Parameter Estimation from Time-Series Data with Correlated Errors: A Wavelet-Based Method and its Application to Transit Light Curves

We consider the problem of fitting a parametric model to time-series data that are afflicted by correlated noise. The noise is represented by a sum of two stationary Gaussian processes: one that is uncorrelated in time, and another that has a power spectral density varying as $1/f^\gamma$. We present an accurate and fast [O(N)] algorithm for parameter estimation based on computing the likelihood in a wavelet basis. The method is illustrated and tested using simulated time-series photometry of exoplanetary transits, with particular attention to estimating the midtransit time. We compare our method to two other methods that have been used in the literature, the time-averaging method and the residual-permutation method. For noise processes that obey our assumptions, the algorithm presented here gives more accurate results for midtransit times and truer estimates of their uncertainties.

💡 Research Summary

The paper addresses the challenge of fitting parametric models to time‑series data that contain correlated (red) noise in addition to ordinary white noise. The authors model the total noise as the sum of two stationary Gaussian processes: an uncorrelated white component and a colored component whose power spectral density follows a 1/f^γ law. Traditional maximum‑likelihood approaches require the inversion of the full N × N covariance matrix, an O(N³) operation that quickly becomes prohibitive for large astronomical light curves.
To overcome this bottleneck, the authors exploit the multiresolution properties of the discrete wavelet transform. In a wavelet basis, the covariance matrix of the combined white‑plus‑red noise becomes (approximately) diagonal because each wavelet coefficient samples the noise at a distinct time‑scale and is statistically independent under the Gaussian‑stationary assumptions. Consequently, the log‑likelihood can be expressed as a simple sum over the squared, scale‑dependent coefficients, allowing its evaluation in O(N) time. The derivation of the wavelet‑based likelihood, the treatment of the γ exponent, and the handling of the white‑noise variance are presented in detail, demonstrating that the method retains the full statistical power of the original Gaussian model while dramatically reducing computational cost.
The authors validate the technique using simulated exoplanet transit photometry. They generate synthetic light curves with known transit parameters, adding white noise and red noise with various γ values (0.5–2.0) and different white‑to‑red amplitude ratios. The primary quantity of interest is the mid‑transit time (Tc), a parameter whose precise determination is essential for transit‑timing variation studies. Three estimation strategies are compared: (1) the new wavelet‑based maximum‑likelihood method, (2) the time‑averaging (or “beta‑factor”) approach, and (3) the residual‑permutation (or “prayer‑bead”) bootstrap. For each method the authors compute the bias and the reported uncertainties of Tc across many Monte‑Carlo realizations.
Results show that the wavelet method consistently yields smaller biases and uncertainty estimates that more accurately reflect the true scatter of the recovered Tc values. In contrast, the time‑averaging method tends to over‑inflate uncertainties when red noise dominates, while the residual‑permutation method can underestimate them, especially for steep γ spectra. Moreover, because the wavelet likelihood scales linearly with data length, the algorithm remains fast even for light curves containing thousands of points—a regime where the O(N³) matrix inversion would be infeasible.
The paper also discusses limitations. The diagonalization in the wavelet domain relies on the assumptions of Gaussianity and stationarity; non‑Gaussian systematics, abrupt trends, or time‑varying noise properties would break the independence of wavelet coefficients and require more sophisticated modeling or preprocessing. Nevertheless, within its domain of applicability, the method offers a powerful combination of speed and statistical rigor.
In summary, the authors present a wavelet‑based O(N) maximum‑likelihood framework for parameter estimation in the presence of 1/f^γ correlated noise. Through extensive simulations they demonstrate that it outperforms two widely used alternatives in both accuracy of the estimated parameters (particularly transit mid‑times) and realism of the associated error bars. The technique holds promise for a broad range of astronomical time‑series analyses where red noise is a dominant source of uncertainty.