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Mathematical Modelling of the Lomb–Scargle Method in Astrophysics
Abstract
In astrophysics, the Lomb–Scargle method is widely used to analyse time series observations of stellar objects. The method allows us to detect periodic variations in the light intensity of a star, which may be due to its rotation, pulsations or interaction with a companion. In this paper, we describe the mathematical modelling of the Lomb–Scargle method and its applications in astrophysics. The Lomb–Scargle method is based on a spectral analysis of the time series of the star’s light intensity. In particular, it uses a periodogram, a function of the power of the spectrum of the time series, which reflects the contribution of each frequency to the total variance of the series. The periodogram can be obtained from the time series using a Fourier transform.
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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.
The Kepler Mission, launched on 2009 March 6, was designed with the explicit capability to detect Earth-size planets in the habitable zone of solar-like stars using the transit photometry method. Results from just 43 days of data along with ground-based follow-up observations have identified five new transiting planets with measurements of their masses, radii, and orbital periods. Many aspects of stellar astrophysics also benefit from the unique, precise, extended, and nearly continuous data set for a large number and variety of stars. Early results for classical variables and eclipsing stars show great promise. To fully understand the methodology, processes, and eventually the results from the mission, we present the underlying rationale that ultimately led to the flight and ground system designs used to achieve the exquisite photometric performance. As an example of the initial photometric results, we present variability measurements that can be used to distinguish dwarf stars from red giants.
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
Randomly distributed nonoverlapping perfectly conducting n spheres of radii rk (k=1,2,…,n) are embedded in a conducting matrix occupying a large ball of the normalized unit radius. The potential and the normal flux are given on the boundary of large ball. The locations of inclusions ak are not known. A perturbation term induced by inclusions is constructed in general case and studied up to O(R4) for equal spheres when R=rk. It includes the unknown centers of inclusions in symbolic form. The inverse problem is reduced to determination of the centers ak by fitting of the given perturbation term on the unit sphere.
At a distance of 1.295 parsecs, the red dwarf Proxima Centauri (α Centauri C, GL 551, HIP 70890 or simply Proxima) is the Sun's closest stellar neighbour and one of the best-studied low-mass stars. It has an effective temperature of only around 3,050 kelvin, a luminosity of 0.15 per cent of that of the Sun, a measured radius of 14 per cent of the radius of the Sun and a mass of about 12 per cent of the mass of the Sun. Although Proxima is considered a moderately active star, its rotation period is about 83 days (ref. 3) and its quiescent activity levels and X-ray luminosity are comparable to those of the Sun. Here we report observations that reveal the presence of a small planet with a minimum mass of about 1.3 Earth masses orbiting Proxima with a period of approximately 11.2 days at a semi-major-axis distance of around 0.05 astronomical units. Its equilibrium temperature is within the range where water could be liquid on its surface. © 2016 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
The inception of e-learning technologies has led to a tremendous increase in the use of e-learning systems to support blended learning in Universities by providing a mix of face-to-face classroom teaching, live e-learning, self-paced e-learning and distance learning. Despite the existing benefits of using e-learning, some higher education institutions have not utilized e-learning to its full potential and yet there are limited studies that offer a comprehensive framework for effectively using e-learning systems. It is therefore imperative that learning technologists understand the factors that influence the effectiveness of blended e-learning. An expert survey was conducted to establish which factors are important for evaluating the effectiveness of e-learning systems. This paper describes a methodological framework consisting of factors necessary for assessing the effectiveness of e-learning within Universities.
Context. Frequency analyses are very important in astronomy today, not least
in the ever-growing field of exoplanets, where short-period signals in stellar
radial velocity data are investigated. Periodograms are the main (and powerful)
tools for this purpose. However, recovering the correct frequencies and
assessing the probability of each frequency is not straightforward.
Aims. We provide a formalism that is easy to implement in a code, to describe
a Bayesian periodogram that includes weights and a constant offset in the data.
The relative probability between peaks can be easily calculated with this
formalism. We discuss the differences and agreements between the various
periodogram formalisms with simulated examples.
Methods. We used the Bayesian probability theory to describe the probability
that a full sine function (including weights derived from the errors on the
data values and a constant offset) with a specific frequency is present in the
data.
Results. From the expression for our Baysian generalised Lomb-Scargle
periodogram (BGLS), we can easily recover the expression for the non-Bayesian
version. In the simulated examples we show that this new formalism recovers the
underlying periods better than previous versions. A Python-based code is
available for the community.
We consider the estimation of the Fourier transform of multidimensional deterministic signals from a finite number of random samples. First, we consider a scenario where the sampling instants are taken from a continuous-time observation window. Under this class of Fourier transform estimation we analyse three estimation schemes, i.e. the total random estimation, stratified estimation and antithetical stratified estimation. We compare the derived estimators in terms of the mean-square error they introduce to the estimated Fourier transform. Also, we compare the rates of convergence of the estimates with respect to the number of random samples. Second, we examine two Fourier transform estimation schemes where the sampling points are selected from a predefined dense and uniformly distributed grid of time instants. The schemes are named as the total random on grid estimation and stratified on grid estimation. Accuracy of these estimates is shown and compared with each other.
A technique is presented for detecting the presence and significance of
a period in unequally sampled time series data. The calculation of the
modified periodogram for unevenly sampled data is reviewed. The proper
definition of the variance that is used to normalize the power of the
modified periodogram is clarified. It is proven that the probability
that a peak in the periodogram is noise or signal can be easily assessed
by the method given here only when the total variance of the data is
used to normalize the periodogram power. The crucial choice of
independent frequencies in calculating both the periodogram and the
false alarm probability from unevenly sampled data is discussed. An
empirical formula for estimating the number of independent frequencies
is derived. In addition, the formula for the uncertainty of a frequency
identified in the periodogram is reviewed. A method for detecting the
presence of an alias frequency caused by the interaction of the window
and signal is prescribed. With some examples of periodic signals, the
minimum number of points required to measure reliably a signal are
shown. The signal-to-noise ratio and the number of points required to
extract signals when one or two periodicities are present in the time
series are investigated.
Statistical properties of two broad classes of methods used in period search, namely, phase binning and model function methods, are compared. We employ hypothesis-testing theory to study these methods and present closed analytical formulae for evaluation of the sensitivity of period search, for different kinds of signals. Based on this theory, we draw two conclusions: (1) the methods using smooth model functions are generally more sensitive than those using phase binning and (2) the resolution of the model functions should match structures in the detected signal. Both excess and insufficient resolution result in decreased detection sensitivity. Finally, we demonstrate that within the broad class of the methods discussed, methods utilizing the same models but different statistics generally are equally sensitive. Our considerations apply to most existing period-search methods, which enable formulation of statistical detection criteria.
The statistical properties of least-squares frequency analysis of unequally spaced data are examined. It is shown that, in the least-squares spectrum of gaussian noise, the reduction in the sum of squares at a particular frequency is aX
2
2
variable. The reductions at different frequencies are not independent, as there is a correlation between the height of the spectrum at any two frequencies,f
1 andf
2, which is equal to the mean height of the spectrum due to a sinusoidal signal of frequencyf
1, at the frequencyf
2. These correlations reduce the distortion in the spectrum of a signal affected by noise. Some numerical illustrations of the properties of least-squares frequency spectra are also given.
In this paper, we present a comprehensive review of methods for spectral analysis of nonuniformly sampled data. For a given finite set of nonuniformly sampled data, a reasonable way to choose the Nyquist frequency and the resampling time are discussed. The various existing methods for spectral analysis of nonuniform data are grouped and described under four broad categories: methods based on least squares; methods based on interpolation techniques; methods based on slotted resampling; methods based on continuous time models. The performance of the methods under each category is evaluated on simulated data sets. The methods are then classified according to their capabilities to handle different types of spectrum, signal models and sampling patterns. Finally the performance of the different methods is evaluated on two real life nonuniform data sets. Apart from the spectral analysis methods, methods for exact signal reconstruction from nonuniform data are also reviewed.
The least-squares (or Lomb–Scargle) periodogram is a powerful tool that is routinely used in many branches of astronomy to
search for periodicities in observational data. The problem of assessing the statistical significance of candidate periodicities
for a number of periodograms is considered. Based on results in extreme value theory, improved analytic estimations of false
alarm probabilities are given. These include an upper limit to the false alarm probability (or a lower limit to the significance).
The estimations are tested numerically in order to establish regions of their practical applicability.
Kepler mission design, realized photometric performance, and early science
- T M Brown
- TM Brown