False alarm and detection probabilities

4.5 False alarm and detection probabilities – Gaussian case

4.5.1 Statistical properties of the $ℱ$ -statistic

We first present the false alarm and detection pdfs when the intrinsic parameters of the signal are known. In this case the statistic $ℱ$ is a quadratic form of the random variables that are correlations of the data. As we assume that the noise in the data is Gaussian and the correlations are linear functions of the data, $ℱ$ is a quadratic form of the Gaussian random variables. Consequently $ℱ$ -statistic has a distribution related to the $2 χ$ distribution. One can show (see Section III B in [49 ]) that for the signal given by Equation (14 ), $2 ℱ$ has a $2 χ$ distribution with 4 degrees of freedom when the signal is absent and noncentral $χ2$ distribution with 4 degrees of freedom and non-centrality parameter equal to signal-to-noise ratio $(h|h)$ when the signal is present.

As a result the pdfs $p0$ and $p1$ of $ℱ$ when the intrinsic parameters are known and when respectively the signal is absent and present are given by

&#x2131;n&#x2215;2&#x2212;1 p0&#x0028;&#x2131; &#x0029; = ----------exp &#x0028;&#x2212; &#x2131; &#x0029;, &#x0028;46 &#x0029; &#x0028;n&#x2215;2 &#x2212; 1&#x0029;! &#x0028; &#x0029; &#x0028;2&#x2131;-&#x0029;&#x0028;n&#x2215;2&#x2212;1&#x0029;&#x2215;2- &#x0028; &#x221A; ---&#x0029; 1-2 p1&#x0028;&#x03C1;,&#x2131; &#x0029; = &#x03C1;n&#x2215;2&#x2212;1 In&#x2215;2&#x2212;1 &#x03C1; 2&#x2131; exp &#x2212; &#x2131; &#x2212; 2&#x03C1; , &#x0028;47 &#x0029;

where $n$ is the number of degrees of freedom of $χ2$ distributions and $In∕2−1$ is the modified Bessel function of the first kind and order $n ∕2 − 1$ . The false alarm probability $PF$ is the probability that $ℱ$ exceeds a certain threshold $ℱ0$ when there is no signal. In our case we have

The probability of detection $PD$ is the probability that $ℱ$ exceeds the threshold $ℱ0$ when the signal-to-noise ratio is equal to $ρ$ :

The integral in the above formula can be expressed in terms of the generalized Marcum $Q$ -function [94 , 44 ], $Q (α,β ) = PD(α, β2∕2)$ . We see that when the noise in the detector is Gaussian and the intrinsic parameters are known, the probability of detection of the signal depends on a single quantity: the optimal signal-to-noise ratio $ρ$ .

4.5.2 False alarm probability

Next we return to the case when the intrinsic parameters $ξ$ are not known. Then the statistic $ℱ(ξ )$ given by Equation (35 ) is a certain generalized multiparameter random process called the random field (see Adler’s monograph [4 ] for a comprehensive discussion of random fields). If the vector $ξ$ has one component the random field is simply a random process. For random fields we can define the autocovariance function $𝒞$ just in the same way as we define such a function for a random process:

where $ξ$ and $ξ′$ are two values of the intrinsic parameter set, and $E0$ is the expectation value when the signal is absent. One can show that for the signal (14

) the autocovariance function $𝒞$ is given by

where

We have $𝒞 (ξ, ξ) = 1$ .

One can estimate the false alarm probability in the following way [50 ]. The autocovariance function $𝒞$ tends to zero as the displacement $′ Δ ξ = ξ − ξ$ increases (it is maximal for $Δ ξ = 0$ ). Thus we can divide the parameter space into elementary cells such that in each cell the autocovariance function $𝒞$ is appreciably different from zero. The realizations of the random field within a cell will be correlated (dependent), whereas realizations of the random field within each cell and outside the cell are almost uncorrelated (independent). Thus the number of cells covering the parameter space gives an estimate of the number of independent realizations of the random field. The correlation hypersurface is a closed surface defined by the requirement that at the boundary of the hypersurface the correlation $𝒞$ equals half of its maximum value. The elementary cell is defined by the equation

for $ξ$ at cell center and $′ ξ$ on cell boundary. To estimate the number of cells we perform the Taylor expansion of the autocorrelation function up to the second-order terms:

As $𝒞$ attains its maximum value when $′ ξ − ξ = 0$ , we have

Let us introduce the symmetric matrix

Then the approximate equation for the elementary cell is given by

It is interesting to find a relation between the matrix $G$ and the Fisher matrix. One can show (see [60

], Appendix B) that the matrix $G$ is precisely equal to the reduced Fisher matrix $˜Γ$ given by Equation (43

Let $K$ be the number of the intrinsic parameters. If the components of the matrix $G$ are constant (independent of the values of the parameters of the signal) the above equation is an equation for a hyperellipse. The $K$ -dimensional Euclidean volume $Vcell$ of the elementary cell defined by Equation (57 ) equals

where $Γ$ denotes the Gamma function. We estimate the number $N c$ of elementary cells by dividing the total Euclidean volume $V$ of the $K$ -dimensional parameter space by the volume $Vcell$ of the elementary cell, i.e. we have

The components of the matrix $G$ are constant for the signal $h(t;A0,φ0, ξμ) = A0 cos(φ(t;ξμ) − φ0)$ when the phase $φ (t;ξμ)$ is a linear function of the intrinsic parameters $ξμ$ .

To estimate the number of cells in the case when the components of the matrix $G$ are not constant, i.e. when they depend on the values of the parameters, we write Equation (59 ) as

This procedure can be thought of as interpreting the matrix $G$ as the metric on the parameter space. This interpretation appeared for the first time in the context of gravitational-wave data analysis in the work by Owen [74

], where an analogous integral formula was proposed for the number of templates needed to perform a search for gravitational-wave signals from coalescing binaries.

The concept of number of cells was introduced in [50 ] and it is a generalization of the idea of an effective number of samples introduced in [36 ] for the case of a coalescing binary signal.

We approximate the probability distribution of $ℱ (ξ)$ in each cell by the probability $p0(ℱ )$ when the parameters are known [in our case by probability given by Equation (46 )]. The values of the statistic $ℱ$ in each cell can be considered as independent random variables. The probability that $ℱ$ does not exceed the threshold $ℱ 0$ in a given cell is $1 − P (ℱ ) F 0$ , where $P (ℱ ) F 0$ is given by Equation (48 ). Consequently the probability that $ℱ$ does not exceed the threshold $ℱ0$ in all the $Nc$ cells is $[1 − PF(ℱ0 )]Nc$ . The probability $PFT$ that $ℱ$ exceeds $ℱ0$ in one or more cell is thus given by

This by definition is the false alarm probability when the phase parameters are unknown. The number of false alarms $NF$ is given by

A different approach to the calculation of the number of false alarms using the Euler characteristic of level crossings of a random field is described in [49

It was shown (see [29 ]) that for any finite $ℱ0$ and $Nc$ , Equation (61 ) provides an upper bound for the false alarm probability. Also in [29 ] a tighter upper bound for the false alarm probability was derived by modifying a formula obtained by Mohanty [68 ]. The formula amounts essentially to introducing a suitable coefficient multiplying the number of cells $Nc$ .

4.5.3 Detection probability

When the signal is present a precise calculation of the pdf of $ℱ$ is very difficult because the presence of the signal makes the data random process $x(t)$ non-stationary. As a first approximation we can estimate the probability of detection of the signal when the parameters are unknown by the probability of detection when the parameters of the signal are known [given by Equation (49 )]. This approximation assumes that when the signal is present the true values of the phase parameters fall within the cell where $ℱ$ has a maximum. This approximation will be the better the higher the signal-to-noise ratio $ρ$ is.

4.5 False alarm and detection probabilities – Gaussian case

4.5.1 Statistical properties of the -statistic

4.5.2 False alarm probability

4.5.3 Detection probability

4.5.1 Statistical properties of the $ℱ$ -statistic