Optimal Prediction of Level Crossings in Gaussian Processes and Sequences

Abstract

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.