4.3 False alarm and detection probabilities

4.3.1 False alarm and detection probabilities for known intrinsic parameters

We first present the false alarm and detection probabilities when the intrinsic parameters ξ of the signal are known. In this case the ℱ-statistic 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 the ℱ-statistic has a distribution related to the 2 χ distribution. One can show (see Section III B in [65Jump To The Next Citation Point]) that for the signal given by Eq. (30View Equation), 2ℱ has a χ2 distribution with n degrees of freedom when the signal is absent and noncentral χ2 distribution with n degrees of freedom and non-centrality parameter equal to the square of the signal-to-noise ratio when the signal is present.

As a result the pdfs p0 and p1 of the ℱ-statistic, when the intrinsic parameters are known and when respectively the signal is absent or present in the data, are given by

n∕2−1 p (ℱ ) = --ℱ-------exp (− ℱ ), (84 ) 0 (n∕2 − 1)! ( ) (2ℱ-)(n∕2−1)∕2- ( √ ---) 1-2 p1(ρ,ℱ ) = ρn∕2−1 In∕2−1 ρ 2ℱ exp − ℱ − 2ρ , (85 )
where n is the number of degrees of freedom of χ2 distribution 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
∫ ∞ n∕∑2−1 ℱ k PF (ℱ0 ) := p0(ℱ )d ℱ = exp(− ℱ0) -0-. (86 ) ℱ0 k=0 k!
The probability of detection PD is the probability that ℱ exceeds the threshold ℱ0 when a signal is present and the signal-to-noise ratio is equal to ρ:
∫ ∞ PD(ρ,ℱ0 ) := ℱ p1(ρ,ℱ )dℱ . (87 ) 0
The integral in the above formula can be expressed in terms of the generalized Marcum Q-function [132, 58], √ ---- PD (ρ,ℱ0 ) = Q (ρ, 2ℱ0 ). 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.

4.3.2 False alarm probability for unknown intrinsic parameters

Next we return to the case in which the intrinsic parameters ξ are not known. Then the statistic ℱ [x;ξ] given by Eq. (62View Equation) is a certain multiparameter random process called the random field (see monographs [5, 6] 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 define the autocovariance function 𝒞 just in the same way as we define such a function for a random process:

′ [ ′] [ ] [ ′ ] 𝒞 (ξ,ξ ) := E0 ℱ[x;ξ ]ℱ [x;ξ ] − E0 ℱ [x;ξ] E0 ℱ [x;ξ ] , (88 )
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 (30View Equation) the autocovariance function 𝒞 is given by
′ 1 ( ′ ′−1 ′T −1) 𝒞 (ξ,ξ ) = -Tr Q(ξ,ξ ) ⋅ M (ξ) ⋅ Q (ξ,ξ ) ⋅ M (ξ) , (89 ) 2
where Q is an n × n matrix with components
′ ′ Q(ξ, ξ)ij := (hi(t;ξ)|hj (t;ξ )), i,j = 1,...,n. (90 )
Obviously Q(ξ, ξ) = M (ξ ), therefore 𝒞(ξ,ξ) = n∕2.

One can estimate the false alarm probability in the following way [68Jump To The Next Citation Point]. 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 of 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.

We choose the elementary cell with its origin at the point ξ to be a compact region with boundary defined by the requirement that the autocovariance 𝒞(ξ,ξ ′) between the origin ξ and any point ξ ′ at the cell’s boundary equals half of its maximum value, i.e., 𝒞 (ξ, ξ)∕2. Thus, the elementary cell is defined by the inequality

1 n 𝒞(ξ,ξ ′) ≤ --𝒞(ξ,ξ) = --, (91 ) 2 4
with ξ at the cell’s center and ξ′ on the cell’s boundary.

To estimate the number of cells we perform the Taylor expansion of the autocovariance function up to the second-order terms:

∑m ′ || ∑m 2 ′|| 𝒞(ξ,ξ′) ∼= n-+ ∂-𝒞(ξ,ξ-)| Δ ξ𝒜 + 1- ∂-𝒞(ξ,ξ-)| Δξ𝒜 Δ ξℬ. (92 ) 2 𝒜=1 ∂ξ′𝒜 |ξ′=ξ 2 𝒜,ℬ=1 ∂ξ′𝒜∂ ξ′ℬ |ξ′= ξ
As 𝒞 attains its maximum value when ′ ξ − ξ = 0, we have
∂ 𝒞(ξ,ξ′)|| -----′---|| = 0, 𝒜 = 1,...,m. (93 ) ∂ξ𝒜 ξ′=ξ
Let us introduce the symmetric matrix G with components
2 ′|| G 𝒜ℬ(ξ) := − ---1----∂-𝒞(ξ,-ξ)| , 𝒜, ℬ = 1, ...,m. (94 ) 2𝒞(ξ,ξ ) ∂ξ′𝒜 ∂ξ′ℬ |ξ′=ξ
Then, the inequality (91View Equation) for the elementary cell can approximately be written as
m ∑ 1- G 𝒜 ℬ(ξ)Δ ξ𝒜Δ ξℬ ≤ 2 . (95 ) 𝒜,ℬ=1
It is interesting to find a relation between the matrix G and the Fisher matrix. One can show (see [78Jump To The Next Citation Point], Appendix B) that the matrix G is precisely equal to the reduced Fisher matrix &tidle;Γ given by Eq. (81View Equation).

If the components of the matrix G are constant (i.e., they are independent of the values of the intrinsic parameters ξ of the signal), the above equation defines a hyperellipsoid in m-dimensional (m is the number of the intrinsic parameters) Euclidean space ℝm. The m-dimensional Euclidean volume Vcell of the elementary cell given by Eq. (95View Equation) equals

(π∕2)m∕2 Vcell = -----------√------, (96 ) Γ (m ∕2 + 1) detG
where Γ denotes the Gamma function. We estimate the number N cells of elementary cells by dividing the total Euclidean volume V of the m-dimensional intrinsic parameter space by the volume Vcell of one elementary cell, i.e., we have
V Ncells = ----. (97 ) Vcell
The components of the matrix G are constant for the signal h(t;A0,ϕ0,ξ ) = A0cos (ϕ(t;ξ) − ϕ0), provided the phase ϕ(t;ξ) is a linear function of the intrinsic parameters ξ.

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 intrinsic parameters ξ, one replaces Eq. (97View Equation) by

∫ Γ (m-∕2-+-1) ∘ -------- Ncells = (π∕2 )m∕2 V det G(ξ)dV. (98 )
This formula 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 [102Jump To The Next Citation Point], 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 [68] and it is a generalization of the idea of an effective number of samples introduced in [46] for the case of a coalescing binary signal.

We approximate the pdf of the ℱ-statistic in each cell by the pdf p0(ℱ ) of the ℱ-statistic when the parameters are known [it is given by Eq. (84View Equation)]. The values of the ℱ-statistic 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 Eq. (86View Equation). Consequently the probability that ℱ does not exceed the threshold ℱ0 in all the Ncells cells is [1 − PF(ℱ0 )]Ncells. Thus, the probability P TF that ℱ exceeds ℱ0 in one or more cells is given by

PTF (ℱ0 ) = 1 − [1 − PF (ℱ0)]Ncells. (99 )
By definition, this is the false alarm probability when the phase parameters are unknown. The number of false alarms NF is given by
T NF = NcellsP F (ℱ0 ). (100 )
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 [65Jump To The Next Citation Point].

It was shown (see [39Jump To The Next Citation Point]) that for any finite ℱ0 and Ncells, Eq. (99View Equation) provides an upper bound for the false alarm probability. Also in [39] a tighter upper bound for the false alarm probability was derived by modifying a formula obtained by Mohanty [92Jump To The Next Citation Point]. The formula amounts essentially to introducing a suitable coefficient multiplying the number Ncells of cells.

4.3.3 Detection probability for unknown intrinsic parameters

When the signal is present in the data a precise calculation of the pdf of the ℱ-statistic is very difficult because the presence of the signal makes the data’s random process non-stationary. As a first approximation we can estimate the probability of detection of the signal when the intrinsic parameters are unknown by the probability of detection when these parameters are known [it is given by Eq. (87View Equation)]. This approximation assumes that when the signal is present the true values of the intrinsic parameters fall within the cell where the ℱ-statistic has a maximum. This approximation will be the better the higher the signal-to-noise ratio ρ is.


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