimator. In addition, the a posteriori probability density function, p(x(k)λk), is also given. Finally, a suboptimal procedure that reduces the computational requirements is presented. The results of extensive digital computer simulations including Monte Carlo study have been presented to establish that the non-linear filter presented here is far superior to the best linear Kalman filter. A practical filter design criterion for utilizing this non-linear filter with reduced data processing requirements is also given.

On the Number of Exits Across the Boundary of a Region by a Vector Stochastic Process

On the Number of Exits Across the Boundary of a Region by a Vector Stochastic Process