Discovering New Worlds: A review of signal processing methods for detecting exoplanets from astronomical radial velocity data [Applications Corner]
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paper_title: Metallicities & Activities of Southern Stars

paper_content:

Aims. We present the results from high-resolution spectroscopic measurements to determine metallicities and activities of bright stars in the southern hemisphere. Methods. We measured the iron abundances ((Fe/H)'s) and chromospheric emission indices (loghRHK ) of 353 solar-type stars with V = 7.5−9.5. (Fe/H) abundances are determined using a custom χ 2 fitting procedure within a large grid of Kurucz model atmospheres. The chromospheric activities were determined by measuring the amount of emission in the cores of the strong Caii HK lines. Results. Our comparison of the metallicity sample to other (Fe/H) determinations was found to agree at the ±0.05 dex level for spectroscopic values and at the ±0.1 dex level for photometric values. The distribution of chromospheric activities is described by a bimodal distribution, agreeing with the conclusions from other works. Also an analysis of Maunder minimum status was attempted, and it was found that 6 ± 4 stars in the sample could be in a Maunder minimum phase of their evolution and hence the Sun should only spend a few per cent of its main sequence lifetime in Maunder minimum.
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paper_title: ELODIE: A spectrograph for accurate radial velocity measurements

paper_content:

The bre{fed echelle spectrograph of Observatoire de Haute{Provence, ELODIE, is presented. This instrument has been in operation since the end of 1993 on the 1.93 m telescope. ELODIE is designed as an updated version of the cross{correlation spectrometer CORAVEL, to perform very accurate radial velocity measurements such as needed in the search, by Doppler shift, for brown{dwarfs or giant planets orbiting around nearby stars. In one single exposure a spectrum at a resolution of 42000 (=) ranging from 3906 A to 6811 Ai s recorded on a 10241024 CCD. This performance is achieved by using a tan = 4 echelle grating and a combination of a prism and a grism as cross{disperser. An automatic on{line data treatment reduces all the ELODIE echelle spectra and computes cross{correlation functions. The instrument design and the data reduction algorithms are described in this paper. The eciency and accuracy of the instrument and its long term instrumental stability allow us to measure radial velocities with an accuracy better than 15 m s 1 for stars up to 9th magnitude in less than 30 minutes exposure time. Observations of 16th magnitude stars are also possible to measure velocities at about 1 km s 1 accuracy. For classic spectroscopic studies (S=N>100) 9th magnitude stars can be observed in one hour exposure time.
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paper_title: Estimation of periods from unequally spaced observations

paper_content:

A better estimation of the power spectrum of a time series formed with unequally spaced observations may be obtained by means of a data-compensated discrete Fourier transform. This transform is defined so as to include the uneven spacing of the dates of observation and weighting of the corresponding data. The accurate determination of the peak heights allows one to design harmonic filters and thus to make a more certain choice among peaks of similar height and also to discriminate peaks that are just aliases of other peaks. The theory is applied to simulated time series and also to true observational data.
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paper_title: Bayesian Spectrum Analysis and Parameter Estimation

paper_content:

1 Introduction.- 2 Single Stationary Sinusoid Plus Noise.- 3 The General Model Equation Plus Noise.- 4 Estimating the Parameters.- 5 Model Selection.- 6 Spectral Estimation.- 7 Applications.- 8 Summary and Conclusions.- A Choosing a Prior Probability.- B Improper Priors as Limits.- C Removing Nuisance Parameters.- D Uninformative Prior Probabilities.- E Computing the "Student t-Distribution".
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paper_title: Studies in astronomical time series analysis. II - Statistical aspects of spectral analysis of unevenly spaced data

paper_content:

Detection of a periodic signal hidden in noise is frequently a goal in astronomical data analysis. This paper does not introduce a new detection technique, but instead studies the reliability and efficiency of detection with the most commonly used technique, the periodogram, in the case where the observation times are unevenly spaced. This choice was made because, of the methods in current use, it appears to have the simplest statistical behavior. A modification of the classical definition of the periodogram is necessary in order to retain the simple statistical behavior of the evenly spaced case. With this modification, periodogram analysis and least-squares fitting of sine waves to the data are exactly equivalent. Certain difficulties with the use of the periodogram are less important than commonly believed in the case of detection of strictly periodic signals. In addition, the standard method for mitigating these difficulties (tapering) can be used just as well if the sampling is uneven. An analysis of the statistical significance of signal detections is presented, with examples
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paper_title: Keplerian periodogram for Doppler exoplanets detection: optimized computation and analytic significance thresholds

paper_content:

We consider the so-called Keplerian periodogram, in which the putative detectable signal is modelled by a highly non-linear Keplerian radial velocity function, appearing in Doppler exoplanetary surveys. We demonstrate that for planets on high-eccentricity orbits the Keplerian periodogram is far more efficient than the classic Lomb-Scargle periodogram and even the multiharmonic periodograms, in which the periodic signal is approximated by a truncated Fourier series. We provide new numerical algorithm for computation of the Keplerian periodogram. This algorithm adaptively increases the parameteric resolution where necessary, in order to uniformly cover all local optima of the Keplerian fit. Thanks to this improvement, the algorithm provides more smooth and reliable results with minimized computing demands. We also derive a fast analytic approximation to the false alarm probability levels of the Keplerian periodogram. This approximation has the form $(P z^{3/2} + Q z) W \exp(-z)$, where $z$ is the observed periodogram maximum, $W$ is proportional to the settled frequency range, and the coefficients $P$ and $Q$ depend on the maximum eccentricity to scan.
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paper_title: Period04: A software package to extract multiple frequencies from real data

paper_content:

Period04, a reworked and extended version of Period98 (Sperl 1998) and PERIOD/ PERDET (Breger 1990), is a new software package especially dedicated to the statistical analysis of large astronomical data sets containing gaps. It offers tools to extract the individual frequencies from the multiperiodic content of time series and provides a flexible interface to perform multiple-frequency fits. A review of the functions of Period04 is given.
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paper_title: A dynamically-packed planetary system around GJ667C with three super-Earths in its habitable zone

paper_content:

Guillen Anglada-Escude, et al, 'A dynamically-packed planetary system around GJ 667J with three super-Earths in its habitable zone', A&A, Vol. 556, A126, first published online 7 August 2013. The version of record is available at doi: http://doi.org/10.1051/0004-6361/201321331. © ESO 2013. Reproduced with permission from Astronomy & Astrophysics. Published by EDP Sciences
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paper_title: PlanetPack: a radial-velocity time-series analysis tool facilitating exoplanets detection, characterization, and dynamical simulations

paper_content:

Abstract We present PlanetPack, a new software tool that we developed to facilitate and standardize the advanced analysis of radial velocity (RV) data for the goal of exoplanets detection, characterization, and basic dynamical N -body simulations. PlanetPack is a command-line interpreter, that can run either in an interactive mode or in a batch mode of automatic script interpretation. Its major abilities include: (i) advanced RV curve fitting with the proper maximum-likelihood treatment of unknown RV jitter; (ii) user-friendly multi-Keplerian as well as Newtonian N -body RV fits; (iii) use of more efficient maximum-likelihood periodograms that involve the full multi-planet fitting (sometimes called as “residual” or “recursive” periodograms); (iv) easily calculatable parametric 2D likelihood function level contours, reflecting the asymptotic confidence regions; (v) fitting under some useful functional constraints is user-friendly; (vi) basic tasks of short- and long-term planetary dynamical simulation using a fast Everhart-type integrator based on Gauss–Legendre spacings; (vii) fitting the data with red noise (auto-correlated errors); (viii) various analytical and numerical methods for the tasks of determining the statistical significance. It is planned that further functionality may be added to PlanetPack in the future. During the development of this software, a lot of effort was made to improve the calculational speed, especially for CPU-demanding tasks. PlanetPack was written in pure C++ (standard of 1998/2003), and is expected to be compilable and useable on a wide range of platforms.
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paper_title: A new cold sub-Saturnian candidate planet orbiting GJ 221

paper_content:

Mikko Tuomi, 'A new cold sub-Saturnian candidate planet orbiting GJ 22', Monthly Notices of the Royal Astronomical Society, Letters, Vol 440: L1-L5, advanced access publication 11 February 2014. The version of record is available at doi: 10.1093/mnrasl/slu014 © 2014 The Author. Published by Oxford University Press on behalf of the Royal Astronomical Society.
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paper_title: An Adaptive Metropolis algorithm

paper_content:

A proper choice of a proposal distribution for Markov chain Monte Carlo methods, for example for the Metropolis-Hastings algorithm, is well known to be a crucial factor for the convergence of the algorithm. In this paper we introduce an adaptive Metropolis (AM) algorithm, where the Gaussian proposal distribution is updated along the process using the full information cumulated so far. Due to the adaptive nature of the process, the AM algorithm is non-Markovian, but we establish here that it has the correct ergodic properties. We also include the results of our numerical tests, which indicate that the AM algorithm competes well with traditional Metropolis-Hastings algorithms, and demonstrate that the AM algorithm is easy to use in practical computation.
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paper_title: The curious case of HD41248. A pair of static signals buried behind red-noise

paper_content:

Gaining a better understanding of the effects of stellar induced radial velocity noise is critical for the future of exoplanet studies, since the discovery of the lowest-mass planets using this method will require us to go below the intrinsic stellar noise limit. An interesting test case in this respect is that of the southern solar analogue HD41248. The radial velocity time series of this star has been proposed to contain either a pair of signals with periods of around 18 and 25 days, that could be due to a pair of resonant super-Earths, or a single and varying 25 day signal that could arise due to a complex interplay between differential rotation and modulated activity. In this letter we build-up more evidence for the former scenario, showing that the signals are still clearly significant even after more than 10 years of observations and they likely do not change in period, amplitude, or phase as a function of time, the hallmarks of static Doppler signals. We show that over the last two observing seasons this star was more intrinsically active and the noise reddened, highlighting why better noise models are needed to find the lowest amplitude signals, in particular models that consider noise correlations. This analysis shows that there is still sufficient evidence for the existence of two super-Earths on the edge of, or locked into, a 7:5 mean motion resonance orbiting HD41248.
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paper_title: Radial Velocity Fitting Challenge. I. Simulating the data set including realistic stellar radial-velocity signals

paper_content:

Stellar signals are the main limitation for precise radial-velocity (RV) measurements. These signals arise from the photosphere of the stars. The m/s perturbation created by these signals prevents the detection and mass characterization of small-mass planetary candidates such as Earth-twins. Several methods have been proposed to mitigate stellar signals in RV measurements. However, without precisely knowing the stellar and planetary signals in real observations, it is extremely difficult to test the efficiency of these methods. The goal of the RV fitting challenge is to generate simulated RV data including stellar and planetary signals and to perform a blind test within the community to test the efficiency of the different methods proposed to recover planetary signals despite stellar signals. In this first paper, we describe the simulation used to model the measurements of the RV fitting challenge. Each simulated planetary system includes the signals from instrumental noise, stellar oscillations, granulation, supergranulation, stellar activity, and observed and simulated planetary systems. In addition to RV variations, this simulation also models the effects of instrumental noise and stellar signals on activity observables obtained by HARPS-type high-resolution spectrographs, that is, the calcium activity index log(R'hk) and the bisector span and full width at half maximum of the cross-correlation function. We publish the 15 systems used for the RV fitting challenge including the details about the planetary systems that were injected into each of them (data available at CDS and here: https://rv-challenge.wikispaces.com.
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paper_title: Detecting multiple periodicities in observational data with the multifrequency periodogram - II. Frequency Decomposer, a parallelized time-series analysis algorithm

paper_content:

Abstract This is a parallelized algorithm performing a decomposition of a noisy time series into a number of sinusoidal components. The algorithm analyses all suspicious periodicities that can be revealed, including the ones that look like an alias or noise at a glance, but later may prove to be a real variation. After the selection of the initial candidates, the algorithm performs a complete pass through all their possible combinations and computes the rigorous multifrequency statistical significance for each such frequency tuple. The largest combinations that still survived this thresholding procedure represent the outcome of the analysis. The parallel computing on a graphics processing unit (GPU) is implemented through CUDA and brings a significant performance increase. It is still possible to run FREDEC solely on CPU in the traditional single-threaded mode, when no suitable GPU device is available. To verify the practical applicability of our algorithm, we apply it to an artificial time series as well as to some real-life exoplanetary radial-velocity data. We demonstrate that FREDEC can successfully reveal several known exoplanets. Moreover, it detected a new 9.8-day variation in the Lick data for the five-planet system of 55 Cnc. It might indicate the existence of a small sixth planet in the 3:2 commensurability with the planet 55 Cnc b, although this detection is model-dependent and still needs a detailed verification.
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paper_title: A Gaussian process framework for modelling stellar activity signals in radial velocity data

paper_content:

To date, the radial velocity (RV) method has been one of the most productive techniques for detecting and confirming extrasolar planetary candidates. Unfortunately, stellar activity can induce RV variations which can drown out or even mimic planetary signals - and it is notoriously difficult to model and thus mitigate the effects of these activity-induced nuisance signals. This is expected to be a major obstacle to using next-generation spectrographs to detect lower mass planets, planets with longer periods, and planets around more active stars. Enter Gaussian processes (GPs) which, we note, have a number of attractive features that make them very well suited to disentangling stellar activity signals from planetary signals. We present here a GP framework we developed to model RV time series jointly with ancillary activity indicators (e.g. bisector velocity spans, line widths, chromospheric activity indices), allowing the activity component of RV time series to be constrained and disentangled from e.g. planetary components. We discuss the mathematical details of our GP framework, and present results illustrating its encouraging performance on both synthetic and real RV datasets, including the publicly-available Alpha Centauri B dataset.
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paper_title: Detecting multiple periodicities in observational data with the multi-frequency periodogram. I. Analytic assessment of the statistical significance

paper_content:

We consider the "multi-frequency" periodogram, in which the putative signal is modelled as a sum of two or more sinusoidal harmonics with idependent frequencies. It is useful in the cases when the data may contain several periodic components, especially when their interaction with each other and with the data sampling patterns might produce misleading results. ::: Although the multi-frequency statistic itself was already constructed, e.g. by G. Foster in his CLEANest algorithm, its probabilistic properties (the detection significance levels) are still poorly known and much of what is deemed known is unrigourous. These detection levels are nonetheless important for the data analysis. We argue that to prove the simultaneous existence of all $n$ components revealed in a multi-periodic variation, it is mandatory to apply at least $2^n-1$ significance tests, among which the most involves various multi-frequency statistics, and only $n$ tests are single-frequency ones. ::: The main result of the paper is an analytic estimation of the statistical significance of the frequency tuples that the multi-frequency periodogram can reveal. Using the theory of extreme values of random fields (the generalized Rice method), we find a handy approximation to the relevant false alarm probability. For the double-frequency periodogram this approximation is given by an elementary formula $\frac{\pi}{16} W^2 e^{-z} z^2$, where $W$ stands for a normalized width of the settled frequency range, and $z$ is the observed periodogram maximum. We carried out intensive Monte Carlo simulations to show that the practical quality of this approximation is satisfactory. A similar analytic expression for the general multi-frequency periodogram is also given in the paper, though with a smaller amount of numerical verification.
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paper_title: Model Order Selection for Complex Sinusoids in the Presence of Unknown Correlated Gaussian Noise

paper_content:

We consider the problem of detecting and estimating the amplitudes and frequencies of an unknown number of complex sinusoids based on noisy observations from an unstructured array. In parametric detection problems like this, information theoretic criteria such as minimum description length (MDL) and Akaike information criterion (AIC) have previously been used for joint detection and estimation. In our paper, model selection based on extreme value theory (EVT), which has previously been used for enumerating real sinusoidal components from one-dimensional observations, is generalized to the case of multidimensional complex observations in the presence of noise with an unknown spatial correlation matrix. Unlike the previous work, the likelihood ratios considered in the mutlidimensional case cannot be addressed using Gaussian random fields. Instead, chi-square random fields associated with the generalized likelihood ratio test are encountered and EVT is used to analyze the model order overestimation probability for a general class of likelihood penalty terms including MDL and AIC, and a novel likelihood penalty term derived based on EVT. Since the exact EVT penalty term involves a Lambert-W function, an approximate penalty term is also derived that is more tractable. We provide threshold signal-to-noise ratios (SNRs) and show that the model order underestimation probability is asymptotically vanishing for EVT and MDL. We also show that MDL and EVT are asymptotically consistent while AIC is not, and that with finite samples, the detection performance of EVT outperforms MDL and AIC. Finally, the accuracy of the derived threshold SNRs is also demonstrated.
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paper_title: The impact of red noise in radial velocity planet searches: Only three planets orbiting GJ581?

paper_content:

We perform a detailed analysis of the latest HARPS and Keck radial velocity data for the planet-hosting red dwarf GJ581, which attracted a lot of attention in recent time. We show that these data contain important correlated noise component ("red noise") with the correlation timescale of the order of 10 days. This red noise imposes a lot of misleading effects while we work in the traditional white-noise model. To eliminate these misleading effects, we propose a maximum-likelihood algorithm equipped by an extended model of the noise structure. We treat the red noise as a Gaussian random process with exponentially decaying correlation function. Using this method we prove that: (i) planets b and c do exist in this system, since they can be independently detected in the HARPS and Keck data, and regardless of the assumed noise models; (ii) planet e can also be confirmed independently by the both datasets, although to reveal it in the Keck data it is mandatory to take the red noise into account; (iii) the recently announced putative planets f and g are likely just illusions of the red noise; (iv) the reality of the planet candidate GJ581 d is questionable, because it cannot be detected from the Keck data, and its statistical significance in the HARPS data (as well as in the combined dataset) drops to a marginal level of $\sim 2\sigma$, when the red noise is taken into account. Therefore, the current data for GJ581 really support existence of no more than four (or maybe even only three) orbiting exoplanets. The planet candidate GJ581 d requests serious observational verification.
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paper_title: Bayesian Re-analysis of the Gliese 581 Exoplanet System

paper_content:

A re-analysis of Gliese 581 HARPS and HIRES precision radial velocity data was carried out with a Bayesian multi-planet Kepler periodogram (from 1 to 6 planets) based on a fusion Markov chain Monte Carlo algorithm. In all cases the analysis included an unknown parameterized stellar jitter noise term. For the HARPS data set the most probable number of planetary signals detected is 5 with a Bayesian false alarm probability of 0.01. These include the $3.1498\pm0.0005$, $5.3687\pm0.0002$, $12.927_{-0.004}^{+0.006}$, and $66.9\pm0.2$d periods reported previously plus a $399_{-16}^{+14}$d period. The orbital eccentricities are $0.0_{-0.0}^{+0.2}$, $0.00_{-0.00}^{+0.02}$, $0.10_{-0.10}^{+0.06}$, $0.33_{-0.10}^{+0.09}$, and $0.02_{-0.02}^{+0.30}$, respectively. The semi-major axis and $M sin i$ of the 5 planets are ($0.0285\pm0.0006$ au, $1.9\pm0.3$M$_{\earth}$), ($0.0406\pm0.0009$ au, $15.7\pm0.7$M$_{\earth}$), ($0.073\pm0.002$ au, $5.3\pm0.4$M$_{\earth}$), ($0.218\pm0.005$ au, $6.7\pm0.8$M$_{\earth}$), and ($0.7\pm0.2$ au, $6.6_{-2.7}^{+2.0}$M$_{\earth}$), respectively. The analysis of the HIRES data set yielded a reliable detection of only the strongest 5.37 and 12.9 day periods. The analysis of the combined HIRES/HARPS data again only reliably detected the 5.37 and 12.9d periods. Detection of 4 planetary signals with periods of 3.15, 5.37, 12.9, and 66.9d was only achieved by including an additional unknown but parameterized Gaussian error term added in quadrature to the HIRES quoted errors. The marginal distribution for the sigma of this additional error term has a well defined peak at $1.8\pm0.4$m s$^{-1}$. It is possible that this additional error arises from unidentified systematic effects. We did not find clear evidence for a fifth planetary signal in the combined HIRES/HARPS data set.
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paper_title: Studies in astronomical time series analysis. II - Statistical aspects of spectral analysis of unevenly spaced data

paper_content:

Detection of a periodic signal hidden in noise is frequently a goal in astronomical data analysis. This paper does not introduce a new detection technique, but instead studies the reliability and efficiency of detection with the most commonly used technique, the periodogram, in the case where the observation times are unevenly spaced. This choice was made because, of the methods in current use, it appears to have the simplest statistical behavior. A modification of the classical definition of the periodogram is necessary in order to retain the simple statistical behavior of the evenly spaced case. With this modification, periodogram analysis and least-squares fitting of sine waves to the data are exactly equivalent. Certain difficulties with the use of the periodogram are less important than commonly believed in the case of detection of strictly periodic signals. In addition, the standard method for mitigating these difficulties (tapering) can be used just as well if the sampling is uneven. An analysis of the statistical significance of signal detections is presented, with examples
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paper_title: A Hot Uranus Orbiting the Super Metal-rich Star HD77338 and the Metallicity - Mass Connection

paper_content:

We announce the discovery of a low-mass planet orbiting the super metal-rich K0V star HD77338 as part of our on-going Calan-Hertfordshire Extrasolar Planet Search. The best fit planet solution has an orbital period of 5.7361\pm0.0015 days and with a radial velocity semi-amplitude of only 5.96\pm1.74 m/s, we find a minimum mass of 15.9+4.7-5.3 Me. The best fit eccentricity from this solution is 0.09+0.25-0.09, and we find agreement for this data set using a Bayesian analysis and a periodogram analysis. We measure a metallicity for the star of +0.35\pm0.06 dex, whereas another recent work (Trevisan et al. 2011) finds +0.47\pm0.05 dex. Thus HD77338b is one of the most metal-rich planet host stars known and the most metal-rich star hosting a sub-Neptune mass planet. We searched for a transit signature of HD77338b but none was detected. We also highlight an emerging trend where metallicity and mass seem to correlate at very low masses, a discovery that would be in agreement with the core accretion model of planet formation. The trend appears to show that for Neptune-mass planets and below, higher masses are preferred when the host star is more metal-rich. Also a lower boundary is apparent in the super metal-rich regime where there are no very low-mass planets yet discovered in comparison to the sub-solar metallicity regime. A Monte Carlo analysis shows that this, low-mass planet desert, is statistically significant with the current sample of 36 planets at around the 4.5\sigma\ level. In addition, results from Kepler strengthen the claim for this paucity of the lowest-mass planets in super metal-rich systems. Finally, this discovery adds to the growing population of low-mass planets around low-mass and metal-rich stars and shows that very low-mass planets can now be discovered with a relatively small number of data points using stable instrumentation.
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paper_title: Parameter Estimation from Time-Series Data with Correlated Errors: A Wavelet-Based Method and its Application to Transit Light Curves

paper_content:

We consider the problem of fitting a parametric model to time-series data that are afflicted by correlated noise. The noise is represented by a sum of two stationary Gaussian processes: one that is uncorrelated in time, and another that has a power spectral density varying as $1/f^\gamma$. We present an accurate and fast [O(N)] algorithm for parameter estimation based on computing the likelihood in a wavelet basis. The method is illustrated and tested using simulated time-series photometry of exoplanetary transits, with particular attention to estimating the midtransit time. We compare our method to two other methods that have been used in the literature, the time-averaging method and the residual-permutation method. For noise processes that obey our assumptions, the algorithm presented here gives more accurate results for midtransit times and truer estimates of their uncertainties.
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Title: Discovering New Worlds: A review of signal processing methods for detecting exoplanets from astronomical radial velocity data [Applications Corner]

Section 1: Introduction
Description 1: Introduce the concept of exoplanets and the importance of their detection using signal processing methods, particularly focusing on the radial velocity method.

Section 2: Radial Velocity Method
Description 2: Explain how the radial velocity method works for detecting exoplanets, including the challenges involved and the process of obtaining and analyzing radial velocity data.

Section 3: LS Periodogram
Description 3: Discuss the Lomb-Scargle (LS) periodogram method used in signal detection, including its implementation, advantages, and limitations.

Section 4: Keplerian Periodogram
Description 4: Describe the Keplerian periodogram method, highlighting its differences from the LS method, its robustness, and its application in detecting exoplanetary systems.

Section 5: Prewhitening Method
Description 5: Outline the prewhitening method for searching Doppler signals in radial velocity time series, including how it works and its strengths and weaknesses.

Section 6: ML Periodograms
Description 6: Explain the Maximum-Likelihood (ML) periodogram method, including its process, advantages, and how it accommodates multiple signals and correlated noise components.

Section 7: Bayesian Analysis
Description 7: Discuss the Bayesian analysis approach for signal detection, its robustness, flexibility, and the use of Markov chains to assess parameter space.

Section 8: MMSE-based Method
Description 8: Describe the Minimum Mean Square Error (MMSE)-based method for detecting signals in nonuniformly sampled radial velocity data, including its application and advantages.

Section 9: Statistical Significance of Signal Detection
Description 9: Elaborate on the importance of statistical validation in signal detection, including methods like False Alarm Probability (FAP) and bootstrap analysis.

Section 10: Potential Directions for Future Research
Description 10: Highlight future research directions for improving the detection of exoplanets, such as better calibration, understanding stellar activity impacts, and advancements in signal processing methods.