A Fast Chi-squared Technique For Period Search of Irregularly Sampled Data

A new, computationally- and statistically-efficient algorithm, the Fast $\chi^2$ algorithm, can find a periodic signal with harmonic content in irregularly-sampled data with non-uniform errors. The algorithm calculates the minimized $\chi^2$ as a function of frequency at the desired number of harmonics, using Fast Fourier Transforms to provide $O (N \log N)$ performance. The code for a reference implementation is provided.

💡 Research Summary

The paper introduces a novel algorithm, termed the Fast χ² technique, for detecting periodic signals in irregularly sampled time‑series data where each measurement may have a distinct uncertainty. Traditional period‑search methods such as the Lomb‑Scargle periodogram, Phase‑Dispersion‑Minimization, or unweighted least‑squares fitting either assume uniform errors, are computationally intensive for large data sets, or cannot readily incorporate harmonic content beyond a single sinusoid. The Fast χ² method addresses both statistical and computational shortcomings by formulating the problem as a weighted least‑squares fit of a multi‑harmonic model and then exploiting the Fast Fourier Transform (FFT) to evaluate the necessary sums efficiently.

Model formulation
Given observations ((t_i, y_i, \sigma_i)) with weights (w_i = 1/\sigma_i^2), the signal model is expressed as a sum of K harmonics: \