Astronomy & Celestial Mechanics
the sky, computed
Concept

Look elsewhere effect — where it appears

The inflation of false detections that comes from searching many places for a signal: the tallest of many noise peaks is taller than any one. A significance must be corrected for the number of independent frequencies or positions searched, which the data's baseline sets, not the grid.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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

Named alongside it

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

False-alarm probabilityPeriodogramRed noiseBayes factorBootstrapDetection thresholdGaussian processIndependent frequenciesLindley's paradoxMultiple testingNull hypothesisPrior

All concepts