The Annals of Statistics

Time Series Discrimination by Higher Order Crossings

Benjamin Kedem and Eric Slud

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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.

Article information

Ann. Statist., Volume 10, Number 3 (1982), 786-794.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62M07: Non-Markovian processes: hypothesis testing 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20]

Stationary time series goodness of fit discrimination higher-order crossings level crossings clipped signal


Kedem, Benjamin; Slud, Eric. Time Series Discrimination by Higher Order Crossings. Ann. Statist. 10 (1982), no. 3, 786--794. doi:10.1214/aos/1176345871.

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