© ESO 2019

1. Introduction

In essence, the matched filter (MF) is a linear low-pass filter that maximizes the signal to noise ratio of the detected signal. When the noise can be assumed as the realization of an isotropic Gaussian random field (GRF), this filter provides the greatest probability of detection for a fixed probability of false detection or false alarm (PFA) (Kay 1998). This property makes the MF a very popular detection technique. However, MF requires the a priori knowledge of the position of the signal within the GRF (e.g. an emission line in a spectrum or a point source in an astronomical map). In most practical applications this is not the case. For this reason, the MF is used assuming that, if present, the position of a signal corresponds to a peak in the matched filtered data. A detection can be assigned to the peaks exceeding a prefixed threshold. In other words, the detection procedure is not based on the entire area of the GRF but only on the subset of the points (pixels in the case of discrete data) corresponding to the position of a peak. Recently Vio & Andreani (2016) and Vio et al. (2018; hereafter VA16 and VVA17) have shown that, when based on the standard but wrong assumption that the probability density function (PDF) of the amplitudes of the peaks of a GRF is a Gaussian, this approach may lead to a severe underestimation of the PFA. Moreover, the PFA does not provide the probability that a given detection is spurious but the probability that a generic peak can exceed, by chance, a fixed threshold. The critical point is that the threshold has to depend on the number of peaks. The greater the number of peaks the higher the threshold. In statistics this is a well-known problem called the multiple comparisons, multiple testing or multiple hypotheses problem, whereas in other fields it is known as the look-elsewhere effect. The majority of the proposed solutions are essentially of a non-parametric type, such as the methods that control the family-wise error rate (Lehmann & Romano 2005) and the procedures to control the false discovery rate (Benjamini & Hochberg 1995), with the latter which has been proposed for the astronomical applications (Miller et al. 2001; Hopkins et al. 2002). The main limitation of these methods is that, unlike the parametric approaches, they are not able to exploit all the available information. Parametric methods have been produced, but in general they are not easy to use. An example is the procedure proposed by Vitells & Gross (2001), based on the computation of the Euler characteristic of a GRF (a quantity more difficult to compute than the peak amplitudes), which needs numerical simulations to fix the value of some fundamental parameters. For this reason, VA16 and VVA17 have introduced an efficient parametric approach able to provide a reliable estimate, called specific probability of false alarm (SPFA), of the probability of a false detection. Such a quantity is computed on the basis of the PDF of the peak amplitudes of a smooth and isotropic GRF¹. In a recent work, Pavesi et al. (2018) reach the same conclusions as VA16 and VVA17 but adopting a different approach based on the Gumbel distribution, which represents the asymptotic PDF of the extreme (i.e. the greatest value) of an isotropic GRF.

In this paper we show that, although the two approaches produce almost the same results, the method based on the PDF of the peak amplitudes (PAM) is more flexible than the method based on the Gumbel distribution (GDM) and moreover it permits an easy check of the conditions of its applicability. Hence, the detection procedure is more effective when PAM is used.

In Sect. 2 the main characteristics of MF are reviewed. The reason why it underestimates the PFA when used in the standard way is explained in Sect. 3. In the same section, the method suggested by VA16 and VVA17 to correctly compute this quantity is illustrated and the quantity SPFA introduced. The GDM approach is presented in Sect. 4. Finally, in Sects. 5 and 6 the PAM and GDM performances are tested on a set of simulated GRFs as well on an interferometric map obtained with the Atacama Large Millimeter/submillimeter Array (ALMA) and the final remarks are deferred to Sect. 7.

2. Matched filter

Given a discrete observed signal x, the model assumed in the MF approach is x = s + n, where s is the deterministic signal to detect and n a zero-mean GRF with known covariance matrix,

$$\begin{array}{c}\hfill \mathit{C}=E[\mathit{n}{\mathit{n}}^T].\end{array}$$(1)

Here, symbols E[.] and ^T denote the expectation operator and the vector or matrix transpose, respectively.

Under these conditions, according to the Neyman–Pearson theorem (Neyman & Pearson 1933; Kay 1998), a detection is claimed when

$$\begin{array}{c}\hfill \mathcal{T}(\mathit{x})={\mathit{x}}^T\mathit{f}>\gamma ,\end{array}$$(2)

with γ a real constant and

$$\begin{array}{c}\hfill {\mathit{f}}_s={\mathit{C}}^{-1}\mathit{s}.\end{array}$$(3)

Here f_s represents the matched filter. The main characteristic of the MF is that it maximizes the probability of detection under the constraint of a fixed PFA.

Matched filter works properly under two assumptions. The first assumption is that the signal s is known. Actually, in practical applications only the template g of the signal s = ag is available but not its amplitude a. However, this does not represent a true problem since the MF in the form

$$\begin{array}{c}\hfill {\mathit{f}}_g={\mathit{C}}^{-1}\mathit{g},\end{array}$$(4)

does not affect the PFA but only the probability of detection (VA16). The second assumption requires that x and s have the same size. This implicitly means that the position of s within x is known. Problems come up when this last assumption is relaxed and the sizes of s are smaller than the sizes of x (e.g. an emission line in a spectrum). If the amplitude a is also unknown, the standard approach consists in cross-correlating x with the MF given by Eq. (4) obtaining signal 𝔵 typically standardized to zero-mean and unit variance. A detection is claimed when a peak in 𝔵 exceeds a threshold set to u. If the number of signals s present in x is also unknown, this procedure has to be applied to all the most significant peaks. It is a widespread practice that the corresponding PFA is given by

$$\begin{array}{c}\hfill \alpha ={\mathrm{\Phi }}_c(u),\end{array}$$(5)

where Φ_c(u) = 1 − Φ(u), with Φ(u) the standard Gaussian cumulative distribution function (CDF).

3. Detection procedure based on the PDF of the amplitude of the peaks

In VA16 and VVA17 it has been shown that the computation of the PFA by means of Eq. (5) can lead to a severe underestimation of the latter quantity. This is because the PDF of the peaks of a GRF is not a Gaussian as implicitly assumed in Eq. (5). For this reason, the correct PFA has to be estimated by means of

$$\begin{array}{c}\hfill \alpha ={\mathrm{\Psi }}_c(u),\end{array}$$(6)

where

$$\begin{array}{c}\hfill {\mathrm{\Psi }}_c(u)=1-\mathrm{\Psi }(u),\end{array}$$(7)

with

$$\begin{array}{c}\hfill \mathrm{\Psi }(u)={\displaystyle {\int }_{-\infty }^u}\psi (z)\mathrm{d}z,\end{array}$$(8)

and

$$\begin{array}{c}\hfill \psi (z)=\frac{\sqrt{3-{\kappa }^2}}{\sqrt{6\pi }}{\mathrm{e}}^{-\frac{3z^2}{2(3-{\kappa }^2)}}+\frac{2\kappa z\sqrt{\pi }}{\sqrt{6}}\varphi (z)\mathrm{\Phi }\left(\frac{\kappa z}{\sqrt{3-{\kappa }^2}}\right)\end{array}$$(9)

for the 1D case, and

$$\begin{array}{cc}\hfill \psi (z)& =\sqrt{3}{\kappa }^2(z^2-1)\varphi (z)\mathrm{\Phi }\left(\frac{\kappa z}{\sqrt{2-{\kappa }^2}}\right)+\frac{\kappa z\sqrt{3(2-{\kappa }^2)}}{2\pi }{\mathrm{e}}^{-\frac{z^2}{2-{\kappa }^2}}\hfill \\ \hfill & +\frac{\sqrt{6}}{\sqrt{\pi (3-{\kappa }^2)}}{\mathrm{e}}^{-\frac{3z^2}{2(3-{\kappa }^2)}}\mathrm{\Phi }\left(\frac{\kappa z}{\sqrt{(3-{\kappa }^2)(2-{\kappa }^2)}}\right)\hfill \end{array}$$(10)

for the 2D case. These expressions represent the PDF of the local maxima of a zero-mean, unit-variance homogeneous GRF (Cheng & Schwartzman 2015a,b)².

Here, κ is a parameter given by

$$\begin{array}{c}\hfill \kappa =-\frac{\varrho \prime (0)}{\sqrt{\varrho ″(0)}},\end{array}$$(11)

where 𝜚′(0) and 𝜚″(0) are, respectively, the first and second derivative with respect to r² of the autocorrelation function 𝜚(r) of the GRF at r = 0, with r the inter-point distance of the random field. As shown in VVA17, this parameter can be estimated by means of a maximum likelihood approach,

$$\begin{array}{c}\hfill \widehat{\kappa }=\underset{\kappa }{argmax}{\displaystyle \sum _{i=1}^{N_{\mathrm{p}}}}log(\psi (z_i;\kappa )),\end{array}$$(12)

where {z_i}, i = 1, 2, …, N_p, are the local maxima in 𝔵³. In this way, it is possible to avoid the use of Eq. (11), which requires knowledge of 𝜚(r). The condition of validity of these expressions is that 𝜚(r) is differentiable at least six times with respect to r, but it is conjectured that four times should be sufficient (Cheng, priv. comm.). Especially in astronomy, this is a weak condition because due to the point spread function of the instruments, the MF often has a Gaussian-like shape. Hence, the resulting 𝔵 are characterized by very smooth 𝜚(r).

It is necessary to stress that the PFA given by Eq. (6) does not provide the probability α that a specific detection is spurious, but the probability that a generic peak due to the noise in 𝔵 can exceed, by chance, the threshold u. If the number of such peaks is N_p, then a number α × N_p among them is expected to exceed the prefixed detection threshold. As a consequence, in spite of a low PFA, the reliability of a detection could actually be small. The solution proposed in VVA17 consists of a preselection based on the PFA and then in the computation of the specific probability of false alarm (SPFA) for each detection. This quantity can be computed by means of the order statistics, in particular by exploiting the statistical characteristics of the greatest value of a finite sample of identical and independently distributed (iid) random variable from a given PDF (Hogg et al. 2013). Under the iid condition, the PDF υ(z_max) of the largest value among a set of N_p peaks {z_i} is given by

$$\begin{array}{c}\hfill \upsilon (z_{max})=N_{\mathrm{p}}{[\mathrm{\Psi }(z_{max})]}^{N_{\mathrm{p}}-1}\psi (z_{max}).\end{array}$$(13)

Hence, the SPFA can be evaluated by means of

$$\begin{array}{c}\hfill \alpha ={\displaystyle {\int }_{z_{max}}^{\infty }}\upsilon (z\prime )\mathrm{d}z\prime .\end{array}$$(14)

The numerical evaluation of this integral does not present particular difficulties since

$$\begin{array}{cc}\hfill \alpha & =N_{\mathrm{p}}{\displaystyle {\int }_{z_{max}}^{\infty }}{[\mathrm{\Psi }(z)]}^{N_{\mathrm{p}}-1}\mathrm{d}\mathrm{\Psi }(z);\hfill \end{array}$$(15)

$$\begin{array}{cc}& ={[\mathrm{\Psi }(z)]}^{N_{\mathrm{p}}}{|}_{z_{\max }}^{\infty };\hfill \end{array}$$(16)

$$\begin{array}{cc}& =1-\mathrm{{\rm Y}}(z_{\max }),\hfill \end{array}$$(17)

with

$$\begin{array}{c}\hfill \mathrm{{\rm Y}}(z_{\max })={[\mathrm{\Psi }(z_{\max })]}^{N_{\mathrm{p}}}.\end{array}$$(18)

This procedure is implicitly based on the assumption that there is only a signal s in x. If the actual number is unknown, it has to be cyclically applied to all the remaining most prominent peaks in order of decreasing amplitude and stopped when the estimated PFA is greater than a prefixed α^*. In principle, N_p should be lowered by one unit after any detection. This is because N_p represents the number of peaks due to the noise. However, since in the practical applications N_p is on the order of thousands if not tens of thousands, this step is irrelevant.

As a final note, it is necessary to stress that the peaks of a isotropic GRF, with 𝜚(r) typical of many astronomical observations, usually have a spatial distribution different from the spatial pattern characteristic of a complete spatial random point process (CSRPP). This is visible in the top panels of Fig. 1 where the PDF of the nearest neighbor distances of the peaks of a simulated GRF with 𝜚(r) given by a two-dimensional circular Gaussian (typical of many astronomical images) with dispersion σ_G = 3 in pixel units is compared with the corresponding PDF of a simulated CSRPP. From this figure it is clear that the PDF related to the peaks lacks small values. Hence, the two processes are different on short spatial scales. This means that the iid condition for the peak amplitudes is not necessarily valid. However, on greater spatial scales, when 𝜚(r) is narrower than the area spanned by the data (a basic situation for the application of the MF), this condition can be expected to hold with good accuracy. The rationale is that two generic points of a GRF with a distance r such that 𝜚(r) ≈ 0 are essentially independent. The same holds for two generic peaks. This is confirmed by the bottom-left panel of Fig. 1 where the sample pair correlation function⁴ρ(r) of the peaks indicates that, for r ≥ 7, their spatial distribution is compatible with a CSRPP. For comparison, the bottom-right panel displays ρ(r) for a true CSRPP. The different behavior on small scales makes the peaks of the GRF show a more uniform spatial distribution than the spatial distribution of the points of the CSRPP (see Fig. 2). In conclusion, since 𝜚(7) ≈ 0, it means that, for r ≥ 7, the peak amplitudes can be considered iid. As a result, most of the peak amplitudes of a GRF with 𝜚(r) typical of many astronomical applications (essentially Gaussian-like) can be expected to be approximately iid. Hence, Eq. (13) is still applicable but possibly with an effective number smaller than N_p (see Sect. 6 in Majumdar & Comtet 2005). This last point is due to the dependence among a set of random variables, which lowers their number of degrees of freedom⁵.

Fig. 1. Statistical characteristics of the spatial distribution of a set of points obtained by means of a GRF and a complete spatial random point process (CSRPP). Top-left panel: histogram of nearest neighbor distances of the peaks of a simulated zero-mean unit-variance GRF with size 500 × 500 pixels and autocorrelation function given by a circular Gaussian with dispersion set to three pixels. Since in this GRF the peaks have coordinates given by integer numbers, in order to mimic a continuous spatial distribution, they have been added to a uniform random number taking its value in the range (−0.5, +0.5]. Bottom-left panel: sample pair correlation function ρ(r) of the peaks in the same GRF. Top-right and bottom-right panels: figures corresponding to the left panels for a CSRPP with the same sizes and containing a number of points equal to the number of peaks as in the GRF. The red lines provide the corresponding theoretical PDFs due to a CSRPP.

Fig. 2. Numerical simulation of the two processes used in Fig. 1.

4. Detection procedure based on the Gumbel distribution

Independently of their PDF, the CDF of the greatest value x_max of a set of N → ∞ iid random variables x is given by

$$\begin{array}{c}\hfill G(x_{\max })=exp[-{(1+{\gamma }_{g}y)}^{1/{\gamma }_g}],\end{array}$$(19)

where

$$\begin{array}{c}\hfill y=\frac{x_{\max }-a}{b},\end{array}$$(20)

with a and b the location and the scale parameter, respectively. Three cases are possible, according to γ_g <  0, γ_g = 0 and γ_g >  0 (Castillo et al. 2004). Assuming that x_max corresponds to a peak, that is, x_max = z_max, and in the case of an isotropic GRF, it has been shown (Colombi et al. 2011) that the CDF of z_max is given by Eq. (19) with γ_g = 0,

$$\begin{array}{c}\hfill G(z_{\max })=exp[-{\mathrm{e}}^{-y}],\end{array}$$(21)

which represents the Gumbel distribution. With a two-dimensional, zero-mean, unit-variance GRF and in the regime of slight clustering of the peaks, Eq. (21) can be approximated by

$$\begin{array}{c}\hfill G(z_{\max })\approx exp[-\frac{{\gamma }^{2}A{z}_{\max }}{{(2\pi )}^{3/2}{R}_{\ast }^2}{\mathrm{e}}^{-z_{\max }^2/2}],\end{array}$$(22)

where A is the area of the GRF,

$$\begin{array}{c}\hfill \gamma =\frac{{\sigma }_1^2}{{\sigma }_0{\sigma }_2},\end{array}$$(23)

and

$$\begin{array}{c}\hfill R_{\ast }=\frac{{\sigma }_1}{{\sigma }_2}\sqrt{2}.\end{array}$$(24)

The SPFA for a specific z_max is given by α = 1 − G(z_max).

If the GRF is obtained by filtering a noise process characterized by a scale-free power spectrum P(k),

$$\begin{array}{c}\hfill P(k)=B{k}^n,\end{array}$$(25)

with a filter w_l(k) (l a scale parameter), it is

$$\begin{array}{c}\hfill {\sigma }_j^2=\frac{1}{2\pi }{\displaystyle {\int }_0^{\infty }}{k}^{2j+1}P(k)w_l^2(k)\mathrm{d}k.\end{array}$$(26)

Details of the three-dimensional case can be found in Colombi et al. (2011).

5. Comparison with simulated data

Given the different derivation of PAM and GDM, it is necessary to compare their performances. With this aim, a numerical experiment was carried out that is based on 5 × 10³ numerical simulations of 500 × 500 pixels GRFs characterized by a circular Gaussian autocorrelation function with dispersion σ_G = 3. This kind of GRF corresponds to what is obtained by filtering a two-dimensional white-noise with a circular Gaussian w_l(x) with standard deviation $l=3/\sqrt{2}$. The resulting G(z_max) can be expressed in the analytical form

$$\begin{array}{c}\hfill G(z_{\max })\approx exp[-\frac{r}{4\sqrt{2\pi }}{z}_{\max }{\mathrm{e}}^{-z_{\max }^2/2}],\end{array}$$(27)

where N_* is the ratio of the total area of the GRF to the window area πl² (Pavesi et al. 2018).

Concerning the PAM, κ = 1 and N_p = 2552 are the theoretical values obtained as explained in VVA17, whereas the corresponding mean value of the maximum likelihood parameter $\widehat{\kappa }$ and the number of peaks N_p obtained from each of the simulated GRFs are $\overline{\kappa }=1.01\pm 0.01$ and N̄_p = 2435 ± 26. The left panel of Fig. 3 shows the PDFs υ_Th(z_max) and υ_Mean(z_max) corresponding to the theoretical value N_p and the mean value N̄_p, respectively. Both slightly differ from the histogram H(z_max). Given the small statistical fluctuation of $\overline{\kappa }$ and N̄_p, such a discrepancy can be explained by the condition of complete spatial randomness assumed for the spatial distribution of the peaks. In particular, the number of effectively iid peaks has to be expected to be smaller than both N and N̄_p. Indeed, a maximum likelihood fit υ_ML(z_max) of the PDF given by Eq. (13) to the set of simulated {z_max} with N_p as free parameter provides N̄_p = 2210. The right panel of Fig. 3 shows that something similar holds for the GDM since the theoretical value N_* = 17 684 is greater than the number Ñ_∗ = 15330 corresponding to the maximum likelihood fit g_ML(z_max). Here, it is worth stressing that υ_Th(z_max) and g_Th(z_max), and hence the corresponding CDFs Υ_Th(z_max) and G_Th(z_max) shown in Fig. 4 are almost identical. The same holds for υ_ML(z_max) and g_ML(z_max) and the corresponding CDFs Υ_ML(z_max) and G_ML(z_max) again shown in Fig. 4. As expected the two approaches are practically equivalent, given that both methods are essentially based on the same assumptions. Moreover, in Fig. 4 all these CDFs appear close to the sample CDF. This indicates that the value of the parameters N_p and N_* is not of critical importance.

Fig. 3. Left panel: histogram H(zmax) vs. the theoretical PDF υTh(zmax), mean PDF υMean(zmax) and the maximum likelihood PDF υML(zmax) of the value zmax of the highest peak of a zero-mean unit-variance GRF with autocorrelation given by a circular Gaussian with dispersion set to three pixels. The numerical experiment is based on the simulation of 5 × 103 GRF’s of size 500 × 500 pixels. υTh(zmax) has been computed using Eqs. (18) and (8) with Np obtained as explained in Vio17, whereas υmean(zmax) has been computed with Np given by the mean number of peaks in the simulated GRF’s. In the case of υML(zmax) the number Np comes out from a maximum likelihood method (see text). Right panel: corresponding gTh(zmax), computed by means of Eq. (27) with N* given by the ratio of the total area of the GRF to the window area πl2 with $l=3/\sqrt{2}$ (see text). Also the PDF gML(zmax) has been computed with N* resulting from a maximum likelihood method. To notice that υTh(zmax) and gTh(zmax) are almost identical. The same holds for υML(zmax) and gML(zmax).

Fig. 4. Cumulative distribution functions corresponding to the PDFs in Fig. 3.

A further comparison of the two methods concerns the computational burden. In this respect the GDM is superior in situations where the autocorrelation function has circular symmetry (i.e., it is characterized by only a scale parameter), since the more expensive step consists in the computation of the quantities σ_j, j = 0, 1, 2, which requires the numerical computation of three one-dimensional integrals. On the other hand, PAM requires the numerical computation of a number of integrals equal to the number of peaks selected for the detection. In general the difference in computational time is on the order of a few seconds even in the case of large images. The computational superiority of GDM vanishes in the presence of autocorrelation functions when they are not circularly symmetric. This is because the three one-dimensional integrals become two-dimensional for a two-dimensional GRF and three-dimensional for a three-dimensional GRF, which is computationally expensive. Finally, as shown in Eq. (26), the GDM requires knowledge of the Fourier transform w(k) of the filter used to obtain the GRF. If this is not available, it must be evaluated numerically with an additional computational cost. In addition, the PAM is fully automatic given that the values of the parameters κ and N_p are estimated only from the peaks. Hence, contrary to the GDM, there is no necessity to write a specific code for a given autorrelation function.

A final aspect to consider is the easiness with PAM to check if the PDF of the peak amplitudes corresponding to $\widehat{\kappa }$ is compatible with the sample PDF. This check is fundamental for the reliability of the detection. With GDM only the Gaussianity of the entries of the random field can be checked, hence the algorithm has to be used as a black box. In this respect, Fig. 5 (to compare to Fig. 3) shows the result of a numerical experiment similar to that presented above, but where the dispersion of the circular Gaussian autocorrelation function is σ_G = 1. It is clear that both PAM and GDM work really badly. The reason is that, again, both methods are developed in the context of a continuous GRF. With σ_G = 1 the effects of the discretization become important. This point can be observed in Fig. 6, which compares ψ(z) for a typical realization of the GRF when σ_G = 3 (left panel) to that when σ_G = 1 (right panel). As a one-sample Kolmogorov-Smirnvov test with a 99% confidence level indicates, only when σ_G = 3 is the sample distribution of the peak amplitudes compatible with the PDF expected for a GRF and hence only in this case can the two methods be safely used. The conclusion is that a blind application of both methods can lead to wrong results.

Fig. 5. As in Fig. 3 but with a circular Gaussian autocorrelation function with dispersion set to one pixel.

6. Comparison with an ALMA map

The peak amplitudes method and GDM are here applied to the standardized zero-mean unit-variance interferometric ALMA map in Fig. 7 with the aim of detecting point sources. The data are taken from the ALMA project 2012.1.00173.S, a mosaicing of the Hubble Ultra-Deep Field (HUDF) in continuum (Dunlop et al. 2017). The HUDF was observed using a 45-pointing mosaic, with each pointing separated by 0.8 times the antenna beamsize. The details of the observing and data reduction procedures can be found in Dunlop et al. (2017). For the purpose of this work, in order to simplify the analysis, we have cut the outer edges of the total image and produced a symmetric, square-shaped image of size 1075 × 1075 pixels (corresponding to 110 × 110 arcsec²). On this inner part we applied our algorithm and found 11 959 peaks.

Fig. 6. Check for the applicability of the proposed detection procedure for two different situations (see text). Left panel: histogram H(z) vs. ψ(z) from Eq. (10) for a numerical realization of a zero-mean unit-variance Gaussian random field with autocorrelation given by a circular Gaussian with dispersion set to three pixels. Right panel: as in the left panel but with the dispersion of the Gaussian autocorrelation function set to one pixel.

Fig. 7. Interferometric 1075 × 1075 pixels ALMA map used for testing the detection performances of PAM and GDM. Six point sources have been detected by both methods with a high level of confidence. They are labeled with a number in order of decreasing intensity.

As explained in Vio16, in ALMA maps the point sources and the blob-shaped structures due to the noise have a similar aspect and the MF cannot be applied. Consequently, the detection test becomes a thresholding test, where a peak in the map is assigned to a point source if it exceeds a given threshold. Before applying the detection algorithm, it is necessary to check whether the map meets the requirements mentioned above. In particular, the Gaussianity and isotropy of the noise background, the compatibility of the sample PDF of the peak amplitudes with the theoretical PDF ψ(z), and the fact that the spatial distribution of the peaks is compatible with a CSRPP.

The results of these checks are presented in Fig. 8. In particular, the top-left panel shows that, although similar to a Gaussian, the histogram of the entries of the map indicates a PDF slightly leptokurtik⁶. The excess of values close to the mean can be understood taking into account that in general a zero-mean, isotropic GFR with a smooth autocorrelation function shows a lack of entries with high absolute value⁷. More importantly, from the top-right panel of the same figure the histogram of the peak amplitudes appears compatible with the ψ(z) corresponding to a maximum likelihood estimate $\widehat{\kappa }=0.98$. This is confirmed by a Kolmogorov–Smirnov test at a confidence level of 99%.

Fig. 8. Checks of the conditions of applicability of the detection procedure for the interferometric ALMA map (see text). Top-left panel: histogram of the values of the pixels vs. the standard Gaussian PDF ϕ(x). Top-right panel: histogram of the peak values vs. the theoretical PDF ψ(z) given by Eq. (10). Bottom-left panel: slices along the X and the Y directions of the sample autocorrelation function vs. the corresponding slices of a least-square fit with a two-dimensional Gaussian function. Bottom-right panel: sample pair correlation function ρ(r) of the peaks. The red line provides the theoretical ρ(r) due to a CSRPP.

The isotropy of the noise background is supported by the bottom-left panel of Fig. 8, which displays the sample autocorrelation functions along the X and Y directions vs. the corresponding slices of a two-dimensional circular Gaussian resulting from a least-squares fit. The good agreement indicates that the ALMA map is compatible with an isotropic Gaussian GRF whose autocorrelation function is a two-dimensional circular Gaussian with dispersion σ_G ≈ 3.

Finally, as is visible in the bottom-right panel of the same figure, the sample pair correlation function ρ(r) is compatible with a CSRPP for r ≥ 7. Since 𝜚(7) ≈ 0, this means that, as seen in Sect. 5, the peak amplitudes do not present a relevant mutual statistical dependence, hence most of them can be considered iid.

Both PAM and GDM produce six statistically significant detections, which are labeled in Fig. 7 with a number in order of decreasing intensity and shown in more detail in Fig. 9. The SPFAs coming from the two methods are very similar, since for PAM the values are 0, 4.65E−11, 1.51E−06, 5.76E−06, 3.68E−02, 7.05E−02, whereas for the GDM they become 0, 0, 1.54E−06, 5.84E−06, 3.74E−02, 7.16E−02. This result confirms the equivalence of the two methods as it concerns the detection performance. Table 1 reports the coordinates of the detected sources and their identification with the sources reported in Dunlop et al. (2017).

Fig. 9. Sub-maps corresponding to the areas of the point sources detected in the map in Fig. 7.

7. Conclusions

In this paper the performances of two techniques, the PDF of the peak amplitudes method (PAM) and the Gumbel distribution method (GDM), have been compared in the context of the detection of weak signal embedded in noise. The two methods have been applied to simulated signals and to observations taken with the ALMA interferometer. We have shown that the two approaches are almost perfectly equivalent in their detection capability, but PAM proves to be more flexible and, at the same time, allows for an easy control of the condition of applicability of the technique. Hence, it appears more appropriate in the search for weak sources in observations dominated by noise.

References

All Tables

All Figures

Fig. 1. Statistical characteristics of the spatial distribution of a set of points obtained by means of a GRF and a complete spatial random point process (CSRPP). Top-left panel: histogram of nearest neighbor distances of the peaks of a simulated zero-mean unit-variance GRF with size 500 × 500 pixels and autocorrelation function given by a circular Gaussian with dispersion set to three pixels. Since in this GRF the peaks have coordinates given by integer numbers, in order to mimic a continuous spatial distribution, they have been added to a uniform random number taking its value in the range (−0.5, +0.5]. Bottom-left panel: sample pair correlation function ρ(r) of the peaks in the same GRF. Top-right and bottom-right panels: figures corresponding to the left panels for a CSRPP with the same sizes and containing a number of points equal to the number of peaks as in the GRF. The red lines provide the corresponding theoretical PDFs due to a CSRPP.

In the text

	Fig. 2. Numerical simulation of the two processes used in Fig. 1.
In the text

Fig. 3. Left panel: histogram H(zmax) vs. the theoretical PDF υTh(zmax), mean PDF υMean(zmax) and the maximum likelihood PDF υML(zmax) of the value zmax of the highest peak of a zero-mean unit-variance GRF with autocorrelation given by a circular Gaussian with dispersion set to three pixels. The numerical experiment is based on the simulation of 5 × 103 GRF’s of size 500 × 500 pixels. υTh(zmax) has been computed using Eqs. (18) and (8) with Np obtained as explained in Vio17, whereas υmean(zmax) has been computed with Np given by the mean number of peaks in the simulated GRF’s. In the case of υML(zmax) the number Np comes out from a maximum likelihood method (see text). Right panel: corresponding gTh(zmax), computed by means of Eq. (27) with N* given by the ratio of the total area of the GRF to the window area πl2 with $l=3/\sqrt{2}$ (see text). Also the PDF gML(zmax) has been computed with N* resulting from a maximum likelihood method. To notice that υTh(zmax) and gTh(zmax) are almost identical. The same holds for υML(zmax) and gML(zmax).

In the text

	Fig. 4. Cumulative distribution functions corresponding to the PDFs in Fig. 3.
In the text

	Fig. 5. As in Fig. 3 but with a circular Gaussian autocorrelation function with dispersion set to one pixel.
In the text

Fig. 6. Check for the applicability of the proposed detection procedure for two different situations (see text). Left panel: histogram H(z) vs. ψ(z) from Eq. (10) for a numerical realization of a zero-mean unit-variance Gaussian random field with autocorrelation given by a circular Gaussian with dispersion set to three pixels. Right panel: as in the left panel but with the dispersion of the Gaussian autocorrelation function set to one pixel.

In the text

	Fig. 7. Interferometric 1075 × 1075 pixels ALMA map used for testing the detection performances of PAM and GDM. Six point sources have been detected by both methods with a high level of confidence. They are labeled with a number in order of decreasing intensity.
In the text

Fig. 8. Checks of the conditions of applicability of the detection procedure for the interferometric ALMA map (see text). Top-left panel: histogram of the values of the pixels vs. the standard Gaussian PDF ϕ(x). Top-right panel: histogram of the peak values vs. the theoretical PDF ψ(z) given by Eq. (10). Bottom-left panel: slices along the X and the Y directions of the sample autocorrelation function vs. the corresponding slices of a least-square fit with a two-dimensional Gaussian function. Bottom-right panel: sample pair correlation function ρ(r) of the peaks. The red line provides the theoretical ρ(r) due to a CSRPP.

In the text

	Fig. 9. Sub-maps corresponding to the areas of the point sources detected in the map in Fig. 7.
In the text