Math 2.0

Statistics Definition

Lomb–Scargle periodogram

Builds on PeriodogramLeast squaresTime series

Definition

Let (tj,yj)(t_j, y_j), j=1,…,Nj = 1, \dots, N, be observations of a real time series at times tj∈ℝt_j \in \R. Let yˉ\bar y be the mean of the yjy_j. For an angular frequency ω>0\omega > 0, choose τ∈ℝ\tau \in \R with ∑jsin⁡(2ω(tj−τ))=0\sum_{j} \sin\big(2\omega(t_j - \tau)\big) = 0. Write cj=cos⁡ω(tj−τ)c_j = \cos\omega(t_j - \tau) and sj=sin⁡ω(tj−τ)s_j = \sin\omega(t_j - \tau). Assume ∑jcj2>0\sum_j c_j^2 > 0 and ∑jsj2>0\sum_j s_j^2 > 0. The Lomb–Scargle periodogram at ω\omega is P(ω)=12[(∑j(yj−yˉ) cj)2∑jcj2+(∑j(yj−yˉ) sj)2∑jsj2].P(\omega) = \frac{1}{2}\left[\frac{\big(\sum_j (y_j - \bar y)\, c_j\big)^2}{\sum_j c_j^2} + \frac{\big(\sum_j (y_j - \bar y)\, s_j\big)^2}{\sum_j s_j^2}\right].

Remarks

The condition on τ\tau is tan⁡(2ωτ)=∑jsin⁡2ωtj/∑jcos⁡2ωtj\tan(2\omega\tau) = \sum_j \sin 2\omega t_j / \sum_j \cos 2\omega t_j, and it makes ∑jcjsj=0\sum_j c_j s_j = 0. So the two columns (cj)(c_j) and (sj)(s_j) are orthogonal. Then 2P(ω)2P(\omega) is the drop in the residual sum of squares when yj−yˉ≈acj+bsjy_j - \bar y \approx a c_j + b s_j is fitted by least squares. P(ω)P(\omega) does not change when all tjt_j are shifted by the same amount. Example: for yj=cos⁡(ω0tj)y_j = \cos(\omega_0 t_j) at many well-spread times, PP has a sharp peak near ω=ω0\omega = \omega_0. Non-example: the squared modulus of the discrete Fourier transform of unevenly sampled data is not the Lomb–Scargle periodogram; it lacks the shift τ\tau and the normalisations. For evenly spaced tjt_j and Fourier frequencies, PP agrees with the classical periodogram up to a constant factor. Many texts divide PP by the sample variance of the yjy_j.

Reading path

To understand this page from the ground up, read these 22 pages in order. The foundations come first.

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Matrix, definition, no evidence levelMatrixdefinitionPeriodogram, concept, no evidence levelPeriodogramconceptTime series, concept, no evidence levelTime seriesconceptVector space, definition, no evidence levelVector spacedefinitionTranspose, definition, no evidence levelTransposedefinitionInner product, definition, no evidence levelInner productdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionNorm, definition, no evidence levelNormdefinitionInner product space, definition, no evidence levelInner product spacedefinitionOrthogonal vectors, definition, no evidence levelOrthogonal vectorsdefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionSum of subspaces, definition, no evidence levelSum of subspacesdefinitionNorm induced by an inner product, definition, no evidence levelNorm induced by an in…definitionOrthogonal complement, definition, no evidence levelOrthogonal complementdefinitionDirect sum of subspaces, definition, no evidence levelDirect sum of subspac…definitionA space is the direct sum of a subspace and its orthogonal complement, statement, no evidence levelA space is the direct…statementProjection, definition, no evidence levelProjectiondefinitionOrthogonal projection, definition, no evidence levelOrthogonal projectiondefinitionOrthogonal projection is the closest point, statement, no evidence levelOrthogonal projection…statementLeast squares, concept, no evidence levelLeast squaresconceptLomb–Scargle periodogram, definition, no evidence levelLomb–Scargle periodog…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Matrix definition
  2. Periodogram concept
  3. Time series concept
  4. Vector space definition
  5. Transpose definition
  6. Inner product definition
  7. Linear map definition
  8. Linear subspace definition
  9. Norm definition
  10. Inner product space definition
  11. Orthogonal vectors definition
  12. Endomorphism definition
  13. Sum of subspaces definition
  14. Norm induced by an inner product definition
  15. Orthogonal complement definition
  16. Direct sum of subspaces definition
  17. A space is the direct sum of a subspace and its orthogonal complement statement
  18. Projection definition
  19. Orthogonal projection definition
  20. Orthogonal projection is the closest point statement
  21. Least squares concept
  22. Lomb–Scargle periodogram definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Least-squares_spectral_analysis
About this page
TagZ68Y2VSGVM
URIm2://Z68Y2VSGVM
Kinddefinition
WritingW0 stub
MSC62M15, 85A04
DefinesLomb–Scargle periodogram
UsesPeriodogram, Least squares, Time series
Changed3 November 2026

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