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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 [34Jump To The Next Citation Point]. The likelihood ratio Λ is the ratio of the pdf when the signal is present to the pdf when it is absent:
Λ(x ) := p1(x)-. (19 ) p0(x)
We accept the hypothesis H1 if Λ > k, where k is the threshold that is calculated from the costs Cij, priors πi, 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). (20 )
In addition, if the noise is a zero-mean, stationary, and Gaussian random process, the log likelihood function is given by
1 log Λ = (x |h ) −--(h|h), (21 ) 2
where the scalar product ( ⋅|⋅) is defined by
∫ ∞ ˜x(f)˜y∗(f)- (x |y) := 4ℜ ˜ df. (22 ) 0 S (f)
In Equation (22View 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 δ(f − f )S(f), (23 )
where E denotes the expectation value.

From the expression (21View 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 ρ defined by

∫ ∞ ˜ 2 ρ2 := (h|h) = 4ℜ |h-(f)|-df. (24 ) 0 ˜S(f )
We see in the following that ρ 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 [34]. This property is independent of the probability distribution of the noise.


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