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August, 1985 Optimal Prediction of Level Crossings in Gaussian Processes and Sequences
Georg Lindgren
Ann. Probab. 13(3): 804-824 (August, 1985). DOI: 10.1214/aop/1176992909


Let $\eta(t)$ be stationary, Gaussian, and suppose one wants to predict the future upcrossings of a certain level $u$. The paper investigates criteria for a good level crossing predictor, and restates in simple form a result by de Mare on optimal prediction. Write $\hat{\eta}_t(t + h)$ for the mean square predictor of $\eta(t + h)$ at time $t$, and let $\hat{\xi}_t(t + h)$ be the conditional expectation of $\eta'(t + h)$ given observed data and given $\eta(t + h) = u$. It is shown that an alarm which predicts an upcrossing at $t + h$ if $\hat{\eta}_t(t + h)$ differs from $u$ by a quantity that is a certain function of $\hat{\zeta}_t(t + h)$ is optimal in the sense that it maximizes the detection probability for a fixed total alarm time. Explicit formulas are given for the upcrossing risks at alarm, detection probability, and total alarm time. In an example the optimal alarm is compared to a naive alarm which predicts an upcrossing if $\hat{\eta}_t(t + h)$ differs from $u$ by a fixed proportion of the residual standard deviation. The optimal alarm locates the upcrossings more precisely and at an earlier stage than the naive alarm, which has a tendency to give late alarms.


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Georg Lindgren. "Optimal Prediction of Level Crossings in Gaussian Processes and Sequences." Ann. Probab. 13 (3) 804 - 824, August, 1985.


Published: August, 1985
First available in Project Euclid: 19 April 2007

zbMATH: 0582.60050
MathSciNet: MR799423
Digital Object Identifier: 10.1214/aop/1176992909

Primary: 60G25
Secondary: 60G15

Keywords: Alarm , Conditional distributions , level crossings , likelihood ratio , Palm probability , prediction , reliability , surveillance

Rights: Copyright © 1985 Institute of Mathematical Statistics


Vol.13 • No. 3 • August, 1985
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