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3.4 Likelihood ratio test

It is remarkable that all three very different approaches - Bayesian, minimax, and Neyman-Pearson - lead to the same test called the likelihood ratio test   [28Jump To The Next Citation Point] . The likelihood ratio /\ is the ratio of the pdf when the signal is present to the pdf when it is absent:
/\(x) := p1(x)-. (17) p0(x)
We accept the hypothesis H1 if /\ > k, where k is the threshold that is calculated from the costs Cij, priors pi, or the significance of the test depending on what approach is being used.

3.4.1 Gaussian case - The matched filter

Let h be the gravitational-wave signal and let n be the detector noise. For convenience we assume that the signal h is a continuous function of time t and that the noise n is a continuous random process. Results for the discrete time data that we have in practice can then be obtained by a suitable sampling of the continuous-in-time expressions. Assuming that the noise is additive the data x can be written as

x(t) = n(t) + h(t). (18)
In addition, if the noise is a zero-mean, stationary, and Gaussian random process, the log likelihood function is given by
1 log /\ = (x |h) - --(h|h), (19) 2
where the scalar product ( .|.) is defined by
integral oo ~x(f)~y*(f)- (x |y) := 4 R ~ df. (20) 0 S(f )
In Equation (20View Equation) R denotes the real part of a complex expression, the tilde denotes the Fourier transform, the asterisk is complex conjugation, and ~S is the one-sided spectral density of the noise in the detector, which is defined through equation
* ' 1- ' ~ E [~n(f )~n (f )] = 2 d(f - f )S(f), (21)
where E denotes the expectation value.

From the expression (19View Equation) we see immediately that the likelihood ratio test consists of correlating the data x with the signal h that is present in the noise and comparing the correlation to a threshold. Such a correlation is called the matched filter . The matched filter is a linear operation on the data.

An important quantity is the optimal signal-to-noise ratio r defined by

integral oo ~ 2 r2 := (h|h) = 4 R |h(f-)|-df. (22) 0 ~S(f )
We see in the following that r determines the probability of detection of the signal. The higher the signal-to-noise ratio the higher the probability of detection.

An interesting property of the matched filter is that it maximizes the signal-to-noise ratio over all linear filters  [28] . This property is independent of the probability distribution of the noise.


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