Significance Tests for Periodogram Peaks

We discuss methods currently in use for determining the significance of peaks in the periodograms of time series. We discuss some general methods for constructing significance tests, false alarm probability functions, and the role played in these by independent random variables and by empirical and theoretical cumulative distribution functions. We also discuss the concept of "independent frequencies" in periodogram analysis. We propose a practical method for estimating the significance of periodogram peaks, applicable to all time series irrespective of the spacing of the data. This method, based on Monte Carlo simulations, produces significance tests that are tailor-made for any given astronomical time series.Comment: 22 pages, 11 Encapsulated Postscript figures, AAS LaTeX v5.2 Submitted to Ap

Artifacts with uneven sampling of red noise

The vast majority of sampling systems operate in a standard way: at each tick of a fixed-frequency master clock a digitizer reads out a voltage that corresponds to the value of some physical quantity and translates it into a bit pattern that is either transmitted, stored, or processed right away. Thus signal sampling at evenly spaced time intervals is the rule: however this is not always the case, and uneven sampling is sometimes unavoidable. While periodic or quasi-periodic uneven sampling of a deterministic signal can reasonably be expected to produce artifacts, it is much less obvious that the same happens with noise: here I show that this is indeed the case only for long-memory noise processes, i.e., power-law noises $1/f^\alpha$ with $\alpha > 2$. The resulting artifacts are usually a nuisance although they can be eliminated with a proper processing of the signal samples, but they could also be turned to advantage and used to encode information.Comment: 5 figure

Algorithms for Spectral Analysis of Irregularly Sampled Time Series

In this paper, we present a spectral analysis method based upon least square approximation. Our method deals with nonuniform sampling. It provides meaningful phase information that varies in a predictable way as the samples are shifted in time. We compare least square approximations of real and complex series, analyze their properties for sample count towards infinity as well as estimator behaviour, and show the equivalence to the discrete Fourier transform applied onto uniformly sampled data as a special case. We propose a way to deal with the undesirable side effects of nonuniform sampling in the presence of constant offsets. By using weighted least square approximation, we introduce an analogue to the Morlet wavelet transform for nonuniformly sampled data. Asymptotically fast divide-and-conquer schemes for the computation of the variants of the proposed method are presented. The usefulness is demonstrated in some relevant applications.

Multi-wavelength variability study of the classical BL Lac object PKS 0735+178 on timescales ranging from decades to minutes

We present the results of our power spectral analysis for the BL Lac object PKS 0735+178 utilizing the Fermi-LAT survey at high-energy $\gamma$-rays, several ground-based optical telescopes, and single-dish radio telescopes operating at GHz frequencies. The novelty of our approach is that, by combining long-term and densely sampled intra-night light curves in the optical regime, we were able to construct for the first time the optical power spectrum of the blazar for a time domain extending from 23 years down to minutes. Our analysis reveals that: (i) the optical variability is consistent with a pure red noise, for which the power spectral density can well be approximated by a single power-law throughout the entire time domain probed; (ii) the slope of power spectral density at high-energy $\gamma$-rays ($\sim 1$), is significantly flatter than that found at radio and optical frequencies ($\sim 2$) within the corresponding time variability range; (iii) for the derived power spectra we did not detect any low-frequency flattening, nor do we see any evidence for cut-offs at the highest frequencies down to the noise floor levels due to measurement uncertainties. We interpret our findings in terms of a model where the blazar variability is generated by the underlying single stochastic process (at radio and optical frequencies), or a linear superposition of such processes (in the $\gamma$-ray regime). Along with the detailed PSD analysis, we also present the results of our extended (1998-2015) intra-night optical monitoring program and newly acquired optical photo-polarimetric data for the source.Comment: The Astrophysical Journal, in pres