A general theory on frequency and time-frequency analysis of irregularly sampled time series based on projection methods. I. Frequency analysis

We develop a general framework for the frequency analysis of irregularly sampled time series. It is based on the Lomb-Scargle periodogram, but extended to algebraic operators accounting for the presence of a polynomial trend in the model for the data, in addition to a periodic component and a background noise. Special care is devoted to the correlation between the trend and the periodic component. This new periodogram is then cast into the Welch overlapping segment averaging (WOSA) method in order to reduce its variance. We also design a test of 5 significance for the WOSA periodogram, against the background noise. The model for the background noise is a stationary Gaussian continuous autoregressive-moving-average (CARMA) process, more general than the classical Gaussian white or red noise processes. CARMA parameters are estimated following a Bayesian framework. We provide algorithms computing the confidence levels for the WOSA periodogram that fully take into account the uncertainty on the CARMA noise parameters. Alternatively, a theory using point estimates of CARMA parameters 10 provides analytical confidence levels for the WOSA periodogram, which are more accurate than Markov chain Monte Carlo (MCMC) confidence levels and, below some threshold for the number of data points, less costly in computing time. We then estimate the amplitude of the periodic component with least squares methods, and derive an approximate proportionality between the squared amplitude and the periodogram. This proportionality leads to a new extension for the periodogram: the weighted WOSA periodogram, that we recommend for most spectral analyses 15 with irregularly sampled data. The estimated signal amplitude also permits filtering in a frequency band. Our results generalize and unify methods developed in the fields of geosciences, engineering, astronomy and astrophysics. They also constitute the starting point for an extension to the continuous wavelet transform developed in a companion article (Lenoir and Crucifix, 2017). All the methods presented in this paper are available to the reader in the Python package WAVEPAL. 20 1 Nonlin. Processes Geophys. Discuss., https://doi.org/10.5194/npg-2017-26 Manuscript under review for journal Nonlin. Processes Geophys. Discussion started: 4 July 2017 c © Author(s) 2017. CC BY 4.0 License.