Class Of Stationary Gaussian Processes Research Articles

Data-Driven Reconstruction of Dynamical Forces and Responses through Bayesian Expectation-Maximization and Modal Reduction Methods

This paper presents a new Bayesian approach for estimating input and state in linear structures using response-only measurements. The proposed approach benefits from a modally reduced state-space model, which circumvents the dimensionality of dynamical responses in complex structures through a low-dimensional subspace. It also substitutes unknown physical forces with a set of equivalent modal forces, which is beneficial when the magnitude and location of input forces are unknown. In this work, these forces are characterized through the conventional random walk model and a class of stationary Gaussian processes. Subsequently, an augmented state-space model is constructed to describe modal states and input loads. Based on this model, a Bayesian expectation-maximization (BEM) methodology is developed to identify the input, state, and noise parameters. This noise identification perspective activates uncertainty quantification in joint input-state estimation problems and enables quantifying the degree of confidence in the estimated quantities. When the proposed method is tested using numerical and experimental examples, accurate estimations and reasonable uncertainty bounds are acquired for the dynamical state and input forces. Although the literature reports the superiority of the Gaussian process latent force model over the random walk model without using a unified noise calibration strategy, this study, to our best knowledge, is the first effort to compare and interpret the results on a consistent basis where the noise and input characteristics are all identified from the data through BEM.

Anomalous scaling of dynamical large deviations of stationary Gaussian processes.

Employing the optimal fluctuation method, we study the large deviation function of long-time averages (1/T)∫_{-T/2}^{T/2}x^{n}(t)dt,n=1,2,⋯, of centered stationary Gaussian processes. These processes are correlated and, in general, non-Markovian. We show that the anomalous scaling with time of the large-deviation function, recently observed for n>2 for the particular case of the Ornstein-Uhlenbeck process, holds for a whole class of stationary Gaussian processes.

Simulating a class of stationary Gaussian processes using the Davies–Harte algorithm, with application to long memory processes

We demonstrate that the fast and exact Davies–Harte algorithm is valid for simulating a certain class of stationary Gaussian processes – those with a negative autocovariance sequence for all non‐zero lags. The result applies to well known classes of long memory processes: Gaussian fractionally differenced (FD) processes, fractional Gaussian noise (fGn) and the nonstationary fractional Brownian Motion (fBm).

On the Asymptotic Behavior of First Passage Time Densities for Stationary Gaussian Processes and Varying Boundaries

Making use of a Rice-like series expansion, for a class of stationary Gaussian processes the asymptotic behavior of the first passage time probability density function through certain time-varying boundaries, including periodic boundaries, is determined. Sufficient conditions are then given such that the density asymptotically exhibits an exponential behavior when the boundary is either asymptotically constant or asymptotically periodic.

On asymptotic normality of estimates for correlation functions of stationary Gaussian processes in the space of continuous functions

We establish conditions of the weak convergence of the empirical correlogram of a stationary Gaussian process to some Gaussian process in the space of continuous functions. We prove that such a convergence holds for a broad class of stationary Gaussian processes with square integrable spectral density.

On the strong information singularity of certain stationary processes (Corresp.)

In an exploratory paper, T. Berger studied discrete random processes which generate information slower than linearly with time. One of his objectives was to provide a physically meaningful definition of a deterministic process, and to this end he introduced the notion of strong information singularity. His work is supplemented by demonstrating that a large class of convariance stationary processes are strongly information singular with respect to a class of stationary Gaussian processes. One important consequence is that for a large class of covariance stationary processes the information rate equals that of the process associated with the Brownian motion component of the spectral representation. In an exploratory paper, T. Berger studied discrete random processes which generate information slower than linearly with time. One of his objectives was to provide a physically meaningful definition of a deterministic process, and to this end he introduced the notion of strong information singularity. His work is supplemented by demonstrating that a large class of convariance stationary processes are strongly information singular with respect to a class of stationary Gaussian processes. One important consequence is that for a large class of covariance stationary processes the information rate equals that of the process associated with the Brownian motion component of the spectral representation. In an exploratory paper, T. Berger studied discrete random In an exploratory paper, T. Berger studied discrete random processes which generate information slower than linearly with time. One of his objectives was to provide a physically meaningful definition of a deterministic process, and to this end he introduced the notion of strong information singularity. His work is supplemented by demonstrating that a large class of convariance stationary processes are strongly information singular with respect to a class of stationary Gaussian processes. One important consequence is that for a large class of covariance stationary processes the information rate equals that of the process associated with the Brownian motion component of the spectral representation.

Conditions for a Class of Stationary Gaussian Processes to be Kolmogorov Mixing

H2 = fE L2(RI, dP): f V(x) =fr)e-iYr dr = 0 for x < 1 V\,T1 ~~ -0 for x < n e Let p denote the orthogonal projection of L2(RI, dr) onto H2, let q = and let X(r) = eir. It is known (see, e.g., Sections 3.3-3.6 and 4.12 of Dym-McKean [1] and the references cited there) that X is mixing in the sense of Rosenblatt if and only if p(T) = JlqXT(h*/h)p41 = o(1), as T { oo, and that X is mixing in the sense of Kolmogorov if and only if lc(T) = IqX'(h*/h)pl2 = o(1 as T { oo. Here I I denotes the operator norm, I I2 denotes the Hilbert-Schmidt norm and h*: r -* [h(T)]* in which [h(r)]* stands for the complex conjugate of the complex number h(r); 11 112 will denote the standard L2 norm. The present objective is to establish

On the differentiability of a class of stationary gaussian processes

For a stationary Gaussian process either almost all sample paths are almost everywhere differentiable or almost all sample paths are almost nowhere differentiable. In this paper it is shown by means of an example involving a random lacunary trigonometric series that “almost everywhere differentiable” and “almost nowhere differentiable” cannot in general be replaced by “everywhere differentiable” and “nowhere differentiable”, respectively.

Randon-Nikodym Derivatives of Stationary Gaussian Measures

A problem of considerable importance in time series analysis is that of determining whether two Gaussian processes are equivalent, i.e., mutually absolutely continuous with respect to each other. If we are given $\{\Omega, \mathscr{a}, P_k\}, k = 0, 1$, where $\Omega$ is a set of real valued functions on some interval $\lbrack a, b\rbrack, \mathscr{a}$ is a Borel field of subsets of $\Omega$ and $P_k$ is a Gaussian probability measure on $\mathscr{a}$, then the problem is to determine the conditions under which $P_0$ is equivalent to $P_1$. Feldman (1958) has shown that a certain dichotomy exists in this problem in the following sense. If $\Omega$ is a linear space, then either $P_0$ and $P_1$ are equivalent or they are perpendicular, i.e., mutually singular. In addition, Feldman (1958) has shown, using some results of Segal (1958), that if $\Lambda$ is the linear span of $\Omega$ and the real constants, then $P_0$ and $P_1$ are equivalent iff the $P_0$-equivalence classes of $\Lambda$ are the same as the $P_1$-equivalence classes of $\Lambda$ and the identity correspondence between the $L_2(P_0)$ closure of $\Lambda$ and the $L_2(P_1)$ closure of $\Lambda$ is a bounded invertible operator $T$ such that $(T^\ast T)^{\frac{1}{2}} - I$ is a Hilbert-Schmidt operator. It is well known that a Gaussian process, and hence its probability measure, is uniquely determined by its covariance function, $R(s, t)$, assuming, of course, that the mean value functions is zero. It should, therefore, be possible to determine when $P_0$ and $P_1$ are equivalent in terms of the covariance functions of the two processes, $R_0(s, t), R_1(s, t)$. Indeed, it is desirable to have a theorem which states the conditions for equivalence of $P_0$ and $P_1$ in terms of the covariance functions, since in most applications the only information concerning the Gaussian processes consists of the covariance functions. In this sense, Feldman's (1958) theorem is in a form which is not too suitable for practical applications. This situation has been corrected by Feldman (1960) who, using his earlier results, has given the necessary and sufficient conditions for equivalence of $P_0$ and $P_1$ in terms of the covariance functions for a rather wide class of stationary Gaussian processes with zero mean value functions. However, Feldman (1960) has not computed the Radon-Nikodym derivative (RND) $dP_1/dP_0$ for pairs of equivalent Gaussian processes. The RND, or likelihood ratio, is of considerable importance in statistical inference for stochastic processes, as is well known. Recently, Parzen (1961), (1962) has described a new approach to time series analysis which is based on the notion of a reproducing kernel Hilbert space (RKHS). The theory of RKHS has been given by Aronszajn (1950). Using some results due to Aronszajn (1950) and Hajek (1958a), (1958b) Parzen (1962) has derived a necessary and sufficient condition for equivalence of pairs of Gaussian processes which have the same covariance function but have different mean value functions. In addition, he gives an explicit expression for the RND associated with pairs of equivalent Gaussian processes. The present work stems from a desire to derive the necessary and sufficient conditions for equivalence of pairs of stationary Gaussian processes with different covariance functions and zero mean value functions and to compute the RND for pairs of certain equivalent Gaussian processes. The method of proof is based on the RKHS approach used by Parzen. Thus, part of the results obtained will serve as an alternate derivation of Feldman's (1960) result. However, Feldman's proof involves a large number of complicated manipulations, while the present proof has the advantage of requiring only a few elementary manipulations. This simplification is achieved by employing convergence theorems of Aronszajn (1950), Hajek (1958a), (1958b) and Hormander (1963) and an expression for an inner product due to Parzen (1961).