A general theory on frequency and time–frequency analysis of irregularly sampled time series based on projection methods – Part 1: Frequency analysis

Abstract

Abstract. 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 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 that compute the confidence levels for the WOSA periodogram and fully take into account the uncertainty in the CARMA noise parameters. Alternatively, a theory using point estimates of CARMA parameters 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, which we recommend for most frequency analyses with irregularly sampled data. The estimated signal amplitude also permits filtering in a frequency band. Our results generalise 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, 2018). All the methods presented in this paper are available to the reader in the Python package WAVEPAL.

A general theory on frequency and time–frequency analysis of irregularly sampled time series based on projection methods – Part 2: Extension to time–frequency analysis

“…In this article, we extend the analysis tools that we derived in the first part of this study (Lenoir and Crucifix, 2018) in the case of the frequency analysis of irregularly sampled time series. They are based on a similar model, which is a locally periodic component in the time-frequency plane, plus a polynomial trend, plus a stationary Gaussian continuous autoregressive-moving-average (CARMA) process.…”

“…Eighth, we illustrate, in Sect. 7, the theory on the same palaeoclimate data set as in our first article (Lenoir and Crucifix, 2018). Finally, a Python package named WAVEPAL is available to the reader and is presented in Sect.…”

“…6 of Part 1, and define, in Sect. 3.6 of this article, the weighted smoothed scalogram as the time-frequency analogue of the weighted WOSA periodogram defined in the first part of this study (Lenoir and Crucifix, 2018). Fourth, we define in Sect.…”

“…Most of the mathematical concepts and notations are introduced in the first part of this study (Lenoir and Crucifix, 2018), and the reader is invited to revise them. Throughout this article, we will denote the equations of the preceding paper by, for example, "Eq.…”

“…To this end, we define the scalogram as the continuous-wavelet-transform equivalent of the extended Lomb-Scargle periodogram defined in Part 1 of this study (Lenoir and Crucifix, 2018). The signal being analysed is modelled as the sum of a locally periodic component in the time-frequency plane, a polynomial trend, and a background noise.…”