Gauss-Markov Process Research Articles

Kalman filtering for adaptive nonuniformity correction in infrared focal-plane arrays.

A novel statistical approach is undertaken for the adaptive estimation of the gain and bias nonuniformity in infrared focal-plane array sensors from scene data. The gain and the bias of each detector are regarded as random state variables modeled by a discrete-time Gauss-Markov process. The proposed Gauss-Markov framework provides a mechanism for capturing the slow and random drift in the fixed-pattern noise as the operational conditions of the sensor vary in time. With a temporal stochastic model for each detector's gain and bias at hand, a Kalman filter is derived that uses scene data, comprising the detector's readout values sampled over a short period of time, to optimally update the detector's gain and bias estimates as these parameters drift. The proposed technique relies on a certain spatiotemporal diversity condition in the data, which is satisfied when all detectors see approximately the same range of temperatures within the periods between successive estimation epochs. The performance of the proposed technique is thoroughly studied, and its utility in mitigating fixed-pattern noise is demonstrated with both real infrared and simulated imagery.

Stochastic models for heterogeneous relaxation: application to inhomogeneous optical lineshapes

Dynamic heterogeneity has often been modeled by assuming that a single-particle observable, fluctuating at a molecular scale, is influenced by its coupling to environmental variables fluctuating on a second, perhaps slower, time scale. Starting from the most simple Gaussian Markov process we model the exchange between `slow' and `fast' environments by treating the fluctuating single-particle variable as a projection from a higher-dimensional Markov process. The moments of the resulting stochastic process are calculated from the corresponding master equations or Langevin equations, depending on the model. The calculations show the importance of the way to treat exchange processes. The resulting stochastic process is non-Markovian for all models. However, the deviations from a Gaussian behavior depend on the details of the models. A comparison of our results with other model treatments and experiments should provide further insight into the concept of dynamic heterogeneity.

Large-deviation functions for nonlinear functionals of a Gaussian stationary Markov process.

We introduce a general method, based on a mapping onto quantum mechanics, for investigating the large-T limit of the distribution P(r,T) of the nonlinear functional r[V]=(1/T)integral(T)(0)dT' V[X(T')], where V(X) is an arbitrary function of the stationary Gaussian Markov process X(T). For T-->infinity at fixed r we obtain P(r,T) approximately exp[-theta(r)T], where theta(r) is a large-deviation function. We present explicit results for a number of special cases including V(X)=XH(X) [where H(X) is the Heaviside function], which is related to the cooling and the heating degree days relevant to weather derivatives.

Markovianity of a subset of components of a Markov process

We consider the necessary and sufficient conditions for a group of the components of a stationary vector Gaussian Markov process to possess Markov property. The representation by a linear Itô stochastic differential equation is also given.

Optimal estimation of diffusion processes hidden by general obstacles

Let X t be an n-dimensional diffusion process and S(t) be a set-valued function. Suppose X t is invisible when it is hidden by S(t), but we can see the process exactly otherwise. In this paper, we derive the optimal estimator E[f(X 1) | X s 1 X s ∉S(s), 0 ≤ s ≤ 1] for a bounded Borel function f. We illustrate some computations for Gauss-Markov processes.

Optimal estimation of diffusion processes hidden by general obstacles

Let Xt be an n-dimensional diffusion process and S(t) be a set-valued function. Suppose Xt is invisible when it is hidden by S(t), but we can see the process exactly otherwise. In this paper, we derive the optimal estimator E[f(X1) | Xs1Xs∉S(s), 0 ≤ s ≤ 1] for a bounded Borel function f. We illustrate some computations for Gauss-Markov processes.

Statistical modelling of the backscattered field in a one-dimensional non-stationary stochastic problem

Within the framework of an exact wave approach in the spatial-time domain, the one-dimensional stochastic problem of sound pulse scattering by a layered random medium is considered. On the basis of a unification of methods which has been developed by the authors, previously applied to the investigation of non-stationary deterministic wave problems and stochastic stationary wave problems, an analytical-numerical simulation of the behaviour of the backscattered field stochastic characteristics was carried out. Several forms of incident pulses and signals are analysed. We assume that random fluctuations of a medium are described by virtue of the Gaussian Markov process with an exponential correlation function. The most important parameters appearing in the problem are discussed; namely, the time scales of diffusion, pulse durations, the medium layer thickness or the largest observation time scale in comparison with the time scale of one correlation length for the case of a half-space. An exact pattern of the pulse backscattering processes is obtained. It is illustrated by the behaviour of the backscattered field statistical moments for all observation times which are of interest. It is shown that during the time interval when the main part of the pulse energy leaves the medium, the backscattered field is a substantially non-stationary process, having a non-zero mean value and an average intensity that decays according to a power law. There are various power indices for the different duration incident pulses, however, they are not the same as those of previous papers, which were obtained on the basis of an approximate and asymptotic analysis. We have also verified that the Gaussian law is valid for the probability density function of the backscattered field in the case of any incident pulse duration.

Georegistration of airborne hyperspectral image data

A suite of geometric sensor and platform modeling tools has been developed which have achieved consistent subpixel accuracy in orthorectification experiments. Aircraft platforms in turbulent atmospheric conditions present unique challenges and have required creative modeling approaches. The geometric relationship between an image point and a ground object has been modeled by rigorous photogrammetric methods. First and second order Gauss-Markov processes have been used to estimate the platform trajectory. These methods have been successfully applied to HYDICE and HyMap data sets. The most important contributors to the subpixel rectification accuracy have been the first order Gauss-Markov model with control linear features.

A computational approach to first-passage-time problems for Gauss–Markov processes

A new computationally simple, speedy and accurate method is proposed to construct first-passage-time probability density functions for Gauss–Markov processes through time-dependent boundaries, both for fixed and for random initial states. Some applications to Brownian motion and to the Brownian bridge are then provided together with a comparison with some computational results by Durbin and by Daniels. Various closed-form results are also obtained for classes of boundaries that are intimately related to certain symmetries of the processes considered.

Small ball probabilities for Gaussian Markov processes under the Lp-norm

Let {X(t); 0⩽t⩽1} be a real-valued continuous Gaussian Markov process with mean zero and covariance σ( s, t)= EX( s) X( t)≠0 for 0<s, t<1 . It is known that we can write σ( s, t)= G(min( s, t)) H(max( s, t)) with G>0, H>0 and G/ H nondecreasing on the interval (0,1). We show that for the L p -norm on C[0,1], 1⩽ p⩽∞ lim ε→0 ε 2 log P(||X(t)|| p<ε)=−κ p ∫ 0 1 (G′H−H′G) p/(2+p) dt (2+p)/p and its various extensions.

Statistics of the occupation time for a class of Gaussian Markov processes

We revisit the work of Dhar and Majumdar [Phys. Rev. E 59, 6413 (1999)] on the limiting distribution of the temporal mean $M_{t}=t^{-1}\int_{0}^{t}du \sign y_{u}$, for a Gaussian Markovian process $y_{t}$ depending on a parameter $\alpha $, which can be interpreted as Brownian motion in the scale of time $t^{\prime}=t^{2\alpha}$. This quantity, for short the mean `magnetization', is simply related to the occupation time of the process, that is the length of time spent on one side of the origin up to time t. Using the fact that the intervals between sign changes of the process form a renewal process in the time scale t', we determine recursively the moments of the mean magnetization. We also find an integral equation for the distribution of $M_{t}$. This allows a local analysis of this distribution in the persistence region $(M_t\to\pm1)$, as well as its asymptotic analysis in the regime where $\alpha$ is large. We finally put the results thus found in perspective with those obtained by Dhar and Majumdar by another method, based on a formalism due to Kac.

Relaxation of probability characteristics of dynamical systems, described by Ferhulst stochastic equation with «pink» noise

Probability characteristics relaxation of а solution of Ferhulst stochastic equation is considered. It was supposed, that random influence on the system is Gaussian Markov process with arbitrary values of correlation time. Two models of nonstationary probability distributions with exact stationary athymptotic are proposed on the basis of numerical solutions of relaxation equations for moments: in the case of delta—correlated random force and quasistationary force. Relaxation of main probability characteristics is examined in dependence оп nonlinearity, initial conditions, noise intensity and kind of noise spectrum. Evolution of the model probability distributions has been constructed.

A computational approach to first-passage-time problems for Gauss–Markov processes

A new computationally simple, speedy and accurate method is proposed to construct first-passage-time probability density functions for Gauss–Markov processes through time-dependent boundaries, both for fixed and for random initial states. Some applications to Brownian motion and to the Brownian bridge are then provided together with a comparison with some computational results by Durbin and by Daniels. Various closed-form results are also obtained for classes of boundaries that are intimately related to certain symmetries of the processes considered.

The worst additive noise under a covariance constraint

The maximum entropy noise under a lag p autocorrelation constraint is known by Burg's theorem to be the pth order Gauss-Markov process satisfying these constraints. The question is, what is the worst additive noise for a communication channel given these constraints? Is it the maximum entropy noise? The problem becomes one of extremizing the mutual information over all noise processes with covariances satisfying the correlation constraints R/sub 0/,..., R/sub p/. For high signal powers, the worst additive noise is Gauss-Markov of order p as expected. But for low powers, the worst additive noise is Gaussian with a covariance matrix in a convex set which depends on the signal power.

Matrices with banded inverses: inversion algorithms and factorization of Gauss-Markov processes

The paper considers the inversion of full matrices whose inverses are L-banded. We derive a nested inversion algorithm for such matrices. Applied to a tridiagonal matrix, the algorithm provides its explicit inverse as an element-wise product (Hadamard product) of three matrices. When related to Gauss-Markov random processes (GMrp), this result provides a closed-form factored expression for the covariance matrix of a first-order GMrp. This factored form leads to the interpretation of a first-order GMrp as the product of three independent processes: a forward independent-increments process, a backward independent-increments process, and a variance-stationary process. We explore the nonuniqueness of the factorization and design it so that the forward and backward factor processes have minimum energy. We then consider the issue of approximating general nonstationary Gaussian processes by Gauss-Markov processes under two optimality criteria: the Kullback-Leibler distance and maximum entropy. The problem reduces to approximating general covariances by covariance matrices whose inverses are banded. Our inversion result is an efficient algorithmic solution to this problem. We evaluate the information loss between the original process and its Gauss-Markov approximation.

The effect upon channel capacity in wireless communications of perfect and imperfect knowledge of the channel

We present a model for time-varying communication single-access and multiple-access channels without feedback. We consider the difference between mutual information when the receiver knows the channel perfectly and mutual information when the receiver only has an estimate of the channel. We relate the variance of the channel measurement error at the receiver to upper and lower bounds for this difference in mutual information. We illustrate the use of our bounds on a channel modeled by a Gauss-Markov process, measured by a pilot tone. We relate the rate of time variation of the channel to the loss in mutual information due to imperfect knowledge of the measured channel.

The marginal likelihood for parameters in a discrete Gauss-Markov process

We use Laplace's method to approximate the marginal likelihood for parameters in a Gauss-Markov process. This approximation requires the determinant of a matrix whose dimensions are equal to the number of state variables times the number of time points. We reduce this to sequential evaluation of determinants and inverses of smaller matrices, we show this is a numerically stable method.

PHASE-SPACE LINEARIZATION FOR NON-LINEAR OSCILLATORS: DETERMINISTIC AND STOCHASTIC SYSTEMS

A new and efficient semi-analytical phase-space linearization (PSL) scheme for a class of non-linear oscillators is developed in this paper. The method is based on replacement of the non-linear vector field by a set of linear ones, each valid over a short segment of the evolving trajectory or over sufficiently small interval of time. Based on this concept, a few explicit and implicit integration schemes are first proposed and applied to a class of low-dimensional non-linear dynamical systems to accurately determine their response trajectories. This approach of local linearization is further extended to non-linear oscillators excited by formal derivatives of one or a combination of Gauss–Markov processes. Since the present methodology reduces the non-linear operator by a set of linear operators, it is also demonstrated that the principles of linear random vibration may be suitable exploited to arrive at a faster and semi-analytical Monte-Carlo scheme for computing the response statistics, both in stationary and non-stationary regimes. Limited examples are presented and compared with exact solutions whenever available, to illustrate the efficiency and versatility of the proposed schemes.

Markov symmetric stable processes

Using the method of generative processes, we construct a model of a random linear symmetric stable Markov process, which is a natural generalization of the Markov Gaussian process and preserves its main property: invariance with respect to arbitrary linear transformations. Methods for analyzing such processes are developed. In particular, it is proposed to use the information correlation function as a characteristic of the pair dependence between the process values at different times. This function is used to calculate the determination interval of the process.

Adaptive soft-input soft-output algorithms for iterative detection with parametric uncertainty

The soft-input soft-output (SISO) module is the basic building block for established iterative detection (ID) algorithms for a system consisting of a network of finite state machines. The problem of performing ID for systems having parametric uncertainty has received relatively little attention in the open literature. Previously proposed adaptive SISO (A-SISO) algorithms are either based on an oversimplified channel model, or have a complexity that grows exponentially with the observation length N (or the smoothing lag D). In this paper, the exact expressions for the soft metrics in the presence of parametric uncertainty modeled as a Gauss-Markov process are derived in a novel way that enables the decoupling of complexity and observation length. Starting from these expressions, a family of suboptimal (practical) algorithms is motivated, based on forward/backward adaptive processing with linear complexity in N. Previously proposed A-SISO algorithms, as well as existing adaptive hard decision algorithms are interpreted as special cases within this framework. Using a representative application-joint iterative equalization-decoding for trellis-based codes over frequency-selective channels-several design options are compared and the impact of parametric uncertainty on previously established results for ID with perfect channel state information is assessed.