Sliding Lomb–Scargle Periodogram

Updated 29 July 2025

Sliding Lomb–Scargle periodogram is a time–frequency analysis tool that extends the classical LSP using a sliding-window approach with localized noise whitening.
It computes Fourier coefficients and local covariance in moving windows, thereby enhancing the detection of transient and evolving periodic signals.
The method is pivotal in astrophysics and geophysics for distinguishing genuine periodicities from sampling artifacts and nonstationary noise.

The sliding Lomb–Scargle periodogram (sLSP) is an advanced time–frequency analysis tool designed for detecting and tracking periodic signals in unevenly sampled time-series data. Functionally, it extends the classical Lomb–Scargle periodogram (LSP), which is widely adopted for irregular sampling, by applying the periodogram over a sequence of moving (often overlapping) time windows. This approach enables temporal monitoring of spectral features, robust identification of transient and time-evolving periodicities, and critical discrimination between intrinsic signals and sampling artifacts. The sLSP formalism, as formalized in recent research, is essential in contemporary astrophysics, geophysics, and other domains where nonuniform cadence, colored/nonstationary noise, or aliasing strongly affect spectral analysis.

1. Mathematical and Statistical Foundations

The sLSP leverages the matrix-algebra formalism developed for the general LSP (1006.2473). For unevenly sampled data, let $x$ be an $M$ -length column vector of observations at times $\{ t_j \}_{j=0}^{M-1}$ . Select $N$ test frequencies, and construct the generalized Fourier matrix $F_{kj} = (1/\sqrt{M})\exp[-i2\pi k\tilde{t}_j/N]$ , with $\tilde{t}_j = t_j / \Delta_{mt}$ and $\Delta_{mt} = \gamma \cdot \min (t_{j+1}-t_j)$ . The “Fourier transform” is $X = F x$ . Unlike in evenly sampled data, $F$ is not unitary, causing correlations among the real and imaginary components of $X$ even under white noise.

The periodogram at frequency $k$ is conventionally

$P_k = X_{R,k}^2 + X_{I,k}^2$

where $X_{R,k}, X_{I,k}$ are the real and imaginary parts. However, with uneven sampling and colored/nonstationary noise, these components must be decorrelated by transforming $X_k^* = [X_{R,k}, X_{I,k}]^\mathrm{T}$ through the whitening operation

$\hat{Y} = (C_k^*)^{-1/2} U_k^* X_k^*$

where $C_k^*$ is the local $2\times2$ covariance matrix estimated from the noise properties, and $U_k^*$ contains its eigenvectors. The periodogram is then computed as

$P_k = \hat{Y}_1^2 + \hat{Y}_2^2$

which restores the desired $\chi_2^2$ (exponential) statistics for $P_k$ under the noise hypothesis (1006.2473).

When sliding the analysis, this procedure is localized to each window, with all matrices and statistics recomputed on the corresponding data subset.

2. Computational Approach and Sliding-Window Construction

The core sLSP algorithm proceeds as follows (1006.2473):

Select a time window of length $T$ (windows may be overlapping, step size and window size tune time and frequency resolution, respectively).
For each window $w$ with $M_w$ points at $\{ t_j, j \in J \subset [0, M-1] \}$ , define the local Fourier matrix $F^{(w)}$ as in Eq. (1) above.
Compute local “Fourier” coefficients $X^{(w)} = F^{(w)} x^{(w)}$ for the window data $x^{(w)}$ .
Estimate the local covariance $C^{(w)}$ from the noise properties within the current window. For colored or time-varying noise, this requires local modeling or empirical estimation.
Apply the whitening transformation and decorrelation within each frequency/window as in Eq. (8).
Obtain the time–frequency map of powers $P_k^{(w)}$ over all window positions and frequencies.

This entire procedure generates a data product $P_k^{(w)}$ that encodes the temporal evolution of the signal spectrum, robustly standardized even under highly nonstationary or correlated noise.

3. Detection Theory and Statistical Thresholds

A rigorous statistical framework for peak detection in temporally resolved periodograms must adapt to changing noise properties and sample sizes across windows. The matrix-based whitening ensures that the periodogram ordinates approximate the $\chi_2^2$ distribution (exponential cumulative distribution function) at each frequency and window. The false–alarm probability (FAP) for a detection threshold $L_{FA}$ is

$1 - [1 - \exp(-L_{FA})]^{N_f} \leq \alpha$

where $N_f$ is the number of statistically independent frequencies (typically estimated as $M_w/2$ for each window), and $\alpha$ is the desired significance level (1006.2473). By setting FAPs relative to this standard, detection thresholds remain meaningful even for varying data cadence and noise characteristics within each window.

4. Robustness to Non-Uniform Sampling, Colored, and Nonstationary Noise

A central motivation for the matrix sLSP approach is its capacity to decouple signal detection from the assumptions of white, stationary, Gaussian noise and uniformly sampled data. If the original noise is colored or nonstationary, the local covariance matrix $C_n$ is used to perform a whitening transformation of the data $y = C_n^{-1/2} x$ (or its windowed variant), followed by the generalized Fourier analysis (1006.2473). In scenarios where $C_n$ is (locally) circulant, the diagonalization is performed via the DFT; for fully general covariance, eigen-decomposition is used (see equations (10)-(12)). Thus, at each window and for each frequency, the covariance structure is explicitly corrected, and the detection process remains valid and optimally sensitive irrespective of underlying noise color or properties.

5. Temporal Tracking and Applications

The sLSP produces a time–frequency representation capable of tracking nonstationary phenomena, mode switching, and transient periodicities. Key scientific applications include:

Detection and monitoring of time-varying stellar pulsations, as in $\delta$ Scuti stars where the sLSP can distinguish genuine frequencies from super-Nyquist aliasing by revealing stable versus modulated features in the time–frequency matrix (Yang et al., 24 Jul 2025).
Discrimination of persistent versus transient periodicities in stochastic light curves of quasars, and assessment of periodic signal stability amidst red noise backgrounds (DRW) and sampling artifacts (1012.3779).
Mapping of evolution (appearance, growth, decay) of periodic signatures due to starspots, accretion, or orbital phenomena, with significance analysis adapted to the local data conditions (Santos et al., 2016, Lu et al., 2022).
Correction for changes in noise properties and data quality (heteroscedasticity, variable background noise) within large surveys or multi-instrument datasets.

The sLSP is therefore particularly advantageous for astrophysical time-series analysis, enabling robust, localized detection in conditions where both the underlying processes and noise characteristics are time-dependent.

While powerful, the sLSP as constructed requires window-by-window estimation of data statistics and noise covariances, entailing higher computational costs than static periodograms. Window length must be selected to balance time resolution (shorter windows) against frequency resolution and statistical stability (longer windows). Too short a window reduces sensitivity to long-period signals and increases false-alarm rates; too long a window may average over essential nonstationary behavior (Santos et al., 2016, Yang et al., 24 Jul 2025).

The sLSP’s matrix whitening procedure assumes that the noise within each window is sufficiently approximated as locally stationary, and that local covariance estimation is statistically stable (i.e., windows contain enough data points for meaningful estimation).

Direct simulation studies and method comparisons show that even in challenging regimes (irregular cadence, colored noise, few cycles per window), the matrix sLSP maintains standard detection statistics and enables the use of uniform, frequency-independent thresholds for peak identification (1006.2473). In contrast, classical LSP and sliding implementations that do not account for noise correlations may yield biased or misleading results in these regimes.

7. Implications for Asteroseismic and Time-Domain Surveys

In large-scale asteroseismic analyses, such as the paper of Kepler $\delta$ Scuti stars (Yang et al., 24 Jul 2025), the sLSP enables clean separation of intrinsic, stable pulsations from aliasing artifacts due to instrumental cadence and super-Nyquist signal reflection. By monitoring the amplitude modulation within narrow frequency bands as a function of time, sLSP detects temporally periodic modulations characteristic of aliases and confirms or refutes the status of candidate frequencies. This increases the reliability and “cleanliness” of modal frequency lists, which are critical for robust stellar modeling and interpretation.

In time-domain surveys (e.g., LSST), the sLSP is a valuable diagnostic for understanding the physical origin of identified periodicities, for tracking variability sources, and for distinguishing genuine periodicities from features caused by nonstationary noise, windowing effects, or red noise backgrounds (1012.3779). It also provides a pathway for consistent false-alarm calibration across diverse datasets.

In summary, the sliding Lomb–Scargle periodogram, as formalized in the matrix-algebra framework, is a rigorous and robust technique for local spectral analysis of unevenly sampled, noisy, and nonstationary time series. By combining localized whitening transformations, covariance modeling, and standardized detection thresholds, it enables scientifically reliable detection and temporal monitoring of periodic signals under the realistic complexities present in contemporary astronomical and geophysical datasets.