Upcrossings Of Level Research Articles

The limit properties of point processes of upcrossings in nonstationary strongly dependent Gaussian models

Let {X(t),t≥0} be a nonstationary strongly dependent Gaussian process. Under some conditions related to the correlation function of the Gaussian process, the point processes formed by the upcrossings of level u by {X(t),t≥0} converge weakly to a Poisson process N with random intensity, as u→∞.

Prediction of catastrophes in space over time

Predicting rare events, such as high level up-crossings, for spatio-temporal processes plays an important role in the analysis of the occurrence and impact of potential catastrophes in, for example, environmental settings. Designing a system which predicts these events with high probability, but with few false alarms, is clearly desirable. In this paper an optimal alarm system in space over time is introduced and studied in detail. These results generalize those obtained by de Mare (Ann. Probab. 8, 841–850, 1980) and Lindgren (Ann. Probab. 8, 775–792, 1980, Ann. Probab. 13, 804–824, 1985) for stationary stochastic processes evolving in continuous time and are applied here to stationary Gaussian random fields.

Outcrossings of safe regions by generalized hyperbolic processes

We present a simple Gaussian mixture model in space and time with generalized hyperbolic marginals. Starting with Rice’s celebrated formula for level upcrossings and outcrossings of safe regions we investigate the consequences of the mean–variance mixture model on such quantities. We obtain explicit formulas and also present some explicit examples and limit results.

Asymptotic Poisson Character of Extremes in Non-Stationary Gaussian Models

Let X be a non-stationary Gaussian process, asymptotically centered with constant variance. Let u be a positive real. Define Ru(t) as the number of upcrossings of level u by the process X on the interval (0, t]. Under some conditions we prove that the sequence of point processes (Ru)u>0 converges weakly, after normalization, to a standard Poisson process as u tends to infinity. In consequence of this study we obtain the weak convergence of the normalized supremum to a Gumbel distribution.

Maxima of the continuous wavelet transform of correlated data

The study of continuous wavelet transform (CWT) of signals through the behavior of its local maxima is a well-developed field that has already led to useful applications in signal and image analysis. Meanwhile, the study of level upcrossings of random field is based on expected values of random quantities related to local maxima of the field. Generalizing the notion of level upcrossings from one dimension to higher-dimensional spaces leads to the problem of evaluating the expected value of the Euler characteristic of excursion sets on those fields. This has been done by Adler [The Geometry of Random Fields, Wiley, New York, 1981] and further extended by Siegmund and Worsley “Testing for a signal with unknown location and scale in a stationary Gaussian random field” [Ann. Stat. 23 (2) (1995) 608–639], who proposed an extension of the method to test for signals not only of unknown location but of unknown scale as well, using an approach quite similar to the CWT. Even for an “irregular” field which does not respect Adler's condition, a proper use of the CWT leads to a representation where the field becomes regular. We first show that this allows us to apply Adler's method to a more general family of irregular random fields as, for instance, a fractional Brownian motion. Then, we introduce a fast implementation based on the discrete dyadic wavelet decomposition that allows us to perform the analysis with fewer operations than the method originally proposed by Siegmund and Worlsey. Finally, we apply this method in order to detect a sharp but continuous signal in a background noise.

Extrapolation of rainflow matrices

An important area in mechanical engineering is fatigue of materials. The load process is an essential parameter in fatigue design, and is often summarized in the so-called rainflow cycle matrix. Extrapolation of a load measurement to longer distances is needed, and hence one is interested in estimating the limiting rainflow matrix (i.e. the ergodic distribution). An interesting fact is that counting rainflow cycles is equivalent to counting crossings of intervals. The main result is an asymptotic expression for the intensity of interval upcrossings as the upper and lower levels tend to ±∞, where it is assumed that the point processes of upcrossings of the upper and lower levels converge to two independent Poisson processes. The asymptotic expression only involves the intensity of upcrossings of the upper and lower levels. When estimating the limiting rainflow matrix from a load measurement, the asymptotic result is used where it is valid, i.e. for large interval crossings, and kernel smoothing is used elsewhere. To use the asymptotic result, one needs to extrapolate the level upcrossings for high and for low levels, where it is suggested to use peak over threshold methods. An algorithm for the estimation is given in detail. The usefulness and validity of the asymptotic result is illustrated by several examples, both from simulated processes and measured signals from an automobile.

Franchissements de niveaux pour un processus à trajectoires régulières et estimation adaptative à partir d'observations discrétisées

We estimate the mean number of level upcrossings for a continuous stationary process with regular trajectories from discretized observations. We apply a non parametric estimator such that the joint probability density function of the process and its derivative is approximated by a polynomial perturbation of the product of the marginal densities. We evaluate the bias and the variance of this estimator with respect to the number of observations, the discretization step and the dimension of the polynomial space.

Slepian models for X 2-processes with dependent components with application to envelope upcrossings

A Slepian model for the local behaviour near the level upcrossings of a x 2-process with dependent Gaussian components is presented. In case of independent components, this model is shown to take on a rather simple form, thereby simplifying earlier results by Aronowich and Adler. The Slepian model is applied to the envelope of a stationary Gaussian process and used to approximate the probability of ‘empty' envelope upcrossings, i.e. the probability that an envelope upcrossing is not followed by a level crossing in the original process.

Slepian models for X2-processes with dependent components with application to envelope upcrossings

A Slepian model for the local behaviour near the level upcrossings of a x2-process with dependent Gaussian components is presented. In case of independent components, this model is shown to take on a rather simple form, thereby simplifying earlier results by Aronowich and Adler.The Slepian model is applied to the envelope of a stationary Gaussian process and used to approximate the probability of ‘empty' envelope upcrossings, i.e. the probability that an envelope upcrossing is not followed by a level crossing in the original process.

Extreme value theory for continuous parameter stationary processes

In this paper the central distributional results of classical extreme value theory are obtained, under appropriate dependence restrictions, for maxima of continuous parameter stochastic processes. In particular we prove the basic result (here called Gnedenko's Theorem) concerning the existence of just three types of non-degenerate limiting distributions in such cases, and give necessary and sufficient conditions for each to apply. The development relies, in part, on the corresponding known theory for stationary sequences. The general theory given does not require finiteness of the number of upcrossings of any levelx. However when the number per unit time is a.s. finite and has a finite meanμ(x), it is found that the classical criteria for domains of attraction apply whenμ(x) is used in lieu of the tail of the marginal distribution function. The theory is specialized to this case and applied to give the general known results for stationary normal processes for whichμ(x) may or may not be finite). A general Poisson convergence theorem is given for high level upcrossings, together with its implications for the asymptotic distributions ofr th largest local maxima.