Must observations be interpolated onto a regular grid?

No; using their actual times is a key advantage.

Does a floating mean equal subtracting the sample mean once?

Not generally. The offset is fitted jointly with the sinusoid at each frequency, which matters with uneven sampling.

Does a peak determine phase and amplitude automatically?

Those can be estimated from the fitted sinusoidal model, but their uncertainty and conventions still need attention.

AI Glossary / Time series and signal processing / Intermediate

Lomb-Scargle Periodograms

Definition

Lomb-Scargle periodograms assess periodic structure by fitting sinusoidal models across a range of frequencies, including when observations are unevenly spaced in time. The periodogram records how strongly each candidate frequency is supported under a specified fitting and normalization scheme. Generalized versions can include measurement weights and an independently fitted offset at each frequency.

What it means

An ordinary frequency analysis can become awkward when measurements arrive at irregular times. Lomb-Scargle methods use the actual observation times instead of pretending that neighboring records are equally spaced or first filling every gap with invented values.
A large peak identifies a frequency that fits the observed pattern well under the chosen model. It does not automatically prove a genuine physical period, identify a unique period, or establish a causal explanation. Sampling patterns and noise can create competing peaks.

How it works

For each candidate frequency f, fit a model such as y(t) = a cos(2πft) + b sin(2πft) + c by least squares, optionally weighting observations by their uncertainty. The floating-mean version estimates c along with the sine and cosine coefficients.
Compare the fit with a reference model, using a documented power normalization. Repeat over a suitable frequency grid. Search range, frequency spacing, observation duration, measurement errors, and the sampling window all affect interpretation. Some software expects ordinary frequency while other software expects angular frequency omega = 2πf, so units must be checked explicitly.

Three practical examples

An astronomy team studies unevenly timed brightness measurements for evidence of periodic variability.
An industrial analyst explores a sensor record with irregular acquisition gaps while retaining the actual timestamps.
A researcher examines recurring patterns in an irregularly observed process, then uses independent data and domain knowledge to assess whether a candidate period is credible.

Think of it this way

Think of listening to fragments of a tune at known times and testing which repeating rhythm best matches those fragments. Missing stretches do not prevent comparison, but several rhythms may fit the same fragments. The observation schedule is therefore part of the evidence, not merely an inconvenience to ignore.

Worked example

Consider an original noiseless record observed at times [0, 1, 2, 4, 5, 7] days, with values [12, 10, 8, 12, 10, 10]. The gaps are unequal. At candidate frequency 0.25 cycles per day, the model y(t) = 10 + 2 cos(2π × 0.25 × t) reproduces every value exactly, with sine coefficient zero.
The constant-only reference fit is the sample mean, 62/6 = 31/3. Its residual sum of squares is 34/3, approximately 11.3333. The sinusoidal fit has residual sum of squares zero. Under the explicitly chosen standard normalization P = (reference sum of squares − sinusoidal sum of squares)/(reference sum of squares), this candidate has power one.
The candidate period is 1/0.25 = four days. However, frequency 1.25 cycles per day reproduces the same observations because every observation time is an integer number of days and the additional full cycles are invisible at those times. Its period is 0.8 days. Unequal gaps have therefore not removed this alias ambiguity.
The example verifies fit improvement and an exact alias, not statistical significance. It contains only six ideal observations and supplies no noise model from which to compute a false-alarm probability. Different software normalizations can report different numeric power values for the same fit.

Illustrative case study

Illustrative fictional case: Pine Observatory finds two prominent candidate periods in a gapped measurement record. Instead of selecting the tallest peak and announcing a discovery, the team examines the sampling window and simulates candidate signals at the actual observation times.
It schedules additional observations at times that help distinguish the alternatives and reassesses the signal with a documented noise model. This fictional case illustrates how periodogram results can guide further evidence collection rather than settle an ambiguous periodicity claim.

Common misconceptions and mistakes

Irregular sampling does not eliminate aliasing. A single average sampling interval also does not provide a universally valid Nyquist limit for every irregular schedule. Frequency-search limits require consideration of timing precision, measurement duration, and the sampling pattern.
Do not confuse normalized power with a probability that a signal is real. A false-alarm probability depends on the null model and the searched frequency range, and is not the posterior probability that a proposed period is wrong. Correlated noise, unmodeled trends, or systematic offsets can invalidate simplistic significance calculations. A sinusoidal model may also poorly represent a strongly nonsinusoidal cycle.

Business use

The method can support exploratory monitoring when timestamped observations are incomplete or irregular. Its business value comes from identifying candidate cycles worth testing, rather than producing a visually convincing peak alone.
Retain timestamps, uncertainty estimates, preprocessing, frequency grid, normalization, and offset settings. Check whether an apparent period reflects operating schedules or data collection practices. Validate important conclusions with additional observations or an appropriate independent analysis, especially when decisions depend on a claimed recurring mechanism.

Check your understanding

What period corresponds to 0.25 cycles per day?

Answer: Four days.

Why is normalized power one in the example?

Answer: The sinusoid explains all variation relative to the constant reference under the stated normalization.

Why is the period still ambiguous?

Answer: Another frequency produces identical sampled values. What angular frequency corresponds to 0.25 cycles per day? π/2 radians per day.

Related terms

Spectral Analysis
Aliasing (factorial experiments)
periodogram
irregular sampling
sinusoidal regression
floating mean
aliasing
sampling window
frequency grid
false-alarm probability

Learn next

fit the same candidate sinusoid using actual timestamps, then inspect how changing the observation schedule affects aliases. Compare normalization conventions before interpreting numerical power and study false-alarm calculations only with an explicit noise and search model.

Sources

Sources: SciPy, official lombscargle documentation: https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.lombscargle.html . Astropy, official Lomb-Scargle Periodograms guide: https://docs.astropy.org/en/stable/timeseries/lombscargle.html . Primary documentation consulted. The exact fit, reference sum of squares, and alias demonstration are original; the case is fictional.

Previous: Logrank test Next: Long-distance dependencies