Astronomy & Celestial Mechanics
the sky, computed
Concept

False-alarm probability — where it appears

The chance that noise alone would produce a periodogram peak at least as tall as the one observed. It depends on how many independent frequencies were searched and on what the noise actually is, and the standard formula assumes white noise that astronomical data rarely have.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

A true peak at 0.3103 and a false one at 0.6897, from the same data. The Lomb–Scargle periodogram of 162 simulated observations taken over 419 nights from one site, of a star carrying a 4.2-unit sinusoid at 0.3103 cycles per day under 3 units of Gaussian noise per point. The injected signal is recovered at 0.3103 cycles per day, within the 0.0024 resolution element the baseline allows. The second peak, at 0.6899, is 87 per cent as tall and corresponds to nothing: it is 1 − f, the signal reflected in the one-cycle-per-day spike of the sampling. Neither peak is more real than the other in this picture — deciding between them needs a second site at a different longitude, or a run long enough for the seasonal window to separate them. The dashed line is the power a pure-noise series would exceed once in 1000 trials, computed from 586 independent frequencies rather than from the 4691 grid points searched; using the grid count instead would put the line 2.08 higher and reject a real detection.

A period found in the gaps

Astronomical time series are sampled when the sky is dark and clear and the target is up, which is a schedule with a spectrum of its own. That spectrum is convolved with the real one, so a single sinusoid produces several peaks — and the tallest is not always the true one.

starlight · Periodograms A peak worth 10.8 in a narrow search is worth nothing in a wide one. The probability that noise alone produces a peak at least as tall as a given power, for searches over four different numbers of independent frequencies. A single frequency examined in isolation gives a one-per-cent chance at a power of 4.6; searching fifty thousand frequencies for the same one-per-cent chance requires 15.4. The threshold rises as the logarithm of the width of the search, which is why the penalty is survivable — but it is a penalty, it is often not applied, and the number of independent frequencies in an unevenly sampled time series is not the number of frequencies on the grid. Overestimating that count is conservative and underestimating it is not, which is the one asymmetry worth remembering.

The tallest peak in nothing at all

A periodogram of pure noise has peaks in it, and the tallest is not small. How tall it has to be before it means something depends on how many frequencies were searched and on what the noise actually is — and astronomical noise is almost never the white noise the standard formula assumes.

starlight · Periodograms The evidence for a period, one trial period at a time. The natural logarithm of the Bayes factor for a sinusoid at each trial period against no signal, for 120 radial velocities spread over 359 days with 2.0 m/s white noise per point and a planet of semi-amplitude 1.0 m/s at 11.3 days. The amplitude prior is Gaussian with scale 3.0 m/s in each component. At the planet's period the evidence is 14.4 — odds of about 1.8·10⁶ to one — and almost everywhere else it is near zero or negative, because a sinusoid that fits nothing is penalised for the amplitude range it wasted. The quantity that answers the question "is there a planet at some period" is not the peak but the average of the Bayes factor over the prior on the period, here uniform in the logarithm of the period from 1.5 to 1,000 days, and that average is ln B = 7.7: 6.7 lower than the peak, the Bayesian form of the look-elsewhere penalty, paid because the data had to find the period as well as the amplitude.

Odds for a planet that depend on a guess about its size

A false-alarm probability asks how often noise alone would produce a peak this tall. A Bayes factor asks something better — how much more probable the data are with a planet than without one — and pays for it with two numbers nobody measured. The width allowed for the period barely matters. The width allowed for the amplitude can turn strong evidence into none, from the same velocities.

starlight · Periodograms The same data searched with a box and with a sine. The signal-to-noise ratio of the best fit at each trial period, for the light curve of the previous figure, searched two ways: with a box of three trial durations at every phase, and with a sinusoid. The box search peaks at 3.85 days with a signal-to-noise ratio of 7.3, against a true period of 3.85. The sinusoid's best peak anywhere is 3.3, at 1.59 days, and at the true period it reaches only 1.5 — no higher than the peaks the noise makes elsewhere in the search. The planet is a clear detection to one method and invisible to the other, in the same numbers.

A periodogram shaped like a shadow

A periodogram that fits sinusoids sees a transiting planet only through the fundamental harmonic of its box-shaped dip, which for a short transit is a few per cent of the signal. Folding the data at each trial period and fitting a box recovers all of it — five times the signal-to-noise for a transit two per cent of the orbit long, eighteen times for an Earth crossing the Sun. The box pays for its extra parameters in trials, and the price is about one unit of signal-to-noise.

starlight · Periodograms

Named alongside it

The objects these essays reach for when they reach for this one.

PeriodogramRed noiseBox least squaresIndependent frequenciesLook elsewhere effectAliasBayes factorBootstrapDetection thresholdGaussian processLindley's paradoxThe Lomb–Scargle periodogram

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