Second-order Stationary Processes Research Articles

Adaptive neural control for a class of random nonlinear Markov jump multi-agent systems with full state constraints

This paper proposes a consensus control protocol based on the adaptive backstepping technique for a class of random Markov jump multi-agent systems (MASs) with full state constraints. Each agent is described by the high-order random nonlinear uncertain system driven by random differential equations, where the random noise is the second-order stationary stochastic process. First, a distributed tracking controller is designed for Markov jump MASs, effectively handling the interaction and coupling terms between agents. Second, an appropriate tan-type barrier Lyapunov function is selected to keep the agents’ states from the violation of constraint boundaries, and the unknown nonlinear terms with Markov jump parameters are approximated by neural networks (NNs) theory. Moreover, to address the differential explosion problem, the extended state observer (ESO) is first introduced instead of employing NNs approximation or command filtering techniques. Finally, the exponentially noise-to-state stability in mean square is proved rigorously through the Lyapunov method, which guarantees all signals of the closed-loop system are bounded in probability and the tracking error converges to a small neighborhood around zero. Simulations are given to demonstrate the feasibility of theoretical results.

Adaptive Neural Control for a Class of Random Fractional-Order Multi-Agent Systems with Markov Jump Parameters and Full State Constraints

Based on an adaptive neural control scheme, this paper investigates the consensus problem of random Markov jump multi-agent systems with full state constraints. Each agent is described by the fractional-order random nonlinear uncertain system driven by random differential equations, where the random noise is the second-order stationary stochastic process. First, in order to deal with the unknown functions with Markov jump parameters, a radial basis function neural network (RBFNN) structure is introduced to achieve approximation. Second, for the purpose of keeping the agents’ states from violating the constraint boundary, the tan-type barrier Lyapunov function is employed. By using the stochastic stability theory and adopting the backstepping technique, a novel adaptive neural control design method is presented. Furthermore, to cope with the differential explosion problem in the design course, the extended state observer (ESO) is developed instead of neural network (NN) approximation or command filtering techniques. Finally, the exponentially noise-to-state stability in the mean square is analyzed rigorously by the Lyapunov method, which guarantees the consensus of the considered multi-agent systems and all the agents’ outputs are bounded in probability. Two simulation examples are provided to verify the effectiveness of the suggested control strategy.

Distributed convex optimization for nonlinear multi-agent systems disturbed by a second-order stationary process over a digraph

Distributed convex optimization for nonlinear multi-agent systems disturbed by a second-order stationary process over a digraph

Statistics of the Stokes Parameters for the Signal Scattered by Layered Structures With an Arbitrary Number of Slightly Rough Interfaces

Horizontally stratified structures are commonly used to represent naturally occurring structures, such as soils. The electromagnetic signature of such a medium illuminated by a radar and the polarization state of the scattered wave are fully determined by the knowledge of the four Stokes parameters. In this paper, we determine the statistics of the four Stokes parameters for the signal scattered by layered structures with an arbitrary number of slightly rough interfaces. The rough interfaces are realizations of second-order stationary centered Gaussian stochastic processes and the layered structure is illuminated by an elliptically polarized monochromatic wave. The zenithal and azimuthal components of the far scattered electric field are derived from the first-order small perturbation method. The derivation leads to a multivariate Gaussian model for the underlying complex scattered amplitudes and we establish the closed-form expressions of the probability density function, the cumulative density function and the first- and second-order moments for the four Stokes parameters. For an observation direction outside the incidence plane, we establish the condition on the incidence wave parameters for which the zenithal and azimuthal components are uncorrelated. For an air / snow cover / frozen soil / unfrozen soil structure, we analyze the marginal probabilities and validate the theory by comparison with Monte-Carlo simulations. More generally, when the two complex components of the field scattered by the illuminated zone are Gaussian random variables, these statistics offer possibilities for in-depth investigating the polarization of scattering processes from random media.

New Robust Cross-Variogram Estimators and Approximations of Their Distributions Based on Saddlepoint Techniques

Let Z(s)=(Z1(s),…,Zp(s))t be an isotropic second-order stationary multivariate spatial process. We measure the statistical association between the p random components of Z with the correlation coefficients and measure the spatial dependence with variograms. If two of the Z components are correlated, the spatial information provided by one of them can improve the information of the other. To capture this association, both within components of Z(s) and across s, we use a cross-variogram. Only two robust cross-variogram estimators have been proposed in the literature, both by Lark, and their sample distributions were not obtained. In this paper, we propose new robust cross-variogram estimators, following the location estimation method instead of the scale estimation one considered by Lark, thus extending the results obtained by García-Pérez to the multivariate case. We also obtain accurate approximations for their sample distributions using saddlepoint techniques and assuming a multivariate-scale contaminated normal model. The question of the independence of the transformed variables to avoid the usual dependence of spatial observations is also considered in the paper, linking it with the acceptance of linear variograms and cross-variograms.

On Hyperbolic Decay of Prediction Error Variance for Deterministic Stationary Sequences

One of the main problems in prediction theory of second-order stationary processes, called direct prediction problem, is to describe the asymptotic behavior of the best linear mean squared one-step ahead prediction error variance in predicting the value $$X(0)$$ of a stationary process $$X(t)$$ by the observed past of finite length $$n$$ as $$n$$ goes to infinity, depending on the regularity nature (deterministic or non-deterministic) of the underlying observed process $$X(t)$$ . In this paper, we obtain sufficient conditions for hyperbolic decay of prediction error variance for deterministic stationary sequences, generalizing a result obtained by Rosenblatt [31].

A spatial concordance correlation coefficient with an application to image analysis

In this work we define a spatial concordance coefficient for second-order stationary processes. This problem has been widely addressed in a non-spatial context, but here we consider a coefficient that for a fixed spatial lag allows one to compare two spatial sequences along a 45°line. The proposed coefficient was explored for the bivariate Matérn and Wendland covariance functions. The asymptotic normality of a sample version of the spatial concordance coefficient for an increasing domain sampling framework was established for the Wendland covariance function. To work with large digital images, we developed a local approach for estimating the concordance that uses local spatial models on non-overlapping windows. Monte Carlo simulations were used to gain additional insights into the asymptotic properties for finite sample sizes. As an illustrative example, we applied this methodology to two similar images of a deciduous forest canopy. The images were recorded with different cameras but similar fields-of-view and within minutes of each other. Our analysis showed that the local approach helped to explain a percentage of the non-spatial concordance and provided additional information about its decay as a function of the spatial lag.

Universal kriging of corn plant data under isotropic power type variogram model

Kriging is a well known notion in geostatistics which aim at the construction of optimal prediction for unobserved quantities of interest. In this paper we derive optimal prediction of a second-order stationary spatial process by minimizing the quadratic risk or mean square error of the predictor subject to some additional conditions. The coefficients of the linear predictor are calculated by a formula which is obtained by applying Lagrange method. The spatial process under study is assumed to be isotropic with a variogram model belongs to quadratic family. Least squares estimation to the parameters of the variogram model is calculated by graphical method. The validity of the mean function is investigated by utilizing asymptotic test based on the partial sums process of weighted least squares residuals. The application of the method to spatial process of rate of growth of corn plants results in the kriging map of the process. The maps are generated under two different variogram models: power and linear (Tent) models.

The debiased Whittle likelihood

The Whittle likelihood is a widely used and computationally efficient pseudolikelihood. However, it is known to produce biased parameter estimates with finite sample sizes for large classes of models. We propose a method for debiasing Whittle estimates for second-order stationary stochastic processes. The debiased Whittle likelihood can be computed in the same O(n log n) operations as the standard Whittle approach. We demonstrate the superior performance of our method in simulation studies and in application to a large-scale oceanographic dataset, where in both cases the debiased approach reduces bias by up to two orders of magnitude, achieving estimates that are close to those of the exact maximum likelihood, at a fraction of the computational cost. We prove that the method yields estimates that are consistent at an optimal convergence rate of n(-1/2) for Gaussian processes and for certain classes of non-Gaussian or nonlinear processes. This is established under weaker assumptions than in the standard theory, and in particular the power spectral density is not required to be continuous in frequency. We describe how the method can be readily combined with standard methods of bias reduction, such as tapering and differencing, to further reduce bias in parameter estimates.

Statistical inference for quantiles in the frequency domain

For second-order stationary processes, the spectral distribution function is uniquely determined by the autocovariance function of the process. We define the quantiles of the spectral distribution function in frequency domain. The estimation of quantiles for second-order stationary processes is considered by minimizing the so-called check function. The quantile estimator is shown to be asymptotically normal. We also consider a hypothesis testing for quantiles in frequency domain and propose a test statistic associated with our quantile estimator, which asymptotically converges to standard normal under the null hypothesis. The finite sample performance of the quantile estimator is shown in our numerical studies.

Space-time mesh adaptation for solute transport in randomly heterogeneous porous media

We assess the impact of an anisotropic space and time grid adaptation technique on our ability to solve numerically solute transport in heterogeneous porous media. Heterogeneity is characterized in terms of the spatial distribution of hydraulic conductivity, whose natural logarithm, Y, is treated as a second-order stationary random process. We consider nonreactive transport of dissolved chemicals to be governed by an Advection Dispersion Equation at the continuum scale. The flow field, which provides the advective component of transport, is obtained through the numerical solution of Darcy's law. A suitable recovery-based error estimator is analyzed to guide the adaptive discretization. We investigate two diverse strategies guiding the (space-time) anisotropic mesh adaptation. These are respectively grounded on the definition of the guiding error estimator through the spatial gradients of: (i) the concentration field only; (ii) both concentration and velocity components. We test the approach for two-dimensional computational scenarios with moderate and high levels of heterogeneity, the latter being expressed in terms of the variance of Y. As quantities of interest, we key our analysis towards the time evolution of section-averaged and point-wise solute breakthrough curves, second centered spatial moment of concentration, and scalar dissipation rate. As a reference against which we test our results, we consider corresponding solutions associated with uniform space-time grids whose level of refinement is established through a detailed convergence study. We find a satisfactory comparison between results for the adaptive methodologies and such reference solutions, our adaptive technique being associated with a markedly reduced computational cost. Comparison of the two adaptive strategies tested suggests that: (i) defining the error estimator relying solely on concentration fields yields some advantages in grasping the key features of solute transport taking place within low velocity regions, where diffusion-dispersion mechanisms are dominant; and (ii) embedding the velocity field in the error estimator guiding strategy yields an improved characterization of the forward fringe of solute fronts which propagate through high velocity regions.

Comparison of Some Approaches to Determine Spatial Dependence of Soil Properties

Knowledge of variability and spatial structure of soil properties is essential for optimal design for collecting soil samples and effectively applying management decisions in the field. The objective of this study is to compare some approaches for characterizing, and comparing spatial dependence of isotropic second-order stationary processes. The evaluated approaches are the nugget to sill ratio (NR), normalized (by fitted sill) semivariogram, correlograms, and two integral scales. Soil samples, collected at a regular 50 m × 50 m grid from 0-15 cm depths, were analyzed for sand and clay, bulk density (b), saturated hydraulic conductivity (Ks), wilting point, available water content (AWC), pH, electrical conductivity (EC), nitrate-nitrogen (NO3- N), and chloride (Cl) were determined. Geostatistical software (GS+, Gamma Design Software, Plainwell, MI) was used to estimate the variance structure of various measured soil properties. Analysis include using data on the spatial variability of various properties from four published studies. NR displayed spatial dependence ignoring the influence of range, normalized semivariogram and correlogram provided the visual comparison, and both integral scales incorporated the influence of range and provided single number spatial dependence summaries. Either of the integral scale formulations can be used to characterize the spatial dependence of soil properties from agricultural fields.

STATIONARY RESPONSES OF GEAR SYSTEMS WITH COMBINED PERIODICALLY TIME-VARYING STIFFNESS AND STOCHASTIC EXTERNAL EXCITATIONS

The stationary dynamic response of a geared system with time-varying mesh stiffness subjected to stationary stochastic external excitation is examined in this paper. To this end, a spur gear pair model with periodical time-varying mesh stiffness is extended to include stochastic external force excitations. These random forces are introduced as second-order stationary and ergodic processes. In order to compute the stationary response, a very efficient method, called the iterative spectral method is used. This method is derived in the frequency domain and provides the explicit power spectral density of the response. The PSD response is expressed as a function of the bispectrum which is the bilinear Fourier transform of the bitemporal impulse response of the parametric system.

Robust Estimation of Sparse Narrowband Spectra from Neuronal Spiking Data.

Characterizing the spectral properties of neuronal responses is an important problem in computational neuroscience, as it provides insight into the spectral organization of the underlying functional neural processes. Although spectral analysis techniques are widely used in the analysis of noninvasive neural recordings such as EEG, their application to spiking data is limited due to the binary and nonlinear nature of neuronal spiking. In this paper, we address the problem of estimating the power spectral density of the neural covariate driving the spiking statistics of a neuronal population from binary observations. We consider a neuronal ensemble spiking according to Bernoulli statistics, for which the conditional intensity function is given by the logistic map of a harmonic second-order stationary process with sparse narrowband spectra. By employing sparsity-promoting priors, we compute the maximum a posteriori estimate of the power spectral density of the process from the binary spiking observations. Furthermore, we construct confidence intervals for these estimates by an efficient posterior sampling procedure. We provide simulation studies which reveal that our method outperforms the existing methods for extracting the frequency content of spiking data. Application of our method to clinically recorded spiking data from a patient under general anesthesia reveals a striking resemblance between our estimated power spectral density and that of the local field potential signal. This result corroborates existing findings regarding the salient role of the local field potential as a major neural covariate of rhythmic cortical spiking activity under anesthesia. Our technique allows us to analyze the harmonic structure of spiking activity in a robust fashion, independently of the local field potentials, and without any prior assumption of the spectral spread and content of the underlying neural processes. Other than its usage in the spectral analysis of neuronal spiking data, our technique can be applied to a wide variety of binary data, such as heart beat data, in order to obtain a robust spectral representation.

The Uncertainty of Storm Season Changes: Quantifying the Uncertainty of Autocovariance Changepoints

In oceanography, there is interest in determining storm season changes for logistical reasons such as equipment maintenance scheduling. In particular, there is interest in capturing the uncertainty associated with these changes in terms of the number and location of them. Such changes are associated with autocovariance changes. This article proposes a framework to quantify the uncertainty of autocovariance changepoints in time series motivated by this oceanographic application. More specifically, the framework considers time series under the locally stationary wavelet (LSW) framework, deriving a joint density for scale processes in the raw wavelet periodogram. By embedding this density within a hidden Markov model (HMM) framework, we consider changepoint characteristics under this multiscale setting. Such a methodology allows us to model changepoints and their uncertainty for a wide range of models, including piecewise second-order stationary processes, for example, piecewise moving average processes.

Penalized contrast estimation in functional linear models with circular data

Our aim is to estimate the unknown slope function in the functional linear model when the response Y is real and the random function X is a second-order stationary and periodic process. We obtain our estimator by minimizing a standard (and very simple) mean-square contrast on linear finite dimensional spaces spanned by trigonometric bases. Our approach provides a penalization procedure which allows to automatically select the adequate dimension, in a non-asymptotic point of view. In fact, we can show that our penalized estimator reaches the optimal (minimax) rate of convergence in the sense of the prediction error. We complete the theoretical results by a simulation study and a real example that illustrates how the procedure works in practice.

Robust sampling designs for a possibly misspecified stochastic process

We address the problem of finding robust sampling designs for the esti- mation of a discrete time second-order stationary process when its autocorrelation function is only approximately specified and has a spectral density belonging to a neighbourhood of a specified 'base' density. The value of the stochastic process is predicted by the best - for the assumed autocorrelation function - linear unbiased predictor on the basis of a finite sample of observations. Following the approach of minimax robustness, we find the least favourable - in the sense of maximizing the average mean squared error (amspe) of these predictions - spectral density. We then obtain, through a genetic algorithm, robust sampling designs which minimize this maximum amspe. Several examples are discussed and assessed, on the basis of which we conclude that the robust designs can offer substantial protection against model errors, at a minimal cost in efficiency at the base model. The techniques are illustrated in a case study, using a series of interest in statistical climatology.

Gaussian pseudo-likelihood estimation for stationary processes on a lattice

For a wide class of second-order stationary spatial processes on a lattice, the statistical properties of the maximum Gaussian pseudo-likelihood estimators are studied. The estimators are natural as they imitate the theoretical prototypes of spatial best linear prediction. Under certain conditions, their asymptotic normality is established with the elements of the asymptotic variance matrix being simple functions of the variable auto-covariances. A short simulation study and a data example favor the use of the Gaussian pseudo-likelihood when the spatial covariance dependence is to be estimated.

On entropy rate for the complex domain.

We derive the entropy rate formula for a complex Gaussian random process by using a widely linear model. The resulting expression is general and applicable to both circular and noncircular Gaussian processes, since any second-order stationary process can be modeled as the output of a widely linear system driven by a circular white noise. Furthermore, we demonstrate application of the derived formula to an order selection problem. We extend a scheme for independent and identically distributed (i.i.d.) sampling to the complex domain to improve the estimation performance of information-theoretic criteria when samples are correlated. We show the effectiveness of the approach for order selection for simulated and actual functional magnetic resonance imaging (fMRI) data that are inherently complex valued.

Generalized Levinson–Durbin sequences, binomial coefficients and autoregressive estimation

For a discrete time second-order stationary process, the Levinson–Durbin recursion is used to determine the coefficients of the best linear predictor of the observation at time k + 1 , given k previous observations, best in the sense of minimizing the mean square error. The coefficients determined by the recursion define a Levinson–Durbin sequence. We also define a generalized Levinson–Durbin sequence and note that binomial coefficients form a special case of a generalized Levinson–Durbin sequence. All generalized Levinson–Durbin sequences are shown to obey summation formulas which generalize formulas satisfied by binomial coefficients. Levinson–Durbin sequences arise in the construction of several autoregressive model coefficient estimators. The least squares autoregressive estimator does not give rise to a Levinson–Durbin sequence, but least squares fixed point processes, which yield least squares estimates of the coefficients unbiased to order 1 / T , where T is the sample length, can be combined to construct a Levinson–Durbin sequence. By contrast, analogous fixed point processes arising from the Yule–Walker estimator do not combine to construct a Levinson–Durbin sequence, although the Yule–Walker estimator itself does determine a Levinson–Durbin sequence. The least squares and Yule–Walker fixed point processes are further studied when the mean of the process is a polynomial time trend that is estimated by least squares.