Higher Order Crossings Research Articles

Higher-order crossings: a new acoustic emission signal processing method: National Bureau of Standards, Gaithersburg, PB89-173488, 8 pp. (1988)

Higher-order crossings: a new acoustic emission signal processing method: National Bureau of Standards, Gaithersburg, PB89-173488, 8 pp. (1988)

Higher-Order Crossings in Time Series Model Identification

Higher-Order Crossings in Time Series Model Identification

Higher-Order Crossings in Time Series Model Identification

The oscillation in time series can be represented by counts of axis crossings in the series and its differences. These counts are called higher-order crossings (HOC's) and display a monotone property whose rate of increase discriminates between processes. Very few of these counts are needed for effective discrimination as shown by plots of HOC's obtained from real and simulated data. It is shown how to use HOC plots in testing the oscillation characteristics in time series.

DETECTION OF PERIODICITIES BY HIGHER‐ORDER CROSSINGS

Abstract. The axis‐crossing counts in a time series and in its successive differences are called higher‐order crossings (HOC). Under the Gaussian assumption the sequence of expected HOC is monotone increasing and admits a spectral representation which establishes a clear connection between HOC and the spectrum. In particular the normalized number of axis‐crossings (first HOC) tends to admit values at or near a dominant frequency, while the sequence of normalized HOC converges to the highest frequency in the spectrum. When the series is first lowpass filtered, the resulting normalized HOC tend to ‘visit’ true discrete frequencies on their way to the highest frequency. If the successive differences are replaced by successive summation, the resulting modified normalized HOC converge monotonically to the lowest frequency in the spectrum.

Spectral analysis and discrimination by zero-crossings

We advance a coherent development of zero-crossing-based methods and theory appropriate for fast signal analysis. Quite a few ideas pertaining to zero-crossing counts found in the literature can be expressed and interpreted with the help of this more general setup. A central issue addressed in some detail is the fruitful connection which exists between zero-crossing counts and linear filtering. This connection is explored and interpreted with the help of a certain zero-crossing spectral representation, and is then applied in spectral analysis, detection, and discrimination. Zero-crossing counts in filtered time series are called higher order crossings. The theme of this work is that higher order crossings analysis provides a useful descriptive as well as an analytical tool that can in many respects match spectral analysis. To a great extent these two types of analysis are, in fact, equivalent, but each emphasizes a different point of view. Advantages offered by higher order crossings are great simplicity and a drastic data reduction.

On the asymptotic variance of higher order crossings with special reference to a fast white noise test

SUMMARY The rate of decrease in the variance of higher order crossings is obtained for stationary m-dependent Gaussian sequences. This result provides a quick approximation to the variance of the number of axis-crossings of m-dependent processes, and is applied in determining approximate probability limits for the higher order crossings of white noise. The limits are used in testing for white noise in simulated and also real data examples.

On the Sinusoidal Limit of Stationary Time Series

We show that the Sinusoidal Limit Theorem of Slutsky in the Gaussian case is a consequence of the equality of certain higher order crossings.

A two-dimensional higher order crossings theorem (Corresp.)

The limiting forms of binary processes obtained via sequential filtering followed by clipping are discussed in the cases of one- and two-dimensional processes. In particular, repeated differencing followed by clipping displays a convergence phenomenon shared by stationary stochastic processes and fields. For this case, the higher order crossings are monotone under some conditions.

Time Series Discrimination by Higher Order Crossings

A new methodology is proposed for discrimination among stationary time-series. The time series are transformed into binary arrays by clipping (retaining only the signs of) the $j$th difference series, $j = 0, 1, 2, \cdots$. The degeneracy of clipped $j$th differences is studied as $j$ becomes large. A new goodness of fit statistic is defined as a quadratic form in the counts of axis-crossings by each of the first $k$ differences of the series. Simulations and the degeneracy of high-order differences justify fixing $k$ no larger than 10 for many processes. Empirical simulated distributions (with $k = 9$) of the goodness of fit statistic suggest a gamma approximation for its tail probabilities. Illustrations are given of discrimination between simple models with the new statistic.

Asymptotic Normality of the Number of Crossings of Level Zero by a Gaussian Process

Asymptotic Normality of the Number of Crossings of Level Zero by a Gaussian Process