Non-stationary Sequences Research Articles

Sharp filters for short sequences

This paper describes a technique for constructing frequency-selective linear filters that can be applied to short nonstationary sequences. Such filters are well adapted to the task of extracting the trend from a climatological or an econometric times series whenever it is capable of being described in terms of a well-defined range of frequencies with a firm upper limit.

On a certain class of nonstationary sequences in Hilbert space

We study the functions of correlation K(n, m) = 〈X(n), X(m)〉 of certain sequences: X(n) = Tnx0, x0 ∈ H where T is a contraction in Hilbert space H. By using the spectral methods of the nonunitary operators we give the general form of K(n, m) and its asymptotic behaviour limp→+∞ K(n + p, m + p).

Filters for short non‐stationary sequences

AbstractThis paper describes a methodology for implementing bidirectional frequency‐selective filters in cases where the data sequence is short and non‐stationary. A simple method is proposed for dealing with the start‐up problem. The method has a firm theoretical basis and it is computationally efficient. Copyright © 2001 John Wiley & Sons, Ltd.

Filters for Short Nonstationary Sequences

This paper describes a methodology for implementing bidirectional frequency-selective filters in cases where the data sequence is short and nonstationary. A simple method is proposed for dealing with the start-up problem. The method has a firm theoretical basis and it is computationally efficient.

Algorithms for accelerated convergence of adaptive PCA

We derive and discuss new adaptive algorithms for principal component analysis (PCA) that are shown to converge faster than the traditional PCA algorithms due to Oja, Sanger, and Xu. It is well known that traditional PCA algorithms that are derived by using gradient descent on an objective function are slow to converge. Furthermore, the convergence of these algorithms depends on appropriate choices of the gain sequences. Since online applications demand faster convergence and an automatic selection of gains, we present new adaptive algorithms to solve these problems. We first present an unconstrained objective function, which can be minimized to obtain the principal components. We derive adaptive algorithms from this objective function by using: 1) gradient descent; 2) steepest descent; 3) conjugate direction; and 4) Newton-Raphson methods. Although gradient descent produces Xu's LMSER algorithm, the steepest descent, conjugate direction, and Newton-Raphson methods produce new adaptive algorithms for PCA. We also provide a discussion on the landscape of the objective function, and present a global convergence proof of the adaptive gradient descent PCA algorithm using stochastic approximation theory. Extensive experiments with stationary and nonstationary multidimensional Gaussian sequences show faster convergence of the new algorithms over the traditional gradient descent methods.We also compare the steepest descent adaptive algorithm with state-of-the-art methods on stationary and nonstationary sequences.

SEGMENT: identifying compositional domains in DNA sequences.

DNA sequences are formed by patches or domains of different nucleotide composition. In a few simple sequences, domains can simply be identified by eye; however, most DNA sequences show a complex compositional heterogeneity (fractal structure), which cannot be properly detected by current methods. Recently, a computationally efficient segmentation method to analyse such nonstationary sequence structures, based on the Jensen-Shannon entropic divergence, has been described. Specific algorithms implementing this method are now needed. Here we describe a heuristic segmentation algorithm for DNA sequences, which was implemented on a Windows program (SEGMENT). The program divides a DNA sequence into compositionally homogeneous domains by iterating a local optimization procedure at a given statistical significance. Once a sequence is partitioned into domains, a global measure of sequence compositional complexity (SCC), accounting for both the sizes and compositional biases of all the domains in the sequence, is derived. SEGMENT computes SCC as a function of the significance level, which provides a multiscale view of sequence complexity.

Random variables generated by ranks in dependent schemes

Some nonstationary sequences having independent vector of ranks and vector of order statistics are under consideration. We extend some characterizations in a class of independent r.v.'s to a class of Archimedean copula processes and construct the interpretation which gives us a simple way for simulating Archimedean copula processes.

Stationarization of stochastic sequences with wide-sense stationary increments or jumps by discrete wavelet transforms

Owing to most physical phenomena observed as nonstationary processes and the form of discrete sequences, it becomes realistic to process the nonstationary sequences in the laboratory if there exists a bi jective transformation for stationarization. In this work, our study is emphasized on the class of nonstationary one-dimensional random sequences with wide-sense stationary increments (WSSI), wide-sense stationary jumps (WSSJ) and a famous case, the fractional Brownian motion (FBM) process. Also, the concept of linear algebra is applied to process the stationarization concisely. Our goal is to derive a stationarization theorem developed by linear operators such that a nonstationary sequence with WSSI/WSSJ may be stationarized by an easily realizable perfect reconstruction-quadrature mirror filter structure of the discrete wavelet transform. Some examples for FBM processes and nonstationary signals generated by autoregressive integrated moving average models are provided to demonstrate the stationarization.

Optimal linear extrapolation algorithm for realization of a vector random sequence observed without errors

A number of linear extrapolation algorithms for scalar random sequences have been proposed in [1-3]. However, as a rule, the state of real technical systems is characterized by more than one independent parameter. This leads to the vector optimal extrapolation problem considered in this article. The complexity of the problem is largely determined by the fact that, in general, we cannot make any simplifying assumptions about the vector sequence being investigated. As a rule, this is a nonstationary, nonmonotone, non-Markovian, and non-Gaussian sequence generated by a nonlinear system. It is in this highly general framework that we solve our problem.

A bootstrap procedure in linear regression with nonstationary errors

AbstractIn the context of linear regression with dependent and nonstationary errors, the classical moving‐block bootstrap (MBB) fails to capture the nonstationarity of the errors. A new bootstrap procedure called the blocking external bootstrap (BEB) is proposed to overcome the problem. The consistency of the BEB in estimating the variance of the least‐squares estimator is studied in the case of α‐mixing and nonstationary sequence of errors. It is shown that the BEB only achieves partial correction if the block size is fixed. Complete consistency is achieved by the BEB when the block size is allowed to go to infinity. We also study the first‐order consistency of the least squares estimator based on the BEB. A simulation study is carried out to assess the performance of the BEB versus the MBB in estimating the variance of the least‐squares estimator. Finally, some open problems are discussed.

Constitutional "Rules" and Intergenerational Fiscal Policy

This paper analyzes the impact of alternative political institutions on sustainable fiscal policies. We study the choice of intergenerational transfers as outcomes of an infinite social security game among successive selfish median voters. Majoritarian systems accord the current median voter maximum fiscal discretion but no direct influence over future policy. This political arrangement sustains, among others, dynamically inefficient transfers and volatile, non-stationary sequences. Constitutional “rules” award to the minorities veto power over fiscal policy changes proposed by the majority. This unanimity provision is equivalent to partial precommitment. Under constitutional “rules,” sustainable fiscal policies feature Pareto efficient, non decreasing transfer sequences.

On a theorem of K. Schmidt

Schmidt (1977) proved that if the family of distributions of the partial sums of a strictly stationary (say ergodic) random sequence is tight, then that sequence is a “coboundary”. Here that result is generalized to nonstationary sequences.

Optimal estimation of the parameters of a nonstationary random pulse train in discrete time

The problem of optimal estimation of the parameters of a nonstationary sequence of random pulses observed additively with the noise is considered with the step change in average pulse repetition rate taken into account. A recurrent algorithm is derived for optimal estimation of the pulse repetition rate and time of its jump in real time. The optimal assessment of the Bernoulli trial parameters is shown as an example. The results of computer simulation experiments are presented.

Asymptotic distributions of records in nonstationary schemes

Abstract We investigate an asymptotic behaviour of record values X ( n ) for nonstationary sequences of independent random variables. The so-called F α -scheme is considered. It is shown that the set of all possible asymptotic distributions for X ( n ) under some conditions on parameters of the F α -scheme coincides with the corresponding set of asymptotic distributions in the stationary scheme.

Inductive methods and rates of Rth-mean convergence in adaptive filtering

An inductive method for establishing rates of rth-mean convergence of linear, decreasing-gain adaptive algorithms is introduced. This inductive method does not require any sophisticated probability theory or mathematics and allows such weak forms of dependence as weak-multiplicativity, strong mixing and decaying covariance. In exposing our method, we establish rates of rth-mean convergence for the process , generated by the basic linear adaptive filtering algorithm with and being weakiy-stationary stochastic processes which drive the algorithm and being a sequence of the form k x for some 0 ≥ α ≥ 1. Specifically, we show in most cases that , where r is a real constant greater than or equal to one and . Finally, our method takes advantage of moment bounds for partial sums of non-stationary random sequences, some of which are established within this note

Law of the iterated logarithm for perturbed empirical distribution functions evaluated at a random point for nonstationary random variables

We consider perturbed empirical distribution functions\(\hat F_n (x) = 1/n\sum\nolimits_{i = 1}^n {G_n (x - X_i )} \), where {Ginn, n≥1} is a sequence of continuous distribution functions converging weakly to the distribution function of unit mass at 0, and {Xi, i≥1} is a non-stationary sequence of absolutely regular random variables. We derive the almost sure representation and the law of the iterated logarithm for the statistic\(\hat F_n (U_n )\) whereUn is aU-statistic based onX1,...,Xn. The results obtained extend or generalize the results of Nadaraya,(7) Winter,(16) Puri and Ralescu,(9,10) Oodaira and Yoshihara,(8) and Yoshihara,(19) among others.

Moment bounds for non-stationary dependent sequences

We provide a unified approach for establishing even-moment bounds for partial sums for a class of weakly dependent random variables satisfying a stationarity condition. As applications, we discuss moment bounds for various types of mixing sequences. To obtain even-moment bounds, we use a ‘combinatorial argument' developed by Cox and Kim (1990).

A note on exceedances and rare events of non-stationary sequences

Exceedances of a non-stationary sequence above a boundary define certain point processes, which converge in distribution under mild mixing conditions to Poisson processes. We investigate necessary and sufficient conditions for the convergence of the point process of exceedances, the point process of upcrossings and the point process of clusters of exceedances. Smooth regularity conditions, as smooth oscillation of the non-stationary sequence, imply that these point processes converge to the same Poisson process. Since exceedances are asymptotically rare, the results are extended to triangular arrays of rare events.

A note on exceedances and rare events of non-stationary sequences

Exceedances of a non-stationary sequence above a boundary define certain point processes, which converge in distribution under mild mixing conditions to Poisson processes. We investigate necessary and sufficient conditions for the convergence of the point process of exceedances, the point process of upcrossings and the point process of clusters of exceedances. Smooth regularity conditions, as smooth oscillation of the non-stationary sequence, imply that these point processes converge to the same Poisson process. Since exceedances are asymptotically rare, the results are extended to triangular arrays of rare events.

Limiting Distribution of Sums of Certain Types of Nonstationary Sequences

Article DataHistoryPublished online: 28 July 2006Publication DataISSN (print): 0040-585XISSN (online): 1095-7219Publisher: Society for Industrial and Applied MathematicsCODEN: tprbau