Stationary Ergodic Process Research Articles

The Shannon’s mutual information of a multiple antenna time and frequency dependent channel: An ergodic operator approach

Consider a random non-centered multiple antenna radio transmission channel. Assume that the deterministic part of the channel is itself frequency selective and that the random multipath part is represented by an ergodic stationary vector process. In the Hilbert space l2(ℤ), one can associate to this channel a random ergodic self-adjoint operator having a so-called Integrated Density of States (IDS). Shannon’s mutual information per receive antenna of this channel coincides then with the integral of a log function with respect to the IDS. In this paper, it is shown that when the numbers of antennas at the transmitter and at the receiver tend to infinity at the same rate, the mutual information per receive antenna tends to a quantity that can be identified and, in fact, is closely related to that obtained within the random matrix approach [I. Telatar, Eur. Trans. Telecommun. 10, 585 (1999)]. This result can be obtained by analyzing the behavior of the Stieltjes transform of the IDS in the regime of the large numbers of antennas.

Simulation of Basis Expansion Model for Channel Fading Using AR1 Process

This correspondence presents an alternate approach for the simulation of basis expansion model (BEM) for channel fading, in which each time-varying BEM coefficient is considered to be governed by the first-order autoregressive (AR1) process. To introduce a high degree of uncorrelation among the BEM coefficients, the Markov parameter of each AR1 process is also assumed to be time-varying according to another independent stationary ergodic AR1 process, which forms the base of BEM---AR1---AR1 paradigm. The simulation results manifest that the proposed BEM---AR1---AR1 scheme is in close agreement with the ideal BEM for slow as well as fast channel fading, which may find applications in the time-varying channel estimation due to its compatibility with model based adaptive algorithms.

Algorithm for Efficient Entropy Estimation

We consider the problem of the nonparametric entropy estimation of a stationary ergodic process. Our approach is based on the nearest-neighbor distances. We propose a broad class of metrics on the space Ω = A N of right-sided infinite sequences drawn from a finite alphabet A. The new metric has a parameter which is a non-increasing function. We apply this metrics to nearest-neighbor entropy estimators. We prove that, under certain conditions, the estimators has a small variance. We show that a special selection of the metric parameters reduction of the estimator’s bias. The article is published in the author’s wording.

Nonparametric statistical inference for the context tree of a stationary ergodic process

We consider the problem of estimating the context tree of a stationary ergodic process with finite alphabet without imposing additional conditions on the process. As a starting point we introduce a Hamming metric in the space of irreducible context trees and we use the properties of the weak topology in the space of ergodic stationary processes to prove that if the Hamming metric is unbounded, there exist no consistent estimators for the context tree. Even in the bounded case we show that there exist no two-sided confidence bounds. However we prove that one-sided inference is possible in this general setting and we construct a consistent estimator that is a lower bound for the context tree of the process with an explicit formula for the coverage probability. We develop an efficient algorithm to compute the lower bound and we apply the method to test a linguistic hypothesis about the context tree of codified written texts in European Portuguese.

Selection of a Metric for the Nearest Neighbor Entropy Estimators

We consider the problem of improving the efficiency of the nonparametric entropy estimation for a stationary ergodic process. Our approach is based on the nearest-neighbor distances. We propose a broad class of metrics on the space of right-sided infinite sequences drawn from a finite alphabet. The new metric has a parameter that is a nonincreasing function. We prove that, under certain conditions, our estimators have a small variance and show that a special selection of the metric parameters reduces the estimator’s bias.

Stationary and Transition Probabilities in Slow Mixing, Long Memory Markov Processes

We observe a length-n sample generated by an unknown, stationary ergodic Markov process (model) over a finite alphabet A. Given any string w of symbols from A we want estimates of the conditional probability distribution of symbols following w, as well as the stationary probability of w. Two distinct problems that complicate estimation in this setting are: 1) long memory and 2) slow mixing, which could happen even with only one bit of memory. Any consistent estimator in this setting can only converge pointwise over the class of all ergodic Markov models. Namely, given any estimator and any sample size n, the underlying model could be such that the estimator performs poorly on a sample of size n with high probability. But can we look at a length-n sample and identify if an estimate is likely to be accurate? Since the memory is unknown a-priori, a natural approach is to estimate a potentially coarser model with memory k <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> = O(log n). As n grows, pointwise consistent estimates that hold eventually almost surely (e.a.s.) are known so long as the scaling of k <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> is not superlogarithmic in n. Here, rather than e.a.s. convergence results, we want the best answers possible with a length-n sample. Combining results in universal compression with Aldous' coupling arguments, we obtain sufficient conditions on the lengthn sample (even for slow mixing models) to identify when naive: 1) estimates of the conditional probabilities and 2) estimates related to the stationary probabilities are accurate, and also bound the deviations of the naive estimates from true values.

Bus vibration study - theoretical response analysis and experimental verification

Random road undulations are given as the input to the finite element model of a bus and the response (acceleration and stress) of the bus is obtained analytically using random vibration concepts. The results obtained are compared with the experimental values reported earlier. For the analysis, a 180 degrees of freedom model of the bus has been used. The road has been treated as a stationary ergodic random process with 100% correlation between the left and right tracks. Appropriate space correlation has been assumed taken between the inputs to the front and rear wheels.

Vector-on-function quantile regression for stationary ergodic processes

This paper deals with quantile regression of a vector response (Y∈Rq, q≥2) on a functional covariate X that takes values in an infinite dimensional space. The main purpose is to introduce a kernel-type estimator of the conditional geometric quantiles whenever functional strictly stationary ergodic data are considered. We established the strong consistency with rate of the proposed estimator as well as the asymptotic distribution that leads to build a confidence region for the multivariate quantile regression. We gave also an application to joint horizon time series forecasting.

Comments on: Extensions of some classical methods in change point analysis

First of all, I would like to compliment the authors for writing this very fine survey paper on some recent developments in change point analysis. There is a lot of current research activity devoted to change point analysis, and many articles have appeared since the publication of the seminal book by Csorgő and Horvath (1997). Among the topics covered in the present paper are empirical process techniques, Darling-Erdős laws, changes in correlations, changes in regression parameters, sequential testing, panel models, and functional data. Horvath and Rice provide an excellent survey of some of the recent results, which will be helpful to all researchers in this area. In my discussion, I will complement the paper by Horvath and Rice by presenting some recent results on U-statistics based robust change-point tests for time series. In a series of papers, see Dehling and Fried (2012), Dehling, Fried, Garcia and Wendler (2013), Dehling, Rooch and Taqqu (2013a, 2013b), and Betken (2014), we have investigated such tests, and derived their asymptotic distribution, both in the case of short range as well as long range dependent time series. We consider a model where the data are generated by Xi = μi + i, where μi is an unknown signal, and where i is a stationary ergodic noise process with E( i) = 0. Given the observations X1, . . . , Xn, we wish to test the hypothesis of no change, i.e. H : μ1 = . . . = μn, against the alternative

Nearest-neighbor entropy estimators with weak metrics

A problem of improving the accuracy of nonparametric entropy estimationfor a stationary ergodic process is considered. New weak metrics are introduced and relations between metrics,measures, and entropy are discussed. A new nonparametric entropy estimator is constructed based on weak metricsand has a parameter with which the estimator is optimized to reduce its bias.It is shown that estimator's variance is upper-bounded by a nearly optimal Cramér-Rao lower bound.

Effect of railway roughness on soil vibrations due to moving trains by 2.5D finite/infinite element approach

The railway irregularity and dynamic properties of the moving train is newly incorporated in the 2.5D finite/infinite element simulation of the soil vibrations caused by subway trains. For soils that are uniform along the railway, only a 2D profile of the soil perpendicular to the railway needs to be considered, of which the near field is simulated by finite elements and the far field by infinite elements. Besides the in-plane degrees of freedom (DOFs) for plane elements, an extra DOF is needed to account for the out-of-plane vibration. The train is modeled by a series of mass-spring-dashpot units moving over rails with the irregularity profile simulated as a stationary ergodic Gaussian random process. By summing up the load distribution function for each axle load moving over an infinite elastically supported beam, the wheel-rail interaction forces are obtained in the Fourier domain. The reliability of the proposed approach is verified by comparing the results obtained with the existing ones. Besides, the effect of floating slabs on the vibration isolation of railway tracks with respect to rail irregularity is also investigated. It is concluded that the velocity and acceleration responses of the soil are largely amplified due to the presence of rail irregularity, and that the high-frequency components induced by rail irregularity can be effectively reduced via installation of the floating slabs.

Universal Estimation of Directed Information

Four estimators of the directed information rate between a pair of jointly stationary ergodic finite-alphabet processes are proposed, based on universal probability assignments. The first one is a Shannon--McMillan--Breiman type estimator, similar to those used by Verd\'u (2005) and Cai, Kulkarni, and Verd\'u (2006) for estimation of other information measures. We show the almost sure and $L_1$ convergence properties of the estimator for any underlying universal probability assignment. The other three estimators map universal probability assignments to different functionals, each exhibiting relative merits such as smoothness, nonnegativity, and boundedness. We establish the consistency of these estimators in almost sure and $L_1$ senses, and derive near-optimal rates of convergence in the minimax sense under mild conditions. These estimators carry over directly to estimating other information measures of stationary ergodic finite-alphabet processes, such as entropy rate and mutual information rate, with near-optimal performance and provide alternatives to classical approaches in the existing literature. Guided by these theoretical results, the proposed estimators are implemented using the context-tree weighting algorithm as the universal probability assignment. Experiments on synthetic and real data are presented, demonstrating the potential of the proposed schemes in practice and the utility of directed information estimation in detecting and measuring causal influence and delay.

Permutation Complexity and Coupling Measures in Hidden Markov Models

Recently, the duality between values (words) and orderings (permutations) has been proposed by the authors as a basis to discuss the relationship between information theoretic measures for finite-alphabet stationary stochastic processes and their permutatio nanalogues. It has been used to give a simple proof of the equality between the entropy rate and the permutation entropy rate for any finite-alphabet stationary stochastic process and to show some results on the excess entropy and the transfer entropy for finite-alphabet stationary ergodic Markov processes. In this paper, we extend our previous results to hidden Markov models and show the equalities between various information theoretic complexity and coupling measures and their permutation analogues. In particular, we show the following two results within the realm of hidden Markov models with ergodic internal processes: the two permutation analogues of the transfer entropy, the symbolic transfer entropy and the transfer entropy on rank vectors, are both equivalent to the transfer entropy if they are considered as the rates, and the directed information theory can be captured by the permutation entropy approach.

Fatigue of girth butt weld for cast steel node connection in tower structure under wave loadings

SUMMARYThe application of cast steel node shows a popular tendency in long‐span space structures. The girth butt welds in the joint are the critical parts under fatigue loading. This study presents a fatigue life evaluation of the girth butt welds for the cast steel node connections, which are in a steel tower structure on an offshore platform under heavy wave fatigue loadings. The refined node FEM model is built including the welds. The random waves are treated as ergodic stationary stochastic process; the node structure is considered as a linear dynamic amplification system to decide its output of the stress amplitude response while the random wave is inputted into the system. The welds are designed as different details of bilateral weld, single‐sided weld with a sleeve, single‐sided weld with a backing plate or without a backing plate. The fatigue life of every weld detail is studied by hot‐spot stress method, and the design of weld detail combination is analyzed. Copyright © 2013 John Wiley &amp; Sons, Ltd.

Divergence rates of Markov order estimators and their application to statistical estimation of stationary ergodic processes

Stationary ergodic processes with finite alphabets are estimated by finite memory processes from a sample, an n-length realization of the process, where the memory depth of the estimator process is also estimated from the sample using penalized maximum likelihood (PML). Under some assumptions on the continuity rate and the assumption of non-nullness, a rate of convergence in $\bar{d}$-distance is obtained, with explicit constants. The result requires an analysis of the divergence of PML Markov order estimators for not necessarily finite memory processes. This divergence problem is investigated in more generality for three information criteria: the Bayesian information criterion with generalized penalty term yielding the PML, and the normalized maximum likelihood and the Krichevsky-Trofimov code lengths. Lower and upper bounds on the estimated order are obtained. The notion of consistent Markov order estimation is generalized for infinite memory processes using the concept of oracle order estimates, and generalized consistency of the PML Markov order estimator is presented.

Symbolic transfer entropy rate is equal to transfer entropy rate for bivariate finite-alphabet stationary ergodic Markov processes

Transfer entropy is a measure of the magnitude and the direction of information flow between jointly distributed stochastic processes. In recent years, its permutation analogues are considered in the literature to estimate the transfer entropy by counting the number of occurrences of orderings of values, not the values themselves. It has been suggested that the method of permutation is easy to implement, computationally low cost and robust to noise when applying to real world time series data. In this paper, we initiate a theoretical treatment of the corresponding rates. In particular, we consider the transfer entropy rate and its permutation analogue, the symbolic transfer entropy rate, and show that they are equal for any bivariate finite-alphabet stationary ergodic Markov process. This result is an illustration of the duality method introduced in [T. Haruna, K. Nakajima, Physica D 240, 1370 (2011)]. We also discuss the relationship among the transfer entropy rate, the time-delayed mutual information rate and their permutation analogues.

Nonparametric [formula omitted]-regression for functional ergodic data

Let (Xi,Yi)i=1,…,n be a sequence of stationary ergodic processes valued in F×R, where F is a semi-metric space. We consider the problem of estimating the regression function of Yi given Xi by the robust M-estimation method. The principal aim of this work is to prove the almost complete convergence (with rate) for the proposed estimator. This result is obtained under a stationary ergodic process assumption, without using traditional mixing conditions.

The Function Space to Describe the Dynamics of Linear Systems

There are two models of a random stationary process with a continuous spectrum of Wiener and the discrete spectrum of E. Slutsky. The first model of Wiener is often used. It is shown that this assumption leads to the absurdity of the equation Wiener-Hopf. The contradictions disappear in the theory of random stationary processes in the transition to a different model of a random process with discrete spectrum. Random ergodic stationary processes have on the mean and do not have the ergodicity of the variance and correlation functions. The struggle is based on the correlation functions of random processes with interference on the Wiener. Correct correlation functions cannot be obtained. Therefore, frequency analysis of random processes is proposed instead of the correlation analysis. The frequency analysis results in a more effective method of combating additive noise. The new method was used to identify the dynamic characteristics of the Airbus on the data obtained during the automatic landing.

Inefficiency of MaxWeight scheduling in spatial wireless networks

MaxWeight scheduling has gained enormous popularity as a powerful paradigm for achieving queue stability and maximum throughput in a wide variety of scenarios. The maximum-stability guarantees however rely on the fundamental premise that the system consists of a fixed set of flows with stationary ergodic traffic processes. In the present paper we examine networks where the population of active flows varies over time, as flows eventually end while new flows occasionally start. We show that MaxWeight policies may fail to provide maximum stability due to persistent inefficient spatial reuse. The intuitive explanation is that these policies tend to serve flows with large backlogs, even when the resulting spatial reuse is not particularly efficient, and fail to exploit maximum spatial reuse patterns involving flows with smaller backlogs. These results indicate that instability of MaxWeight scheduling can occur due to spatial inefficiency in networks with fixed transmission rates, which is fundamentally different from the inability to fully exploit time-varying rates shown in prior work. We discuss how the potential instability effects can be countered by spatial traffic aggregation, and describe some of the associated challenges and performance trade-offs.

Asymptotic Stability Region of Slotted Aloha

We analyze the stability of standard, buffered, slotted-Aloha systems. Specifically, we consider a set of $N$ users, each equipped with an infinite buffer. Packets arrive into user $i$ 's buffer according to some stationary ergodic Markovian process of intensity $\lambda_{i}$ . At the beginning of each slot, if user $i$ has packets in its buffer, it attempts to transmit a packet with fixed probability $p_{i}$ over a shared resource/channel. The transmission is successful only when no other user attempts to use the channel. The stability of such systems has been open since their very first analysis in 1979 by Tsybakov and Mikhailov. In this paper, we propose an approximate stability condition that is provably exact when the number of users $N$ grows large. We provide theoretical evidence and numerical experiments to explain why the proposed approximate stability condition is extremely accurate even for systems with a restricted number of users (even two or three).