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(a spacebor years) high to with benefits (months observations long and field [10] detectors This noise extrem low decades. allowing improvements two technological constant since rese from active Astrophysics the extremely in an www.exoplanets.org), is field 2016, planets extrasolar September of detection (in candidates td tepooe eeto prahmycvrother cover may approach detection prese proposed however). the applications, is (the motivated it that situations, application such study some the reviewing introducing Before worth because met. (i)-(iv) be assumptions research all) not of (or may field some situations active practical an in remains signals periodic te hnGusa,seT.4o 9) ndtcin such ordinates periodogram detection, on In based f of tests [9]). probabilities of the of detection obtain and 4 to alarm exploited Th. be see can fre- properties relevant proce Gaussian, linear the than to at extends other PSD actually assumption noise (this the quencies on dependent distributeparameters exponentially and independent asymptotically vleof -value ihalmost With of detection the subject, the on literature large a Despite 0 . 02% 000 3 n ic hnsbett otoes [17], controversy to subject then since and ) ofimdeolnt n nearly and exoplanets confirmed sl rrdsrbto osreso it,or lists, or servers to redistribution or esale rn rftr ei,including media, future or rrent α etuiBb Centauri with d above 500 2 and s ment etec- arch odic om- sses alse in s sue 5]. hat ely . ne th nt n d s 1 . l l 2 simulations in a reliable way [19]. This suggests the possibility problem [45]. In the particular field of exoplanet detection of using such simulations to calibrate the detection process, as using RV, we find similar families of techniques [17], [46]– RV time series are strongly affected by this stellar noise. The [48]. present study shows that one such calibration indeed leads to In conclusion, regarding the problem of assessing tests’ improved control of the statistical significance and to detection significance levels for multiple sinusoids detection in noise, tests with increased power. an inspection of the literature shows that: For white noise of unknown variance, several studies provide accurate• results for standardized test statistics of the form (2), B. Unknown noise statistics: related works e.g., [21], [22], [27]–[29]. Turning back to the deviations encountered in practice For colored noise with unknown PSD, we are not aware w.r.t. assumptions (i) (iv) above, this work will be primarily • − of works studying the false alarm rate when AR/ARMA or interested in the case of an incomplete knowledge of the noise other models are used for test standardization as in (2). The statistics under the null hypothesis (i.e., relaxing condition (i); difficulty in this case is the dependence of the distribution of we mention in the last section perspectives to relax the other SE on estimated parameters, which complicates the analytical conditions). In this situation, the distribution of P under the characterization of the distribution of P . null is not known and consequently the significance level (the Theb procedures described in [5], [32], [37] provide asymptotic size) of the test is not known either. Constant false alarm control of the false alarm rate. These procedurese do not operate rate (CFAR) detectors have been devised when the noise is explicitly on test statistics of the form (2) but rather on white [20]–[22]. When the noise is colored, the detection windowed periodograms that depend on several parameters. problem is more complicated. In practice, two approaches can We found that these tests are in practice sensitive to parameter be followed. The first approach is simply to ignore possible setting and that estimating these parameters impacts the noise correlations and to apply tests designed for white noise. significance level at which the tests are conducted. This However, as will be illustrated in this study, the statistical level can of course be approximated by simply neglecting behavior of the resulting testing procedure may be hazardous, the influence of such a ‘preprocessing stage’ (dealing, e.g., with unpredictable significance level and poor power (see also with model order selection, filtering, adaptive window design, [23] on this point). or standardization). For instance, we might pretend that A more sophisticated approach consists of estimating the SE = SE in (2). As will be highlighted in Sec. VI and VII, the noise PSD (called SE below) so that condition (i) above is actual significance level obtained when doing so may however considered to hold approximately. This estimate can then be beb far from the assumed one. This leaves open the question of used to calibrate the periodogram of the data P (ν), leading to designing both powerful and CFAR tests for unknown colored a frequency-wise standardized periodogram of the form noise and we propose such tests in this paper. P (ν) Before closing this literature survey, we mention a few P (ν SE) := . (2) tests designed for a particularly interesting configuration of | S (ν) E the detection problem, which is the so-called rare and weak Note that the classicale Fisher’sb test [20] standardizes the setting. In this setting, the sinusoids are both of small am- b periodogram ordinates by the estimated PSD of a white noise. plitudes w.r.t. the noise level and in small (and unknown) Standardization (2) can be seen as a generalization of this number w.r.t. the number of samples N. When viewed in the approach (see [24] for a recent review). Fourier domain, sinusoid detection can be casted as a sparse The estimate SE(ν) can be parametric or non-parametric. Non heterogeneous mixtures problem, which has attracted much parametric approches originate from seminal works of Whittle attention in the last decade [49]–[53]. We will see that while [25] and Bartlettb [26]. Parametric methods often proceed by fixed and adaptive (in the number of sinusoids) procedures fitting AutoRegressive (AR) or ARMA (AR Moving Average) lead to inconsistencies when noise correlations are ignored processes to the time series. (because then the statistics under the null hypothesis are A further complication arises when multiple sinusoids are wrongly specified), such tests keep their nominal properties, present under the alternative, as they perturb the estimation with the CFAR property added, when applied to periodograms of the noise PSD [5], [27]. Standardized tests for this case suitably standardized with training data sets. In the particular can be found in [21], [22], [24], [27]–[29]. These tests are case of the Higher Criticism [49], standardization (2) is an however non adaptive in the number of sinusoids (which must alternative approach to that of [54]. be set a priori) and designed for white noise. For adaptive procedures for colored noise see Chap. 8 of [5], and [30]– [38]. Techniques reducing the influence of signal peaks under C. This study the alternative are proposed in [39], [40]. Different approaches, The present study (an extended version of [55]) focuses on related to standardization (2), can be found in [41]–[44]. the statistical characterization of test statistics when both the When following Generalized Likelihood Ratio (GLR) ap- PSD of the colored noise and the parameters of the sinusoids proaches for detecting multiple sinusoids in unknown number, are unknown. We propose a detailed analysis of the effects the GLR must be combined with model selection procedures. of periodogram standardization by means of, say, L training While sharp model selection criteria and CFAR detectors time series, which are independent realizations of the noise exist under white noise, the correlated case remains an open process alone, and of the gain that can be expected by using 3

TABLE I such training signals in a detection framework. TABLE OF NOTATIONS We, of course, make the important assumption that such a training data set is available. Beyond the case of exoplanet N Number of data points detection considered here, one may imagine various situations X(tj ) Regularly sampled time series where training signals can be obtained. In astronomical instru- E(tj ) Zero-mean stationary Gaussian colored noise ments for instances, secondary optical paths are often devoted ν,νk Continuous frequency, Fourier frequency to monitor ‘empty’ regions of the sky or calibration stars [56]. SE (ν) Noise PSD Note that training noise vectors are routinely used for detection rE Noise autocorrelation function in radar systems, with however, an important difference. Ns, αq ,fq ,ϕq Parameters of model (3): Number of sines, sines’ amplitude, frequency and phase Adaptive test statistics in radar typically use estimates of the N Number of exoplanets orbiting target star covariance matrix of the training vectors and therefore require p Tp,Kp,Mp Planet period and its six Keplerian parameters L > N for this matrix to be nonsingular. This is a very ep,ωp, t0, γ0 different regime from that considered here, where L N. NC Proxy for Ns ≪ In the present study, we assume that the training data set is P (ν) Classical periodogram unbiased, in the sense that an averaged periodogram obtained L Number of available training data sets from an infinitely large batch would converge uniformly to the P L(ν) Periodogram averaged with L training data sets true noise PSD. In practice, finite (possibly small) batch sizes e P (ν|P L) Periodogram standardized by P L can be encountered. For this reason, we say below that the Ω Indices set of considered Fourier frequencies noise is partially unknown and we address the effect of small λk := λ(νk) Non centrality parameter values of L on the detection performances. Fλk (d1, d2) Non central Fisher-Snedecor distribution Because one important objective of this study is to obtain with d1 and d2 degrees of freedom analytical characterization of the test performances, we con- Z,z Scalar random variable, one realization of Z sider here a regular sampling. Comments on how to relax this vZ (z) Pr(Z>z) (observed p-value) assumption are discussed at the end of the paper. Also, our VZ P-value as a random variable (VZ ∼U[0 1]) ⊤ results are asymptotic in the number of samples N, which Z =[Z1,...,ZN ] Vector of random variables is characteristic of time series analysis. However, we will VZ,k P-value associated to component Zk of Z also pay attention to whether asymptotic theory accurately VZ,(k) k-th ordered p-value of Z describes reality for finite sample sizes through simulations. In the considered application framework of exoplanet detection in RV data the working hypotheses are justified because R∗+ accurate simulations of stellar noise can be produced to form Under the alternative, the Ns amplitudes αq , R∗+ ∈ training data sets. These simulations are however compu- frequencies fq and phases ϕq [0, 2π[ of the ∈ ∈ tationally demanding. Obtaining a simulation of 100 days, deterministic part are unknown. In RV exoplanet detection, for a star similar to the one shown in [19], takes about 3 this deterministic part represents the planetary signature(s). months of computing time on 120 cores on modern clusters. Note that model (3) can actually be useful for the detection Consequently, realistic values of L are in the range of one to, of periodic signals more general than only pure sinusoids. In say, a hundred at most. This motivates the study of the impact such cases, the Fourier spectrum may contain many harmonics. of estimation noise in the proposed standardization approach. Because the fundamental frequency has zero probability to We proceed as follows. Sec. II presents the model and the coincide with a Fourier frequency, the corresponding number detection approach. Sec. III recalls classical results regarding of nonzero Fourier coefficients (i.e., of deviations under the periodogram’s distribution. Sec. IV and V derive false alarm alternative) always equals N. However, most of the energy and detection rates for several tests. Sec. VII is a numerical of periodic signals is captured by a small fraction of Fourier study. Table I summarizes the main notations used in the paper. coefficients, so that model (3) would often be accurate for such signals with some Ns N. In the case of RV signals≪ for instance, a study of the influence II. STATISTICAL MODEL AND DETECTION APPROACH of the Keplerian parameters (planets’ orbital parameters) We consider the two hypotheses: shows that this is indeed the case [57]. In all cases except perhaps very rare and exotic systems, the RV spectral signa- : X(t )= E(t ) H0 j j tures exhibit only a small fraction of significant harmonics N  s (3) because the planets tend to have low eccentricities and are in 1 : X(tj)= αq sin(2πfqtj + ϕq)+ E(tj )  H small number (Np). In short, the spectrum is sparse (though  q=1 X not strictly sparse) and RV signals can be modeled by a sum  where X(tj = j∆t), j = 1,...,N, is an evenly sampled of a small number of pure sinusoids. We call this number Ns, data time series, E(t ) is a zero-mean second-order stationary and we say that N N. When there is one planet with j s ≪ Gaussian noise, with inf(SE(ν)) > 0 and u∈R rE(u) < frequency close to the Fourier grid, Ns will be essentially 1. . This is the noise of which we assume a set of training| time| In our simulations for Sec. VII, we found that multiplanetary series∞ is available. To simplify the presentation,P we consider systems with 5 eccentric planets behave like model (3) with for the rest of the paper a unit sampling step ∆t =1 in (1). Ns not exceeding, say, 20 at most. 4

III. PERIODOGRAMS’ STATISTICS: ASYMPTOTICS The effect of stochastic estimation noise caused by the A. Classical (Schuster's) periodogram finiteness of is encapsulated in L. This clearly impacts the distribution ofT P in (7) and in turn the efficiency of the The frequencies considered in (1) will be Fourier frequen- L k subsequent standardization by P L. cies νk := N k=0,...,N−1 and N is considered even. For simplicity{ but without} loss of generality we will often consider N C. Periodograms standardized with P L the subset of ( 2 1) Fourier frequencies corresponding to −N k Ω := 1,..., 2 1 . Asymptotically, the periodogram We now turn to statistical properties of standardized peri- P ∈in (1){ is an unbiased− } but inconsistent estimate of the odograms of the form (2). When the averaged periodogram PSD [2]. Under the above assumptions on E, the periodogram P L is used, this yields: ordinates at different frequencies νk and νk′ are asymptotically P (νk) independent [24]. P (νk P L) := . (8) | P L(νk) Under 0, the asymptotic distribution of P is (Th. 5.2.6, [2]): H As the numeratore and denominator are independent vari- S (ν ) E k χ2, k Ω, ables with known asymptotic distributions, assessing the dis- 2 2 ∀ ∈ P (νk 0)  (4) tribution of their ratio is straightforward. The ratio of two |H ∼ 2 N 2 2  SE(νk)χ , for k =0, . independent random variables (r.v.) V1 χ and V2 χ  1 2 ∼ d1 ∼ d2 follows a Fisher-Snedecor law noted F (d1, d2) with (d1, d2) Under 1, the distribution of P is known when cisoids are V1/d1  degrees of freedom: F (d1, d2) [59]. H V2/d2 present in (3) ( [24], Corollary 6.2(b)). The real case of model Consequently, from (4) and∼ (7), the asymptotic distribution of (3) can be treated similarly (see Appendix A). This leads to this standardized periodogram under 0 is: the asymptotic distribution: H 2 SE(νk)χ2/2 SE(νk) 2 2 F (2, 2L), k Ω, χ , k Ω, SE(νk)χ /2L ∼ ∀ ∈ 2,λk P (ν P , ) 2L P (ν ) 2 ∀ ∈ (5) k L 0  2 k 1  | H ∼ SE(νk)χ1 N |H ∼ 2 N  F (1,L), for k =0, .  SE(νk)χ1,λ , for k =0, . 2  k 2 e SE(νk)χL/L ∼ 2  (9) The λ := λ(ν ) are non centrality parameters given for  k k  Similarly, from (5) and (7), we have under 1: k {Ω by } H ∈ χ2 /2 2,λk Ns Ns F (2, 2L), k Ω, N 2 λk 2 2 (6) χ2L/2L ∼ ∀ ∈ λk = αqκq +2αqκq αℓκℓ cos(θq θℓ) , P (νk P L, 1)  (10) 2SE(νk) − 2 q=1 ℓ=q+1 | H ∼ χ1,λ N X h X i  k F (1,L), for k =0, , χ2 /L ∼ λk 2 and for k =0, N this expression is halved. The terms κ and e L 2 q  where F denotes a non-central F distribution with non θq, given by (39) and (40) in Appendix A, arise from spectral λk  leakage through the spectral window KN (36). Owing to the centrality parameter λk given by (6). fast decay of KN (ν), the proportion of parameters λk that Under the null hypothesis, (9) shows that the distribution of significantly differ from 0 is small if Ns N. the standardized periodogram is independent of the nuisance ≪ signal, i.e., of the partially unknown noise PSD. This property is important as it will be inherited by some of the test statistics B. Averaged periodogram investigated in Sec. VI, leading to CFAR tests. We assume that a training data set of independent realisations of the colored noise is available.T This set is IV. CONSIDERED TESTS obtained by L independent simulations corresponding to L A. Preliminary notations time series Xℓ sampled on the same grid as the observations: . A straightforward estimate These tests are better presented using P -values1 and order = Xℓ(tj ) j=1,...,N ℓ=1,...,L ofT the noise{ PSD} is the averaged periodogram [3]: statistics. When necessary the notation will distinguish be-  tween a r.v. Z and its realization z. We recall that the observed P -value vZ is defined as: L N 2 1 1 −i2πνkj P L(νk 0) := Xℓ(tj )e . |H L N vZ (z) := Pr (Z>z) j=1 Xℓ=1 X and vZ is one realization of the r.v. VZ , which is uniformly dis- Using (4), the asymptotic distribution of P L is: ⊤ tributed. Similarly, for a vector of r.v. Z = [Z1,Z2,...,ZN ] S (ν ) of which z = [z ,z ,...,z ]⊤ is one realization we will E k χ2 , k Ω, 1 2 N 2L 2L ∀ ∈ denote by P L(νk 0)  (7) |H ∼ SE(νk) 2 N  χL, for k =0, . min zk := z(1)

the ordered values of z and by Z(1),...,Z(N) the order estimates of the noise variance. TC is a simplification of statistics of Z. The observed P -values corresponding to z will these tests: the normalization is ignored, because it will be denoted by vZ,k (with vZ,k := Pr (Zk > zk)) and the not be necessary to yield a CFAR detector once applied to observed ordered P -values by vZ,(k). The corresponding r.v. periodograms standardized by P L. The false alarm rate of will be denoted by VZ,k and VZ,(k). The ordered P -values test TC for white noise can be deduced from the expression VZ,(k) are not uniform, because they are order statistics from obtained in Sec. V-C. a uniform distribution, and obviously dependent [60]. As for the tests [21], [22], we expect TC to have decreasing We now present some tests discussed in the Introduction and power in case of strong mismatch between the value of selected for reference as they cover different classical models. parameter NC and the number of detectable deviations under Under assumptions specified below on the distributions of the (roughly speaking, N ). Not fixing N in advance but H1 s C variates Zi , the properties of these tests are known and we estimating this parameter from the data may lead to more shall summarize{ } them. powerful tests, but at the cost of a more difficult control of the FA rate (as NC becomes random). This suggests to B. Test statistics consider other approaches that are adaptive in the number of sinusoids, which is the case of the last two tests (14) and (16). All tests below are of the form T(z) ≷H1 γ, with z the H0 data (which below will be a set of ordinates of one of the 4) Higher Criticism: This test statistic is defined by [49]: periodograms discussed above), T( ) the test statistic and + · γ R a threshold that determines the false alarm rate. √N(k/N vZ,(k)) ∈ HC⋆(Z) := max − , (14) 1≤k≤α0N vZ,(k)(1 vZ,(k)) 1) Test of the maximum: − 1 where vZ are ordered P -valuesp and parameter α [ , 1]. Z ,(k) 0 ∈ N TM( ) := Z(N). (11) HC is designed under the assumption that under the null For independent variates Z of Cumulative Distribution hypothesis the ordinates Zk are i.i.d. with known marginal i { 2P} distribution. When Z = 2 , with P the periodogram of Function (CDF) ΦZi , the false alarm of the test is σE N a white noise of known variance σ2 under the null, this PFA(γ) = 1 ΦZi (γ). For a model involving E − i=1 2 under 1 a single sinusoid with unknown frequency distribution is given by (4) with SE = σE. The ordered P - H Q (but on the Fourier grid) and under 0 a white Gaussian values involved in (14) are thus ordered values of noise (WGN) of known variance σ2H, the ordinates of the 1 Φχ2 (P (νk)), k Ω, periodogram P evaluated at successive Fourier frequencies − 2 ∀ ∈ v2P/σ2 , k N (15) E  are under 0 independent and identically distributed (i.i.d.) 1 Φχ2 (P (νk)), for k =0, . H 2  − 1 2 with distributions given by (4), where SE = σ . For this Under the alternative, a fraction of the ordinates contain a model TM(P) corresponds to the GLR test [61].  2 deterministic part, and hence follow (5), with SE = σE (see 2) Fisher's test: Sec. 1.7 of [49]). The frequencies at which these deviations occur are unknown. Their magnitudes ( related to λk) are also Z(N) TF(Z) := . (12) unknown and weak, in the sense that they are comparable to Z(k) the expected magnitude of the periodogram maximum under Xk the null hypothesis. When applied to periodogram of WGN of unknown Optimality2 results of HC are asymptotic and established variance, Fisher’s test is CFAR (the distribution of the for a specific sparsity vs amplitude model. When the deviation test statistics is established in [20]). This test is actually occurs at a single frequency or for extremely sparse signatures the GLR test under the model of a single sinusoid on the in the Fourier domain, [49] showed that the test based on Fourier grid and WGN of unknown variance (see [62], who the maximum (11) is asymptotically optimal in the sense that also covers the case of more than one sinusoid). Examples whenever the Neyman-Pearson test has full power, test (11) of works using this test in Astronomy are [46], [47], [63]–[65]. has full power as well. In such cases, the largest periodogram ordinate (or, equivalently, the smallest P -value) is the most 3) A test inspired by the tests of Chiu and Shimshoni: powerful test statistic to discriminate against the null. As discussed about the tests [21], [22] and TC above, better Z (13) TC( ) := Z(N−NC +1), regions than the maximum ordinate/smallest P -value may be useful for sparse but not extremely sparse signatures. [49] with NC 1 a parameter related to the number of sinusoids. This test≥ statistic is justified by the observation made in [21], showed that there exists sparse signals of weak amplitude [22] that for multiple sinusoids, order statistics different from that can be optimally detected (in the sense of full power) by HC but not by TM. This makes HC particularly interesting in the maximum may be more discriminative than Z(N) against the null. These tests are designed for periodograms of white the present study. noise of unknown variance and their asymptotic false alarm rates are given in [21], [22]. As Fisher’s test, they involve 2A test is said optimal in [49] if the sum of the probability of error and denominators whose purpose is normalization by consistent the probability of missed detection tends to zero. 6

5) Berk-Jones : A test statistic related to HC is that of indicating that departure to the independence assumption may Berk-Jones [52], [66]–[69] defined by: be more pronounced under 1 than 0. Consequently, we are using the independence asH an operationalH assumption to BJ(Z) := max I (N k +1, k), (16) 1−vZ,(k) quantify the tests’ performances, and the validity of the re- 1≤k≤α0 N − sulting expressions below should be checked against numerical where denotes the regularized incomplete beta function [59]. I simulations. Sec. VII provides several examples. The deviations in form of Z scores in (14) for HC are established using the asymptotic− convergence of a binomial distribution to a Gaussian distribution. In the tails, however, A. TM. this convergence is very slow (see [52], [67], [70] for illustra- Under and in the asymptotic regime, T (P P ) tions). For this reason the test statistics (16), which compares H0 M | L favorably to HC and other goodness-of-fit (GOF) tests in some is the maximum of independent variables given by (9). For k Ω these N 1 variables follow an F (2, 2L) distributione cases, was recently (and almost simultaneously) proposed by ∈ 2 − [52], [66], [68], [69]. As noted in [52], [53], this test was with general density given in [59]. Using the beta function 1 L−1 1 initially proposed by Berk and Jones (and called +) in [71]. (1,L) := (1 t) dt = , this density ϕF (γ, 2, 2L) Mn B 0 − L This test is based on the exact significance reflected by can be expressed as R the P -values, that is, on the P -values of the ordered P - 1 1 γ −L−1 γ −L−1 values. Since the ordered P -values are Beta distributed, with ϕ (γ, 2, 2L)= 1+ = 1+ . F (1,L) · L · L L V2P/σ2 ,(k) Beta(k,N k + 1), their P -values involve the B E ∼ −     CDF of Beta variables, which is an incomplete Beta function: ∞ It can be checked that 0 ϕF (γ, 2, 2L)dγ = 1. The corre- Pr(Beta(k,N k + 1) x) = Ix(k,N k + 1) and leads sponding CDF ΦF (γ, 2, 2L) is obtained by integration of ϕF : to test (16) with− the P -values≤ computed− as in (15). R BJ presents the same adaptive optimality as HC for sparse L γ L mixture detection, and the asymptotic distribution of the BJ Φ (γ, 2, 2L)= ϕ (γ, 2, 2L)dγ =1 . (17) F F − L + γ test statistic can be found in Th. 4.1 of [52]. As for HC, con- Z0 ! vergence to the asymptotic distribution may be slow. Efficient The probability of false alarm (P ) can be computed thanks algorithms for computing significance levels (and hence the FA to the asymptotic independence of the ordinates of P PL: function γ PFA(γ)) of HC and BJ for finite (but possibly | 7→ large) values of N can be found in [52], [72]. BJ and HC are P P P P PFA(TM( L),γ) := Pr (TM( L)>γ 0) e studied from the viewpoint of local levels in [53]. | | |H N 2 −1 =1 ePr P (νk) γ 0, P Le =1 ΦF (γ, 2, 2L) Z P P − ≤ |H − V. TESTS APPLIED TO = L k∈Ω     | Y N eL L 2 −1 To simplify the presentation of the results we restrict in the =1 1 . following to the frequency set Ω, i.e., to standardizede vectors − − γ + L     (18) ⊤ The probability of false alarm is (asymptotically) indepen- P (ν ) P (ν N −1) P P 1 2 L := ,..., . dent of the noise PSD, which makes TM(P PL) a CFAR | P L(ν1) P L(ν N −1) | " 2 # detector.

Extensione of the results to ν0,ν N can be obtained using the Under 1, using (10) and the approximatee independence of 2 H distributions (9) and (10). periodogram ordinates ( [24], Theorem 6.5), the probability of In the following we evaluate false alarm and detection detection (PDET) of TM(P PL) can be approximated as: | rates by postulating independence of the considered ordi- nates. While the asymptotic independence of the periodograms P (T (P P ),γ) :=e Pr T (P P )>γ DET M | L M | L |H1 ordinates at positive Fourier frequencies is well established   (19) 1 ΦF (γ, 2, 2L). (e.g. [73], Th. 2.14), theoretical results regarding the joint e ≈ − eλk distribution of periodogram ordinates under departures from kY∈Ω whiteness (and Gaussian) assumptions are lacking, however. With (18), the relationship γ(PFA) for TM can be derived as: For some results on the largest order statistics, see [74], who − 1 consider MA Gaussian processes, and [75], who consider non- 1 L γ(T (P P ), P )=L 1 (1 P ) η 1 , (20) Gaussian sequences. For finite values of N, the approaches M | L FA − − FA − of [76], [77] might be followed to better characterize the h  i where η e:= N 1. With (19) and (20), we deduce P (P ) performances of some tests considered below. 2 − DET FA In practical situations, the marginal distributions of the con- which can be used to compute ROC (Receiver Operating sidered ordinates are only approximated by their asymptotic Characteristics) curves: distribution (this is visible in Eq. (29) for instance). Besides, P (T (P P ), P ) as grows the correlation between the periodogram ordinates DET M L FA N | 1 1 η − L (21) at the signal frequencies approaches zero at a slower rate than 1 ΦFλ (L[(1 (1 PFA) ) 1], 2, 2L) ≈ − e k − − − the correlation between other ordinates ([24], Theorem 6.5), kY∈Ω 7

⋆ B. TF. D. HC and BJ. P P When applied to P P , the P -values involved in the HC Under 0, the Fisher’s test TF applied to L looks for | L the maximumH of identically distributed variables,| which from test (14) and in the BJ test (16) should be computed according (9) and (12) correspond to the ratio of a Fevariable over a to the distribution undere the null given by (9). Hence, (15) is sum of F variables. To our knowledge, there is no analytical replaced by: P (νk) characterization of the resulting distribution for finite values 1 ΦF , 2, 2L , k Ω, − P L(νk) ∀ ∈ of L. Hence, although this test is CFAR, computing the ve :=   P|PL, k   P (νk) N false alarm rate is problematic. Resorting to Monte Carlo  1 ΦF , 1,L , for k =0, . − P (ν ) 2 simulations to evaluate the function γ PFA is not possible,  L k  7→ The properties of the two tests are otherwise left unchanged, owing to the limited number of available noise realizations.  Similar remarks can be made about standardized versions of with the CFAR property added: thanks to the standardization P P other tests like [21], [22], [27]. of by L, the P -values are independent of the noise PSD.

VI. TESTS APPLIED TO Z = P S | E C. . TC Estimates of SE different from the averaged periodogram e b P L can be used for standardization in (8). Sec. I-B has We turn to TC(P PL,NC), where parameter NC allows to focus on the regions| of order statistics where deviations reviewed someb methods to obtain such estimates. For the purpose of comparing P SE with P PL, we opt for parametric under the alternativee are likely to be significant. The PFA can | | be obtained by observing that, owing to (9), the number K estimates allowing for an automatic parameter setting, as this approach is commonlye usedb in practice.e For instance, when of standardized ordinates larger than γ under 0 follows a H using an estimate based on an AR process of order o, this binomial distribution: K Bin(N/2 1, 1 ΦF(γ, 2, 2L)), ∼ − − order can be estimated using many criteria, e.g. [78]–[82]. whose CDF is: Pr (K k) = IΦF (γ,2,2L)(N/2 NC,NC ), see [59]. Using (13), (17)≤ and noting − In our studies, we found that the selected order is often different from the true order (as expected, see e.g. p. 211 L L of [78] and [83] for similar conclusions) but, as far as u := 1 Φ (γ, 2, 2L)= , − F γ + L detection results are concerned, these criteria have very similar   behaviour for sufficiently large N. the P of this test applied to P P can be expressed as: In any such method, let us denote respectively by o , FA | L AR c b and σ2(o ) the selected order, AR coeffi- { j}j=,1,··· ,oAR AR PFA(TC(P PL, NC),γ):=Pr (Te C(P PL, NC)>γ 0) cients and corresponding estimated prediction error variance. | | |H b NC −1 The PSD estimate and resulting standardized periodogram are N b b b =1 e Pr (K = k 0)=1 e I1−u( NC ,NC) 2 − | H − 2 − σ (oAR) P (νk) k=0 SE,AR(ν):= b , P (νk SE,AR):= . X oAR N 2 | SE,AR(νk) = I (N , N ), 1+ c e−2πijν u C 2 − C b b bj e b (22) j=1 b X (24) where the last equation uses Ix(a,b)=1 I1−x(b,a) (cf Prop. b − 6.6.3 in [59]). As TM(P PL), this test is CFAR. Even if such approaches are relatively straightforward to | As a side remark, note that the false alarm rate of the test implement, characterizing the distribution of P (ν S ) is k| E,AR TC applied to the periodograme of a white noise of known more difficult than in the case of P (ν P ), owing essentially k| L variance is obtained as above, by replacing ΦF with the CDF to the stochastic nature of oAR in (24). Ine practice,b the 2 of χ variables according to (4). ‘whitening’ effect of SE,AR is efficiente because the selection The function γ PDET(γ) of TC(P PL,NC ) can be procedures have good fitting properties (they are approxi- 7→ | b ≈ deduced similarly to (22), with the difference that K is no mately consistent, i.e.b, Sˆ S as N + ). E,AR −→ E −→ ∞ longer binomial owing to the λk . Appendixe B shows that This leads to consider as reasonable the assumptions that, { } P (νk) P (νk ) P (νk ) in effect, b S (ν ) and, with (4), that S (ν ) is SE,AR(νk ) ≈ E k E k PDET(TC(P PL, NC),γ) := Pr (TC(P PL, NC)>γ 1) 2 2 N | | |H approximately a χ2/2 r.v. for k Ω and a χ1 r.v. for k =0, 2 . N −1−i ∈ NC −1 i 2 An approximate false alarm rate can then be evaluated from e γ, , Le 1 1 ϕF ( 2 2 ) ϕFλ (γ,2,2L), these assumptions. For example, for the T test applied to λ (i) (i) C ≈ − − Ω Ω ′ P i=0 Ω(i)∈Ωi k=1 k k′ =1 k   Sb , following the lines of (22) leads to: X X Y Y (23) E,AR which can be used with (22) to compute ROC curves. The P S P S non centrality parameters λ (i) := λ(ν (i) ) and λ (i) := PFA(TC( E,AR,NC),γ) PFA(TC( E,NC),γ) Ω Ω Ω ′ | ≈ | { k k } { k N λ(ν (i) ) are given by (6) with the notation defined in (43). Ω = Iu(NC, NC ). k′ } e b 2e− Note that the tests TM and TC are both working on order (25) −γ statistics of P -values, which are Beta random variables, so in with u := Φ 2 (2γ)=e . We will evaluate the reliability χ2 both cases the PFA is given by the tail of a Beta distribution. of this approximation by numerical simulations in Sec.VII-D. 8

VII. NUMERICAL STUDY as the estimation noise decreases with the increasing size of A. Simulation setting the training data set.

We consider under 0 two PSD models for the noise E. The first model comesH from real RV data of the Sun obtained from the GOLF spectrophotometer on board SoHO satellite [84]. This instrument has been observing the Sun for 18 years with a sampling rate of 20 s. As several gaps are present in the resulting time series, we selected some (158) regularly sampled data blocks of T 23 days, of which we averaged and smoothed the periodograms≈ (see Fig. 1, left panel). (Note that the data we used are filtered at low frequencies so that the resulting PSD estimate does probably not accurately reflect the solar PSD at low frequencies). The second model (Fig. 1, right panel) corresponds to a zero- mean second-order stationary Gaussian AR(6) process. The coefficients were chosen to yield a correlated process exhibit- ing higher energy at low frequencies and local variations, as in some stars, but the choice of this PSD is not intended to reflect the reality of a particular star.

GOLF AR(6)

1 10 2 ] 10 1 − z H

. 0

2 10 − s . 2 0 10-1 10 e e Fig. 2. Theoretical vs empirical results for TM(P | PL) and TC(P | PL) ) [m − ν 1 by MC simulations. Parameters: ∆t = 30 min, Ns = 3, αq = 0.2 m.s -2 10 for the three RV signatures of respective periods 11 h, 2.45 d and 6.61 d. The

log P( -2 curves in dashed lines show the inconsistency of ignoring noise correlations -3 10 10 (Sec. VII-C).

10-4 -4 -3 10-6 10-5 10-4 10 10 ν [Hz] ν [Hz] C. Effects of standardization by PL We compare the detection performances of the tests T Fig. 1. Left: Estimated PSD of the solar noise with part of GOLF data M and T applied to the standardized periodogram (P P ) with (blue) and periodogram of one of the 158 data blocks (grey). Parameters: C | L N = 1110, ∆t = 30 min. This PSD is used to generate the noise for Figs. their unstandardized versions TM(P) and TC(P), as described 2 and 4. Right: Theoretical PSD of the AR(6) noise (blue) and one noise in Sec. IV-B. We evaluate first how accurate woulde be the false periodogram (grey). This PSD is used for Fig. 3. Parameters:N = 1024, ∆t = 1 min. alarm rate obtained by neglecting noise correlation for tests TC and TM. For this we assume the detectors consider the noise Under 1, several cases of exoplanetary RV signatures will 2 H is white and have knowledge of σE , the exact variance of E. be considered. The tests’ performances will be illustrated by Z P 2 When = 2 /σE, it is easy to show that the false alarm ROC curves representing PDET as a function of PFA. For rates assumed by these two tests are given by Monte Carlo (MC) simulations, 104 realizations have been N −1 used. P 2 2 PFA TM(2 /σE),γ =1 Φχ2 (γ), − 2  N P T (2P/σ2 ,N
),γ = I (N , N ). P P  FA C E C Φχ2 γ C C B. Analytical expressions for tests based on L  2( ) 2 − | (26) P P P P
We first consider the tests TM( L) and TC( L). These expressions are compared to the true false alarm rates in The first panel of Fig.2 compares to empirical| e results obtain| ed the top left panel of Fig. 2. The cyan dashed and blue dotted from MC simulations to the expressionse obtainede for the curves show respectively approximation (26) and empirical for both tests (see (18) and (22)). The second panel PFA(γ) P for T , while the yellow dashed and red dotted curves regards the corresponding expressions for ((19) and FA M PDET(γ) show respectively approximation (26) and empirical P for (23)) and the last panel the expression (21) for . FA PDET(PFA) T . Clearly, the correspondence between the thresholds values All the theoretical expressions are shown by color dots and C and the target false alarm rates is destroyed because of noise the empirical results from MC simulations are plotted in full correlation in absence of standardization. lines. Different values of L are considered to illustrate the improvement brought by larger training data sets. The figure S shows a fair agreement between theoretical and empirical D. Effects of standardization by E,AR results, even for the not so large value of N considered here As discussed in Sec.VI, a way to deal with the frequency (N = 1110). The test performances logically increase with L, dependence of the noise is to estimateb its PSD by parametric 9

models. A difficulty with such methods is the injection of E. TC(P PL,NC) and TM(P PL) vs adaptative approaches estimation noise in the detection process. A standard approach | | We do not attempt to apply HC⋆ and BJ to P in the discussed in Sec.VI is to consider that the estimates are correlatede case as this leadse to the same inconsistencies as sufficiently accurate for their intrinsic error to be negligible. those illustrated in Sec. VII-C and Fig. 2, top left panel, dotted We study the performances of this approach here. For this we curves. We compare here the adaptative approaches HC⋆ and consider the case of five sinusoidal signals with frequencies BJ standardized by P as in Sec.V-D with the corresponding falling into a ‘valley’ of the noise PSD (Fig.1, right panel). We L tests T and T for L = 1 and L = 50. (Note that when assume the noise PSD follows an AR model (which is indeed M C N =1, T reduces to T , cf (13)). the case here) and we estimate as described in Sec.VI. C C M SE,AR For this comparison, we consider two different cases of The question is the reliability of tests using P SE,AR, and in Keplerian signals under 1, which are typical of ‘super-Earth’ particular how accurate is expressionb (25) with| this approach. H planets (Fig. 4): Np =1 planet with null eccentricity in Case e b Fig.3 compares, as a function of the test threshold, the PFA 1 (left column), and Np =5 eccentric planets in Case 2 (right assumed by approximation (25) (blue curve) with the actual column). false alarm rates obtained for 1000 MC simulations. Case 1 Case 2 1 1

0.8 0.8

0.6 0.6 e T (P|P1) DET M Pe P P TC( | 1, NC = 2Np) 0.4 ⋆ e 0.4 HC (P|P1) e BJ(P|P1) e TM(P|P50) 0.2 e 0.2 TC(P|P50, NC = 2Np) ⋆ e HC (P|P50) e BJ(P|P50) 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 PFA PFA Hz] /

2 Signal for L = 1 s

/ Signal for L = 50 2 0 Fig. 3. Comparison of the approximated PFA (25) (blue curve) with 100 10 true PFA for L = 1, 20 and 100. The solid lines with dots represent the ) [m average actual P . The shaded regions are the corresponding 3σ enveloppes. ν 1.5 2 2.5 3 3.5 5 10 15 20 25 30

FA P( The right panel is a zoom in log-scale on the violet square in left panel. ν [µHz] ν [µHz] −1 Parameters: Ns = 5, αq = 0.07 m.s , fq = [5.0, 5.5, 5.75, 6, 6.5] mHz. Fig. 4. Top panels: Empirical ROC curves comparing classical and adaptative tests for different signals. Case 1: Np = 1, Mp = 3.5M⊕ (for L = 1), Tp = 5.7813 d (signal frequency 1/Tp on-grid), ep = 0, In each such simulation, an estimate SE,AR(L) was ob- ωp = 0, T0 = 0, γ0 = 0. For L = 50, Mp = 0.8M⊕. Case tained with Akaike’s Final Prediction Error (FPE) [78] from 2: Np = 5, Mp = [0.15, 0.15, 0.25, 0.25, 0.25]M⊕ (for L = 1), T = [11.21 h, 1.33 d, 2.45 d, 6.91 d, 9.25 d] (signal frequencies off-grid), L noise time series and used to calibrate the periodogram. For p b ep = 0.9, ωp = π, T0 = 0, γ0 = 0. For L = 50, Mp(Tp = 11.21 h) = each such estimate, the true PFA of test TC(P SE,AR,NC = 0.07M⊕. In the adaptative tests, α0 = 1/2. Bottom panels: periodograms | (logscale) of the signals under H . Ns) was evaluated using 100 MC simulations. The figure 1 plots, respectively for L = 1, 20 and 100 etheb average PFA, respectively in black, green and red solid lines with dots. The lower panels illustrate the periodogram of the (noise- We see that (25) is accurate, on average, only when L is less) Keplerian signatures for the two cases. We see the ap- large. This figure also indicates the variability of the true parition of significant harmonics in the case of off-grid signal false alarm rate. For each value of L, the figure shows the 3σ frequencies and highly eccentric orbits. For each case, the dispersion of the true PFA w.r.t. its empirical average (shaded planet masses have been adapted depending on the considered regions in grey for L = 1, green for L = 20 and red for value of L for a better display of the ROC curves. L = 100). Even when N is large, the true significance levels From L = 1 (dashed lines) to L = 50 (solid lines), the at which such tests are conducted can undergo wild (and in performances of all tests increase in both cases. When the practice unknown) variations. The right panel is a zoom on signal is extremely sparse in the Fourier domain (Case 1), the 3σ region for L = 100. For a threshold γ = 5.9 for TM is more powerful than the considered adaptive approaches. instance, the PFA 0.013 from (25). In the right panel, This situation changes when the spectrum is less sparse (Case we see that the true≈ false alarm rate for this threshold varies 2, compare the bottom panels), which is expected. in reality in the range [0.001 0.3]. For smaller values of L, Results of [49] show that for a proportion of deviations in N 1 N 1 the excursions of the true false alarm rates are so large that the range [( 1) 4 ( 1) 2 ] an adaptive procedure such 2 − 2 − (25) is simply useless. Conclusions drawn from tests based as HC may have better asymptotic power than TM. For the on parametric estimation may thus be very hazardous, even case N = 1110 considered here, this corresponds to the range when the parametric model is true and when large data sets [5 23] (considering Ω). The situation should be opposite for are available for PSD estimation. very sparse signals (in the range [1 4] here). It turns out that 10 the superiority of adaptive procedures is confirmed in Fig. probability of detection of 0.9 while ensuring a false alarm rate 4, where the spectrum is 1-sparse in Case 1 and about 10- of 0.01. This situation is indicated by the black square. With sparse in Case 2. Note, however, that while the theory of [49] only L = 5 training time series, the probability of detection may be used as a guideline for guessing the sparsity range in would fall to about 0.1, all other parameters equal (red square). which each test should work better, we should not expect a too tight agreement with this theory. Indeed, in the framework 1 of [49], all deviations under the alternative have the same PFA = 50%, L → ∞ amplitude, while RV signals lead to different amplitudes in 0.9 L = 100 general. Moreover, these theoretical results are asymptotical 0.8 L = 20 (while N is not so large here). 0.7 L = 5 P = 10%, L → ∞ ⋆ 0.6 FA An interesting point is the comparison of HC and BJ with L = 100 P P 0.5 TC( L,NC ), for which NC is a proxy for the number DET L = 20

| P of significant deviations in the Fourier spectrum (here we 0.4 L = 5 considerede NC = 1 in Case 1 and NC = 10 in Case 2). 0.3 PFA = 1%, L → ∞ TC represents a kind of Oracle, which knows in which region 0.2 L = 100 L = 20 of the P -values to look at in order to ‘make the case’ against 0.1 L = 5 0. The right panel shows that adaptive procedures have better 0 2 3 4 H 10 10 1500 10 power than TC, yet without prior knowledge. N

F. A detectability study Fig. 5. Example of a detectability study for a planet (relevant Keplerian −1 A direct application of the previous results is detectability parameters: K = 0.54 m.s , Tp = 3.23 d and e = 0; Mp = 1.241M⊕, orbit inclination of 90°, semi-major axis a = 0.0425 Astronomical Unit) studies, which can be used for the design of observational orbiting a G2 type star. The plot shows the achievable PDET for different strategies for instance. To illustrate this, we consider under configurations of PFA budgets and numbers of available training light curves. 1 a planet with the parameters of α Centauri B’s exoplanet Hcandidate as estimated in [16] (see the legend of Fig.5). As the eccentricity is supposed null, the signal can be considered very VIII. SUMMARY AND PERSPECTIVES sparse in the Fourier domain. We consider the TM test and a This paper has investigated the possibility of using training time sampling of ∆t =4 hours. It was allowed to slightly vary data sets to standardize periodograms in order to improve from one value of N to another in order to guarantee that the the control of the resulting false alarm rate. The paper first planet’s period yields a frequency exactly on the Fourier grid, provided an extended (though unavoidably selective) overview in which case the spectrum is 1-sparse on Ω and TM is the of classical and recent techniques in sinusoid detection, with test that yields the best performances. emphasis on the problem of designing CFAR detectors for the As for the noise PSD, we used a model based on HD composite hypotheses of multiple sinusoids and colored noise. simulations of a star with similar spectral type as that of We proposed an asymptotic analysis of the periodograms αCenB. There is some mismatch between the simulated RV statistics after standardization for a model involving an un- time series of the spectral type (G2) and the true spectral known number of sinusoids with unknown parameters in type (K1) of αCenB. Because spectral type affects the noise partially unknown colored noise. This analysis allowed to properties [85], these results should not be considered to reflect characterize the performances of some standardized tests in perfectly the case of the candidate planet orbiting αCenB. terms of false alarm and detection rates. We showed that In Fig.5, we illustrate the feasible performance compromises when standardization is performed with a simple averaged (PDET, PFA) as a function of N, for three target PFA (0.5, 0.1 periodogram as a noise PSD estimate these tests are CFAR and 0.01, indicated respectively by the dotted, solid and dashed for all sizes of training data sets. In contrast, we pointed out lines) and for different sizes of available training data sets L that standardization based on parametric estimates of the noise ( , 100, 20, 5, shown respectively in black, green, blue and ∞ PSD may present actual false alarm rates that may be very far red). These curves were built using the expressions (20) and from the assumed ones, even for large data sets. (21) for the TM test and we checked that they are accurate The tests we considered include classical approaches and also in separate MC simulations (not shown). In the case L more recent adaptive tests designed for the rare and weak 2 → + , we use the fact that F (2, 2L) χ2 to calculate the P ∞ L→−→+∞ setting. For the latter tests, the standardization (by L) offers theoretical PDET(PFA). the same benefits as if the statistics of the noise were known The study presented in Fig.5 allows to quantify interesting a priori, with the CFAR property added. We also showed that facts. First, of course, PDET is larger if the allowed PFA is such tests can present better power comparing to procedures larger. Second, for a fixed PFA, PDET is larger for a larger for which the number of sinusoids would be known in advance. value of L. Going to specific cases, we see that if a planet In practical situations, some of the assumptions (i-iv) in Sec. similar to the considered candidate was orbiting a star of the I-A may not be met. This can be the case in exoplanet detec- considered spectral type (G2), of which 100 training time tion using RV time series, owing for instance to astrophysical series are available, it would require 250 days (1500 4h) effects linked to magnetic activity (like spots, affecting (i)) or of observations with 1 point every 4 hours to guarantee× a instrumental defects / observational constraints (affecting (ii) 11

and (iii)). Comparing theory to pratice, the present study is The DFT is composed of a deterministic part, µk, and a useful to make feasibility studies (as in Sec. VII-F), which then stochastic part, ǫk, defined as describe best achievable performances (i.e., around “quiet” 1 H 1 H stars and in absence of other unmodelled perturbations) for µk := f (νk)R and ǫk := f (νk)E. (28) a regular sampling. N N An important extension to the considered framework regards Because E is a zero mean Gaussian process, the random the case when, for periodicity analysis, the time series is not variable yk = µk + ǫk is Gaussian with mean µk and variance 2 correlated to orthogonal exponentials. This case encompasses σk. This is a complex variable for all Fourier frequencies situations where (a) the sampling is irregular, (b) P is not eval- except for ν0 and ν N because f(ν0) and f(ν N ) are real. 2 2 uated on the Fourier grid (as in oversampled periodograms) The distribution of the periodogram requires to investigate (c) P is modified in the form of “generalized periodograms”, the variance of ǫk which, with (28), writes: which correlate the time series with highly redundant dictio- 1 naries of specific features [28], [86]–[93]. In such cases, the Nvar ǫ = E f H (ν )EE⊤f(ν ) k N k k considered ordinates exhibit strong dependencies. With the 1 N
additional complication of partially unknown colored noise, = r (t s)e−i2πνk (t−s) N E − analytical evaluation of the false alarm rate for the considered t,s=1 tests seems out of reach. We conducted however preliminary X 1 −i2πνku studies suggesting that it might be possible to obtain accurate = (N u )rE(u)e N − | | estimates of the false alarm rate when the noise is colored, the |uX|

DERIVATION OF EXPRESSIONS (5) AND (6) 2 −1 −1 σk := var ǫk = N SE(νk)+ (N rN ). (30) We prove here that for model (3) the periodogram is asymp- O totically distributed as in (5) with non centrality parameters as By Lemma 12.2.1(b) of [24], we obtain in (6). The proof is adapted from Theorem 6.2 of [24], which 1 2 considers the complex case. We first prove (5) and then turn χ µ 2 , k Ω, 2,2 | k | 2 σ2 ∀ ∈ to (6). The time series of model (3) can also be written as k y 2/σ2  k k  N N | | ∼  2 s  χ µ 2 , for k =0, .  1, | k| σ2 2 X(j)= αq sin(2πfqj + ϕq)+ E(j)= R(j)+ E(j), k q=1  X Hence, for the periodogram this implies with R(j) := Ns α sin(2πf j + ϕ ) a deterministic part, q=1 q q q P (ν H )= N y 2 = Nσ2( y 2/σ2) which using Euler formulae can be written as k| 1 | k| k | k| k P 1 −1 2 ρ SE(νk)χ , k Ω, Ns 2 k 2,2ρk γk ∀ ∈ (31) αq i(ϕ − π ) 2πif j αq −i(ϕ + π ) −2πif j R(j)= e q 2 e q e q 2 e q . ∼  N 2 − 2  ρ−1S (ν )χ2 , for k =0, , q=1  k E k 1,ρk γk X 2 f i2πν i2πNν ⊤ (ν) := [e ,..., e ] , where  π  2 2 f + i(ϕq− 2 )f By introducing:  (fq) := e (fq), ρk := SE(νk)/(Nσk) and γk := N µk /SE(νk). (32) π | |  f − −i(ϕq+ 2 )f  (fq) := e (fq), With (30), we see that the time series writes in vector form :  SE(νk) N s α ρk = =1+ (rN ) (33) X = q f +(f ) f −(f ) + E = R + E (27) SE(νk)+ (rN ) O 2 q − q O q=1 X   for all Fourier frequencies. Owing to (29), an approximated distribution of (31) can be obtained by neglecting the (r ) and its discrete Fourier transform (DFT) yk at frequency νk O N is in (33). The distribution (5) follows by noting 1 1 1 N y = f H (ν )X = f H (ν )R + f H (ν )E. λ := 2γ for k Ω and λ := γ for k =0, . (34) k N k N k N k k k ∈ k k 2 12

We now turn to the computation of the non centrality param- Consequently, the expression of the γk in (37) becomes eters. We have from (27), (28) and (32) { } Ns Ns N 1 f H R 2 N 2 2 γk = | (νk) | γk = αqκq+2αqκq αℓκℓcos(θq θℓ) (41) SE (νk) N 4SE(νk) − q=1 ℓ=q+1 N Ns Xh X i 1 αq i(ϕ − π ) 2πi(f −ν )j q 2 q k (35) = e e NSE (νk) X X 2 and the non centrality parameters λk of (6) follow from j=1 q=1 { } π 2 (34),with κq and θq given by (39) and (40).  −i(ϕq + ) −2πi(fq +νk)j . . . −e 2 e  . Note that if all signal frequencies f fall on the Fourier { p} Introducing the Dirichlet Kernel (cf Lemma 12.1.3 of [24]): frequency grid, the crossed term in (41) vanish owing to the orthogonality of the Fej´er kernels centered at different signal 1 N sin(Nπν) frequencies. In this case, expression (41) precisely reduces to D (ν) := ei2πνj = ei(N+1)πν, N N N sin(πν) expression given in Remark 6.6 of [24]. j=1 X We finally wish to mention that the expression of the non and the corresponding Fej´er kernel (or spectral window) centrality parameters is erroneously reported in exp. (5) of [55] 2 (sign error and crossed terms missing). 2 sin(Nπν) KN (ν) := DN (ν) = , (36) | | N sin(πν) ! APPENDIX B expression (35) becomes DERIVATION OF EXPRESSION (23) Ns N − π i(ϕq 2 ) γk = αq DN (fq − νk)e Let K denote the number of ordinates of P PL larger than 4SE(νk) X  q=1 | γ under 1, and pi := Pr (K = i 1). From the definition 2 −i(ϕ + π ) H | H q 2 (13), we have: . . . − DN (fq + νk)e  . e

This equation can also be written as : P (T (P P ), γ, N ):=Pr (T (P P , N )>γ ) DET C | L C C | L C |H1 Ns N 2 = Pr (K NC 1) γk = αqzq(νk) , (37) ≥e | H e 4S (ν ) NC −1 NC −1 E k q=1 X =1 Pr (K = i 1)=1 pi. − | H − with i=0 i=0 π π X X (42) i(ϕq − 2 ) −i(ϕq+ 2 ) e zq(νk) := DN (fq νk)e DN (fq + νk)e P (νi) − − Owing to (10) each ordinate (P PL)i := has proba- iθ+ iθ− P L(νi) = x+e x−e | − bility 1 ΦFλ (γ) to be larger than γ. These variates can be − i = (x+ cos θ+ x− cos θ−) + i(x+ sin θ+ x− sin θ−), considered approximately independente but not i.i.d. Hence, the − − (38) variable K is not binomially distributed (as it is under 0) and where the probabilities p require further investigation. WeH proceed { i} sin(Nπ(fq νk)) by induction. In the following, all probabilities are under 1. x+ = x+ (νk, q) := − , H N sin(π(fq νk)) The first probability p0 can simply be approximated as  −  sin(Nπ(fq + νk))  x = x (ν , q) := , N −1 N −1  − − k 2 2  N sin(π(fq + νk))  p0 = Pr (P PL)k γ ΦF .  π λk  θ+ = θ+ (νk, q) := +[(N + 1)π(fq νk) + (ϕq )],  | ≤  ≈ − − 2  k\=1  kY=1  π e  θ− = θ− (νk, q) := [(N + 1)π(fq + νk) + (ϕq + )].  − 2 The probability p1 = Pr(K = 1) is similarly  The modulus κ of z may be written as  q q N 2 −1 1 2 2 2 κ := z = x + x 2x x cos(θ θ ) , (39) p1 = Pr (P PL)k >γ (P PL)j γ q q + − + − + −  | | ≤  | | − − k=1 j6=k 2
[  \  x+ = KN (fq νk), N −1 e N −1 e − 2 2 with x2 = K (f + ν ),    − N q k (1 ΦF ) ΦF . ≈ − λk λj  θ+ θ− =2π(N + 1)fq +2ϕq, k=1 j=1, − X h jY6=k i and  θq := ∠ zq, θq ] π, π], (40)  ∈ − To generalize further, denote by Ω(i) one particular com- the phase of zp obtained from the real and imaginary parts of (i) bination of i indices taken in Ω and Ω := Ω Ω(i) (38) [96]. With these notations, it is easy to show that (i) (i) \ the set of remaining indices. Let Ω ,..., Ω (resp. N N N 1 i s 2 s s (i) (i) { } 2 Ω1 ,..., Ω N −1−i ) denote the indices in two such combi- zq = κq +2κq κℓ cos(θq θℓ) for Ns > 1. { 2 } − nations, and let Ωi be the set of all the Ω(i) . With these q=1 q=1 h ℓ=q+1 i X X X { }

13 notations we obtain for i> 1 : [28] E. B¨olviken, “New tests of significance in periodogram analysis,” Scandinavian J. Stat., vol. 10, no. 1, pp. 1–9, 1983. N −1−i i 2 [29] E. B¨olviken, “The distribution of certain rational functions of order Pr P P (i) P P (i) statistics from exponential distributions,” Scandinavian J. Stat., vol. 10, pi = ( L)Ω >γ ( L)Ω γ  | k | k′ ≤  no. 2, pp. 117–123, 1983. (i) i k=1 k′=1 Ω [∈Ω  \ \  [30] R. Von Sachs, “Estimating the spectrum of a stochastic process in the N i e 2 −1−i e presence of a contaminating signal,” IEEE Trans. Signal Process., vol.   41, no. 1, pp. 323, 1993. γ, , L 1 ΦF ( 2 2 ) ΦFλ (γ,2,2L). λ (i) (i) [31] R. Von Sachs, “Peak-insensitive non-parametric spectrum estimation,” ≈ − Ω Ωk′ Ω(i) k=1 k  k′=1 J. Time Series Anal., vol. 15, no. 4, pp. 429–452, 1994. X Y Y (43) [32] R.J. Bhansali, “A mixed spectrum analysis of the lynx data,” J. R. Stat. Expression (23) follows by combining (42) and (43).  Soc. Ser. A, , no. 142, pp. 199–209, 1979. [33] B. Truong-Van, “A new approach to frequency analysis with amplified harmonics,” J. R. Stat. Soc. Ser. B, vol. 52, pp. 203–221, 1990. ACKNOWLEDGEMENT [34] B.G. Quinn and J.M. Fernandes, “A fast efficient technique for the estimation of frequency,” Biometrika, vol. 78, no. 3, pp. 489–497, 1991. The GOLF instrument onboard SoHO is a cooperative effort [35] B.G. Quinn, “A fast efficient technique for the estimation of frequency: of scientists, engineers, and technicians, to whom we are Interpretation and generalisation,” Biometrika, vol. 86, pp. 213–220, indebted. SoHO is a project of international collaboration 1999. [36] L. Kavalieris and Hannan E.J., “Determining the number of terms in a between ESA and NASA. trigonometric regression.,” J. Time Series Analysis, vol. 15, pp. 613– 625, 1994. REFERENCES [37] E.J. Hannan, “Testing for a jump in the spectral function,” J. R. Stat. Soc. Ser. B, vol. 23, no. 2, pp. 394–404, 1961. [1] A. Schuster, “On the investigation of hidden periodicities,” J. Geophys. [38] D.F. Nicholls, “Estimation of the spectral density function when testing Res., vol. 3, pp. 13, 1898. for a jump in the spectrum,” Austr. J. Stat., vol. 9, pp. 103–108, 1967. [2] D.R. Brillinger, Time Series : Data Analysis and Theory, Holden Day, [39] S.T. Chiu, “Peak-insensitive parametric spectrum estimation,” Stochastic San Francisco, 1981. Processes and their Applications, vol. 35, pp. 121–140, 1990. [3] M.S. Bartlett, “Periodogram analysis and continuous spectra,” [40] J.K. Gryca, “Detection of multiple sinusoids buried in noise via balanced Biometrika, vol. 37, pp. 1–16, 1950. model truncation,” IEEE Instrum. Meas. Conf., pp. 1353–1358, 1998. [4] U. Grenander and M. Rosenblatt, Statistical analysis of stationary time [41] L.B. White, “Detection of sinusoids in unknown coloured noise using series, John Wiley and Sons, 1957. ratios of ar spectrum estimates,” Proc. Inform., Decision and Control, [5] M.B. Priestley, Spectral Analysis and Time Series, Academic Press, pp. 257–262, 1999. San Diego, 1981. [42] N. Lu and D. Zimmerman, “Testing for directional symmetry in spatial [6] P.J. Brockwell and R.A. Davis, Time series : theory and methods, dependence using the periodogram,” J. Stat. Planning and Inference, Springer, 1991. vol. 129, pp. 369–385, 2005. [7] P. Bloomfield, Fourier Analysis of Time Series, Wiley-Intersci., 2000. [43] A.P. Liavas et al., “A periodogram-based method for the detection of [8] P. Stoica and R. Moses, Spectral analysis of signals, Prentice Hall, steady-state visually evoked potentials.,” IEEE Trans. Biom. Eng., vol. 2005. 45, no. 2, pp. 242–248, 1998. [9] B. G. Quinn and E.J. Hannan, The Estimation and Tracking of [44] C. Zheng, “Detection of multiple sinusoids in unknown colored noise Frequency, Cambridge Univ., 2001. using truncated cepstrum thresholding and local signal-to-noise-ratio,” [10] F. Pepe et al., “Instrumentation for the detection and characterization of Applied Acoust., pp. 809–816, 2012. exoplanets,” Nature, vol. 513, pp. 358–366, 2014. [45] B. Nadler and A. Kontorovich, “Model selection for sinusoids in noise: [11] N.M. Batalha, “Exploring exoplanet populations with NASA’s Kepler Statistical analysis and a new penalty term,” IEEE Trans. Signal Process., Mission,” Proc. Nat. Acad. Sci., vol. 111, pp. 12647–12654, Sept. 2014. vol. 59, no. 4, pp. 1333–1345, 2011. [12] M. Auvergne et al., “The CoRoT satellite in flight: description and [46] C. Koen, “The analysis of indexed astronomical time series - xi. the performance,” A&A, vol. 506, pp. 411–424, 2009. statistics of oversampled white noise periodograms,” MNRAS, vol. 449, [13] H. Rauer et al., “The PLATO 2.0 mission,” Experimental Astronomy, no. 1, pp. 1098–1105, 2015. vol. 38, pp. 249–330, 2014. [47] C. Koen, “The analysis of indexed astronomical time series - xii. the [14] D.A. Fischer et al., “Exoplanet Detection Techniques,” Protostars and statistics of oversampled fourier spectra of noise plus a single sinusoid,” Planets VI, pp. 715–737, 2014. MNRAS, vol. 453, no. 2, pp. 1793–1798, 2015. [15] M. Perryman, The exoplanet handbook, Cambridge Univ., 2011. [48] M. Tuomi et al., “Signals embedded in the radial velocity noise. Periodic [16] X. Dumusque et al., “An Earth-mass planet orbiting α Centauri B,” variations in the τ Ceti velocities,” A&A, vol. 551, pp. A79, 2012. Nature, vol. 491, pp. 207–211, 2012. [49] D. Donoho and J. Jin, “Higher criticism for detecting sparse heteroge- [17] A. Hatzes, “The Radial Velocity Detection of Earth-mass Planets in the neous mixtures,” Ann. Stat., 2004. Presence of Activity Noise: The Case of α Centauri Bb,” ApJ, vol. 770, [50] Y.I. Ingster et al., “Detection boundary in sparse regression,” Electron. pp. 133, 2013. J. Statist., vol. 4, pp. 1476–1526, 2010. [18] V. Rajpaul et al., “Ghost in the time series: no planet for Alpha Cen [51] G. Walther, “The Average Likelihood Ratio for Large-scale Multiple B,” MNRAS, vol. 456, pp. L6–L10, 2016. Testing and Detecting Sparse Mixtures,” IMS Collections : From [19] L. Bigot et al., “The diameter of the CoRoT target HD 49933,” A&A, Probability to Stat. and Back: High-Dimensional Models and Processes, vol. 534, no. 3, 2011. vol. 9, pp. 317–326, 2012. [20] R.A. Fisher, “Tests of Significance in Harmonic Analysis,” Proc. R. [52] A. Moscovich et al., “On the exact Berk-Jones statistics and their p-value Soc. London, Ser. A, vol. 125, pp. 54–59, 1929. calculation,” Electron. J. Stat., vol. 10, pp. 2329–2354, 2016. [21] S.T. Chiu, “Detecting periodic components in a white gaussian time [53] V. Gontscharuk et al., “Goodness of fit tests in terms of local levels series,” J. R. Stat. Soc. Series B, vol. 51, no. 2, pp. 249–259, 1989. with special emphasis on higher criticism tests,” Bernoulli, vol. 22, no. [22] M. Shimshoni, “On fisher’s test of significance in harmonic analysis,” 3, pp. 1331–1363, 2016,. Geophys. J. R. Astronom. Soc., pp. 373–377, 1971. [54] P. Hall and J. Jin, “Innovated higher criticism for detecting sparse signals [23] S. Kay, “Adaptive detection for unknown noise power spectral densities,” in correlated noise,” Ann. Stat., vol. 38, pp. 1686–1732, 2010. IEEE Trans. Signal Process, vol. 47, no. 1, pp. 10–21, 1999. [55] S. Sulis, D. Mary, and L. Bigot, “Using hydrodynamical simulations [24] T.H. Li, Time series with mixed spectra, CRC Press, 2014. of stellar atmospheres for periodogram standardization: Application to [25] P. Whittle, “The simultaneous estimation of a time series harmonic exoplanet detection,” in IEEE ICASSP, March 2016, pp. 4428–4432. components and covariance structure,” Trabajos de Estadistica, vol. 3, [56] S.K. Gupta et al., “UPSO three channel fast photometer,” Bulletin no. 1-2, pp. 43–57, 1952. Astron. Soc. of India, vol. 29, pp. 479–486, 2001. [26] M.S. Bartlett, “An introduction to stochastic processes,” Quart. J. R. [57] S. Sulis, D. Mary, and L. Bigot, “Overcoming the stellar noise Meteorological Soc., vol. 81, no. 350, pp. 650, 1955. barrier for the detection of telluric exoplanets: an approach based on [27] A. Siegel, “Testing for periodicity in a time series,” J. Amer. Stat. hydrodynamical simulations,” in EAS Publications Series, D. Mary Assoc., vol. 75, no. 370, pp. 345–348, 1980. et al., Eds., Sept. 2016, vol. 78-79, pp. 247–274. 14

[58] J.G. Proakis and D.G. Manolakis, Digital Signal Processing, Prentice- [91] M. Zechmeister and M. K¨urster, “The generalised Lomb-Scargle Hall, 1996. periodogram. A new formalism for the floating-mean and Keplerian [59] M. Abramowitz et al., Spectral Analysis and Time Series, Dover periodograms,” A&A, vol. 496, pp. 577–584, 2009. Publications, 1972. [92] R.V. Baluev, “Keplerian periodogram for Doppler exoplanet detection: [60] H.A. David and H.N. Nagaraja, Order Statistics, 3rd Ed., Wiley, 2003. optimized computation and analytic significance thresholds,” MNRAS, [61] S.M. Kay, Fundamentals of Statistical signal processing. Vol II : vol. 446, pp. 1478–1492, 2015. Detection theory., Prentice-Hall, Inc, 1998. [93] P.C. Gregory, “An apodized kepler periodogram for separating planetary [62] B. G. Quinn, “Testing for the presence of sinusoidal components,” J. and stellar activity signals,” MNRAS, vol. 458, pp. 2604–2633, 2016. Appl. Probability, vol. 23, pp. 201–210, 1986. [94] A.M. Zoubir, “Bootstrap: theory and applications,” in SPIE Conf., F.T. [63] A. Schwarzenberg-Czerny, “The distribution of empirical periodograms: Luk, Ed., 1993, vol. 2027, pp. 216–235. Lomb–Scargle and PDM spectra,” MNRAS, vol. 301, pp. 831–840, [95] M. S¨uveges et al., “A comparative study of four significance measures 1998. for periodicity detection in astronomical surveys,” MNRAS, vol. 450, [64] T. Aittokallio et al., “Testing for Periodicity in Signals: An Application no. 2, pp. 2052–2066, 2015. to Detect Partial Upper Airway Obstruction during Sleep,,” J. Theoretical [96] H.S. Kasana, Complex variables : theory and applications. 2nd Edition, Medicine, vol. 3, no. 4, pp. 231–245, 2001. Prentice-Hall of India, 2005. [65] J. Guti´errez-Soto et al., “Low-amplitude variations detected by CoRoT in the B8IIIe star HD 175869,” A&A, vol. 506, pp. 133–141, 2009. [66] S. Aldor-Noiman et al., “The power to see: A new graphical test of normality.,” Am. Stat., vol. 68, no. 4, pp. 318–318, 2013. [67] D. Mary and A. Ferrari, “A non-asymptotic standardization of binomial counts in higher criticism,” in Inform. Theory (ISIT), IEEE Int. Symp., June 2014, pp. 561–565. [68] D.M. Kaplan and M. Goldman, “True equality (of pointwise sensitivity) at last: a dirichlet alternative to Kolmogorov-Smirnov inference on distributions.,” Tech. report, 2014. [69] V. Gontscharuk et al., “The intermediates take it all: Asymptotics of higher criticism statistics and a powerful alternative based on equal local levels.,” Biom. J., vol. 57, no. 1, pp. 159–180, 2014. [70] J. Li and D. Siegmund, “Higher criticism: p-values and criticism,” Ann. Statist., vol. 43, no. 3, pp. 1323–1350, 2015. [71] R.H. Berk and D.H. Jones, “Goodness-of-fit test statistics that dominate the Kolmogorov statistics,” Z. Wahrscheinlichkeit., vol. 47, pp. 47–59, 1979. [72] A. Moscovich-Eiger and B. Nadler, “Fast calculation of boundary crossing probabilities for Poisson processes,” ArXiv e-prints (V.3), 2015. [73] J. Fan and Q. Yao, Nonlinear Time Series-Nonparametric and Parametric Methods, Springer-Verlag New York, 2003. [74] K. F. Turkman and A. M. Walker, “On the asymptotic distributions of maxima of trigonometric polynomials with random coefficients,” Advances in Applied Probability, vol. 16, no. 4, pp. 819–842, 1984. [75] R.A Davis and T. Mikosch, “The maximum of the periodogram of a non-gaussian sequence,” Ann. of Prob., vol. 27, pp. 522–536, 1999. [76] D. Rife and R. Boorstyn, “Single tone parameter estimation from discrete-time observations,” IEEE Trans. Inf. Theory, vol. 20, no. 5, pp. 591–598, 1974. [77] B.G. Quinn and P. J. Kootsookos, “Threshold behavior of the maximum likelihood estimator of frequency,” IEEE Trans. Signal Process., vol. 42, no. 11, pp. 3291–3294, 1994. [78] H. Akaike, “Fitting autoregressive models for prediction,” Ann. Inst. Stat. Math., vol. 21, no. 1, pp. 243–247, 1969. [79] H. Akaike, “A new look at the statistical model identification,” IEEE Trans. Automatic Control, vol. 19, no. 6, pp. 716–723, 1974. [80] E. Parzen, “Multiple time series: determining the order of approximating autoregressive schemes,” Tech. Report, , no. 23, pp. 716–723, 1975. [81] E.J. Hannan and B.G. Quinn, “The determination of the order of an autoregression,” J. R. Stat. Soc. Ser. B, vol. 41, pp. 190–195, 1979. [82] J. Rissanen, “Universal coding, information prediction and estimation,” IEEE Trans. Inf. Theory, , no. 30, pp. 629–636, 1984. [83] A. Boardman et al., “A study on the optimum order of autoregressive models for heart rate variability,” Physio. Meas., vol. 23, pp. 325, 2002. [84] R.A. Garcia et al., “Global solar Doppler velocity determination with the GOLF/SoHO instrument,” A&A, vol. 442, pp. 385–395, 2005. [85] N. Meunier et al., “Variability of stellar granulation and convective blueshift with spectral type and magnetic activity. I. K and G main sequence stars,” ArXiv e-prints, 2016. [86] R.V. Baluev, “Assessing the statistical significance of periodogram peaks,” MNRAS, vol. 385, pp. 1279–1285, 2008. [87] M. S¨uveges, “Extreme-value modelling for the significance assessment of periodogram peaks,” MNRAS, vol. 440, no. 3, pp. 2099–2114, 2014. [88] J.D. Scargle, “Studies in astronomical time series analysis. II - Statistical aspects of spectral analysis of unevenly spaced data,” ApJ, vol. 263, pp. 835–853, 1982. [89] G.L. Bretthorst, “Frequency Estimation And Generalized Lomb-Scargle Periodograms.,” Stat. Challenges in Astronomy, pp. 309–329, 2003. [90] T. Thong et al., “Lomb-wech periodogram for non-uniform sampling,” Proc. 26th Annu. Int. Conf. IEEE EMBS, 2004.