Working definition
A periodic function repeats after a period T, while phase identifies position within that cycle. Fourier representations decompose suitable signals into sinusoidal components, but finite, noisy, drifting, or irregularly sampled observations require uncertainty estimates and tests against non-periodic alternatives.
Notation
f(t + T) = f(t)φ(t) = 2πt/T + φ₀Assumptions
- The proposed period is stable over the analyzed interval.
- Sampling can resolve the frequency.
- Trend and seasonality are not conflated.
Invariants
- Phase is equivalent modulo 2π.
- Frequency is the reciprocal of period.
- A time shift produces a phase shift.
Reproducible procedure
- Inspect sampling and detrend only under a declared model.
- Estimate candidate frequency and phase.
- Validate out of sample and test aliases.
Error and boundary controls
- Aliasing can create false periods.
- Short windows yield broad frequency uncertainty.
- Phase drift invalidates fixed-period extrapolation.
What this does not establish
Detecting periodic structure does not identify a celestial cause, and a visually aligned cycle is not evidence of forecast skill.
Explicit applications
2 cross-domain bridges
Periodic-signal search
Estimate candidate periods in irregular observations while testing aliases and phase stability.
Inputs
- observation times
- signal values
- noise model
Outputs
- candidate period
- phase model
- false-alarm assessment
Transformation: Evaluate a declared period estimator against shuffled and simulated null data.
Limit: A periodogram peak is not a physical mechanism and may be an alias.
Open connected system →Transit cycle and repeated crossing model
Represent direct, retrograde, and return crossings without treating mean orbital periods as exact event schedules.
Inputs
- unwrapped longitude series
- target boundary
- motion direction
Outputs
- crossing sequence
- cycle phase
- retrograde loop markers
Transformation: Track phase continuously and segment each distinct crossing.
Limit: Cycle phase is descriptive geometry and does not imply repeated life outcomes.
Open connected system →Authoritative references
- [1]NIST/SEMATECH e-Handbook of Statistical Methods · National Institute of Standards and Technology
Establishes: Methods for uncertainty analysis, calibration, time-series modeling, process monitoring, experimental design, reliability, and statistical comparison.
Boundary: Statistical procedures quantify evidence under a design and model; they do not repair biased sampling, outcome leakage, post-hoc hypotheses, or unmeasured confounding.
- [2]DLMF Chapter 3: Numerical Methods · National Institute of Standards and Technology
Establishes: Reference definitions, algorithms, convergence conditions, and error terms for interpolation, quadrature, differentiation, and nonlinear equation solving.
Boundary: A numerical method is reliable only under its stated regularity, conditioning, precision, and convergence assumptions; the reference does not validate any domain interpretation.