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where

with covariance matrix £rr(j) = E[rj(t)rJ(t—j)] [189]. The corresponding fault detection test statistic is pn{t) = vT(t)S^v(t)

Likelihood Ratio Tests for Change Detection. Consider a simple change detection problem, detecting the change in the mean of an i.i.d. random variable yk from Mo to ni and estimating the time of change (switching or jump time) q. If yk is a sequence of n observations and is a white noise sequence with variance u2

where

The detection problem can be phrased as a hypothesis testing problem [44].

This is an easy case since the new value of the mean (mi) is known and only the change time is investigated. The likelihood ratio between these two hypotheses is where pi(-) is the Gaussian probability density function of y, with mean

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